# Particle Physics Planet

## May 24, 2018

### Emily Lakdawalla - The Planetary Society Blog

### Peter Coles - In the Dark

*High in the midst, surrounded by his peers,*

*Magnus his ample front sublime uprears:*

*Plac’d on his chair of state, he seems a God,*

*While Sophs and Freshmen tremble at his nod;*

*As all around sit wrapt in speechless gloom,*

*His voice, in thunder, shakes the sounding dome;*

*Denouncing dire reproach to luckless fools,*

*Unskill’d to plod in mathematic rules.*

*Happy the youth! in Euclid’s axioms tried,*

*Though little vers’d in any art beside;*

*Who, scarcely skill’d an English line to pen,*

*Scans Attic metres with a critic’s ken.*

*What! though he knows not how his fathers bled,*

*When civil discord pil’d the fields with dead,*

*When Edward bade his conquering bands advance,*

*Or Henry trampled on the crest of France:*

*Though marvelling at the name of Magna Charta,*

*Yet well he recollects the laws of Sparta;*

*Can tell, what edicts sage Lycurgus made,*

*While Blackstone’s on the shelf, neglected laid;*

*Of Grecian dramas vaunts the deathless fame,*

*Of Avon’s bard, rememb’ring scarce the name.*

*Such is the youth whose scientific pate*

*Class-honours, medals, fellowships, await;*

*Or even, perhaps, the declamation prize,*

*If to such glorious height, he lifts his eyes.*

*But lo! no common orator can hope*

*The envied silver cup within his scope:*

*Not that our heads much eloquence require,*

*Th’ ATHENIAN’S glowing style, or TULLY’S fire.*

*A manner clear or warm is useless, since*

*We do not try by speaking to convince;*

*Be other orators of pleasing proud,—*

*We speak to please ourselves, not move the crowd:*

*Our gravity prefers the muttering tone,*

*A proper mixture of the squeak and groan:*

*No borrow’d grace of action must be seen,*

*The slightest motion would displease the Dean;*

*Whilst every staring Graduate would prate,*

*Against what—he could never imitate.*

*The man, who hopes t’ obtain the promis’d cup,*

*Must in one posture stand, and ne’er look up;*

*Nor stop, but rattle over every word—*

*No matter what, so it can not be heard:*

*Thus let him hurry on, nor think to rest:*

*Who speaks the fastest’s sure to speak the best;*

*Who utters most within the shortest space,*

*May, safely, hope to win the wordy race.*

*The Sons of Science these, who, thus repaid,*

*Linger in ease in Granta’s sluggish shade;*

*Where on Cam’s sedgy banks, supine, they lie,*

*Unknown, unhonour’d live—unwept for die:*

*Dull as the pictures, which adorn their halls,*

*They think all learning fix’d within their walls:*

*In manners rude, in foolish forms precise,*

*All modern arts affecting to despise;*

*Yet prizing Bentley’s, Brunck’s, or Porson’s note,*

*More than the verse on which the critic wrote:*

*Vain as their honours, heavy as their Ale,*

*Sad as their wit, and tedious as their tale;*

*To friendship dead, though not untaught to feel,*

*When Self and Church demand a Bigot zeal.*

*With eager haste they court the lord of power,*

*(Whether ’tis PITT or PETTY rules the hour;)*

*To him, with suppliant smiles, they bend the head,*

*While distant mitres to their eyes are spread;*

*But should a storm o’erwhelm him with disgrace,*

*They’d fly to seek the next, who fill’d his place.*

*Such are the men who learning’s treasures guard!*

*Such is their practice, such is their reward!*

*This much, at least, we may presume to say—*

*The premium can’t exceed the price they pay.*

by George Gordon Byron (1788-1824)

Follow @telescoper

## May 23, 2018

### Peter Coles - In the Dark

I took today off on annual leave (as I have to use all my allowance before I depart my job at Cardiff University). My intention was to make the best of the good weather to watch some cricket.

And so it came to pass that this morning I wandered down to Sophia Gardens for the start of the Royal London One-Day Cup (50-over) match between Glamorgan and Middlesex. It also came to pass that about fifteen minutes later I wandered back home again. I hadn’t checked the start time, which was actually 2pm…

The later start screwed up my plans as I had something to do in the evening but I thought I’d at least watch the first team bat (which turned out to be Middlesex).

(I’m not sure what caused the weird stripes on the picture.. .)

It was a lovely afternoon for cricket, and Middlesex got off to a good start in excellent batting conditions. Gradually though Glamorgan’s bowlers established some measure of control. After a mini-collapse of three wickets in three overs (to Ingram’s legspin) it looked like Middlesex might not make 300 (which seems to be the par score in this competition). Unfortunately for Glamorgan, however, de Lange and Wagg were expensive at the death and a flurry of boundaries took Middlesex to 304 for 6 off their 50 overs.

At that point I left Sophia Gardens to get ready to go out.

I’ve just got back to discover that Glamorgan lost by just 2 runs, ending on 302 for 9. It must have been a tense finish, and was a good game overall, but Glamorgan have now lost all three games they have played in this competition..

Follow @telescoper### ZapperZ - Physics and Physicists

The team built their EM drive with the same dimensions as the one that NASA tested, and placed it in a vacuum chamber. Then, they piped microwaves into the cavity and measured its tiny movements using lasers. As in previous tests, they found it produced thrust, as measured by a spring. But when positioned so that the microwaves could not possibly produce thrust in the direction of the spring, the drive seemed to push just as hard.

And, when the team cut the power by half, it barely affected the thrust. So, it seems there’s something else at work. The researchers say the thrust may be produced by an interaction between Earth’s magnetic field and the cables that power the microwave amplifier.

So far, this has only been reported in a conference proceeding, which is linked in the New Scientist article (you will need ResearchGate access).

I'm sure there will be many more tests of this thing soon, but I can't help but chuckle at the apparent conclusion here.

Zz.

### Emily Lakdawalla - The Planetary Society Blog

### Clifford V. Johnson - Asymptotia

A pair of panels from my short story “Resolution” in the Science Fiction anthology Twelve Tomorrows, out on Friday from MITPress! Preorder now, share, and tell everyone about it. See here for ordering, for example.

-cvj Click to continue reading this post

The post Bull appeared first on Asymptotia.

## May 22, 2018

### Christian P. Robert - xi'an's og

**A** mostly traditional question on X validated about the “best” [minimum variance] unbiased estimator of θ² from a Poisson P(θ) sample leads to the Rao-Blackwell solution

and a similar estimator could be constructed for θ³, θ⁴, … With the interesting limitation that this procedure stops at the power equal to the number of observations (minus one?). But, since the expectation of a power of the sufficient statistics S [with distribution P(nθ)] is a polynomial in θ, there is *de facto* no limitation. More interestingly, there is no unbiased estimator of negative powers of θ in this context, while this neat comparison on Wikipedia (borrowed from the great book of counter-examples by Romano and Siegel, 1986, selling for a mere $180 on amazon!) shows why looking for an unbiased estimator of exp(-θ) is particularly foolish: the only solution is -1 to the power S. (There is however a way to circumvent the difficulty if having access to an arbitrary number of generations from the Poisson, since the Forsythe – von Neuman algorithm allows for an unbiased estimation of exp(-F(x)).)

### Peter Coles - In the Dark

In between marking exams and project reports I’ve been doing a little bit of reading in preparation for a talk that I’m due to give next week, which prompted me to share this talk by Justin Khoury of the University of Pennsylvania, which is about framework that unifies the claimed success of Modified Newtonian Dynamics (MOND) on galactic scales with the that of the standard ΛCDM model on cosmological scales. This is achieved through the physics of superfluidity. The dark matter and MOND components have a common origin, representing different phases of a single underlying substance. In galaxies, dark matter thermalizes and condenses to form a superfluid phase. The superfluid phonons couple to baryonic matter particles and mediate a MOND-like force. This framework naturally distinguishes between galaxies (where MOND is successful) and galaxy clusters (where MOND is not): dark matter has a higher temperature in clusters, and hence is in a mixture of superfluid and normal phase. The rich and well-studied physics of superfluidity leads to a number of observational signatures, discussed in the talk.

The idea that dark matter might be in the form of a superfluid is not new (see e.g. this paper) but there has been a recent surge of interest driven largely by Khoury and collaborators. If you want to find out more, can find a review paper about this model here.

Follow @telescoper### Christian P. Robert - xi'an's og

**T**oday, two statisticians (and good friends of mine) from Australia, Noel Cressie and Kerrie Mengersen, got elected at the Australian Academy of Sciences. Congratulations to them!

### Emily Lakdawalla - The Planetary Society Blog

### The n-Category Cafe

Intuitionistic logic, i.e. logic without the principle of excluded middle ($<semantics>P\vee \neg P<annotation\; encoding="application/x-tex">P\; \backslash vee\; \backslash neg\; P</annotation></semantics>$), is important for many reasons. One is that it arises naturally as the internal logic of toposes and more general categories. Another is that it is the logic traditionally used by constructive mathematicians — mathematicians following Brouwer, Heyting, and Bishop, who want all proofs to have “computational” or “algorithmic” content. Brouwer observed that excluded middle is the primary origin of nonconstructive proofs; thus using intuitionistic logic yields a mathematics in which all proofs are constructive.

However, there are other logics that constructivists might have chosen for this purpose instead of intuitionistic logic. In particular, Girard’s *(classical) linear logic* was explicitly introduced as a “constructive” logic that nevertheless retains a form of the law of excluded middle. But so far, essentially no constructive mathematicians have seriously considered replacing intuitionistic logic with any such alternative. I will refrain from speculating on why not. Instead, in a paper appearing on the arXiv today:

*Linear logic for constructive mathematics*, arxiv:1805.07518

I argue that in fact, constructive mathematicians (going all the way back to Brouwer) have *already* been using linear logic without realizing it!

Let me explain what I mean by this and how it comes about — starting with an explanation, for a category theorist, of what linear logic *is* in the first place.

When we first learn about logic, we often learn various tautologies such as de Morgan’s laws $<semantics>\neg (P\wedge Q)\equiv (\neg P\vee \neg Q)<annotation\; encoding="application/x-tex">\backslash neg(P\; \backslash wedge\; Q\; )\; \backslash equiv\; (\backslash neg\; P\; \backslash vee\; \backslash neg\; Q)</annotation></semantics>$ and the law of excluded middle $<semantics>P\vee \neg P<annotation\; encoding="application/x-tex">P\; \backslash vee\; \backslash neg\; P</annotation></semantics>$. (As usual, $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ denotes “and”, while $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$ denotes “or”.) The field of *algebraic semantics* of logic starts with the observation that these same laws can be regarded as axioms on a poset, with $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ denoting the binary meet (greatest lower bound) and $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$ the join (least upper bound). The laws of classical logic correspond to requiring such a poset to be a Boolean algebra. Thus, a proof in propositional logic actually shows an equation that must hold in all Boolean algebras.

This suggests there should be other “logics” corresponding to other ordered/algebraic structures. For instance, a Heyting algebra is a lattice that, considered as a thin category, is cartesian closed. That is, in addition to the meet $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ and join $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$, it has an “implication” $<semantics>P\to Q<annotation\; encoding="application/x-tex">P\backslash to\; Q</annotation></semantics>$ satisfying $<semantics>(P\le Q\to R)\iff (P\wedge Q\le R)<annotation\; encoding="application/x-tex">(P\backslash le\; Q\backslash to\; R)\; \backslash iff\; (P\; \backslash wedge\; Q\; \backslash le\; R)</annotation></semantics>$. Any Boolean algebra is a Heyting algebra (with $<semantics>(P\to Q)\equiv (\neg P\vee Q)<annotation\; encoding="application/x-tex">(P\backslash to\; Q)\; \backslash equiv\; (\backslash neg\; P\; \backslash vee\; Q)</annotation></semantics>$), but not conversely. For instance, the open-set lattice of a topological space is a Heyting algebra, but not generally a Boolean one. The logic corresponding to Heyting algebras is usually called intuitionistic logic, and as noted above it is also the traditional logic of constructive mathematics.

(*Note:* calling this “intuitionistic logic” is unfaithful to Brouwer’s original meaning of “intuitionism”, but unfortunately there seems to be no better name for it.)

Now we can further weaken the notion of Heyting algebra by asking for a closed symmetric *monoidal* lattice instead of a *cartesian* closed one. The corresponding logic is called *intuitionistic linear logic*: in addition to the meet $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ and join $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$, it has a tensor product $<semantics>\otimes <annotation\; encoding="application/x-tex">\backslash otimes</annotation></semantics>$ with internal-hom $<semantics>\u22b8<annotation\; encoding="application/x-tex">\backslash multimap</annotation></semantics>$ satisfying an adjunction $<semantics>(P\le Q\u22b8R)\iff (P\otimes Q\le R)<annotation\; encoding="application/x-tex">(P\backslash le\; Q\backslash multimap\; R)\; \backslash iff\; (P\; \backslash otimes\; Q\; \backslash le\; R)</annotation></semantics>$. Logically, both $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ and $<semantics>\otimes <annotation\; encoding="application/x-tex">\backslash otimes</annotation></semantics>$ are versions of “and”; we call $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ the “additive conjunction” and $<semantics>\otimes <annotation\; encoding="application/x-tex">\backslash otimes</annotation></semantics>$ the “multiplicative conjunction”.

Note that to get from closed symmetric monoidal lattices to Boolean algebras by way of Heyting algebras, we first make the monoidal structure cartesian and then impose self-duality. (A Boolean algebra is precisely a Heyting algebra such that $<semantics>P\equiv (P\to 0)\to 0<annotation\; encoding="application/x-tex">P\; \backslash equiv\; (P\backslash to\; 0)\backslash to\; 0</annotation></semantics>$, where $<semantics>0<annotation\; encoding="application/x-tex">0</annotation></semantics>$ is the bottom element and $<semantics>P\to 0<annotation\; encoding="application/x-tex">P\backslash to\; 0</annotation></semantics>$ is the intuitionistic “not $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$”.) But we can also impose self-duality *first* and *then* make the monoidal structure cartesian, obtaining an intermediate structure called a star-autonomous lattice: a closed symmetric monoidal lattice equipped with an object $<semantics>\perp <annotation\; encoding="application/x-tex">\backslash bot</annotation></semantics>$ (not necessarily the bottom element) such that $<semantics>P\equiv (P\u22b8\perp )\u22b8\perp <annotation\; encoding="application/x-tex">P\; \backslash equiv\; (P\backslash multimap\; \backslash bot)\backslash multimap\; \backslash bot</annotation></semantics>$. Such a lattice has a contravariant involution defined by $<semantics>{P}^{\perp}=(P\u22b8\perp )<annotation\; encoding="application/x-tex">P^\backslash perp\; =\; (P\; \backslash multimap\; \backslash bot)</annotation></semantics>$ and a “cotensor product” $<semantics>(P\u214bQ)\equiv ({P}^{\perp}\otimes {Q}^{\perp}{)}^{\perp}<annotation\; encoding="application/x-tex">(P\; \backslash parr\; Q)\; \backslash equiv\; (P^\backslash perp\; \backslash otimes\; Q^\backslash perp)^\backslash perp</annotation></semantics>$, and its internal-hom is definable in terms of these: $<semantics>(P\u22b8Q)\equiv ({P}^{\perp}\u214bQ)<annotation\; encoding="application/x-tex">(P\; \backslash multimap\; Q)\; \backslash equiv\; (P^\backslash perp\; \backslash parr\; Q)</annotation></semantics>$. Its logic is called (classical) *linear logic*: in addition to the two conjunctions $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ and $<semantics>\otimes <annotation\; encoding="application/x-tex">\backslash otimes</annotation></semantics>$, it has two disjunctions $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$ and $<semantics>\u214b<annotation\; encoding="application/x-tex">\backslash parr</annotation></semantics>$, again called “additive” and “multiplicative”.

Star-autonomous lattices are not quite as easy to come by as Heyting algebras, but one general way to produce them is the Chu construction. (I blogged about this last year from a rather different perspective; in this post we’re restricting it to the case of posets.) Suppose $<semantics>C<annotation\; encoding="application/x-tex">C</annotation></semantics>$ is a closed symmetric monoidal lattice, and $<semantics>\perp <annotation\; encoding="application/x-tex">\backslash bot</annotation></semantics>$ is *any* element of $<semantics>C<annotation\; encoding="application/x-tex">C</annotation></semantics>$ at all. Then there is a star-autonomous lattice $<semantics>\mathrm{Chu}(C,\perp )<annotation\; encoding="application/x-tex">Chu(C,\backslash bot)</annotation></semantics>$ whose objects are pairs $<semantics>({P}^{+},{P}^{-})<annotation\; encoding="application/x-tex">(P^+,P^-)</annotation></semantics>$ such that $<semantics>{P}^{+}\otimes {P}^{-}\le \perp <annotation\; encoding="application/x-tex">P^+\; \backslash otimes\; P^-\; \backslash le\; \backslash bot</annotation></semantics>$, and whose order is defined by
$$<semantics>(({P}^{+},{P}^{-})\le ({Q}^{+},{Q}^{-}))\iff ({P}^{+}\le {Q}^{+})\phantom{\rule{thickmathspace}{0ex}}\text{and}\phantom{\rule{thickmathspace}{0ex}}({Q}^{-}\le {P}^{-}).<annotation\; encoding="application/x-tex">((P^+,P^-)\; \backslash le\; (Q^+,Q^-))\; \backslash iff\; (P^+\; \backslash le\; Q^+)\; \backslash ;\backslash text\{and\}\backslash ;\; (Q^-\; \backslash le\; P^-).\; </annotation></semantics>$$
In other words, $<semantics>\mathrm{Chu}(C,\perp )<annotation\; encoding="application/x-tex">Chu(C,\backslash bot)</annotation></semantics>$ is a full sub-poset of $<semantics>C\times {C}^{\mathrm{op}}<annotation\; encoding="application/x-tex">C\backslash times\; C^\{op\}</annotation></semantics>$. Its lattice operations are pointwise:
$$<semantics>({P}^{+},{P}^{-})\wedge ({Q}^{+},{Q}^{-})\equiv ({P}^{+}\wedge {Q}^{+},{P}^{-}\vee {Q}^{-})<annotation\; encoding="application/x-tex">\; (P^+,P^-)\; \backslash wedge\; (Q^+,Q^-)\; \backslash equiv\; (P^+\; \backslash wedge\; Q^+,\; P^-\; \backslash vee\; Q^-)\; </annotation></semantics>$$
$$<semantics>({P}^{+},{P}^{-})\vee ({Q}^{+},{Q}^{-})\equiv ({P}^{+}\vee {Q}^{+},{P}^{-}\wedge {Q}^{-})<annotation\; encoding="application/x-tex">\; (P^+,P^-)\; \backslash vee\; (Q^+,Q^-)\; \backslash equiv\; (P^+\; \backslash vee\; Q^+,\; P^-\; \backslash wedge\; Q^-)\; </annotation></semantics>$$
while the tensor product is more interesting:
$$<semantics>({P}^{+},{P}^{-})\otimes ({Q}^{+},{Q}^{-})\equiv ({P}^{+}\otimes {Q}^{+},({P}^{+}\u22b8{Q}^{-})\wedge ({Q}^{+}\u22b8{P}^{-}))<annotation\; encoding="application/x-tex">\; (P^+,P^-)\; \backslash otimes\; (Q^+,Q^-)\; \backslash equiv\; (P^+\; \backslash otimes\; Q^+,\; (P^+\; \backslash multimap\; Q^-)\; \backslash wedge\; (Q^+\; \backslash multimap\; P^-))</annotation></semantics>$$
The self-duality is
$$<semantics>({P}^{+},{P}^{-}{)}^{\perp}\equiv ({P}^{-},{P}^{+}).<annotation\; encoding="application/x-tex">\; (P^+,P^-)^\backslash bot\; \backslash equiv\; (P^-,P^+).</annotation></semantics>$$
from which we can deduce the definitions of $<semantics>\u214b<annotation\; encoding="application/x-tex">\backslash parr</annotation></semantics>$ and $<semantics>\u22b8<annotation\; encoding="application/x-tex">\backslash multimap</annotation></semantics>$:
$$<semantics>({P}^{+},{P}^{-})\u214b({Q}^{+},{Q}^{-})\equiv (({P}^{-}\u22b8{Q}^{+})\wedge ({Q}^{-}\u22b8{P}^{+}),{P}^{-}\otimes {Q}^{-}).<annotation\; encoding="application/x-tex">\; (P^+,P^-)\; \backslash parr\; (Q^+,Q^-)\; \backslash equiv\; ((P^-\; \backslash multimap\; Q^+)\; \backslash wedge\; (Q^-\; \backslash multimap\; P^+),\; P^-\; \backslash otimes\; Q^-).\; </annotation></semantics>$$
$$<semantics>({P}^{+},{P}^{-})\u22b8({Q}^{+},{Q}^{-})\equiv (({P}^{+}\u22b8{Q}^{+})\wedge ({Q}^{-}\u22b8{P}^{-}),{P}^{+}\otimes {Q}^{-}).<annotation\; encoding="application/x-tex">\; (P^+,P^-)\; \backslash multimap\; (Q^+,Q^-)\; \backslash equiv\; ((P^+\; \backslash multimap\; Q^+)\; \backslash wedge\; (Q^-\; \backslash multimap\; P^-),\; P^+\; \backslash otimes\; Q^-).\; </annotation></semantics>$$

Where do closed symmetric monoidal lattices come from? Well, they include all Heyting algebras! So if we have any Heyting algebra, all we need to do to get a star-autonomous lattice from the Chu construction is to pick an object $<semantics>\perp <annotation\; encoding="application/x-tex">\backslash bot</annotation></semantics>$. One natural choice is the bottom element $<semantics>0<annotation\; encoding="application/x-tex">0</annotation></semantics>$, since that *would* be the self-dualizing object if our Heyting algebra were a Boolean algebra. The resulting monoidal structure on $<semantics>\mathrm{Chu}(H,0)<annotation\; encoding="application/x-tex">Chu(H,0)</annotation></semantics>$ is actually semicartesian: the monoidal unit coincides with the top element $<semantics>(1,0)<annotation\; encoding="application/x-tex">(1,0)</annotation></semantics>$.

The point, now, is to look at this Chu construction $<semantics>\mathrm{Chu}(H,0)<annotation\; encoding="application/x-tex">Chu(H,0)</annotation></semantics>$ from the *logical* perspective, where elements of $<semantics>H<annotation\; encoding="application/x-tex">H</annotation></semantics>$ are regarded as propositions in intuitionistic logic. The elements of $<semantics>\mathrm{Chu}(H,0)<annotation\; encoding="application/x-tex">Chu(H,0)</annotation></semantics>$ are pairs $<semantics>({P}^{+},{P}^{-})<annotation\; encoding="application/x-tex">(P^+,\; P^-)</annotation></semantics>$ such that $<semantics>{P}^{+}\wedge {P}^{-}=0<annotation\; encoding="application/x-tex">P^+\; \backslash wedge\; P^-\; =\; 0</annotation></semantics>$, i.e. pairs of mutually incompatible propositions. We think of such a pair as a proposition $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ together with information $<semantics>{P}^{+}<annotation\; encoding="application/x-tex">P^+</annotation></semantics>$ about what it means to *affirm* or *prove* it and also information $<semantics>{P}^{-}<annotation\; encoding="application/x-tex">P^-</annotation></semantics>$ about what it means to *refute* or *disprove* it. The condition $<semantics>{P}^{+}\wedge {P}^{-}=0<annotation\; encoding="application/x-tex">P^+\; \backslash wedge\; P^-\; =\; 0</annotation></semantics>$ means that $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ cannot be both proven and refuted; but we might still have propositions such as $<semantics>(0,0)<annotation\; encoding="application/x-tex">(0,0)</annotation></semantics>$ which can be *neither* proven nor refuted.

The above definitions of the operations in a Chu construction similarly translate into “explanations” of the additive connectives $<semantics>\wedge ,\vee <annotation\; encoding="application/x-tex">\backslash wedge,\backslash vee</annotation></semantics>$ and the multiplicative connectives $<semantics>\otimes ,\u214b<annotation\; encoding="application/x-tex">\backslash otimes,\backslash parr</annotation></semantics>$ in terms of affirmations and refutations:

- A proof of $<semantics>P\wedge Q<annotation\; encoding="application/x-tex">P\backslash wedge\; Q</annotation></semantics>$ is a proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ together with a proof of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$. A refutation of $<semantics>P\wedge Q<annotation\; encoding="application/x-tex">P\backslash wedge\; Q</annotation></semantics>$ is either a refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ or a refutation of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$.
- A proof of $<semantics>P\vee Q<annotation\; encoding="application/x-tex">P\backslash vee\; Q</annotation></semantics>$ is either a proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ or a proof of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$. A refutation of $<semantics>P\vee Q<annotation\; encoding="application/x-tex">P\backslash vee\; Q</annotation></semantics>$ is a refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ together with a refutation of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$.
- A proof of $<semantics>{P}^{\perp}<annotation\; encoding="application/x-tex">P^\backslash perp</annotation></semantics>$ is a refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$. A refutation of $<semantics>{P}^{\perp}<annotation\; encoding="application/x-tex">P^\backslash perp</annotation></semantics>$ is a proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$.
- A proof of $<semantics>P\otimes Q<annotation\; encoding="application/x-tex">P\backslash otimes\; Q</annotation></semantics>$ is, like for $<semantics>P\wedge Q<annotation\; encoding="application/x-tex">P\backslash wedge\; Q</annotation></semantics>$, a proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ together with a proof of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$. But a refutation of $<semantics>P\otimes Q<annotation\; encoding="application/x-tex">P\backslash otimes\; Q</annotation></semantics>$ consists of both (1) a construction of a refutation of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$ from any proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$, and (2) a construction of a refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ from any proof of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$.
- A proof of $<semantics>P\u214bQ<annotation\; encoding="application/x-tex">P\backslash parr\; Q</annotation></semantics>$ consists of both (1) a construction of a proof of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$ from any refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$, and (2) a construction of a proof of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ from any refutation of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$. A refutation of $<semantics>P\u214bQ<annotation\; encoding="application/x-tex">P\backslash parr\; Q</annotation></semantics>$ is, like for $<semantics>P\vee Q<annotation\; encoding="application/x-tex">P\backslash vee\; Q</annotation></semantics>$, a refutation of $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ together with a refutation of $<semantics>Q<annotation\; encoding="application/x-tex">Q</annotation></semantics>$.

These explanations constitute a “meaning explanation” for classical linear logic, parallel to the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic. In particular, they justify the characteristic features of linear logic. For instance, the “additive law of excluded middle” $<semantics>P\vee {P}^{\perp}<annotation\; encoding="application/x-tex">P\; \backslash vee\; P^\backslash perp</annotation></semantics>$ fails for the same reason that it fails under the BHK-interpretation: we cannot decide for an arbitrary $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$ whether to prove it or refute it. However, the “multiplicative law of excluded middle” $<semantics>P\u214b{P}^{\perp}<annotation\; encoding="application/x-tex">P\; \backslash parr\; P^\backslash perp</annotation></semantics>$ is a tautology: if we can refute $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$, then by definition we can prove $<semantics>{P}^{\perp}<annotation\; encoding="application/x-tex">P^\backslash perp</annotation></semantics>$, while if we can refute $<semantics>{P}^{\perp}<annotation\; encoding="application/x-tex">P^\backslash perp</annotation></semantics>$, then again by definition we can prove $<semantics>P<annotation\; encoding="application/x-tex">P</annotation></semantics>$. In general, the multiplicative disjunction $<semantics>\u214b<annotation\; encoding="application/x-tex">\backslash parr</annotation></semantics>$ often carries the “constructive content” of what the *classical* mathematician means by “or”, whereas the additive one $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$ carries the constructive mathematician’s meaning. (Note also that the “proof” clauses of $<semantics>P\u214bQ<annotation\; encoding="application/x-tex">P\backslash parr\; Q</annotation></semantics>$ are essentially the disjunctive syllogism.)

I think this is already pretty neat. Linear logic, regarded as a *logic*, has always been rather mysterious to me, especially the multiplicative disjunction $<semantics>\u214b<annotation\; encoding="application/x-tex">\backslash parr</annotation></semantics>$ (and I know I’m not alone in that). But this construction explains both of them quite nicely.

However, there’s more. It’s not much good to have a logic if we can’t do mathematics with it, so let’s do some mathematics in linear logic, translate it into intuitionistic logic along this Chu construction, and see what we get. In this post I won’t be precise about the context in which this happens; the paper formalizes it more carefully as a “linear tripos”. Following the paper, I’ll call this the *standard interpretation*.

To start with, every set should have an equality relation. Thus, a set in linear logic (which I’ll call an “L-set”) will have a relation $<semantics>(x{=}^{L}y)<annotation\; encoding="application/x-tex">(x=^L\; y)</annotation></semantics>$ valued in linear propositions. Since each of these is an element of $<semantics>\mathrm{Chu}(H,0)<annotation\; encoding="application/x-tex">Chu(H,0)</annotation></semantics>$, it is a *pair* of mutually incompatible intuitionistic relations; we will call these $<semantics>(x{=}^{I}y<annotation\; encoding="application/x-tex">(x=^I\; y</annotation></semantics>$ and $<semantics>(x{\ne}^{I}y)<annotation\; encoding="application/x-tex">(x\backslash neq^I\; y)</annotation></semantics>$ respectively.

Now we expect equality to be, among other things, a reflexive, symmetric, and transitive relation. Reflexivity means that $<semantics>(x{=}^{I}x,x{\ne}^{I}x)\equiv (1,0)<annotation\; encoding="application/x-tex">(x=^I\; x,\; x\backslash neq^I\; x)\; \backslash equiv\; (1,0)</annotation></semantics>$, i.e. that $<semantics>x{=}^{I}x<annotation\; encoding="application/x-tex">x\; =^I\; x</annotation></semantics>$ is true (that is, $<semantics>{=}^{I}<annotation\; encoding="application/x-tex">=^I</annotation></semantics>$ is reflexive) and $<semantics>x{\ne}^{I}x<annotation\; encoding="application/x-tex">x\; \backslash neq^I\; x</annotation></semantics>$ is false (that is, $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ is irreflexive). Symmetry means that $<semantics>(x{=}^{I}y,x{\ne}^{I}y)\equiv (y{=}^{I}x,y{\ne}^{I}x)<annotation\; encoding="application/x-tex">(x=^I\; y,\; x\backslash neq^I\; y)\; \backslash equiv\; (y=^I\; x,\; y\backslash neq^I\; x)</annotation></semantics>$, i.e. that $<semantics>(x{=}^{I}y)\equiv (y{=}^{I}x)<annotation\; encoding="application/x-tex">(x=^I\; y)\; \backslash equiv\; (y=^I\; x)</annotation></semantics>$ and $<semantics>(x{\ne}^{I}y)\equiv (y{\ne}^{I}x)<annotation\; encoding="application/x-tex">(x\backslash neq^I\; y)\; \backslash equiv\; (y\backslash neq^I\; x)</annotation></semantics>$: that is, $<semantics>{=}^{I}<annotation\; encoding="application/x-tex">=^I</annotation></semantics>$ and $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ are both symmetric.

Of course, transitivity says that if $<semantics>x=y<annotation\; encoding="application/x-tex">x=y</annotation></semantics>$ and $<semantics>y=z<annotation\; encoding="application/x-tex">y=z</annotation></semantics>$, then $<semantics>x=z<annotation\; encoding="application/x-tex">x=z</annotation></semantics>$. But in linear logic we have two different “and”s; which do we mean here? Suppose first we use the additive conjunction $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$, so that transitivity says $<semantics>(x{=}^{L}y)\wedge (y{=}^{L}z)\u22a2(x{=}^{L}z)<annotation\; encoding="application/x-tex">(x=^L\; y)\; \backslash wedge\; (y=^L\; z)\; \backslash vdash\; (x=^L\; z)</annotation></semantics>$ (here $<semantics>\u22a2<annotation\; encoding="application/x-tex">\backslash vdash</annotation></semantics>$ is the logical equivalent of the inequality $<semantics>\le <annotation\; encoding="application/x-tex">\backslash le</annotation></semantics>$ in our lattices). Using the definition of $<semantics>\wedge <annotation\; encoding="application/x-tex">\backslash wedge</annotation></semantics>$ in $<semantics>\mathrm{Chu}(H,0)<annotation\; encoding="application/x-tex">Chu(H,0)</annotation></semantics>$, this says that $<semantics>(x{=}^{I}y)\wedge (y{=}^{I}z)\u22a2(x{=}^{I}z)<annotation\; encoding="application/x-tex">(x=^I\; y)\; \backslash wedge\; (y=^I\; z)\; \backslash vdash\; (x=^I\; z)</annotation></semantics>$ (that is, $<semantics>{=}^{I}<annotation\; encoding="application/x-tex">=^I</annotation></semantics>$ is transitive) and $<semantics>(x{\ne}^{I}z)\u22a2(x{\ne}^{I}y)\vee (y{\ne}^{I}z)<annotation\; encoding="application/x-tex">(x\backslash neq^I\; z)\; \backslash vdash\; (x\backslash neq^I\; y)\; \backslash vee\; (y\backslash neq^I\; z)</annotation></semantics>$ (this is sometimes called comparison).

Put together, the assertions that $<semantics>{=}^{L}<annotation\; encoding="application/x-tex">=^L</annotation></semantics>$ is reflexive, symmetric, and transitive say that (1) $<semantics>{=}^{I}<annotation\; encoding="application/x-tex">=^I</annotation></semantics>$ is reflexive, symmetric, and transitive, and (2) $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ is irreflexive, symmetric, and a comparison. Part (1) says essentially that an L-set has an underlying I-set, while (2) says that this I-set is equipped with an apartness relation.

Apartness relations are a well-known notion in constructive mathematics; if you’ve never encountered them before, here’s the idea. In classical mathematics, if we need to say that two things are “distinct” we just say that they are “not equal”. However, in intuitionistic logic, being “not equal” is not always a very useful thing. For instance, we cannot prove intuitionistically that if a real number is not equal to zero then it is invertible. However, we can prove that every real number that is *apart* from zero is invertible, where two real numbers are “apart” if there is a positive rational distance between them. This notion of “apart” is an irreflexive symmetric comparison which is stronger than “not equal”, and many other sets in intuitionistic mathematics are similarly equipped with such relations.

The point is that the standard interpretation *automatically* produces the notion of “apartness relation”, which constructive mathematicians using intuitionistic logic were led to from purely practical considerations. This same sort of thing happens over and over, and is what I mean by “constructive mathematicians have been using linear logic without realizing it”: they invented lots of concepts which are invisible to classical mathematics, and which may seem *ad hoc* at first, but actually arise automatically if we do mathematics “naturally” in linear logic and then pass across the standard interpretation.

To convince you that this happens all over the place, here are a bunch more examples.

If $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ are L-sets, then in the cartesian product L-set $<semantics>A\times B<annotation\; encoding="application/x-tex">A\backslash times\; B</annotation></semantics>$ we have $<semantics>({a}_{1},{b}_{1}){=}^{L}({a}_{2},{b}_{2})<annotation\; encoding="application/x-tex">(a\_1,b\_1)\; =^L\; (a\_2,b\_2)</annotation></semantics>$ defined by $<semantics>({a}_{1}{=}^{L}{a}_{2})\wedge ({b}_{1}{=}^{L}{b}_{2})<annotation\; encoding="application/x-tex">(a\_1\; =^L\; a\_2)\; \backslash wedge\; (b\_1\; =^L\; b\_2)</annotation></semantics>$. In the standard interpretation, this corresponds to the usual pairwise equality for ordered pairs and a disjunctive apartness: $<semantics>({a}_{1},{b}_{1}){\ne}^{I}({a}_{2},{b}_{2})<annotation\; encoding="application/x-tex">(a\_1,b\_1)\; \backslash neq^I\; (a\_2,b\_2)</annotation></semantics>$ means $<semantics>({a}_{1}{\ne}^{I}{a}_{2})\vee ({b}_{1}{\ne}^{I}{b}_{2})<annotation\; encoding="application/x-tex">(a\_1\; \backslash neq^I\; a\_2)\; \backslash vee\; (b\_1\; \backslash neq^I\; b\_2)</annotation></semantics>$. That is, two ordered pairs differ if they differ in one component.

If $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ and $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ are sets, the elements of the function L-set $<semantics>A\to B<annotation\; encoding="application/x-tex">A\backslash to\; B</annotation></semantics>$ are I-functions that are

*strongly extensional*: $<semantics>(f(x){\ne}^{I}f(y))\u22a2(x{\ne}^{I}y)<annotation\; encoding="application/x-tex">(f(x)\; \backslash neq^I\; f(y))\; \backslash vdash\; (x\backslash neq^I\; y)</annotation></semantics>$ (this is called “strong” because the apartness $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ may be stronger than “not equal”). This is a common condition on functions between sets with apartness relations.Equality between functions $<semantics>(f{=}^{L}g)<annotation\; encoding="application/x-tex">(f\; =^L\; g)</annotation></semantics>$ is defined by $<semantics>\forall x.(f(x){=}^{L}g(x))<annotation\; encoding="application/x-tex">\backslash forall\; x.\; (f(x)\; =^L\; g(x))</annotation></semantics>$. I didn’t talk about quantifiers $<semantics>\forall /\exists <annotation\; encoding="application/x-tex">\backslash forall/\backslash exists</annotation></semantics>$ above, but they act like infinitary versions of $<semantics>\wedge /\vee <annotation\; encoding="application/x-tex">\backslash wedge/\backslash vee</annotation></semantics>$. Thus we get $<semantics>(f{=}^{I}g)<annotation\; encoding="application/x-tex">(f=^I\; g)</annotation></semantics>$ meaning $<semantics>\forall x.(f(x){=}^{I}g(x))<annotation\; encoding="application/x-tex">\backslash forall\; x.\; (f(x)\; =^I\; g(x))</annotation></semantics>$, the usual pointwise equality of functions, and $<semantics>(f{\ne}^{I}g)<annotation\; encoding="application/x-tex">(f\backslash neq^I\; g)</annotation></semantics>$ meaning $<semantics>\exists x.(f(x){\ne}^{I}g(x))<annotation\; encoding="application/x-tex">\backslash exists\; x.\; (f(x)\; \backslash neq^I\; g(x))</annotation></semantics>$: two functions differ if they differ on at least one input.

Because linear propositions are

*pairs*of intuitionistic propositions, the elements of the L-powerset $<semantics>P(A)<annotation\; encoding="application/x-tex">P(A)</annotation></semantics>$ of an L-set $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ are pairs $<semantics>({U}^{+},{U}^{-})<annotation\; encoding="application/x-tex">(U^+,U^-)</annotation></semantics>$ of I-subsets of the underlying I-set of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$, which must additionally be*strongly disjoint*in that $<semantics>(x\in {U}^{+})\wedge (y\in {U}^{-})\u22a2(x{\ne}^{I}y)<annotation\; encoding="application/x-tex">(x\backslash in\; U^+)\; \backslash wedge\; (y\backslash in\; U^-)\; \backslash vdash\; (x\backslash neq^I\; y)</annotation></semantics>$. Bishop and Bridges introduced such pairs in their book*Constructive analysis*under the name “complemented subset”. The substitution law $<semantics>(x{=}^{L}y)\wedge (x\in U)\u22a2(y\in U)<annotation\; encoding="application/x-tex">(x=^L\; y)\; \backslash wedge\; (x\backslash in\; U)\; \backslash vdash\; (y\backslash in\; U)</annotation></semantics>$ translates to the requirement that $<semantics>{U}^{-}<annotation\; encoding="application/x-tex">U^-</annotation></semantics>$ be*strongly extensional*(also called “$<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$-open”): $<semantics>(y\in {U}^{-})\u22a2(x{\ne}^{I}y)\vee (x\in {U}^{-})<annotation\; encoding="application/x-tex">(y\backslash in\; U^-)\; \backslash vdash\; (x\backslash neq^I\; y)\; \backslash vee\; (x\backslash in\; U^-)</annotation></semantics>$.Equality between L-subsets $<semantics>(U{=}^{L}V)<annotation\; encoding="application/x-tex">(U=^L\; V)</annotation></semantics>$ means $<semantics>\forall x.((x\in U\u22b8x\in V)\wedge (x\in V\u22b8x\in U))<annotation\; encoding="application/x-tex">\backslash forall\; x.\; ((x\backslash in\; U\; \backslash multimap\; x\backslash in\; V)\; \backslash wedge\; (x\backslash in\; V\; \backslash multimap\; x\backslash in\; U))</annotation></semantics>$. In the standard interpretation, $<semantics>({U}^{+},{U}^{-}){=}^{I}({V}^{+},{V}^{-})<annotation\; encoding="application/x-tex">(U^+,U^-)\; =^I\; (V^+,V^-)</annotation></semantics>$ means $<semantics>({U}^{+}={V}^{+})\wedge ({U}^{-}={V}^{-})<annotation\; encoding="application/x-tex">(U^+=V^+)\; \backslash wedge\; (U^-\; =\; V^-)</annotation></semantics>$ as we would expect, while $<semantics>({U}^{+},{U}^{-}){\ne}^{I}({V}^{+},{V}^{-})<annotation\; encoding="application/x-tex">(U^+,U^-)\; \backslash neq^I\; (V^+,V^-)</annotation></semantics>$ means $<semantics>(\exists x\in {U}^{+}\cap {V}^{-})\vee (\exists x\in {U}^{-}\cap {V}^{+})<annotation\; encoding="application/x-tex">(\backslash exists\; x\; \backslash in\; U^+\; \backslash cap\; V^-)\; \backslash vee\; (\backslash exists\; x\backslash in\; U^-\; \backslash cap\; V^+)</annotation></semantics>$. In particular, we have $<semantics>U{\ne}^{L}\varnothing <annotation\; encoding="application/x-tex">U\; \backslash neq^L\; \backslash emptyset</annotation></semantics>$ (where the empty L-subset is $<semantics>(\varnothing ,A)<annotation\; encoding="application/x-tex">(\backslash emptyset,A)</annotation></semantics>$) just when $<semantics>\exists x\in {U}^{+}<annotation\; encoding="application/x-tex">\backslash exists\; x\backslash in\; U^+</annotation></semantics>$. So the obvious linear notion of “nonempty” translates to the intuitionistic notion of inhabited that constructive mathematicians have found to be much more useful than the intuitionistic “not empty”.

An L-group is an L-set with a multiplication $<semantics>m:G\times G\to G<annotation\; encoding="application/x-tex">m:G\backslash times\; G\backslash to\; G</annotation></semantics>$, unit $<semantics>e\in G<annotation\; encoding="application/x-tex">e\backslash in\; G</annotation></semantics>$, and inversion $<semantics>i:G\to G<annotation\; encoding="application/x-tex">i:G\backslash to\; G</annotation></semantics>$ satisfying the usual axioms. In the standard interpretation, this corresponds to an ordinary I-group equipped with an apartness $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ such that multiplication and inversion are strongly extensional: $<semantics>({x}^{-1}{\ne}^{I}{y}^{-1})\u22a2(x{\ne}^{I}y)<annotation\; encoding="application/x-tex">(x^\{-1\}\; \backslash neq^I\; y^\{-1\})\; \backslash vdash\; (x\; \backslash neq^I\; y)</annotation></semantics>$ and $<semantics>(xu{\ne}^{I}yv)\u22a2(x{\ne}^{I}y)\vee (u{\ne}^{I}v)<annotation\; encoding="application/x-tex">(x\; u\; \backslash neq^I\; y\; v)\; \backslash vdash\; (x\; \backslash neq^I\; y)\; \backslash vee\; (u\backslash neq^I\; v)</annotation></semantics>$. Groups with apartness have been studied in intuitionistic algebra going back to Heyting, and similarly for other algebraic structures like rings and modules.

An L-subgroup of an L-group corresponds to a complemented subset $<semantics>({H}^{+},{H}^{-})<annotation\; encoding="application/x-tex">(H^+,H^-)</annotation></semantics>$ as above such that $<semantics>{H}^{+}<annotation\; encoding="application/x-tex">H^+</annotation></semantics>$ is an ordinary I-subgroup and $<semantics>{H}^{-}<annotation\; encoding="application/x-tex">H^-</annotation></semantics>$ is an

*antisubgroup*, meaning a set not containing $<semantics>e<annotation\; encoding="application/x-tex">e</annotation></semantics>$, closed under inversion ($<semantics>(x\in {H}^{-})\u22a2({x}^{-1}\in {H}^{-})<annotation\; encoding="application/x-tex">(x\backslash in\; H^-)\; \backslash vdash\; (x^\{-1\}\backslash in\; H^-)</annotation></semantics>$), and satisfying $<semantics>(xy\in {H}^{-})\u22a2(x\in {H}^{-})\vee (y\in {H}^{-})<annotation\; encoding="application/x-tex">(x\; y\; \backslash in\; H^-)\; \backslash vdash\; (x\backslash in\; H^-)\; \backslash vee\; (y\backslash in\; H^-)</annotation></semantics>$. Antisubgroups have also been studied in constructive algebra; for instance, they enable us to define an apartness on the quotient $<semantics>G/{H}^{+}<annotation\; encoding="application/x-tex">G/H^+</annotation></semantics>$ by $<semantics>[x]{\ne}^{I}[y]<annotation\; encoding="application/x-tex">[x]\backslash neq^I\; [y]</annotation></semantics>$ if $<semantics>x{y}^{-1}\in {H}^{-}<annotation\; encoding="application/x-tex">x\; y^\{-1\}\; \backslash in\; H^-</annotation></semantics>$.An L-poset is an L-set with a relation $<semantics>{\le}^{L}<annotation\; encoding="application/x-tex">\backslash le^L</annotation></semantics>$ that is reflexive, transitive ($<semantics>(x{\le}^{L}y)\wedge (y{\le}^{L}z)\u22a2(x{\le}^{L}z)<annotation\; encoding="application/x-tex">(x\backslash le^L\; y)\; \backslash wedge\; (y\backslash le^L\; z)\; \backslash vdash\; (x\backslash le^L\; z)</annotation></semantics>$) and antisymmetric ($<semantics>(x{\le}^{L}y)\wedge (y{\le}^{L}x)\u22a2(x{=}^{L}y)<annotation\; encoding="application/x-tex">(x\backslash le^L\; y)\; \backslash wedge\; (y\backslash le^L\; x)\; \backslash vdash\; (x\; =^L\; y)</annotation></semantics>$). Under the standard interpretation, this corresponds to two binary relations, which it is suggestive to write $<semantics>{\le}^{I}<annotation\; encoding="application/x-tex">\backslash le^I</annotation></semantics>$ and $<semantics>{<}^{I}<annotation\; encoding="application/x-tex">\backslash lt^I</annotation></semantics>$: then $<semantics>{\le}^{I}<annotation\; encoding="application/x-tex">\backslash le^I</annotation></semantics>$ is an ordinary I-partial-order, $<semantics>{<}^{I}<annotation\; encoding="application/x-tex">\backslash lt^I</annotation></semantics>$ is a “bimodule” over it ($<semantics>(x{\le}^{I}y)\wedge (y{<}^{I}z)\u22a2(x{<}^{I}z)<annotation\; encoding="application/x-tex">(x\backslash le^I\; y)\; \backslash wedge\; (y\; \backslash lt^I\; z)\; \backslash vdash\; (x\backslash lt^I\; z)</annotation></semantics>$ and dually) which is cotransitive ($<semantics>(x{<}^{I}z)\u22a2(x{<}^{I}y)\vee (y{<}^{I}z)<annotation\; encoding="application/x-tex">(x\backslash lt^I\; z)\; \backslash vdash\; (x\backslash lt^I\; y)\; \backslash vee\; (y\backslash lt^I\; z)</annotation></semantics>$) and “anti-antisymmetric” in the sense that $<semantics>(x{\ne}^{I}y)\equiv ((x{<}^{I}y)\vee (y{<}^{I}x))<annotation\; encoding="application/x-tex">(x\backslash neq^I\; y)\; \backslash equiv\; ((x\backslash lt^I\; y)\; \backslash vee\; (y\backslash lt^I\; x))</annotation></semantics>$. Such pairs of strict and non-strict relations are quite common in constructive mathematics, for instance on the real numbers.

In the paper there are even more examples of this sort of thing. However, even *this* is not all! So far, we haven’t made any use of the *multiplicative* connectives in linear logic. It turns out that often, replacing some or all of the additive connectives in a definition by multiplicative ones yields, under the standard interpretation, a different intuitionistic version of the same classical definition that is also useful.

Here are a few examples. Note that by semicartesianness (or “affineness” of our logic), we have $<semantics>P\otimes Q\u22a2P\wedge Q<annotation\; encoding="application/x-tex">P\; \backslash otimes\; Q\; \backslash vdash\; P\backslash wedge\; Q</annotation></semantics>$ and $<semantics>P\vee Q\u22a2P\u214bQ<annotation\; encoding="application/x-tex">P\backslash vee\; Q\; \backslash vdash\; P\; \backslash parr\; Q</annotation></semantics>$ (in a general star-autonomous lattice, there may be no implication either way).

A “$<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$-field” is an L-ring such that $<semantics>(x{=}^{L}0)\vee \exists y.(xy{=}^{L}1)<annotation\; encoding="application/x-tex">(x=^L\; 0)\; \backslash vee\; \backslash exists\; y.\; (x\; y\; =^L\; 1)</annotation></semantics>$. Under the standard interpretation, this corresponds to an I-ring (with apartness) such that $<semantics>(x{=}^{I}0)\vee \exists y.(xy{=}^{I}1)<annotation\; encoding="application/x-tex">(x=^I\; 0)\; \backslash vee\; \backslash exists\; y.\; (x\; y\; =^I\; 1)</annotation></semantics>$, i.e. every element is either zero or invertible. The rational numbers are a field in this sense (sometimes called a

*geometric field*or*discrete field*), but the real numbers are not. On the other hand, a “$<semantics>\u214b<annotation\; encoding="application/x-tex">\backslash parr</annotation></semantics>$-field” is an L-ring such that $<semantics>(x{=}^{L}0)\u214b\exists y.(xy{=}^{L}1)<annotation\; encoding="application/x-tex">(x=^L\; 0)\; \backslash parr\; \backslash exists\; y.\; (x\; y\; =^L\; 1)</annotation></semantics>$. Under the standard interpretation, this corresponds to an I-ring (with apartness) such that $<semantics>(x{\ne}^{I}0)\to \exists y.(xy{=}^{I}1)<annotation\; encoding="application/x-tex">(x\; \backslash neq^I\; 0)\; \backslash to\; \backslash exists\; y.\; (x\; y\; =^I\; 1)</annotation></semantics>$ and $<semantics>(\forall y.(xy{\ne}^{I}1))\to (x{=}^{I}0)<annotation\; encoding="application/x-tex">(\backslash forall\; y.\; (x\; y\; \backslash neq^I\; 1))\; \backslash to\; (x\; =^I\; 0)</annotation></semantics>$, i.e. every element apart from zero is invertible and every “strongly noninvertible” element is zero. The real numbers are a field in this weaker sense (if the apartness is tight, this is called a Heyting field).In classical mathematics, $<semantics>x\le y<annotation\; encoding="application/x-tex">x\backslash le\; y</annotation></semantics>$ means $<semantics>(x<y)\vee (x=y)<annotation\; encoding="application/x-tex">(x\backslash lt\; y)\; \backslash vee\; (x=y)</annotation></semantics>$. Constructively, this is (again) true for integers and rationals but not the reals. However, it is true for the reals in linear logic that $<semantics>(x{\le}^{L}y)\equiv (x{<}^{L}y)\u214b(x{=}^{L}y)<annotation\; encoding="application/x-tex">(x\backslash le^L\; y)\; \backslash equiv\; (x\backslash lt^L\; y)\; \backslash parr\; (x=^L\; y)</annotation></semantics>$.

If in the definition of an L-set we weaken transitivity to $<semantics>(x{=}^{L}y)\otimes (y{=}^{L}z)\u22a2(x{=}^{L}z)<annotation\; encoding="application/x-tex">(x=^L\; y)\; \backslash otimes\; (y=^L\; z)\; \backslash vdash\; (x=^L\; z)</annotation></semantics>$, then in the standard interpretation the comparison condition disappears, so that $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ need only be irreflexive and symmetric, i.e. an inequality relation. (To be precise, the comparison condition is actually replaced by the weaker statement that $<semantics>{\ne}^{I}<annotation\; encoding="application/x-tex">\backslash neq^I</annotation></semantics>$ satisfies substitution with respect to $<semantics>{=}^{I}<annotation\; encoding="application/x-tex">=^I</annotation></semantics>$.) This is useful because not every I-set has an apartness relation, but every I-set does have at least one inequality relation, namely “not equal” (the denial inequality). There are also other inequality relations that are not apartnesses. For instance, the inequality defined above on L-powersets is not an apartness but is still stronger than “not equal”, and in the paper there is an example of a very naturally-occurring normal L-subgroup of a very naturally occurring L-group for which the inequality on the quotient is not an apartness.

Again, there are more examples in the paper, including generalized metric spaces and topological/apartness spaces. (I should warn you that the terminology and notation in the paper is somewhat different; in this post I’ve been constrained in the math symbols available, and also omitted some adjectives for simplicity.)

What does this observation mean for constructive mathematics? Well, there are three levels at which it can be applied. Firstly, we can use it as a “machine” for producing constructive versions of classical definitions: write the classical definition in linear logic, making additive or multiplicative choices for the connectives, and then pass across the standard interpretation. The examples in the paper suggest that this is more “automatic” and less error-prone than the usual process of searching for constructive versions of classical notions.

Secondly, we can also use it as a machine for producing *theorems* about such constructive notions, by writing a proof in linear logic (perhaps by simply translating a classical proof — many classical proofs remain valid without significant changes) and translating across the standard interpretation. This can save effort and prevent mistakes (e.g. we don’t have to manually keep track of all the apartness relations, or worry about forgetting to prove that some function is strongly extensional).

Thirdly, if we start to do that a lot, we may notice that we might as well be *doing constructive mathematics directly in linear logic*. Linear logic has a “computational” interpretation just like intuitionistic logic does — its proofs satisfy “cut-elimination” and the “disjunction and existence properties” — so it should be just as good at ensuring that mathematics has computational content, with the benefit of not having to deal explicitly with apartness relations and so on. And the “meaning explanation” I described above, in terms of proofs and refutations, *could* theoretically have been given by Brouwer or Heyting instead of the now-standard BHK-interpretation. So one might argue that the prevalence of the latter is just a historical accident; maybe in the future a *linear constructive mathematics* will grow up alongside today’s “intuitionistic constructive mathematics”.

## May 21, 2018

### Christian P. Robert - xi'an's og

**O**n July 2-4, 2018, there will be an ISBA sponsored workshop on Bayesian non-parametrics for signal and image processing, in Bordeaux, France. This is a wee bit further than Warwick (BAYsm) or Rennes (MCqMC), but still manageable from Edinburgh with direct flights (if on Ryanair). Deadline for free (yes, free!) registration is May 31.

### Emily Lakdawalla - The Planetary Society Blog

### ZapperZ - Physics and Physicists

It began with a theory -- scientists at the University of California knew graphene could convert light into electricity, and wondered whether that electricity had the capacity to stimulate human cells. Graphene isextremelysensitive to light (1,000 times more than traditional digital cameras and smartphones) and after experimenting with different light intensities, Alex Savchenko and his team discovered that cells could indeed be stimulated via optical graphene stimulation."

I was looking at the microscope's computer screen and I'm turning the knob for light intensity and I see the cells start beating faster," he said. "I showed that to our grad students and they were yelling and jumping and asking if they could turn the knob. We had never seen this possibility of controlling cell contraction."

The source paper can be found here, and it is open-access.

Again, this is why it is vital that funding in basic physics continues at a healthy pace. Even if you do not see the immediate application or benefit from many of these seemingly esoteric research, you just never know when any of the discovery and knowledge that are gained from such areas will turn into something that could save people's lives. We have seen such examples NUMEROUS times throughout history. Unfortunately, people are often ignorant at the origin of many of the benefits that they now take for granted.

Zz.

### Andrew Jaffe - Leaves on the Line

I have the unfortunate duty of using this blog to announce the death a couple of weeks ago of Professor Leon B Lucy, who had been a Visiting Professor working here at Imperial College from 1998.

Leon got his PhD in the early 1960s at the University of Manchester, and after postdoctoral positions in Europe and the US, worked at Columbia University and the European Southern Observatory over the years, before coming to Imperial. He made significant contributions to the study of the evolution of stars, understanding in particular how they lose mass over the course of their evolution, and how very close binary stars interact and evolve inside their common envelope of hot gas.

Perhaps most importantly, early in his career Leon realised how useful computers could be in astrophysics. He made two major methodological contributions to astrophysical simulations. First, he realised that by simulating randomised trajectories of single particles, he could take into account more physical processes that occur inside stars. This is now called “Monte Carlo Radiative Transfer” (scientists often use the term “Monte Carlo” — after the European gambling capital — for techniques using random numbers). He also invented the technique now called smoothed-particle hydrodynamics which models gases and fluids as aggregates of pseudo-particles, now applied to models of stars, galaxies, and the large scale structure of the Universe, as well as many uses outside of astrophysics.

Leon’s other major numerical contributions comprise advanced techniques for interpreting the complicated astronomical data we get from our telescopes. In this realm, he was most famous for developing the methods, now known as Lucy-Richardson deconvolution, that were used for correcting the distorted images from the Hubble Space Telescope, before NASA was able to send a team of astronauts to install correcting lenses in the early 1990s.

For all of this work Leon was awarded the Gold Medal of the Royal Astronomical Society in 2000. Since then, Leon kept working on data analysis and stellar astrophysics — even during his illness, he asked me to help organise the submission and editing of what turned out to be his final papers, on extracting information on binary-star orbits and (a subject dear to my heart) the statistics of testing scientific models.

Until the end of last year, Leon was a regular presence here at Imperial, always ready to contribute an occasionally curmudgeonly but always insightful comment on the science (and sociology) of nearly any topic in astrophysics. We hope that we will be able to appropriately memorialise his life and work here at Imperial and elsewhere. He is survived by his wife and daughter. He will be missed.

### Peter Coles - In the Dark

So here I am in Dublin Airport, waiting for my flight back to Cardiff. It’s been a nice weekend in Ireland, with good weather and lots to do in and around Maynooth. In the course of my perambulations on Saturday I came across a group of people campaigning to Repeal the Eighth Amendment. I A referendum on that issue takes place on Friday this week (25th). I bought a badge from them, which I’m happy to wear in solidarity:

There are lots of posters around supporting supporting one or other side in the campaign. It’s very noticeable that the `Yes’ ones seem to be getting torn down quite regularly. It’s also noticeable that the `No’ ones are frequently rather crude and sometimes offensive. That’s a shame because there is a serious ethical issue at stake, and a grown-up debate is important. Still, past experience suggests that referendums and grown-up debates don’t necessarily go together.

I won’t be in Ireland for the vote, but I hope the ‘yes’ campaign succeeds in removing what I think is a daft piece of law. If it fails then it won’t stop Irish women having terminations, it will just mean that the continue to have to travel abroad (if they can afford to do so) or take terrible risks have an illegal abortion at home (if they can’t) . For me, a vote for `No’ is therefore just a vote for hypocrisy.

Incidentally, a letter arrived at my Cardiff residence a few days ago from the Human Resources Department at Cardiff University, acknowledging my resignation (which I handed in about 6 weeks ago). I noticed that the letter contains the sentence `We have sent this letter to the home address we have on record for you. If this address is incorrect please contact us..’. Hmmm. If the letter had gone to the wrong address how would I know?

Anyway, I’ll be back in Cardiff for the next week, with another set of exams to mark in a few days, then back to Maynooth. And now it’s time to go to the gate.

Follow @telescoper## May 20, 2018

### Christian P. Robert - xi'an's og

A (rather urgent) call for candidates for PhD studentships in AI and Data Science at Warwick. These are part of Warwick’s MRC-Funded Doctoral Training Program in Interdisciplinary Biomedical Research. Selected students would start in Autumn 2018. The deadline is on June 3.

### The n-Category Cafe

My student Brandon Coya finished his thesis, and successfully defended it last Tuesday!

• Brandon Coya, *Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective*, Ph.D. thesis, U. C. Riverside, 2018.

It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.

His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!

However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results. What follows is heavily based on the introduction of his thesis, but I’ve baezified it a little.

### The basic idea

People, and especially scientists and engineers, are naturally inclined to draw diagrams and pictures when they want to better understand a problem. One example is when Feynman introduced his famous diagrams in 1949; particle physicists have been using them ever since. But some other diagrams introduced by engineers are far more important to the functioning of the modern world and its technology. It’s outrageous, but sociologically understandable, that mathematicians have figured out more about Feynman diagrams than these other kinds: circuit diagrams, bond graphs and signal-flow diagrams. This is the problem Brandon aims to fix.

I’ve been unable to track down the early history of circuit diagrams, so if you know about that please tell me! But in the 1940s, Harry Olson pointed out analogies in electrical, mechanical, thermodynamic, hydraulic, and chemical systems, which allowed circuit diagrams to be applied to a wide variety of fields. And on April 24, 1959, Henry Paynter woke up and invented the diagrammatic language of bond graphs to study generalized versions of voltage and current, called ‘effort’ and ‘flow,’ which are implicit in the analogies found by Olson. Bond graphs are now widely used in engineering. On the other hand, control theorists use diagrams of a different kind, called ‘signal-flow diagrams’, to study linear open dynamical systems.

Although category theory predates some of these diagrams, it was not until the 1980s that Joyal and Street showed string digrams can be used to reason about morphisms in any symmetric monoidal category. This motivates Brandon’s first goal: viewing electrical circuits, signal-flow diagrams, and bond graphs as string diagrams for morphisms in symmetric monoidal categories.

This lets us study networks from a *compositional* perspective. That is, we can study a big network by describing how it is composed of smaller pieces. Treating networks as morphisms in a symmetric monoidal category lets us build larger ones from smaller ones by composing and tensoring them: this makes the compositional perspective into precise mathematics. To study a network in this way we must first define a notion of ‘input’ and ‘output’ for the network diagram. Then gluing diagrams together, so long as the outputs of one match the inputs of the other, defines the composition for a category.

Network diagrams are typically assigned data, such as the potential and current associated to a wire in an electrical circuit. Since the relation between the data tells us how a network behaves, we call this relation the ‘behavior’ of a network. The way in which we assign behavior to a network comes from first treating a network as a ‘black box’, which is a system with inputs and outputs whose internal mechanisms are unknown or ignored. A simple example is the lock on a doorknob: one can insert a key and try to turn it; it either opens the door or not, and it fulfills this function without us needing to know its inner workings. We can treat a system as a black box through the process called ‘black-boxing’, which forgets its inner workings and records only the relation it imposes between its inputs and outputs.

Since systems with inputs and outputs can be seen as morphisms in a category we expect black-boxing to be a functor out of a category of this sort. Assigning each diagram its behavior in a functorial way is formalized by functorial semantics, first introduced in Lawvere’s thesis in 1963. This consists of using categories with specific extra structure as ‘theories’ whose ‘models’ are structure-preserving functors into other such categories. We then think of the diagrams as a syntax, while the behaviors are the semantics. Thus black-boxing is actually an example of functorial semantics. This leads us to another goal: to study the functorial semantics, i.e. black-boxing functors, for electrical circuits, signal-flow diagrams, and bond graphs.

Brendan Fong and I began this type of work by showing how to describe circuits made of wires, resistors, capacitors, and inductors as morphisms in a category using ‘decorated cospans’. Jason Erbele and I, and separately Bonchi, Sobociński and Zanasi, studied signal flow diagrams as morphisms in a category. In other work Brendan Fong, Blake Pollard and I looked at Markov processes, while Blake and I studied chemical reaction networks using decorated cospans. In all of these cases, we also studied the functorial semantics of these diagram languages.

Brandon’s main tool is the framework of ‘props’, also called ‘PROPs’, introduced by Mac Lane in 1965. The acronym stands for “products and permutations”, and these operations roughly describe what a prop can do. More precisely, a prop is a strict symmetric monoidal category equipped with a distinguished object $<semantics>X<annotation\; encoding="application/x-tex">X</annotation></semantics>$ such that every object is a tensor power $<semantics>{X}^{\otimes n}.<annotation\; encoding="application/x-tex">X^\{\backslash otimes\; n\}.</annotation></semantics>$ Props arise because very often we think of a network as going between some set of input nodes and some set of output nodes, where the nodes are indistinguishable from each other. Thus we typically think of a network as simply having some natural number as an input and some natural number as an output, so that the network is actually a morphism in a prop.

### Circuits and bond graphs

Now let’s take a quick tour of circuits and bond graphs. Much more detail can be found in Brandon’s thesis, but this may help you know what to picture when you hear terminology from electrical engineering.

Here is an electrical circuit made of only perfectly conductive wires:

This is just a graph, consisting of a set $<semantics>N<annotation\; encoding="application/x-tex">N</annotation></semantics>$ of nodes, a set $<semantics>E<annotation\; encoding="application/x-tex">E</annotation></semantics>$ of edges, and maps $<semantics>s,t:E\to N<annotation\; encoding="application/x-tex">s,t\backslash colon\; E\backslash to\; N</annotation></semantics>$ sending each edge to its source and target node. We refer to the edges as perfectly conductive wires and say that wires go between nodes. Then associated to each perfectly conductive wire in an electrical circuit is a pair of real numbers called ‘potential’, $<semantics>\varphi ,<annotation\; encoding="application/x-tex">\backslash phi,</annotation></semantics>$ and ‘current’, $<semantics>I.<annotation\; encoding="application/x-tex">I.</annotation></semantics>$

Typically each *node* gets a potential, but in the above case the potential at either end of a wire would be the same so we may as well associate the potential to the wire. Current and potential in circuits like these obey two laws due to Kirchoff. First, at any node, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. The other law states that any connected wires must have the same potential.

We say that the above circuit is *closed* as opposed to being *open* because it does not have any inputs or outputs. In order to talk about open circuits and thereby bring the ‘compositional perspective’ into play we need a notion for inputs and outputs of a circuit. We do this using two maps $<semantics>i:X\to N<annotation\; encoding="application/x-tex">i\backslash colon\; X\backslash to\; N</annotation></semantics>$ and $<semantics>o:Y\to N<annotation\; encoding="application/x-tex">o\backslash colon\; Y\; \backslash to\; N</annotation></semantics>$ that specifiy the *inputs* and *outputs* of a circuit. Here is an example:

We call the sets $<semantics>X,Y,<annotation\; encoding="application/x-tex">X,\; Y,</annotation></semantics>$ and the disjoint union $<semantics>X+Y<annotation\; encoding="application/x-tex">X\; +\; Y</annotation></semantics>$ the **inputs**, **outputs**, and **terminals** of the circuit, respectively. To each terminal we associate a potential and current. In total this gives a space of allowed potentials and currents on the terminals and we call this space the ‘behavior’ of the circuit. Since we do this association without knowing the potentials and currents inside the rest of the circuit we call this process ‘black-boxing’ the circuit. This process hides the internal workings of the circuit and just tells us the relation between inputs and outputs. In fact this association is functorial, but to understand the functoriality first requires that we say how to compose these kinds of circuits. We save this for later.

There are also electrical circuits that have ‘components’ such as resistors, inductors, voltage sources, and current sources. These are graphs as above, but with edges now labelled by elements in some set *L*. Here is one for example:

We call this an ** L-circuit**. We may also black-box an

*L*-circuit to get a space of allowed potentials and currents, i.e. the behavior of the

*L*-circuit, and this process is functorial as well. The components in a circuit determine the possible potential and current pairs because they impose additional relationships. For example, a resistor between two nodes has a resistance $<semantics>R<annotation\; encoding="application/x-tex">R</annotation></semantics>$ and is drawn as:

In an *L*-circuit this would be an edge labelled by some positive real number $<semantics>R.<annotation\; encoding="application/x-tex">R.</annotation></semantics>$ For a resistor like this Kirchhoff’s current law says $<semantics>{I}_{1}={I}_{2}<annotation\; encoding="application/x-tex">I\_1=I\_2</annotation></semantics>$ and Ohm’s Law says $<semantics>{\varphi}_{2}-{\varphi}_{1}={I}_{1}R.<annotation\; encoding="application/x-tex">\backslash phi\_2-\backslash phi\_1\; =I\_1R.</annotation></semantics>$ This tells us how to construct the black-boxing functor that extracts the right behavior.

Engineers often work with wires that come in pairs where the current on one wire is the negative of the current on the other wire. In such a case engineers care about the difference in potential more than each individual potential. For such pairs of perfectly conductive wires:

we call $<semantics>V={\varphi}_{2}-{\varphi}_{1}<annotation\; encoding="application/x-tex">V=\backslash phi\_2-\backslash phi\_1</annotation></semantics>$ the ‘voltage’ and $<semantics>I={I}_{1}=-{I}_{2}<annotation\; encoding="application/x-tex">I=I\_1=-I\_2</annotation></semantics>$ the ‘current’. Note the word current is used for two different, yet related concepts. We call a pair of wires like this a ‘bond’ and a pair of nodes like this a ‘port’. To summarize we say that bonds go between ports, and in a ‘bond graph’ we draw a bond as follows:

Note that engineers do not explicitly draw ports at the ends of bonds; we follow this notation and simply draw a bond as a thickened edge. Engineers who work with bond graphs often use the terms ‘effort’ and ‘flow’ instead of voltage and current. Thus a bond between two ports in a bond graph is drawn equipped with an effort and flow, rather than a voltage and current, as follows:

A bond graph consists of bonds connected together using ‘1-junctions’ and ‘0-junctions’. These two types of junctions impose equations between the efforts and flows on the attached bonds. The flows on bonds connected together with a 1-junction are all equal, while the efforts sum to zero, after sprinkling in some signs depending on how we orient the bonds. For 0-junctions it works the other way: the flows are all equal while the efforts sum to zero! The duality here is well-known to engineers but perhaps less so to mathematicians. This is one topic Brandon’s thesis explores.

Brandon explains bond graphs in more detail in Chapter 5 of his thesis, but here is an example:

The arrow at the end of a bond indicates which direction of current flow counts as positive, while the bar is called the ‘causal stroke’. These are unnecessary for Brandon’s work, so he adopts a simplified notation without the arrow or bar. In engineering it’s also important to attach general circuit components, but Brandon doesn’t consider these.

### Outline

In Chapter 2 of his thesis, Brandon provides the necessary background for studying four categories as props:

• the category of finite sets and spans: $<semantics>\mathrm{FinSpan}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinSpan\}</annotation></semantics>$

• the category of finite sets and relations: $<semantics>\mathrm{FinRel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinRel\}</annotation></semantics>$

• the category of finite sets and cospans: $<semantics>\mathrm{FinCospan}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCospan\}</annotation></semantics>$

• the category of finite sets and corelations: $<semantics>\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}</annotation></semantics>$.

In particular, $<semantics>\mathrm{FinCospan}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCospan\}</annotation></semantics>$ and $<semantics>\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}</annotation></semantics>$ are crucial to the study of networks.

In Corollary 2.3.4 he notes that any prop has a presentation in terms of generators and equations. Then he recalls the known presentations for $<semantics>\mathrm{FinSpan},<annotation\; encoding="application/x-tex">\backslash mathrm\{FinSpan\},</annotation></semantics>$ $<semantics>\mathrm{FinCospan},<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCospan\},</annotation></semantics>$ and $<semantics>\mathrm{FinRel}<annotation\; encoding="application/x-tex">FinRel</annotation></semantics>$. Proposition 2.3.7 lets us build props as quotients of other props.

He begins Chapter 3 by showing that $<semantics>\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}</annotation></semantics>$ is ‘the prop for extraspecial commutative Frobenius monoids’, based on a paper he wrote with Brendan Fong. This result also gives a presentation for $<semantics>\mathrm{FinCorel}.<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}.</annotation></semantics>$

Then he defines an “*L*-circuit” as a graph with specified inputs and outputs where all the edge are labeled by elements of some set *L*. *L*-circuits are morphisms in the prop $<semantics>{\mathrm{Circ}}_{L}.<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\_L.</annotation></semantics>$ In Proposition 3.2.8 he uses a result of Rosebrugh, Sabadini and Walters to show that $<semantics>{\mathrm{Circ}}_{L}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\_L</annotation></semantics>$ can be viewed as the coproduct of $<semantics>\mathrm{FinCospan}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCospan\}</annotation></semantics>$ and the free prop on the set *L* of labels.

Brandon then defines $<semantics>\mathrm{Circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}</annotation></semantics>$ to be the prop $<semantics>{\mathrm{Circ}}_{L}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\_L</annotation></semantics>$ where *L* consists of a single element. This example is important, because $<semantics>\mathrm{Circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}</annotation></semantics>$ can be seen as the category whose morphisms are circuits made of only perfectly conductive wires! From any morphism in $<semantics>\mathrm{Circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}</annotation></semantics>$ he extracts a cospan of finite sets and then turns the cospan into a corelation. These two processes are functorial, so he gets a method for sending a circuit made of only perfectly conductive wires to a corelation:

$$<semantics>\mathrm{Circ}\stackrel{{H}^{\prime}}{\u27f6}\mathrm{FinCospan}\stackrel{H}{\u27f6}\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\; \backslash mathrm\{Circ\}\; \backslash stackrel\{H^\{\text{\'}\}\}\{\backslash longrightarrow\}\; \backslash mathrm\{FinCospan\}\; \backslash stackrel\{H\}\{\backslash longrightarrow\}\; \backslash mathrm\{FinCorel\}\; </annotation></semantics>$$

There is also a functor

$$<semantics>K:\mathrm{FinCorel}\to {\mathrm{FinRel}}_{k}<annotation\; encoding="application/x-tex">K\backslash colon\; \backslash mathrm\{FinCorel\}\; \backslash to\; \backslash mathrm\{FinRel\}\_k</annotation></semantics>$$

where $<semantics>{\mathrm{FinRel}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinRel\}\_k</annotation></semantics>$ is the category whose objects are finite dimensional vector spaces and whose morphisms $<semantics>R:U\to V<annotation\; encoding="application/x-tex">R\backslash colon\; U\backslash to\; V</annotation></semantics>$ are linear relations, that is, linear subspaces $<semantics>R\subseteq U\oplus V.<annotation\; encoding="application/x-tex">R\backslash subseteq\; U\; \backslash oplus\; V.</annotation></semantics>$ By composing with the above functors $<semantics>H\prime <annotation\; encoding="application/x-tex">H\text{\'}</annotation></semantics>$ and $<semantics>H<annotation\; encoding="application/x-tex">H</annotation></semantics>$ he associates a linear relation $<semantics>R<annotation\; encoding="application/x-tex">R</annotation></semantics>$ to any circuit made of perfectly conductive wires. On the other hand he gets a subspace for any such circuit by first assigning potential and current to each terminal, and then subjecting these variables to the appropriate physical laws.

It turns out that these two ways of assigning a subspace to a morphism in $<semantics>\mathrm{Circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}</annotation></semantics>$ are the same. So, he calls the linear relation associated to a circuit using the composite $<semantics>KHH\prime <annotation\; encoding="application/x-tex">K\; H\; H\text{\'}</annotation></semantics>$ the “behavior” of the circuit and defines the “black-boxing” functor

$$<semantics>\blacksquare :\mathrm{Circ}\to {\mathrm{FinRel}}_{k}<annotation\; encoding="application/x-tex">\backslash blacksquare\; \backslash colon\; \backslash mathrm\{Circ\}\backslash to\; \backslash mathrm\{FinRel\}\_k</annotation></semantics>$$

to be this composite.

Note that the underlying corelation of a circuit made of perfectly conductive wires completely determines the behavior of the circuit via the functor $<semantics>K.<annotation\; encoding="application/x-tex">K.</annotation></semantics>$

In Chapter 4 he reinterprets the black-boxing functor $<semantics>\blacksquare <annotation\; encoding="application/x-tex">\backslash blacksquare</annotation></semantics>$ as a morphism of props. He does this by introducing the category $<semantics>{\mathrm{LagRel}}_{k},<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k,</annotation></semantics>$ whose objects are “symplectic” vector spaces and whose morphisms are “Lagrangian” relations. In Proposition 4.1.6 he proves that the functor $<semantics>K:\mathrm{FinCorel}\to {\mathrm{FinRel}}_{k}<annotation\; encoding="application/x-tex">K\backslash colon\; \backslash mathrm\{FinCorel\}\; \backslash to\; \backslash mathrm\{FinRel\}\_k</annotation></semantics>$ actually picks out a Lagrangian relation for any corelation and thus determines a morphism of props. So, he redefines $<semantics>K<annotation\; encoding="application/x-tex">K</annotation></semantics>$ to be this morphism

$$<semantics>K:\mathrm{FinCorel}\to {\mathrm{LagRel}}_{k}<annotation\; encoding="application/x-tex">K\backslash colon\; \backslash mathrm\{FinCorel\}\; \backslash to\; \backslash mathrm\{LagRel\}\_k</annotation></semantics>$$

and reinterprets black-boxing as the composite

$$<semantics>\mathrm{Circ}\stackrel{{H}^{\prime}}{\u27f6}\mathrm{FinCospan}\stackrel{H}{\u27f6}\mathrm{FinCorel}\stackrel{K}{\u27f6}{\mathrm{LagRel}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\; \backslash stackrel\{H^\{\text{\'}\}\}\{\backslash longrightarrow\}\; \backslash mathrm\{FinCospan\}\; \backslash stackrel\{H\}\{\backslash longrightarrow\}\; \backslash mathrm\{FinCorel\}\; \backslash stackrel\{K\}\{\backslash longrightarrow\}\; \backslash mathrm\{LagRel\}\_k\; </annotation></semantics>$$

After doing all this hard work for circuits made of perfectly conductive wires — a warmup exercise that engineers might scoff at — Brandon shows the power of his results by easily extending the black-boxing functor to circuits with arbitrary label sets in Theorem 4.2.1. He applies this result to a prop whose morphisms are circuits made of resistors, inductors, and capacitors. Then he considers a more general and mathematically more natural approach to linear circuits using the prop $<semantics>{\mathrm{Circ}}_{k}.<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\_k.</annotation></semantics>$ The morphisms here are open circuits with wires labelled by elements of some chosen field $<semantics>k.<annotation\; encoding="application/x-tex">k.</annotation></semantics>$ In Theorem 4.2.4 he prove the existence of a morphism of props

$$<semantics>\blacksquare :{\mathrm{Circ}}_{k}\to {\mathrm{LagRel}}_{k}<annotation\; encoding="application/x-tex">\backslash blacksquare\; \backslash colon\; \backslash mathrm\{Circ\}\_k\; \backslash to\; \backslash mathrm\{LagRel\}\_k</annotation></semantics>$$

that describes the black-boxing of circuits built from arbitrary linear components.

Brandon then picks up where Jason Erbele’s thesis left off, and recalls how control theorists use “signal-flow diagrams” to draw linear relations. These diagrams make up the category $<semantics>{\mathrm{SigFlow}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{SigFlow\}\_k</annotation></semantics>$, which is the free prop generated by the same generators as $<semantics>{\mathrm{FinRel}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinRel\}\_k</annotation></semantics>$. Similarly he defines the prop $<semantics>{\tilde{\mathrm{Circ}}}_{k}<annotation\; encoding="application/x-tex">\; \backslash widetilde\{\backslash mathrm\{Circ\}\}\_k</annotation></semantics>$ as the free prop generated by the same generators as $<semantics>{\mathrm{Circ}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{Circ\}\_k</annotation></semantics>$. Then there is a strict symmetric monoidal functor $<semantics>T:{\tilde{\mathrm{Circ}}}_{k}\to {\mathrm{SigFlow}}_{k}<annotation\; encoding="application/x-tex">T\; \backslash colon\; \backslash widetilde\{\backslash mathrm\{Circ\}\}\_k\; \backslash to\; \backslash mathrm\{SigFlow\}\_k</annotation></semantics>$ giving a commutative square:

Of course, circuits made of perfectly conductive wires are a special case of linear circuits. We can express this fact using another commutative square:

Combining the diagrams so far, Brandon gets a commutative diagram summarizing the relationship between linear circuits, cospans, corelations, and signal-flow diagrams:

Brandon concludes Chapter 4 by extending his work to circuits with voltage and current sources. These types of circuits define *affine* relations instead of linear relations. The prop framework lets Brandon extend black-boxing to these types of circuits by showing that affine Lagrangian relations are morphisms in a prop $<semantics>{\mathrm{AffLagRel}}_{k}.<annotation\; encoding="application/x-tex">\backslash mathrm\{AffLagRel\}\_k.</annotation></semantics>$ This leads to Theorem 4.4.5, which says that for any field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ and label set *L* there is a unique morphism of props

$$<semantics>\blacksquare :{\mathrm{Circ}}_{L}\to {\mathrm{AffLagRel}}_{k}<annotation\; encoding="application/x-tex">\backslash blacksquare\; \backslash colon\; \backslash mathrm\{Circ\}\_L\; \backslash to\; \backslash mathrm\{AffLagRel\}\_k\; </annotation></semantics>$$

extending the other black-boxing functor and sending each element of *L* to an arbitrarily chosen affine Lagrangian relation between potentials and currents.

In Chapter 5, Brandon study bonies graphs as morphisms in a category. His goal is to define a category $<semantics>\mathrm{BondGraph},<annotation\; encoding="application/x-tex">\backslash mathrm\{BondGraph\},</annotation></semantics>$ whose morphisms are bond graphs, and then assign a space of efforts and flows as behavior to any bond graph using a functor. He also constructs a functor that assigns a space of potentials and currents to any bond graph, which agrees with the way that potential and current relate to effort and flow.

The subtle way he defines $<semantics>\mathrm{BondGraph}<annotation\; encoding="application/x-tex">\backslash mathrm\{BondGraph\}</annotation></semantics>$ comes from two different approaches to studying bond graphs, and the problems inherent in each approach. The first approach leads him to a subcategory $<semantics>{\mathrm{FinCorel}}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}^\backslash circ</annotation></semantics>$ of $<semantics>\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}</annotation></semantics>$, while the second leads him to a subcategory $<semantics>{\mathrm{LagRel}}_{k}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k^\backslash circ</annotation></semantics>$ of $<semantics>{\mathrm{LagRel}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k</annotation></semantics>$. There isn’t a commutative square relating these four categories, but Brandon obtains a pentagon that commutes up to a natural transformation by inventing a new category $<semantics>\mathrm{BondGraph}<annotation\; encoding="application/x-tex">BondGraph</annotation></semantics>$:

This category is a way of formalizing Paynter’s idea of bond graphs.

In his first approach, Brandon views a bond graph as an electrical circuit. He takes advantage of his earlier work on circuits and corelations by taking $<semantics>\mathrm{FinCorel}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}</annotation></semantics>$ to be the category whose morphisms are circuits made of perfectly conductive wires. In this approach a terminal is the object 1 and a wire is the identity corelation from 1 to 1, while a circuit from *m* terminals to *n* terminals is a corelation from *m* to *n*.

In this approach Brandon thinks of a port as the object 2, since a port is a pair of nodes. Then he thinks of a bond as a pair of wires and hence the identity corelation from 2 to 2. Lastly, the two junctions are two different ways of connecting ports together, and thus specific corelations from 2*m* to 2*n*. It turns out that by following these ideas he can equip the object 2 with two different Frobenius monoid structures, which behave very much like 1-junctions and 0-junctions in bond graphs!

It would be great if the morphisms built from these two Frobenius monoids corresponded perfectly to bond graphs. Unfortunately there are some equations which hold between morphisms made from these Frobenius monoids that do not hold for corresponding bond graphs. So, Brandon defines a category $<semantics>{\mathrm{FinCorel}}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}^\backslash circ</annotation></semantics>$ using the morphisms that come from these two Frobenius monoids and moves on to a second attempt at defining $<semantics>\mathrm{BondGraph}<annotation\; encoding="application/x-tex">BondGraph</annotation></semantics>$.

Since bond graphs impose Lagrangian relations between effort and flow, this second approach starts by looking back at $<semantics>{\mathrm{LagRel}}_{k}<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k</annotation></semantics>$. The relations associated to a 1-junction make $<semantics>k\oplus k<annotation\; encoding="application/x-tex">k\backslash oplus\; k</annotation></semantics>$ into yet another Frobenius monoid, while the relations associated to a 0-junction make $<semantics>k\oplus k<annotation\; encoding="application/x-tex">k\backslash oplus\; k</annotation></semantics>$ into a different Frobenius monoid. These two Frobenius monoid structures interact to form a bimonoid! Unfortunately, a bimonoid has some equations between morphisms that do not correspond to equations between bond graphs, so this approach also does not result in morphisms that are bond graphs. Nonetheless, Brandon defines a category $<semantics>{\mathrm{LagRel}}_{k}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k^\backslash circ</annotation></semantics>$ using the two Frobenius monoid structures $<semantics>k\oplus k.<annotation\; encoding="application/x-tex">k\backslash oplus\; k.</annotation></semantics>$

Since it turns out that $<semantics>{\mathrm{FinCorel}}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}^\backslash circ</annotation></semantics>$ and $<semantics>{\mathrm{LagRel}}_{k}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k^\backslash circ</annotation></semantics>$ have corresponding generators, Brandon defines $<semantics>\mathrm{BondGraph}<annotation\; encoding="application/x-tex">\backslash mathrm\{BondGraph\}</annotation></semantics>$ as a prop that also has corresponding generators, but with only the equations found in both $<semantics>{\mathrm{FinCorel}}^{\circ}<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}^\backslash circ</annotation></semantics>$ and $<semantics>{\mathrm{LagRel}}_{k}^{\circ}.<annotation\; encoding="application/x-tex">\backslash mathrm\{LagRel\}\_k^\backslash circ.</annotation></semantics>$ By defining $<semantics>\mathrm{BondGraph}<annotation\; encoding="application/x-tex">\backslash mathrm\{BondGraph\}</annotation></semantics>$ in this way he automatically gets two functors

$$<semantics>F:\mathrm{BondGraph}\to {\mathrm{LagRel}}_{k}^{\circ}<annotation\; encoding="application/x-tex">F\backslash colon\; \backslash mathrm\{BondGraph\}\; \backslash to\; \backslash mathrm\{LagRel\}\_k^\backslash circ</annotation></semantics>$$

and

$$<semantics>G:\mathrm{BondGraph}\to {\mathrm{FinCorel}}^{\circ}<annotation\; encoding="application/x-tex">G\backslash colon\; \backslash mathrm\{BondGraph\}\; \backslash to\; \backslash mathrm\{FinCorel\}^\backslash circ</annotation></semantics>$$

The functor $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ associates effort and flow to a bond graph, while the functor $<semantics>G<annotation\; encoding="application/x-tex">G</annotation></semantics>$ lets us associate potential and current to a bond graph using the previous work done on $<semantics>\mathrm{FinCorel}.<annotation\; encoding="application/x-tex">\backslash mathrm\{FinCorel\}.</annotation></semantics>$ Then the Lagrangian subspace relating effort, flow, potential, and current:

$$<semantics>\{(V,I,{\varphi}_{1},{I}_{1},{\varphi}_{2},{I}_{2})|V={\varphi}_{2}-{\varphi}_{1},I={I}_{1}=-{I}_{2}\}<annotation\; encoding="application/x-tex">\; \backslash \{(V,I,\backslash phi\_1,I\_1,\backslash phi\_2,I\_2)\; |\; V\; =\; \backslash phi\_2-\backslash phi\_1,\; I\; =\; I\_1\; =\; -I\_2\backslash \}</annotation></semantics>$$

defines a natural transformation in the following diagram:

Putting this together with the diagram we saw before, Brandon gets a giant diagram which encompasses the relationships between circuits, signal-flow diagrams, bond graphs, and their behaviors in category theoretic terms:

This diagram is a nice quick road map of his thesis. Of course, you need to understand all the categories in this diagram, all the functors, and also their applications to engineering, to fully appreciate what he has accomplished! But his thesis explains that.

### To learn more

Coya’s thesis has lots of references, but if you want to see diagrams at work in actual engineering, here are some good textbooks on bond graphs:

• D. C. Karnopp, D. L. Margolis and R. C. Rosenberg, *System Dynamics: A Unified Approach*, Wiley, New York, 1990.

• F. T. Brown, *Engineering System Dynamics: A Unified Graph-Centered Approach*, Taylor and Francis, New York, 2007.

and here’s a good one on signal-flow diagrams:

• B. Friedland, *Control System Design: An Introduction to State-Space Methods*, S. W. Director (ed.), McGraw–Hill Higher Education, 1985.

### The n-Category Cafe

The Centre of Australian Category Theory is advertising for a postdoc. The position is for 2 years and the ad is here.

Applications close on 15 June. Most questions about the position would be best directed to Richard Garner or Steve Lack. You can also find out more about CoACT here.

This is a great opportunity to join a fantastic research group. Please help spread the word to those who might be interested!

## May 19, 2018

### John Baez - Azimuth

My student Brandon Coya finished his thesis, and successfully defended it last Tuesday!

• Brandon Coya, *Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective*, Ph.D. thesis, U. C. Riverside, 2018.

It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.

His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!

However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results. What follows is heavily based on the introduction of his thesis, but I’ve baezified it a little.

### The basic idea

People, and especially scientists and engineers, are naturally inclined to draw diagrams and pictures when they want to better understand a problem. One example is when Feynman introduced his famous diagrams in 1949; particle physicists have been using them ever since. But some other diagrams introduced by engineers are far more important to the functioning of the modern world and its technology. It’s outrageous, but sociologically understandable, that mathematicians have figured out more about Feynman diagrams than these other kinds: circuit diagrams, bond graphs and signal-flow diagrams. This is the problem Brandon aims to fix.

I’ve been unable to track down the early history of circuit diagrams, so if you know about that please tell me! But in the 1940s, Harry Olson pointed out analogies in electrical, mechanical, thermodynamic, hydraulic, and chemical systems, which allowed circuit diagrams to be applied to a wide variety of fields. On April 24, 1959, Henry Paynter woke up and invented the diagrammatic language of bond graphs to study generalized versions of voltage and current, called ‘effort’ and ‘flow,’ which are implicit in the analogies found by Olson. Bond graphs are now widely used in engineering. On the other hand, control theorists use diagrams of a different kind, called ‘signal-flow diagrams’, to study linear open dynamical systems.

Although category theory predates some of these diagrams, it was not until the 1980s that Joyal and Street showed string digrams can be used to reason about morphisms in any symmetric monoidal category. This motivates Brandon’s first goal: viewing electrical circuits, signal-flow diagrams, and bond graphs as string diagrams for morphisms in symmetric monoidal categories.

This lets us study networks from a *compositional* perspective. That is, we can study a big network by describing how it is composed of smaller pieces. Treating networks as morphisms in a symmetric monoidal category lets us build larger ones from smaller ones by composing and tensoring them: this makes the compositional perspective into precise mathematics. To study a network in this way we must first define a notion of ‘input’ and ‘output’ for the network diagram. Then gluing diagrams together, so long as the outputs of one match the inputs of the other, defines the composition for a category.

Network diagrams are typically assigned data, such as the potential and current associated to a wire in an electrical circuit. Since the relation between the data tells us how a network behaves, we call this relation the ‘behavior’ of a network. The way in which we assign behavior to a network comes from first treating a network as a ‘black box’, which is a system with inputs and outputs whose internal mechanisms are unknown or ignored. A simple example is the lock on a doorknob: one can insert a key and try to turn it; it either opens the door or not, and it fulfills this function without us needing to know its inner workings. We can treat a system as a black box through the process called ‘black-boxing’, which forgets its inner workings and records only the relation it imposes between its inputs and outputs.

Since systems with inputs and outputs can be seen as morphisms in a category we expect black-boxing to be a functor out of a category of this sort. Assigning each diagram its behavior in a functorial way is formalized by functorial semantics, first introduced in Lawvere’s thesis in 1963. This consists of using categories with specific extra structure as ‘theories’ whose ‘models’ are structure-preserving functors into other such categories. We then think of the diagrams as a syntax, while the behaviors are the semantics. Thus black-boxing is actually an example of functorial semantics. This leads us to another goal: to study the functorial semantics, i.e. black-boxing functors, for electrical circuits, signal-flow diagrams, and bond graphs.

Brendan Fong and I began this type of work by showing how to describe circuits made of wires, resistors, capacitors, and inductors as morphisms in a category using ‘decorated cospans’. Jason Erbele and I, and separately Bonchi, Sobociński and Zanasi, studied signal flow diagrams as morphisms in a category. In other work Brendan Fong, Blake Pollard and I looked at Markov processes, while Blake and I studied chemical reaction networks using decorated cospans. In all of these cases, we also studied the functorial semantics of these diagram languages.

Brandon’s main tool is the framework of ‘props’, also called ‘PROPs’, introduced by Mac Lane in 1965. The acronym stands for “products and permutations”, and these operations roughly describe what a prop can do. More precisely, a prop is a strict symmetric monoidal category equipped with a distinguished object latex X$ such that every object is a tensor power Props arise because very often we think of a network as going between some set of input nodes and some set of output nodes, where the nodes are indistinguishable from each other. Thus we typically think of a network as simply having some natural number as an input and some natural number as an output, so that the network is actually a morphism in a prop.

### Circuits and bond graphs

Now let’s take a quick tour of circuits and bond graphs. Much more detail can be found in Brandon’s thesis, but this may help you know what to picture when you hear terminology from electrical engineering.

Here is an electrical circuit made of only perfectly conductive wires:

This is just a graph, consisting of a set of nodes, a set of edges, and maps sending each edge to its source and target node. We refer to the edges as perfectly conductive wires and say that wires go between nodes. Then associated to each perfectly conductive wire in an electrical circuit is a pair of real numbers called ‘potential’, and ‘current’,

Typically each *node* gets a potential, but in the above case the potential at either end of a wire would be the same so we may as well associate the potential to the wire. Current and potential in circuits like these obey two laws due to Kirchoff. First, at any node, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. The other law states that any connected wires must have the same potential.

We say that the above circuit is *closed* as opposed to being *open* because it does not have any inputs or outputs. In order to talk about open circuits and thereby bring the ‘compositional perspective’ into play we need a notion for inputs and outputs of a circuit. We do this using two maps and that specifiy the *inputs* and *outputs* of a circuit. Here is an example:

We call the sets and the disjoint union the **inputs**, **outputs**, and **terminals** of the circuit, respectively. To each terminal we associate a potential and current. In total this gives a space of allowed potentials and currents on the terminals and we call this space the ‘behavior’ of the circuit. Since we do this association without knowing the potentials and currents inside the rest of the circuit we call this process ‘black-boxing’ the circuit. This process hides the internal workings of the circuit and just tells us the relation between inputs and outputs. In fact this association is functorial, but to understand the functoriality first requires that we say how to compose these kinds of circuits. We save this for later.

There are also electrical circuits that have ‘components’ such as resistors, inductors, voltage sources, and current sources. These are graphs as above, but with edges now labelled by elements in some set *L*. Here is one for example:

We call this an ** L-circuit**. We may also black-box an

*L*-circuit to get a space of allowed potentials and currents, i.e. the behavior of the

*L*-circuit, and this process is functorial as well. The components in a circuit determine the possible potential and current pairs because they impose additional relationships. For example, a resistor between two nodes has a resistance and is drawn as:

In an *L*-circuit this would be an edge labelled by some positive real number For a resistor like this Kirchhoff’s current law says and Ohm’s Law says This tells us how to construct the black-boxing functor that extracts the right behavior.

Engineers often work with wires that come in pairs where the current on one wire is the negative of the current on the other wire. In such a case engineers care about the difference in potential more than each individual potential. For such pairs of perfectly conductive wires:

we call the ‘voltage’ and the ‘current’. Note the word current is used for two different, yet related concepts. We call a pair of wires like this a ‘bond’ and a pair of nodes like this a ‘port’. To summarize we say that bonds go between ports, and in a ‘bond graph’ we draw a bond as follows:

Note that engineers do not explicitly draw ports at the ends of bonds; we follow this notation and simply draw a bond as a thickened edge. Engineers who work with bond graphs often use the terms ‘effort’ and ‘flow’ instead of voltage and current. Thus a bond between two ports in a bond graph is drawn equipped with an effort and flow, rather than a voltage and current, as follows:

A bond graph consists of bonds connected together using ‘1-junctions’ and ‘0-junctions’. These two types of junctions impose equations between the efforts and flows on the attached bonds. The flows on bonds connected together with a 1-junction are all equal, while the efforts sum to zero, after sprinkling in some signs depending on how we orient the bonds. For 0-junctions it works the other way: the efforts are all equal while the flows sum to zero! The duality here is well-known to engineers but perhaps less so to mathematicians. This is one topic Brandon’s thesis explores.

Brandon explains bond graphs in more detail in Chapter 5 of his thesis, but here is an example:

The arrow at the end of a bond indicates which direction of current flow counts as positive, while the bar is called the ‘causal stroke’. These are unnecessary for Brandon’s work, so he adopts a simplified notation without the arrow or bar. In engineering it’s also important to attach general circuit components, but Brandon doesn’t consider these.

### Outline

In Chapter 2 of his thesis, Brandon provides the necessary background for studying four categories as props:

• the category of finite sets and spans:

• the category of finite sets and relations:

• the category of finite sets and cospans:

• the category of finite sets and corelations:

In particular, and are crucial to the study of networks.

In Corollary 2.3.4 he notes that any prop has a presentation in terms of generators and equations. Then he recalls the known presentations for and Proposition 2.3.7 lets us build props as quotients of other props.

He begins Chapter 3 by showing that $\mathrm{FinCorel}$ is ‘the prop for extraspecial commutative Frobenius monoids’, based on a paper he wrote with Brendan Fong. This result also gives a presentation for

Then he defines an “*L*-circuit” as a graph with specified inputs and outputs, together with a labelling set for the edges of the graph. *L*-circuits are morphisms in the prop In Proposition 3.2.8 he uses a result of Rosebrugh, Sabadini and Walters to show that can be viewed as the coproduct of and the free prop on the set *L* of labels.

Brandon then defines to be the prop where *L* consists of a single element. This example is important, because can be seen as the category whose morphisms are circuits made of only perfectly conductive wires! From any morphism in he extracts a cospan of finite sets and then turns the cospan into a corelation. These two processes are functorial, so he gets a method for sending a circuit made of only perfectly conductive wires to a corelation:

There is also a functor

where is the category whose objects are finite dimensional vector spaces and whose morphisms are linear relations, that is, linear subspaces By composing with the above functors and he associates a linear relation to any circuit made of perfectly conductive wires. On the other hand he gets a subspace for any such circuit by first assigning potential and current to each terminal, and then subjecting these variables to the appropriate physical laws.

It turns out that these two ways of assigning a subspace to a morphism in are the same. So, he calls the linear relation associated to a circuit using the composite the “behavior” of the circuit and defines the “black-boxing” functor

to be the composite of these:

Note that the underlying corelation of a circuit made of perfectly conductive wires completely determines the behavior of the circuit via the functor

In Chapter 4 he reinterprets the black-boxing functor as a morphism of props. He does this by introducing the category whose objects are “symplectic” vector spaces and whose morphisms are “Lagrangian” relations. In Proposition 4.1.6 he proves that the functor actually picks out a Lagrangian relation for any corelation and thus determines a morphism of props. So, he redefines to be this morphism

and reinterprets black-boxing as the composite

After doing al this hard work for circuits made of perfectly conductive wires—a warmup exercises that engineers might scoff at—Brandon shows the power of his results by easily extending the black-boxing functor to circuits with arbitrary label sets in Theorem 4.2.1. He applies this result to a prop whose morphisms are circuits made of resistors, inductors, and capacitors. Then he considers a more general and mathematically more natural approach to linear circuits using the prop The morphisms here are open circuits with wires labelled by elements of some chosen field In Theorem 4.2.4 he prove the existence of a morphism of props

that describes the black-boxing of circuits built from arbitrary linear components.

Brandon then picks up where Jason Erbele’s thesis left off, and recalls how control theorists use “signal-flow diagrams” to draw linear relations. These diagrams make up the category which is the free prop generated by the same generators as Similarly he defines the prop as the free prop generated by the same generators as Then there is a strict symmetric monoidal functor giving a commutative square:

Of course, circuits made of perfectly conductive wires are a special case of linear circuits. We can express this fact using another commutative square:

Combining the diagrams so far, Brandon gets a commutative diagram summarizing the relationship between linear circuits, cospans, corelations, and signal-flow diagrams:

Brandon concludes Chapter 4 by extending his work to circuits with voltage and current sources. These types of circuits define *affine* relations instead of linear relations. The prop framework lets Brandon extend black-boxing to these types of circuits by showing that affine Lagrangian relations are morphisms in a prop This leads to Theorem 4.4.5, which says that for any field and label set *L* there is a unique morphism of props

extending the other black-boxing functor and sending each element of *L* to an arbitrarily chosen affine Lagrangian relation between potentials and currents.

In Chapter 5, Brandon studies bond graphs as morphisms in a category. His goal is to define a category whose morphisms are bond graphs, and then assign a space of efforts and flows as behavior to any bond graph using a functor. He also constructs a functor that assigns a space of potentials and currents to any bond graph, which agrees with the way that potential and current relate to effort and flow.

The subtle way he defines comes from two different approaches to studying bond graphs, and the problems inherent in each approach. The first approach leads him to a subcategory of while the second leads him to a subcategory of There isn’t a commutative square relating these four categories, but Brandon obtains a pentagon that commutes up to a natural transformation by inventing a new category :

This category is a way of formalizing Paynter’s idea of bond graphs.

In his first approach, Brandon views a bond graph as an electrical circuit. He takes advantage of his earlier work on circuits and corelations by taking to be the category whose morphisms are circuits made of perfectly conductive wires. In this approach a terminal is the object 1 and a wire is the identity corelation from 1 to 1, while a circuit from *m* terminals to *n* terminals is a corelation from *m* to *n*.

In this approach Brandon thinks of a port as the object 2, since a port is a pair of nodes. Then he thinks of a bond as a pair of wires and hence the identity corelation from 2 to 2. Lastly, the two junctions are two different ways of connecting ports together, and thus specific corelations from 2*m* to 2*n*. It turns out that by following these ideas he can equip the object 2 with two different Frobenius monoid structures, which behave very much like 1-junctions and 0-junctions in bond graphs!

It would be great if the morphisms built from these two Frobenius monoids corresponded perfectly to bond graphs. Unfortunately there are some equations which hold between morphisms made from these Frobenius monoids that do not hold for corresponding bond graphs. So, Brandon defines a category using the morphisms that come from these two Frobenius monoids and moves on to a second attempt at defining

Since bond graphs impose Lagrangian relations between effort and flow, this second approach starts by looking back at The relations associated to a 1-junction make into yet another Frobenius monoid, while the relations associated to a 0-junction make into a different Frobenius monoid. These two Frobenius monoid structures interact to form a bimonoid! Unfortunately, a bimonoid has some equations between morphisms that do not correspond to equations between bond graphs, so this approach also does not result in morphisms that are bond graphs. Nonetheless, Brandon defines a category using the two Frobenius monoid structures

Since it turns out that and have corresponding generators, Brandon defines as a prop that also has corresponding generators, but with only the equations found in both and By defining in this way he automatically gets two functors

and

The functor associates effort and flow to a bond graph, while the functor lets us associate potential and current to a bond graph using the previous work done on Then the Lagrangian subspace relating effort, flow, potential, and current:

defines a natural transformation in the following diagram:

Putting this together with the diagram we saw before, Brandon gets a giant diagram which encompasses the relationships between circuits, signal-flow diagrams, bond graphs, and their behaviors in category theoretic terms:

This diagram is a nice quick road map of his thesis. Of course, you need to understand all the categories in this diagram, all the functors, and also their applications to engineering, to fully appreciate what he has accomplished! But his thesis explains that.

### To learn more

Coya’s thesis has lots of references, but if you want to see diagrams at work in actual engineering, here are some good textbooks on bond graphs:

• D. C. Karnopp, D. L. Margolis and R. C. Rosenberg, *System Dynamics: A Unified Approach*, Wiley, New York, 1990.

• F. T. Brown, *Engineering System Dynamics: A Unified Graph-Centered Approach*, Taylor and Francis, New York, 2007.

and here’s a good one on signal-flow diagrams:

• B. Friedland, *Control System Design: An Introduction to State-Space Methods*, S. W. Director (ed.), McGraw–Hill Higher Education, 1985.

### Tommaso Dorigo - Scientificblogging

### Emily Lakdawalla - The Planetary Society Blog

### Peter Coles - In the Dark

One of the contributors to the `Out Thinkers’ event I went to a couple of weeks ago, Emer Maguire, talked about science and music. During the course of her presentation she mentioned one of the most common sets of chord changes in pop music, the I-V-vi-IV progression. In the key of C major, the chords of this progression would be C, G, Am and F. You will for example find this progression comes up often in the songs of Ed Sheeran (whoever that is).

These four chords include those based on the tonic (I), the dominant (V) and the sub-dominant (IV) – i.e. the three chords of the basic blues progression – as well as the relative minor (vi). The relative minor for a major key is a key with exactly the same notes (i.e. the same sharps and flats) in it, but with a different tonic. With these four chords (shuffled in various ways) you can reproduce the harmonies of a very large fraction of the modern pop repertoire. It’s a comfortable and pleasant harmonic progression, but to my ears it sounds a bit bland and uninteresting.

These thoughts came into my head the other night when I was listening to an album of music by Thelonious Monk. One of my `hobbies’ is to try to figure out what’s going on underneath the music that I listen to, especially jazz. I can’t really play the piano, but I have an electronic keyboard which I play around on while trying to figure out what chord progressions are being used. I usually make a lot of terrible mistakes fumbling around in this way, so my neighbours and I are grateful that I use headphones rather than playing out loud!

I haven’t done a detailed statistical study, but I would guess that the most common chord progression in jazz might well be ii-V-I, a sequence that resolves onto the tonic through a cadence of fifths. I think one of the things some people dislike about modern jazz is that many of the chord progressions eschew this resolution which can make the music rather unsettling or, to put it another way, interesting.

Here’s a great example of a Thelonious Monk composition that throws away the rule book and as a result creates a unique atmosphere; it’s called *Off Minor* and it’s one of my absolute favourite Monk tunes, recorded for Blue Note in 1947:

The composition follows the standard 32 bar format of AABA; the A section ends with a strange D sharp chord extended with a flattened 9th which clashes with a B in the piano melody. This ending is quite a shock given the more conventional changes that precede it.

But it’s the B section (the bridge) where it gets really fascinating. The first bar starts on D-flat, moves up to D, and then goes into a series of unresolved ii-V changes beginning in B-flat. That’s not particularly weird in itself, but these changes don’t take place in the conventional way (one each bar): the first does, but the second is over two bars; and the third over four bars. Moreover, after all these changes the bridge ends on an unresolved D chord. It’s the fact that each set of eight bars ends in mid-air that provides this piece with its compelling sense of forward motion.

There’s much more to it than just the chords, of course. There are Monk’s unique voicings and playful use of time as he states the melody, and then there’s his improvised solo, which I think is one of his very best, especially in the first chorus as he sets out like a brave explorer to chart a path through this curious harmonic landscape..

Ed Sheeran, eat your heart out!

Follow @telescoper## May 18, 2018

### Clifford V. Johnson - Asymptotia

Bay Area! You're up next! The Maker Faire is a wonderful event/movement that I've heard about for years and which always struck me as very much in line with my own way of being (making, tinkering, building, creating, as time permits...) On Sunday I'll have the honour of being on one of the centre stages (3:45pm) talking with Kishore Hari (of the podcast Inquiring Minds) about how I made The Dialogues, and why. I might go into some extra detail about my research into making graphic books, and the techniques I used, given the audience. Why yes, I'll sign books for you afterwards, of course. Thanks for asking.

I recommend getting a day pass and see a ton of interesting events that day! Here's a link to the Sunday schedule and amor there you can see links to the whole faire and tickets!

-cvj Click to continue reading this post

The post Make with Me! appeared first on Asymptotia.

## May 17, 2018

### ZapperZ - Physics and Physicists

This video tries to explain the significance of her work connecting conservation laws with symmetry principles.

However, I think that if I were a layperson, I'd miss the important point in this video. So here is the takeaway message if you want one:

**Everything that we see and every behavior of our universe can be traced to some conservation laws. Each conservation law is a manifestation of some underlying symmetry of our universe.**

This is the insight, and a very important insight, that Noether brought to the table, and it was revolutionary to physics. These symmetries are what we currently have as the most fundamental description of the universe that we live in.

Watch this video, and read the links that I gave above, several times if you must, because you owe it to yourself to know about this person and her immense effect on our understanding of our world.

Zz.

### Jester - Resonaances

In the particle world the LHC still attracts the most attention, but in parallel there is ongoing progress at the low-energy frontier. A new episode in that story is the Qweak experiment in Jefferson Lab in the US, which just published their final results. Qweak was shooting a beam of 1 GeV electrons on a hydrogen (so basically proton) target to determine how the scattering rate depends on electron's polarization. Electrons and protons interact with each other via the electromagnetic and weak forces. The former is much stronger, but it is parity-invariant, i.e. it does not care about the direction of polarization. On the other hand, since the classic Wu experiment in 1956, the weak force is known to violate parity. Indeed, the Standard Model postulates that the Z boson, who mediates the weak force, couples with different strength to left- and right-handed particles. The resulting asymmetry between the low-energy electron-proton scattering cross sections of left- and right-handed polarized electrons is predicted to be at the 10^-7 level. That has been experimentally observed many times before, but Qweak was able to measure it with the best precision to date (relative 4%), and at a lower momentum transfer than the previous experiments.

What is the point of this exercise? Low-energy parity violation experiments are often sold as precision measurements of the so-called Weinberg angle, which is a function of the electroweak gauge couplings - the fundamental parameters of the Standard Model. I don't like too much that perspective because the electroweak couplings, and thus the Weinberg angle, can be more precisely determined from other observables, and Qweak is far from achieving a competing accuracy. The utility of Qweak is better visible in the effective theory picture. At low energies one can parameterize the relevant parity-violating interactions between protons and electrons by the contact term

where v ≈ 246 GeV, and QW is the so-called

*weak charge*of the proton. Such interactions arise thanks to the Z boson in the Standard Model being exchanged between electrons and quarks that make up the proton. At low energies, the exchange diagram is well approximated by the contact term above with QW = 0.0708 (somewhat smaller than the "natural" value QW ~ 1 due to numerical accidents making the Z boson effectively protophobic). The measured polarization asymmetry in electron-proton scattering can be re-interpreted as a determination of the proton weak charge:

**QW = 0.0719 ± 0.0045,**in perfect agreement with the Standard Model prediction.

New physics may affect the magnitude of the proton weak charge in two distinct ways. One is by altering the strength with which the Z boson couples to matter. This happens for example when light quarks mix with their heavier exotic cousins with different quantum numbers, as is often the case in the models from the Randall-Sundrum family. More generally, modified couplings to the Z boson could be a sign of quark compositeness. Another way is by generating new parity-violating contact interactions between electrons and quarks. This can be a result of yet unknown short-range forces which distinguish left- and right-handed electrons. Note that the observation of lepton flavor violation in B-meson decays can be interpreted as a hint for existence of such forces (although for that purpose the new force carriers do not need to couple to 1st generation quarks). Qweak's measurement puts novel limits on such broad scenarios. Whatever the origin, simple dimensional analysis allows one to estimate the possible change of the proton weak charge as

where M* is the mass scale of new particles beyond the Standard Model, and g* is their coupling strength to matter. Thus, Qweak can constrain new weakly coupled particles with masses up to a few TeV, or even 50 TeV particles if they are strongly coupled to matter (g*～4π).

What is the place of Qweak in the larger landscape of precision experiments? One can illustrate it by considering a simple example where heavy new physics modifies only the vector couplings of the Z boson to up and down quarks. The best existing constraints on such a scenario are displayed in this plot:

From the size of the rotten egg region you see that the Z boson couplings to light quarks are currently known with a per-mille accuracy. Somewhat surprisingly, the LEP collider, which back in the 1990s produced tens of millions of Z boson to precisely study their couplings, is not at all the leader in this field. In fact, better constraints come from precision measurements at very low energies: pion, kaon, and neutron decays, parity-violating transitions in cesium atoms, and the latest Qweak results which make a difference too. The importance of Qweak is even more pronounced in more complex scenarios where the parameter space is multi-dimensional.

Qweak is certainly not the last salvo on the low-energy frontier. Similar but more precise experiments are being prepared as we read (I wish the follow up were called SuperQweak, or SQweak in short). Who knows, maybe quarks are made of more fundamental building blocks at the scale of ~100 TeV, and we'll first find it out thanks to parity violation at very low energies.

by Mad Hatter (noreply@blogger.com) at May 17, 2018 12:36 PM

### ZapperZ - Physics and Physicists

*If I'm moving in a spaceship and I turn on my flash light*...."

Here's Don Lincoln's lesson on relativistic velocity addition:

Zz.

## May 16, 2018

### ZapperZ - Physics and Physicists

Renowned condensed matter theorist David Pines passed away on May 3, 2018 at the age of 93. I practically read his text (co-authored by Nozieres) on Fermi Liquid from cover to cover while I was a graduate student. In fact, he was on the cusp of a Nobel Prize when he was working with John Bardeen at UIUC. They published a paper on the electron-phonon interaction in superconductors in 1955, a paper that many thought was the precursor to the subsequent BCS Theory paper in 1957. Unfortunately, he left UIUC, and Bob Schrieffer took over his work on this, which ultimately led to the BCS theory and the Nobel prize.

This did not diminished his body of work throughout his life. He certainly was a main figure during the High-Tc superconductivity craze of the late 80's and 90's. His 1991 PRL paper with Monthoux and Balatsky and the 1992 PRL paper with Monthoux, both on the spin-fluctuation effect as the possible "glue" in the cuprate superconductors, where ground-breaking and highly cited.

His contribution to this body of knowledge will have a lasting impact.

Zz.

### Lubos Motl - string vacua and pheno

TrackML Particle Tracking ChallengeTo make the story short, the data you will have to download include 5 times 15 GB train files plus 1 GB train sample and 1 GB test file. A sample submission has 30 MB, detectors.zip have 175 kB.

Well, readers whose infrastructure is similar to mine have already given up. I don't know what to do with 75 GB. On Windows, there's no trouble to store this much data but I would have to manipulate it with Mathematica and that would clearly be too slow with 75 GB.

On the other hand, I could run a VirtualBox with some Linux, like during the Higgs Kaggle contest, but then I would have to study whether I have to allocate some extra hard disk for the simulated Linux hard disk and face similar problems that I am not experienced with. I just don't want to do that – this dataset is simply too big for me.

If such things aren't trouble for you, you should try. In the first phase of the contest – three months are left – you need to design the most accurate algorithm to reconstruct particles' tracks from the points that the huge datasets are composed of.

There will be another part of the contest that starts in the summer where the speed of the calculation will matter.

The leaderboard shows the first contestant among 222 to have the score of just 0.46 – so I believe that there's a lot of room for improvement. The preliminary leaderboard is based on some 29% of the data, the final one will be based on the remaining 71% of the data so it may be different.

**Most importantly, the prizes are $12k, $8k, $5k (in total, $25k) for the first, second, third place.**

Good luck.

by Luboš Motl (noreply@blogger.com) at May 16, 2018 06:15 AM

## May 15, 2018

### Clifford V. Johnson - Asymptotia

Here's a little montage of some of the wildflowers beginning to emerge in the garden this season. Some months ago I sprinkled the seeds in a fee patches, raked the beds and remembered to keep things moist over the days and weeks that followed. These are some of the results... (Click for a larger view.)

-cvj Click to continue reading this post

The post Wild appeared first on Asymptotia.

### CERN Bulletin

**On 17 April, the Staff Council proceeded to the election of the Executive Committee of the Staff Association and the members of the Bureau.**

First of all, why a new election of the Executive Committee elected in April 2018 after that of December 2017 (*Echo* No. 281)? Quite simply because a Crisis Executive Committee with a provisional Bureau had been elected for a period from 1^{st} January to 16 April 2018 with defined and restricted objectives (*Echo* No. 283).

Therefore, on 17 April, G. Roy presented for election a list of 12 persons, including five members for the Bureau, who agreed to continue their work within the Executive Committee, based on an intensive programme with the following main axes:

- Crèche and School and in particular the establishment of a foundation;
*Concertation*: review and relaunch of the*concertation*process;- Finalisation of the 2015 five-yearly review;
- Preparation and start of the 2020 five-yearly review;
- Actuarial reviews of the Pension Fund and the CHIS;
- Internal enquiries and justice;
- Improving the Association’s internal procedures (secretariat, documentation, protection of personal data).

Following the presentation of the list and the programme, the delegates of the Staff Council showed their support for the proposed list and elected the Executive Committee by ballot, with 24 votes in favour and three votes against.

**Members elected to the Executive Committee one 17 April 2018**

Thank you to the 12 colleagues who agreed to take on responsibility within the Staff Association and stay committed to the CERN personnel.

### CERN Bulletin

**ASCERI is the Association of the Sports Communities of the European Research Institutes and aims to contribute to a united Europe through regular sports meetings, bringing together members of public Research Institutes at European level. The Association's members come from over 42 Research Institutes spanning 15 countries.**

The association was born from the German "**Kernforschungszentrum Karlsruhe**" (KfK) football team who had the idea to play against other teams from institutes also involved in nuclear research. Therefore, six teams from different German centres were invited to take part in a "*Reaktoren Fußballturnier*" in Karlsruhe on 2 July 1966.

Ever since, The Winter-ATOMIADE has taken place every three years and alternating with the Summer-ATOMIADE and a Mini Atomiade in between with numerous sports and leisure activities including football, skiing, golf, athletics, tennis, volleyball to name a few. CERN has been a regular participant in these events and even hosted the mini atomiade in 2016 (*Bulletin *No. 28-29/2016).

Since 1989, regular meetings have been held yearly for ASCERI delegates and this year’s annual ASCERI conference was hosted by JRC-GEEL in Antwerp. For the first time ever a female president, Anne-Françoise Maydew from ESRF Grenoble was elected along with a diverse set of board members, including Rachel Bray, delegate representing CERN.

Along with the election of a new committee and set of board members, one of the bigger topics of the 33 Annual conference was the upcoming Summer Atomiade in June, organised by JRC-ISPRA. CERN will be participating with a team of 60, participating in football, tennis, golf, table tennis, athletics, volleyball and cycling.

ASCERI delegates also gave the outgoing President, Henry Koekenberg, and his team a fine send off the final evening of the conference.

## May 14, 2018

### Sean Carroll - Preposterous Universe

In completely separate video news, someone has (I don’t know how) found videos of lectures I gave a CERN several years ago: “Cosmology for Particle Physicists.” (2005, maybe?) These are slightly technical — at the very least they presume you know calculus and basic physics — but are still basically accurate despite their age.

- Introduction to Cosmology
- Dark Matter
- Dark Energy
- Thermodynamics and the Early Universe
- Inflation and Beyond

### CERN Bulletin

The CERN running club, in collaboration with the Staff Association, is happy to announce the 2018 relay race edition. It will take place on **Thursday, May 24 ^{th}** and will consist as every year in a round trip of the CERN Meyrin site in teams of 6 members. It is a fun event, and you do not have to run fast to enjoy it.

Registrations will be open from May 1^{st} to May 22^{nd} on the running club web site. All information concerning the race and the registration are available there too: __http://runningclub.web.cern.ch/content/cern-relay-race__.

A video of the previous edition is also available here : http://cern.ch/go/Nk7C.

As every year, there will be animations starting at noon on the lawn in front of restaurant 1, and information stands for many CERN associations and clubs will be available. The running club partners will also be participate in the event, namely Berthie Sport, Interfon and Uniqa.

### CERN Bulletin

**Cooperative open to international civil servants. We welcome you to discover the advantages and discounts negotiated with our suppliers either on our website www.interfon.fr or at our information office located at CERN, on the ground floor of bldg. 504, open Monday through Friday from 12.30 to 15.30.**

### CERN Bulletin

**Le GAC organise des permanences avec entretiens individuels qui se tiennent le dernier mardi de chaque mois, sauf en juillet et décembre.**

La prochaine permanence se tiendra le :

**Mardi 29 mai de 13 h 30 à 16 h 00 Salle de réunion de l’Association du personnel**

Les permanences suivantes auront lieu les mardis 26 juin, 28 août, 25 septembre, 30 octobre et 27 novembre 2018.

Les permanences du Groupement des Anciens sont ouvertes aux bénéficiaires de la Caisse de pensions (y compris les conjoints survivants) et à tous ceux qui approchent de la retraite. Nous invitons vivement ces derniers à s’associer à notre groupement en se procurant, auprès de l’Association du personnel, les documents nécessaires.

Nous invitons vivement ces derniers à s’associer à notre groupement en se procurant, auprès de l’Association du personnel, les documents nécessaires.

Informations : http://gac-epa.org/

Formulaire de contact : http://gac-epa.org/Organization/ContactForm/ContactForm-fr.php

### The n-Category Cafe

João Faria Martins and Paul Martin at the University of Leeds are advertising a 2-year research fellowship in geometric topology, topological quantum field theory and applications to quantum computing. This is part of a Leverhulme funded project.

The deadline is Tuesday 29th May. Contact João or Paul with any informal inquiries.

by willerton (S.Willerton@sheffield.ac.uk) at May 14, 2018 09:25 AM

## May 13, 2018

### Tommaso Dorigo - Scientificblogging

## May 12, 2018

### John Baez - Azimuth

*guest post by Christian Williams*

Mike Stay has been doing some really cool stuff since earning his doctorate. He’s been collaborating with Greg Meredith, who studied the π-calculus with Abramsky, and then conducted impactful research and design in the software industry before some big ideas led him into the new frontier of decentralization. They and a great team are developing RChain, a distributed computing infrastructure based on the reflective higher-order π-calculus, the ρ-calculus.

They’ve made significant progress in the first year, and on April 17-18 they held the RChain Developer Conference in Boulder, Colorado. Just five months ago, the first conference was a handful of people; now this received well over a hundred. Programmers, venture capitalists, blockchain enthusiasts, experts in software, finance, and mathematics: myriad perspectives from around the globe came to join in the dawn of a new internet. Let’s just say, it’s a lot to take in. This project is the real deal – the idea is revolutionary, the language is powerful, the architecture is elegant; the ambition is immense and skilled developers are actually bringing it to reality. There’s no need for hype: you’re gonna be hearing about RChain.

Documentation , GitHub , Architecture

Here’s something from the documentation:

The open-source RChain project is building a decentralized, economic, censorship-resistant, public compute infrastructure and blockchain. It will host and execute programs popularly referred to as “smart contracts”. It will be trustworthy, scalable, concurrent, with proof-of-stake consensus and content delivery.

The decentralization movement is ambitious and will provide awesome opportunities for new social and economic interactions. Decentralization also provides a counterbalance to abuses and corruption that occasionally occur in large organizations where power is concentrated. Decentralization supports self-determination and the rights of individuals to self-organize. Of course, the realities of a more decentralized world will also have its challenges and issues, such as how the needs of international law, public good, and compassion will be honored.

We admire the awesome innovations of Bitcoin, Ethereum, and other platforms that have dramatically advanced the state of decentralized systems and ushered in this new age of cryptocurrency and smart contracts. However, we also see symptoms that those projects did not use the best engineering and formal models for scaling and correctness in order to support mission-critical solutions. The ongoing debates about scaling and reliability are symptomatic of foundational architectural issues. For example, is it a scalable design to insist on an explicit serialized processing order for all of a blockchain’s transactions conducted on planet earth?

To become a blockchain solution with industrial-scale utility, RChain must provide content delivery at the scale of Facebook and support transactions at the speed of Visa. After due diligence on the current state of many blockchain projects, after deep collaboration with other blockchain developers, and after understanding their respective roadmaps, we concluded that the current and near-term blockchain architectures cannot meet these requirements. In mid-2016, we resolved to build a better blockchain architecture.

Together with the blockchain industry, we are still at the dawn of this decentralized movement. Now is the time to lay down a solid architectural foundation. The journey ahead for those who share this ambitious vision is as challenging as it is worthwhile, and this document summarizes that vision and how we seek to accomplish it.

We began by admitting the following minimal requirements:

- Dynamic, responsive, and provably correct smart contracts.
- Concurrent execution of independent smart contracts.
- Data separation to reduce unnecessary data replication of otherwise independent tokens and smart contracts.
- Dynamic and responsive node-to-node communication.
- Computationally non-intensive consensus/validation protocol.

Building quality software is challenging. It is easier to build “clever” software; however, the resulting software is often of poor quality, riddled with bugs, difficult to maintain, and difficult to evolve. Inheriting and working on such software can be hellish for development teams, not to mention their customers. When building an open-source system to support a mission-critical economy, we reject a minimal-success mindset in favor of end-to-end correctness.

To accomplish the requirements above, our design approach is committed to:

- A computational model that assumes fine-grained concurrency and dynamic network topology.
- A composable and dynamic resource addressing scheme.
- The functional programming paradigm, as it more naturally accommodates distributed and parallel processing.
- Formally verified, correct-by-construction protocols which leverage model checking and theorem proving.
- The principles of intension and compositionality.

RChain is light years ahead of the industry. Why? It is upholding the principle of correct by construction with the depth and rigor of mathematics. For years, Mike and Greg have been developing original ideas for distributed computation: in particular, logic as a distributive law is an “algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus.” This is a powerful way to integrate operational semantics into a language, and prove soundness with a single natural transformation; it also provides an extremely expressive query language, with which you could search the entire world to find “code that does x”. Mike’s strong background in higher category theory has enabled the formalization of Greg’s ideas, which he has developed over decades of thinking deeply and comprehensively about the world of computing. Of all of these, there is one concept which is the heart and pulse of RChain, which unifies the system as a rational whole: the ρ-calculus.

So what’s the big idea? First, some history: back in the late 80s, Greg developed a classification of computational models called “the 4 C’s”:

completeness,

compositionality,

(a good notion of) complexity, and

concurrency.

He found that there was none which had all four, and predicted the existence of one. Just a few years later, Milner invented the π-calculus, and since then it has reigned as the natural language of network computing. It presents a totally different way of thinking: instead of representing sequential instructions for a single machine, the π-calculus is fundamentally concurrent—processes or agents communicate over names or channels, and computation occurs through the interaction of processes. The language is simple yet remarkably powerful; it is deeply connected with game semantics and linear logic, and has become an essential tool in systems engineering and biocomputing: see mobile process calculi for programming the blockchain.

Here is the basic syntax. The variables x,y are names, and P,Q are processes:

P,Q := 0 | (νx)P | x?(y).P | x!(y).P | P|Q

(do nothing | create new x; run P | receive on x and bind to y; run P | send value y on x; run P | run P and Q in parallel)

The computational engine, the basic reduction analogous to beta-reduction of lambda calculus, is the communication rule:

COMM : x!(y).P|x?(z).Q → P|Q[y/z]

(given parallel output and input processes along the same channel, the value is transferred from the output to the input, and is substituted for all occurrences of the input variable in the continuation process)

The definition of a process calculus must specify structural congruence: these express the equivalences between processes—for example, ({P},|,0) forms a commutative monoid.

The π-calculus reforms computation, on the most basic level, to be a cooperative activity. Why is this important? To have a permanently free internet, we have to be able to support it without reliance on centralized powers. This is one of the simplest points, but there are many deeper reasons which I am not yet knowledgeable enough to express. It’s all about the philosophy of openness which is characteristic of applied category theory: historically, we have developed theories and practices which are isolated from each other and the world, and had to fabricate their interrelation and cooperation ad hoc; this leaves us needlessly struggling with inadequate systems, and limits our thought and action.

Surely there must be a less primitive way of making big changes in the store than by pushing vast numbers of words back and forth through the von Neumann bottleneck. Not only is this tube a literal bottleneck for the data traffic of a problem, but, more importantly, it is an intellectual bottleneck that has kept us tied to word-at-a-time thinking instead of encouraging us to think in terms of the larger conceptual units of the task at hand. Thus programming is basically planning and detailing the enormous traffic of words through the von Neumann bottleneck, and much of that traffic concerns not significant data itself, but where to find it. — John Backus, 1977 ACM Turing Award

There have been various mitigations to these kind of problems, but the cognitive limitation remains, and a total renewal is necessary; the π-calculus completely reimagines the nature of computation in society, and opens vast possibility. We can begin to conceive of the many interwoven strata of computation as a coherent, fluid whole. While I was out of my depth in many talks during the conference, I began to appreciate that this was a truly comprehensive innovation: RChain reforms almost every aspect of computing, from the highest abstraction all the way down to the metal. Coalescing the architecture, as mentioned earlier, is the formal calculus as the core guiding principle. There was some concern that because the ρ-calculus is so different from traditional languages, there may be resistance to adoption; but our era is a paradigm shift, the call is for a new way of thinking, and we must adapt.

So why are we using the reflective higher-order π-calculus, the ρ-calculus? Because there’s just one problem with the conventional π-calculus: it presupposes a countably infinite collection of atomic names. These are not only problematic to generate and manage, but the absence of structure is a massive waste. In this regard, the π-calculus was incomplete, until Greg realized that you can “close the loop” with reflection, a powerful form of self-reference:

Code ←→ Data

The mantra is that names are quoted processes; this idea pervades and guides the design of RChain. There is no need to import infinitely many opaque, meaningless symbols—the code itself is nothing but clear, meaningful syntax. If there is an intrinsic method of reference and dereference, or “quoting and unquoting”, code can be turned into data, sent as a message, and then turned back into code; known as “code mobility”, one can communicate big programs as easily as emails. This allows for metaprogramming: on an industrial level, not only people write programs—programs write programs. This is essential to creating a robust virtual infrastructure.

So, how can the π-calculus be made reflective? By solving for the least fixed point of a recursive equation, which parametrizes processes by names:

P[x] = 0 | x?(x).P[x] | x!(P[x]) | P[x]|P[x] | @P[x] | *x

RP = P[RP]

This is reminiscent of how the Y combinator enables recursion by giving the fixed point of any function, Yf = f(Yf). The last two terms of the syntax are reference and dereference, which turn code into data and data into code. Notice that we did not include a continuation for output: the ρ-calculus is asynchronous, meaning that the sender does not get confirmation that the message has been received; this is important for efficient parallel computation and corresponds to polarised linear logic. We adopt the convention that names are output and processes are input. The last two modifications to complete the official ρ-calculus syntax are multi-input and pattern-matching:

P,Q := 0 null process

| for(p1←x1,…,pn←xn).P input guarded process

| x!(@Q) output a name

| *x dereference, evaluate code

| P|Q parallel composition

x,p := @P name or quoted process

(each ‘pi’ is a “pattern” or formula to collect terms on channel ‘xi’—this is extremely useful and general, and enables powerful functionality throughout the system)

Simple. Of course, this is not really a programming language yet, though it is more usable than the pure λ-calculus. Rholang, the actual language of RChain, adds some essential features:

ρ-calculus + variables + useful ground terms + new name construction + arithmetic + pattern matching = Rholang

Here’s the specification, syntax and semantics, and a visualization; explore code and examples in GitHub and learn the contract design in the documentation—you can even try coding on rchain.cloud! For those who don’t like clicking all these links, let’s see just one concrete example of a contract, the basic program in Rholang: a process with persistent state, associated code, and associated addresses. This is a Cell, which stores a value until it is accessed or updated:

contract Cell( get, set, state ) = {

select {

case rtn <- get; v <- state => {

rtn!( *v ) | state!( *v ) | Cell( get, set, state ) }

case newValue <- set; v <- state => {

state!( *newValue ) | Cell( get, set, state ) }

}}

The parameters are the channels on which the contract communicates. Cell selects from two possibilities: either it is being accessed, i.e. there is data (the return channel) to receive on get, then it outputs on rtn and maintains its state and call; or it is being updated, i.e. there is data (the new value) to receive on set, then it updates state and calls itself again. This shows how the ontology of the language enables natural recursion, and thereby persistent storage: state is Cell’s way of “talking to itself”—since the sequential aspect of Rholang is functional, one “cycles” data to persist. The storage layer uses a similar idea; the semantics may be related to traced monoidal categories.

Curiously, the categorical semantics of the ρ-calculus has proven elusive. There is the general ideology that λ:sequential :: π:concurrent, that the latter is truly fundamental, but the Curry-Howard-Lambek isomorphism has not yet been generalized canonically—though there has been partial success, involving functor-category denotational semantics, linear logic, and session types. Despite its great power and universality, the ρ-calculus remains a bit of a mystery in mathematics: this fact should intrigue anyone who cares about logic, types, and categories as the foundations of abstract thought.

Now, the actual system—the architecture consists of five interwoven layers (all better explained in the documentation):

Storage: based on Special K – “a pattern language for the web.” This layer stores both key-value pairs and continuations through an essential duality of database and query—if you don’t find what you’re looking for, leave what you would have done in its place, and when it arrives, the process will continue. Greg characterizes this idea as the computational equivalent of the law of excluded middle.

[channel, pattern, data, continuation]

RChain has a refined, multidimensional view of resources – compute, memory, storage, and network—and accounts for their production and consumption linearly.

Execution: a Rho Virtual Machine instance is a context for ρ-calculus reduction of storage elements. The entire state of the blockchain is one big Rholang term, which is updated by a transaction: a receive invokes a process which changes key values, and the difference must be verified by consensus. Keys permanently maintain the entire history of state transitions. While currently based on the Java VM, it will be natively hosted.

Namespace: a set of channels, i.e. resources, organized by a logic for access and authority. The primary significance is scalability – a user does not need to deal with the whole chain, only pertinent namespaces. ‘A namespace definition may control the interactions that occur in the space, for example, by specifying: accepted addresses, namespaces, or behavioral types; maximum or minimum data size; or input-output structure.’ These handle nondeterminism of the two basic “race conditions”, contention for resources:

x!(@Q1) | for(ptrn <- x){P} | x!(@Q2)

for(ptrn <- x){P1} | x!(@Q) | for(ptrn <- x){P2}

Contrasted with flat public keys of other blockchains, domains work with DNS and extend them by a compositional tree structure. Each node as a named channel is itself a namespace, and hence definitions can be built up inductively, with precise control.

Consensus: verify partial orders of changes to the one-big-Rholang-term state; the block structure should persist as a directed acyclic graph. The algorithm is Proof of Stake – the capacity to validate in a namespace is tied to the “stake” one holds in it. Greg explains via tangles, and how the complex CASPER protocol works naturally with RChain.

Contracts: ‘An RChain contract is a well-specified, well-behaved, and formally verified program that interacts with other such programs.’ (K Framework) ; (DAO attack) ‘A behavioral type is a property of an object that binds it to a discrete range of action patterns. Behavioral types constrain not only the structure of input and output, but the permitted order of inputs and outputs among communicating and (possibly) concurrent processes under varying conditions… The Rholang behavioral type system will iteratively decorate terms with modal logical operators, which are propositions about the behavior of those terms. Ultimately properties [such as] data information flow, resource access, will be concretized in a type system that can be checked at compile-time. The behavioral type systems Rholang will support make it possible to evaluate collections of contracts against how their code is shaped and how it behaves. As such, Rholang contracts elevate semantics to a type-level vantage point, where we are able to scope how entire protocols can safely interface.’ (LADL)

So what can you build on RChain? Anything.

Decentralized applications: identity, reputation, tokens, timestamping, financial services, content delivery, exchanges, social networks, marketplaces, (decentralized autonomous) organizations, games, oracles, (Ethereum dApps), … new forms of code yet to be imagined. It’s much more than a better internet: RChain is a potential abstract foundation for a rational global society. The system is a minimalist framework of universally principled design; it is a canvas with which we can begin to explore how the world should really work. If we are open and thoughtful, if we care enough, we can learn to do things right.

The project is remarkably unknown for its magnitude, and building widespread adoption may be one of RChain’s greatest challenges. Granted, it is new; but education will not be easy. It’s too big a reformation for a public mindset which thinks of (technological) progress as incrementally better specs or added features; this is conceptual progression, an alien notion to many. That’s why the small but growing community is vital. This article is nothing; I’m clearly unqualified—click links, read papers, watch videos. The scale of ambition, the depth of insight, the lucidity of design, the unity of theory and practice—it’s something to behold. And it’s real. Mercury will be complete in December. It’s happening, right now, and you can be a part of it. Learn, spread the word. Get involved; join the discussion or even the development—the RChain website has all the resources you need.

40% of the world population lives within 100km of the ocean. Greg pointed out that if we can’t even handle today’s refugee crises, what will possibly happen when the waters rise? At the very least, we desperately need better large-scale coordination systems. Will we make it to the next millennium? We can—just a matter of will.

Thank you for reading. You are great.

### John Baez - Azimuth

My course on applied category theory is continuing! After a two-week break where the students did exercises, I’m back to lecturing about Fong and Spivak’s book *Seven Sketches*. Now we’re talking about “resource theories”. Resource theories help us answer questions like this:

- Given what I have,
*is it possible*to get what I want? - Given what I have,
*how much will it cost*to get what I want? - Given what I have,
*how long will it take*to get what I want? - Given what I have,
*what is the set of ways*to get what I want?

Resource theories in their modern form were arguably born in these papers:

• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, Resource convertibility and ordered commutative monoids.

We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory here on this blog:

• Tobias Fritz, Resource convertibility (part 1), *Azimuth*, 7 April 2015.

• Tobias Fritz, Resource convertibility (part 2), *Azimuth*, 10 April 2015.

• Tobias Fritz, Resource convertibility (part 3), *Azimuth*, 13 April 2015.

We’re having fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures so far:

• Lecture 18 – Chapter 2: Resource Theories

• Lecture 19 – Chapter 2: Chemistry and Scheduling

• Lecture 20 – Chapter 2: Manufacturing

• Lecture 21 – Chapter 2: Monoidal Preorders

• Lecture 22 – Chapter 2: Symmetric Monoidal Preorders

• Lecture 23 – Chapter 2: Commutative Monoidal Posets

• Lecture 24 – Chapter 2: Pricing Resources

• Lecture 25 – Chapter 2: Reaction Networks

• Lecture 26 – Chapter 2: Monoidal Monotones

• Lecture 27 – Chapter 2: Adjoints of Monoidal Monotones

• Lecture 28 – Chapter 2: Ignoring Externalities

### Clifford V. Johnson - Asymptotia

Turns out that 100 years ago today, Richard Feynman was born. His contributions to physics - science in general - are huge, and if you dig a little you'll find lots of discussion about him. His beautiful "Lectures on Physics..." books are deservedly legendary, and I wish that my old Imperial College lecturers had spent more time impressing upon us young impressionable undergraduate minds (c1986) to read those instead of urging us at every opportunity to read the famous "Surely You're Joking..." book, which even back then in my naivety, I began to recognise as partly a physicist's user manual for how to be a jerk to those around you. (I know I'm in the minority on this point...)

But anyway, in honour of the occasion, I give you a full page from my book containing a chat about the Feynman diagram. It's an example of how something that's essentially a cartoon can play a central role in understanding our world (something that's of course, not unknown in cartoons...) Click the image above for an enlarged view.

-cvj Click to continue reading this post

The post Feynman Centenary appeared first on Asymptotia.

## May 11, 2018

### Jester - Resonaances

Sometimes progress consists in realizing that you know nothing Jon Snow. The lack of new physics at the LHC invalidates most of the historical motivations for WIMPs. Theoretically, the mass of the dark matter particle could be anywhere between 10^-30 GeV and 10^19 GeV. There are myriads of models positioned anywhere in that range, and it's hard to argue with a straight face that any particular one is favored. We now know that we don't know what dark matter is, and that we should better search in many places. If anything, the small-scale problem of the 𝞚CDM cosmological model can be interpreted as a hint against the boring WIMPS and in favor of light dark matter. For example, if it turns out that dark matter has significant (nuclear size) self-interactions, that can only be realized with sub-GeV particles.

It takes some time for experiment to catch up with theory, but the process is already well in motion. There is some fascinating progress on the front of ultra-light axion dark matter, which deserves a separate post. Here I want to highlight the ongoing developments in direct detection of dark matter particles with masses between MeV and GeV. Until recently, the only available constraint in that regime was obtained by recasting data from the XENON10 experiment - the grandfather of the currently operating XENON1T. In XENON detectors there are two ingredients of the signal generated when a target nucleus is struck:

*ionization electrons*and

*scintillation photons.*WIMP searches require both to discriminate signal from background. But MeV dark matter interacting with electrons could eject electrons from xenon atoms without producing scintillation. In the standard analysis, such events would be discarded as background. However, this paper showed that, recycling the available XENON10 data on ionization-only events, one can exclude dark matter in the 100 MeV ballpark with the cross section for scattering on electrons larger than ~0.01 picobarn (10^-38 cm^2). This already has non-trivial consequences for concrete models; for example, a part of the parameter space of milli-charged dark matter is currently best constrained by XENON10.

It is remarkable that so much useful information can be extracted by basically misusing data collected for another purpose (earlier this year the DarkSide-50 recast their own data in the same manner, excluding another chunk of the parameter space). Nevertheless, dedicated experiments will soon be taking over. Recently, two collaborations published first results from their prototype detectors: one is SENSEI, which uses 0.1 gram of silicon CCDs, and the other is SuperCDMS, which uses 1 gram of silicon semiconductor. Both are sensitive to eV energy depositions, thanks to which they can extend the search region to lower dark matter mass regions, and set novel limits in the virgin territory between 0.5 and 5 MeV. A compilation of the existing direct detection limits is shown in the plot. As you can see, above 5 MeV the tiny prototypes cannot yet beat the XENON10 recast. But that will certainly change as soon as full-blown detectors are constructed, after which the XENON10 sensitivity should be improved by several orders of magnitude.

Should we be restless waiting for these results? Well, for any single experiment the chance of finding nothing are immensely larger than that of finding something. Nevertheless, the technical progress and the widening scope of searches offer some hope that the dark matter puzzle may be solved soon.

by Mad Hatter (noreply@blogger.com) at May 11, 2018 02:35 PM

## May 10, 2018

### Sean Carroll - Preposterous Universe

Some of you might be familiar with the Moving Naturalism Forward workshop I organized way back in 2012. For two and a half days, an interdisciplinary group of naturalists (in the sense of “not believing in the supernatural”) sat around to hash out the following basic question: “So we don’t believe in God, what next?” How do we describe reality, how can we be moral, what are free will and consciousness, those kinds of things. Participants included Jerry Coyne, Richard Dawkins, Terrence Deacon, Simon DeDeo, Daniel Dennett, Owen Flanagan, Rebecca Newberger Goldstein, Janna Levin, Massimo Pigliucci, David Poeppel, Nicholas Pritzker, Alex Rosenberg, Don Ross, and Steven Weinberg.

Happily we recorded all of the sessions to video, and put them on YouTube. Unhappily, those were just unedited proceedings of each session — so ten videos, at least an hour and a half each, full of gems but without any very clear way to find them if you weren’t patient enough to sift through the entire thing.

No more! Thanks to the heroic efforts of Gia Mora, the proceedings have been edited down to a number of much more accessible and content-centered highlights. There are over 80 videos (!), with a median length of maybe 5 minutes, though they range up to about 20 minutes and down to less than one. Each video centers on a particular idea, theme, or point of discussion, so you can dive right into whatever particular issues you may be interested in. Here, for example, is a conversation on “Mattering and Secular Communities,” featuring Rebecca Goldstein, Dan Dennett, and Owen Flanagan.

The videos can be seen on the workshop web page, or on my YouTube channel. They’re divided into categories:

- Introductions
- Consciousness
- Emergence and Reduction
- What Is Real?
- Morality
- Meaning
- Free Will
- Philosophy and Science
- Final Thoughts

A lot of good stuff in there. Enjoy!

## May 09, 2018

### Jester - Resonaances

*K*+ → 𝝿+ 𝜈 𝜈. The Standard Model predicts the branching fraction BR(

*K*+ → 𝝿+ 𝜈 𝜈) = 8.4x10^-11 with a small, 10% theoretical uncertainty (precious stuff in the flavor business). The previous measurement by the BNL-E949 experiment reported BR(

*K*+ → 𝝿+ 𝜈 𝜈) = (1.7 ± 1.1)x10^-10, consistent with the Standard Model, but still leaving room for large deviations. NA62 is expected to pinpoint the decay and measure the branching fraction with a 10% accuracy, thus severely constraining new physics contributions. The wires, pipes, and gory details of the analysis were nicely summarized by Tommaso. Let me jump directly to explaining what is it good for from the theory point of view.

To this end it is useful to adopt the effective theory perspective. At a more fundamental level, the decay occurs due to the strange quark inside the kaon undergoing the transformation

*sbar*→

*dbar*𝜈 𝜈

*bar*. In the Standard Model, the amplitude for that process is dominated by one-loop diagrams with W/Z bosons and heavy quarks. But kaons live at low energies and do not really see the fine details of the loop amplitude. Instead, they effectively see the 4-fermion contact interaction:

The mass scale suppressing this interaction is quite large, more than 1000 times larger than the W boson mass, which is due to the loop factor and small CKM matrix elements entering the amplitude. The strong suppression is the reason why the

*K*+ → 𝝿+ 𝜈 𝜈 decay is so rare in the first place. The corollary is that even a small new physics effect inducing that effective interaction may dramatically change the branching fraction. Even a particle with a mass as large as 1 PeV coupled to the quarks and leptons with order one strength could produce an observable shift of the decay rate. In this sense, NA62 is a microscope probing physics down to 10^-20 cm distances, or up to PeV energies, well beyond the reach of the LHC or other colliders in this century. If the new particle is lighter, say order TeV mass, NA62 can be sensitive to a tiny milli-coupling of that particle to quarks and leptons.

So, from a model-independent perspective, the advantages of studying the

*K*+ → 𝝿+ 𝜈 𝜈 decay are quite clear. A less trivial question is what can the future NA62 measurements teach us about our cherished models of new physics. One interesting application is in the industry of explaining the apparent violation of lepton flavor universality in

*B*→

*K*

*l*+

*l*-, and

*B*→

*D l*𝜈 decays. Those anomalies involve the 3rd generation bottom quark, thus a priori they do not need to have anything to do with kaon decays. However, many of the existing models introduce flavor symmetries controlling the couplings of the new particles to matter (instead of just ad-hoc interactions to address the anomalies). The flavor symmetries may then relate the couplings of different quark generations, and thus predict correlations between new physics contributions to B meson and to kaon decays. One nice example is illustrated in this plot:

The observable RD(*) parametrizes the preference for

*B*→

*D*𝜏 𝜈 over similar decays with electrons and muon, and its measurement by the BaBar collaboration deviates from the Standard Model prediction by roughly 3 sigma. The plot shows that, in a model based on U(2)xU(2) flavor symmetry, a significant contribution to RD(*) generically implies a large enhancement of BR(

*K*+ → 𝝿+ 𝜈 𝜈), unless the model parameters are tuned to avoid that. The anomalies in the

*B*→

*K*(*) 𝜇 𝜇 decays can also be correlated with large effects in

*K*+ → 𝝿+ 𝜈 𝜈, see here for an example. Finally, in the presence of new light invisible particles, such as axions, the NA62 observations can be polluted by exotic decay channels, such as e.g.

*K*+ →

*axion*𝝿+.

The

*K*+ → 𝝿+ 𝜈 𝜈 decay is by no means the magic bullet that will inevitably break the Standard Model. It should be seen as one piece of a larger puzzle that may or may not provide crucial hints about new physics. For the moment, NA62 has analyzed only a small batch of data collected in 2016, and their error bars are still larger than those of BNL-E949. That should change soon when the 2017 dataset is analyzed. More data will be acquired this year, with 20 signal events expected before the long LHC shutdown. Simultaneously, another experiment called KOTO studies an even more rare process where neutral kaons undergo the CP-violating decay

*KL*→ 𝝿0 𝜈 𝜈, which probes the imaginary part of the effective operator written above. As I wrote recently, my feeling is that low-energy precision experiments are currently our best hope for a better understanding of fundamental interactions, and I'm glad to see a good pace of progress on this front.

by Mad Hatter (noreply@blogger.com) at May 09, 2018 07:31 PM

### Jester - Resonaances

AI is also entering the field of science at an accelerated pace, and particle physics is as usual in the avant-garde. It's not a secret that physics analyses for the LHC papers (even if finally signed by 1000s of humans) are in reality performed by

*neural networks*, which are just beefed up versions of Alexa developed at CERN. The hottest topic in high-energy physics experiment is now machine learning, where computers teach humans the optimal way of clustering jets, or telling quarks from gluons. The question is

**when**, not if, AI will become sophisticated enough to perform a creative work of theoreticians.

It seems that the answer is

**now**.

Some of you might have noticed a certain Alan Irvine, affiliated with the Los Alamos National Laboratory, regularly posting on arXiv single-author theoretical papers on fashionable topics such as the ATLAS diphoton excess, LHCb B-meson anomalies, DAMPE spectral feature, etc. Many of us have received emails from this author requesting citations. Recently I got one myself; it seemed overly polite, but otherwise it didn't differ in relevance or substance from other similar requests. During the last two and half years, A. Irvine has accumulated a decent h-factor of 18. His papers have been submitted to prestigious journals in the field, such as the PRL, JHEP, or PRD, and some of them were even accepted after revisions. The scandal broke out a week ago when a JHEP editor noticed that the extensive revision, together with a long cover letter, was submitted within 10 seconds from receiving the referee's comments. Upon investigation, it turned out that A. Irvine never worked in Los Alamos, nobody in the field has ever met him in person, and the IP from which the paper was submitted was that of the well-known Ragnarok Thor server. A closer analysis of his past papers showed that, although linguistically and logically correct, they were merely a compilation of equations and text from the previous literature without any original addition.

Incidentally, arXiv administrators have been aware that, since a few years, all source files in daily hep-ph listings were downloaded for an unknown purpose by automated bots. When you have excluded the impossible, whatever remains, however improbable, must be the truth. There is no doubt that A. Irvine is an AI bot, that was trained on the real hep-ph input to produce genuinely-looking particle theory papers.

The works of A. Irvine have been quietly removed from arXiv and journals, but difficult questions remain. What was the purpose of it? Was it a spoof? A parody? A social experiment? A Facebook research project? A Russian provocation? And how could it pass unnoticed for so long within the theoretical particle community? What's most troubling is that, if there was one, there can easily be more. Which other papers on arXiv are written by AI? How can we recognize them? Should we even try, or maybe the dam is already broken and we have to accept the inevitable? Is Résonaances written by a real person? How can you be sure that you are real?

*Update: obviously, this post is an April Fools' prank. It is absolutely unthinkable that the creative process of writing modern particle theory papers can ever be automatized. Also, the neural network referred to in the LHC papers is nothing like Alexa; it's simply a codename for PhD students. Finally, I assure you that*

*Résonaances is written by a hum*00105e0 e6b0 343b 9c74 0804 e7bc 0804 e7d5 0804 [core dump]

by Mad Hatter (noreply@blogger.com) at May 09, 2018 07:31 PM

### Jester - Resonaances

In this respect, nothing much has changed during the time when the blog was dormant, except that these sentiments are now firmly established. Crisis is no longer a whispered word, but it's openly discussed in corridors, on blogs, on arXiv, and in color magazines. The clear message from the LHC is that the dominant paradigms about the physics at the weak scale were completely misguided. The Standard Model seems to be a perfect effective theory at least up to a few TeV, and there is no indication at what energy scale new particles have to show up. While everyone goes through the five stages of grief at their own pace, my impression is that most are already well past the denial. The open question is what should be the next steps to make sure that exploration of fundamental interactions will not halt.

One possible reaction to a crisis is

*more of the same*. Historically, such an approach has often been efficient, for example it worked for a long time in the case of the Soviet economy. In our case one could easily go on with more models, more epicycles, more parameter space, more speculations. But the driving force for all these SusyWarpedCompositeStringBlackHairyHole enterprise has always been the (small but still) possibility of being vindicated by the LHC. Without serious prospects of experimental verification, model building is reduced to intellectual gymnastics that can hardly stir imagination. Thus the business-as-usual is not an option in the long run: it couldn't elicit any enthusiasm among the physicists or the public, it wouldn't attract new bright students, and thus it would be a straight path to irrelevance.

So, particle physics has to change. On the experimental side we will inevitably see, just for economical reasons, less focus on high-energy colliders and more on smaller experiments. Theoretical particle physics will also have to evolve to remain relevant. Certainly, the emphasis needs to be shifted away from empty speculations in favor of more solid research. I don't pretend to know all the answers or have a clear vision of the optimal strategy, but I see

*three*promising directions.

One is astrophysics where there are much better prospects of experimental progress. The cosmos is a natural collider that is constantly testing fundamental interactions independently of current fashions or funding agencies. This gives us an opportunity to learn more about dark matter and neutrinos, and also about various hypothetical particles like axions or milli-charged matter. The most recent story of the 21cm absorption signal shows that there are still treasure troves of data waiting for us out there. Moreover, new observational windows keep opening up, as recently illustrated by the nascent gravitational wave astronomy. This avenue is of course a non-brainer, already explored since a long time by particle theorists, but I expect it will further gain in importance in the coming years.

Another direction is precision physics. This, also, has been an integral part of particle physics research for quite some time, but it should grow in relevance. The point is that one can probe very heavy particles, often beyond the reach of present colliders, by precisely measuring low-energy observables. In the most spectacular example, studying proton decay may give insight into new particles with masses of order 10^16 GeV - unlikely to be ever attainable directly. There is a whole array of observables that can probe new physics well beyond the direct LHC reach: a myriad of rare flavor processes, electric dipole moments of the electron and neutron, atomic parity violation, neutrino scattering, and so on. This road may be long and tedious but it is bound to succeed: at some point some experiment somewhere must observe a phenomenon that does not fit into the Standard Model. If we're very lucky, it may be that the anomalies currently observed by the LHCb in certain rare B-meson decays are already the first harbingers of a breakdown of the Standard Model at higher energies.

Finally, I should mention formal theoretical developments. The naturalness problem of the cosmological constant and of the Higgs mass may suggest some fundamental misunderstanding of quantum field theory on our part. Perhaps this should not be too surprising. In many ways we have reached an amazing proficiency in QFT when applied to certain precision observables or even to LHC processes. Yet at the same time QFT is often used and taught in the same way as magic in Hogwarts: mechanically, blindly following prescriptions from old dusty books, without a deeper understanding of the sense and meaning. Recent years have seen a brisk development of alternative approaches: a revival of the old S-matrix techniques, new amplitude calculation methods based on recursion relations, but also complete reformulations of the QFT basics demoting the sacred cows like fields, Lagrangians, and gauge symmetry. Theory alone rarely leads to progress, but it may help to make more sense of the data we already have. Could better understanding or complete reformulating of QFT bring new answers to the old questions? I think that is not impossible.

All in all, there are good reasons to worry, but also tons of new data in store and lots of fascinating questions to answer. How will the B-meson anomalies pan out? What shall we do after we hit the neutrino floor? Will the 21cm observations allow us to understand what dark matter is? Will China build a 100 TeV collider? Or maybe a radio telescope on the Moon instead? Are experimentalists still needed now that we have machine learning? How will physics change with the centre of gravity moving to Asia? I will tell you my take on such and other questions and highlight old and new ideas that could help us understand the nature better. Let's see how far I'll get this time ;)

by Mad Hatter (noreply@blogger.com) at May 09, 2018 07:31 PM

## May 08, 2018

### John Baez - Azimuth

*guest post by Matteo Polettini*

Suppose you receive an email from someone who claims “here is the project of a machine that runs forever and ever and produces energy for free!” Obviously he must be a crackpot. But he may be well-intentioned. You opt for not being rude, roll your sleeves, and put your hands into the dirt, holding the Second Law as lodestar.

Keep in mind that there are two fundamental sources of error: either he is not considering certain input currents (“hey, what about that tiny hidden cable entering your machine from the electrical power line?!”, “uh, ah, that’s just to power the “ON” LED”, “mmmhh, you sure?”), or else he is not measuring the energy input correctly (“hey, why are you using a Geiger counter to measure input voltages?!”, “well, sir, I ran out of voltmeters…”).

In other words, the observer might only have partial information about the setup, either in quantity or quality. Because he has been marginalized by society (most crackpots believe they are misunderstood geniuses) we will call such observer “marginal,” which incidentally is also the word that mathematicians use when they focus on the probability of a subset of stochastic variables.

In fact, our modern understanding of thermodynamics as embodied in statistical mechanics and stochastic processes is founded (and funded) on ignorance: we never really have “complete” information. If we actually had, all energy would look alike, it would not come in “more refined” and “less refined” forms, there would not be a differentials of order/disorder (using Paul Valery’s beautiful words), and that would end thermodynamic reasoning, the energy problem, and generous research grants altogether.

Even worse, within this statistical approach we might be missing chunks of information because some parts of the system are invisible to us. But then, what warrants that * we *are doing things right, and *he* (our correspondent) is the crackpot? Couldn’t it be the other way around? Here I would like to present some recent ideas I’ve been working on together with some collaborators on how to deal with incomplete information about the sources of dissipation of a thermodynamic system. I will do this in a quite theoretical manner, but somehow I will mimic the guidelines suggested above for debunking crackpots. My three buzzwords will be: **marginal**, **effective**, and **operational**.

### “Complete” thermodynamics: an out-of-the-box view

The laws of thermodynamics that I address are:

• The good ol’ Second Law (2nd)

• The Fluctuation-Dissipation Relation (FDR), and the Reciprocal Relation (RR) close to equilibrium.

• The more recent Fluctuation Relation (FR)^{1} and its corollary the Integral Fluctuation Relation (IFR), which have been discussed on this blog in a remarkable post by Matteo Smerlak.

The list above is all in the “area of the second law”. How about the other laws? Well, thermodynamics has for long been a phenomenological science, a patchwork. So-called stochastic thermodynamics is trying to put some order in it by systematically grounding thermodynamic claims in (mostly Markov) stochastic processes. But it’s not an easy task, because the different laws of thermodynamics live in somewhat different conceptual planes. And it’s not even clear if they are theorems, prescriptions, or habits (a bit like in jurisprudence^{2}).

Within stochastic thermodynamics, the Zeroth Law is so easy nobody cares to formulate it (I do, so stay tuned…). The Third Law: no idea, let me know. As regards the First Law (or, better, “laws”, as many as there are conserved quantities across the system/environment interface…), we will assume that all related symmetries have been exploited from the offset to boil down the description to a minimum.

This minimum is as follows. We identify a system that is well separated from its environment. The system evolves in time, the environment is so large that its state does not evolve within the timescales of the system^{3}. When tracing out the environment from the description, an uncertainty falls upon the system’s evolution. We assume the system’s dynamics to be described by a stochastic Markovian process.

How exactly the system evolves and what is the relationship between system and environment will be described in more detail below. Here let us take an “out of the box” view. We resolve the environment into several reservoirs labeled by index . Each of these reservoirs is “at equilibrium” on its own (whatever that means^{4}). Now, the idea is that each reservoir tries to impose “its own equilibrium” on the system, and that their competition leads to a flow of currents across the system/environment interface. Each time an amount of the reservoir’s resource crosses the interface, a “thermodynamic cost” has to be to be paid or gained (be it a chemical potential difference for a molecule to go through a membrane, or a temperature gradient for photons to be emitted/absorbed, etc.).

The fundamental quantities of stochastic thermodynamic modeling thus are:

• On the “-dynamic” side: the time-integrated currents , *independent* among themselves^{5}. Currents are stochastic variables distributed with joint probability density

• On the “thermo-” side: The so-called thermodynamic forces or “affinities”^{6} (collectively denoted ). These are tunable parameters that characterize reservoir-to-reservoir gradients, and they are not stochastic. For convenience, we conventionally take them all positive.

Dissipation is quantified by the **entropy production**:

We are finally in the position to state the main results. Be warned that in the following expressions the exact treatment of time and its scaling would require a lot of specifications, but keep in mind that all these relations hold true in the long-time limit, and that all cumulants scale linearly with time.

• **FR**: The probability of observing positive currents is exponentially favoured with respect to negative currents according to

*Comment*: This is not trivial, it follows from the explicit expression of the path integral, see below.

• **IFR**: The exponential of minus the entropy production is unity

*Homework*: Derive this relation from the FR in one line.

• **2nd Law**: The average entropy production is not negative

*Homework*: Derive this relation using Jensen’s inequality.

• **Equilibrium**: Average currents vanish if and only if affinities vanish:

*Homework*: Derive this relation taking the first derivative w.r.t. of the IFR. Notice that also the average depends on the affinities.

• **S-FDR**: At equilibrium, it is impossible to tell whether a current is due to a spontaneous fluctuation (quantified by its variance) or to an external perturbation (quantified by* *the response of its mean). In a symmetrized (S-) version:

*Homework*: Derive this relation taking the mixed second derivatives w.r.t. of the IFR.

• **RR**: The reciprocal response of two different currents to a perturbation of the reciprocal affinities close to equilibrium is symmetrical:

*Homework*: Derive this relation taking the mixed second derivatives w.r.t. of the FR.

Notice the implication scheme: FR ⇒ IFR ⇒ 2nd, IFR ⇒ S-FDR, FR ⇒ RR.

### “Marginal” thermodynamics (still out-of-the-box)

Now we assume that we can only measure a marginal subset of currents (index always has a smaller range than ), distributed with joint marginal probability

Notice that a state where these marginal currents vanish might not be an equilibrium, because other currents might still be whirling around. We call this a *stalling* state.

My central question is: *can we associate to these currents some effective affinity in such a way that at least some of the results above still hold true? And, are all definitions involved just a fancy mathematical construct, or are they operational?*

First the bad news: In general the FR is violated for all choices of effective affinities:

This is not surprising and nobody would expect that. How about the IFR?

• **Marginal IFR**: There are effective affinities such that

Mmmhh. Yeah. Take a closer look this expression: can you see why there actually exists an infinite choice of “effective affinities” that would make that average cross 1? Which on the other hand is just a number, so who even cares? So this can’t be the point.

The fact is, the IFR per se is hardly of any practical interest, as are all “absolutes” in physics. What matters is “relatives”: in our case, response. But then we need to specify how the effective affinities depend on the “real” affinities. And here steps in a crucial technicality, whose precise argumentation is a pain. Basing on reasonable assumptions^{7}, we demonstrate that the IFR holds for the following choice of effective affinities:

,

where is the set of values of the affinities that make marginal currents stall. Notice that this latter formula gives an *operational* definition of the effective affinities that could in principle be reproduced in laboratory (just go out there and tune the tunable until everything stalls, and measure the difference). Obviously:

• **Stalling**: Marginal currents vanish if and only if effective affinities vanish:

Now, according to the inference scheme illustrated above, we can also prove that:

• **Effective 2nd Law**: The average marginal entropy production is not negative

• **S-FDR at stalling**:

Notice instead that the RR is gone at stalling. This is a clear-cut prediction of the theory that can be experimented with basically the same apparatus with which response theory has been experimentally studied so far (not that I actually know what these apparatus are…): *at stalling states, differing from equilibrium states, the S-FDR still holds, but the RR does not*.

### Into the box

You’ve definitely gotten enough at this point, and you can give up here. Please exit through the gift shop.

If you’re stubborn, let me tell you what’s inside the box. The system’s dynamics is modeled as a **continuous-time, discrete configuration-space Markov “jump” process**. The state space can be described by a graph where is the set of configurations, is the set of possible transitions or “edges”, and there exists some incidence relation between edges and couples of configurations. The process is determined by the rates of jumping from one configuration to another.

We choose these processes because they allow some nice network analysis and because the path integral is well defined! A single realization of such a process is a **trajectory**

A “Markovian jumper” waits at some configuration for some time with an exponentially decaying probability with exit rate , then instantaneously jumps to a new configuration with transition probability . The overall probability density of a single trajectory is given by

One can in principle obtain the probability distribution function of any observable defined along the trajectory by taking the marginal of this measure (though in most cases this is technically impossible). Where does this expression come from? For a formal derivation, see the very beautiful review paper by Weber and Frey, but be aware that this is what one would intuitively come up with if one had to simulate with the Gillespie algorithm.

The dynamics of the Markov process can also be described by the probability of being at some configuration at time , which evolves via the **master equation**

.

We call such probability the system’s **state**, and we assume that the system relaxes to a uniquely defined steady state .

A time-integrated current along a single trajectory is a linear combination of the net number of jumps between configurations in the network:

The idea here is that one or several transitions within the system occur because of the “absorption” or the “emission” of some environmental degrees of freedom, each with different intensity. However, for the moment let us simplify the picture and require that only one transition contributes to a current, that is that there exist such that

.

Now, what does it mean for such a set of currents to be “complete”? Here we get inspiration from Kirchhoff’s Current Law in electrical circuits: the continuity of the trajectory at each configuration of the network implies that after a sufficiently long time, *cycle *or *loop * or *mesh *currents completely describe the steady state. There is a standard procedure to identify a set of cycle currents: take a spanning tree of the network; then the currents flowing along the edges left out from the spanning tree form a complete set.

The last ingredient you need to know are the affinities. They can be constructed as follows. Consider the Markov process on the network where the observable edges are removed . Calculate the steady state of its associated master equation , which is necessarily an equilibrium (since there cannot be cycle currents in a tree…). Then the affinities are given by

.

Now you have all that is needed to formulate the complete theory and prove the FR.

*Homework*: (Difficult!) With the above definitions, prove the FR.

How about the marginal theory? To define the effective affinities, take the set of edges where there run observable currents. Notice that now its complement obtained by removing the observable edges, the **hidden** edge set , is not in general a spanning tree: there might be cycles that are not accounted for by our observations. However, we can still consider the Markov process on the hidden space, and calculate its **stalling** steady state , and ta-taaa: The effective affinities are given by

.

Proving the marginal IFR is far more complicated than the complete FR. In fact, very often in my field we will not work with the current’ probability density itself, but we prefer to take its bidirectional Laplace transform and work with the currents’ cumulant generating function. There things take a quite different and more elegant look.

Many other questions and possibilities open up now. The most important one left open is: Can we generalize the theory the (physically relevant) case where the current is supported on several edges? For example, for a current defined like ? Well, it depends: the theory holds provided that the stalling state is not “internally alive”, meaning that if the observable current vanishes on average, then also should and separately. This turns out to be a physically meaningful but quite strict condition.

### Is all of thermodynamics “effective”?

Let me conclude with some more of those philosophical considerations that sadly I have to leave out of papers…

Stochastic thermodynamics strongly depends on the identification of physical and information-theoretic entropies â€” something that I did not openly talk about, but that lurks behind the whole construction. Throughout my short experience as researcher I have been pursuing a program of “relativization” of thermodynamics, by making the role of the observer more and more evident and movable. Inspired by Einstein’s *Gedankenexperimenten*, I also tried to make the theory operational. This program may raise eyebrows here and there: Many thermodynamicians embrace a naive materialistic world-view whereby what only matters are “real” physical quantities like temperature, pressure, and all the rest of the information-theoretic discourse is at best mathematical speculation or a fascinating analog with no fundamental bearings. According to some, information as a physical concept lingers alarmingly close to certain extreme postmodern claims in the social sciences that “reality” does not exist unless observed, a position deemed dangerous at times when the authoritativeness of science is threatened by all sorts of anti-scientific waves.

I think, on the contrary, that making concepts relative and effective and by summoning the observer explicitly is a laic and prudent position that serves as an antidote to radical subjectivity. The other way around—clinging to the objectivity of a preferred observer, which is implied in any materialistic interpretation of thermodynamics, e.g. by assuming that the most fundamental degrees of freedom are the positions and velocities of gas’s molecules—is the dangerous position, expecially when the role of such preferred observer is passed around from the scientist to the technician and eventually to the technocrat, who would be induced to believe there are simple technological fixes to complex social problems…

How do we reconcile observer-dependency and the laws of physics? The object and the subject? On the one hand, much like the position of an object depends on the reference frame, so much so entropy and entropy production do depend on the observer and the particular apparatus that he controls or experiment he is involved with. On the other hand, much like motion is ultimately independent of position and it is agreed upon by all observers that share compatible measurement protocols, so much so the laws of thermodynamics are independent of that particular observer’s quantification of entropy and entropy production (e.g., the effective Second Law holds independently of how much the marginal observer knows of the system, if he operates according to our phenomenological protocol…). This is the case even in the every-day thermodynamics as practiced by energetic engineers *et al.*, where there are lots of choices to gauge upon, and there is no other external warrant that the amount of dissipation being quantified is the “true” one (whatever that means…)—there can only be trust in one’s own good practices and methodology.

So in this sense, I like to think that all observers are marginal, that this effective theory serves as a dictionary by which different observers practice and communicate thermodynamics, and that we should not revere the laws of thermodynamics as “true” idols, but rather as tools of good scientific practice.

### References

• M. Polettini and M. Esposito, Effective fluctuation and response theory, arXiv:1803.03552.

In this work we give the complete theory and numerous references to work of other people that was along the same lines. We employ a “spiral” approach to the presentation of the results, inspired by the pedagogical principle of Albert Baez.

• M. Polettini and M. Esposito, Effective thermodynamics for a marginal observer, *Phys. Rev. Lett.* **119** (2017), 240601, arXiv:1703.05715.

This is a shorter version of the story.

• B. Altaner, M. Polettini and M. Esposito, Fluctuation-dissipation relations far from equilibrium, *Phys. Rev. Lett.* **117** (2016), 180601, arXiv:1604.0883.

An early version of the story, containing the FDR results but not the full-fledged FR.

• G. Bisker, M. Polettini, T. R. Gingrich and J. M. Horowitz, Hierarchical bounds on entropy production inferred from partial information, *J. Stat. Mech.* (2017), 093210, arXiv:1708.06769.

Some extras.

• M. F. Weber and E. Frey, Master equations and the theory of stochastic path integrals, *Rep. Progr. Phys.* **80** (2017), 046601, arXiv:1609.02849.

Great reference if one wishes to learn about path integrals for master equation systems.

### Footnotes

^{1} There are as many so-called “Fluctuation Theorems” as there are authors working on them, so I decided not to call them by any name. Furthermore, notice I prefer to distinguish between a relation (a formula) and a theorem (a line of reasoning). I lingered more on this here.

^{2} *“Just so you know, nobody knows what energy is.”*—Richard Feynman.

I cannot help but mention here the beautiful book by Shapin and Schaffer, *Leviathan and the Air-Pump*, about the Boyle vs. Hobbes diatribe about what constitutes a “matter of fact,” and Bruno Latour’s interpretation of it in *We Have Never Been Modern*. Latour argues that “modernity” is a process of separation of the human and natural spheres, and within each of these spheres a process of purification of the unit facts of knowledge and the unit facts of politics, of the object and the subject. At the same time we live in a world where these two spheres are never truly separated, a world of “hybrids” that are at the same time necessary “for all practical purposes” and unconceivable according to the myths that sustain the narration of science, of the State, and even of religion. In fact, despite these myths, we cannot conceive a scientific fact out of the contextual “network” where this fact is produced and replicated, and neither we can conceive society out of the material needs that shape it: so in this sense “we have never been modern”, we are not quite different from all those societies that we take pleasure of studying with the tools of anthropology. Within the scientific community Latour is widely despised; probably he is also misread. While it is really difficult to see how his analysis applies to, say, high-energy physics, I find that thermodynamics and its ties to the industrial revolution perfectly embodies this tension between the natural and the artificial, the matter of fact and the matter of concern. Such great thinkers as Einstein and Ehrenfest thought of the Second Law as the only physical law that would never be replaced, and I believe this is revelatory. A second thought on the Second Law, a systematic and precise definition of all its terms and circumstances, reveals that the only formulations that make sense are those phenomenological statements such as Kelvin-Planck’s or similar, which require a lot of contingent definitions regarding the operation of the engine, while fetishized and universal statements are nonsensical (such as that masterwork of confusion that is “the entropy of the Universe cannot decrease”). In this respect, it is neither a purely *natural* law—as the moderns argue, nor a purely *social* construct—as the postmodern argue. One simply has to renounce to operate this separation. While I do not have a definite answer on this problem, I like to think of the Second Law as a *practice*, a consistency check of the thermodynamic discourse.

^{3} This assumption really belongs to a time, the XIXth century, when resources were virtually infinite on planet Earth…

^{4} As we will see shortly, we *define* equilibrium as that state where there are no currents at the interface between the system and the environment, so what is the environment’s own definition of equilibrium?!

^{5} This because we have already exploited the First Law.

^{6} This nomenclature comes from alchemy, via chemistry (think of Goethe’s *The elective affinities*…), it propagated in the XXth century via De Donder and Prigogine, and eventually it is still present in language in Luxembourg because in some way we come from the “late Brussels school”.

^{7} Basically, we ask that the tunable parameters are environmental properties, such as temperatures, chemical potentials, etc. and *not* internal properties, such as the energy landscape or the activation barriers between configurations.

## May 06, 2018

### The n-Category Cafe

*Compositionality*

A new journal! We’ve been working on it for a long time, but we finished sorting out some details at Applied Category Theory 2018, and now we’re ready to tell the world!

Here’s the top of the journal’s home page right now:

Here’s the official announcement:

We are pleased to announce the launch of

Compositionality, a new diamond open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition. To learn more about the scope and editorial policies of the journal, please visit our website at www.compositionality-journal.org/

Compositionalityis the culmination of a long-running discussion by many members of the extended category theory community, and the editorial policies, look, and mission of the journal have yet to be finalized. We would love to get your feedback about our ideas on the forum we have established for this purpose:http://reddit.com/r/compositionality

Lastly, the journal is currently receiving applications to serve on the editorial board; submissions are due May 31 and will be evaluated by the members of our steering board: John Baez, Bob Coecke, Kathryn Hess, Steve Lack, and Valeria de Paiva.

https://tinyurl.com/call-for-editors

We will announce a call for submissions in mid-June.

We’re looking forward to your ideas and submissions!

Best regards,

Brendan Fong, Nina Otter, and Joshua Tan

### John Baez - Azimuth

A new journal! We’ve been working on it for a long time, but we finished sorting out some details at ACT2018, and now we’re ready to tell the world!

It’s free to read, free to publish in, and it’s about building big things from smaller parts. Here’s the top of the journal’s home page right now:

Here’s the official announcement:

We are pleased to announce the launch of

Compositionality, a new diamond open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition. To learn more about the scope and editorial policies of the journal, please visit our website at http://www.compositionality-journal.org.

Compositionalityis the culmination of a long-running discussion by many members of the extended category theory community, and the editorial policies, look, and mission of the journal have yet to be finalized. We would love to get your feedback about our ideas on the forum we have established for this purpose:http://reddit.com/r/compositionality

Lastly, the journal is currently receiving applications to serve on the editorial board; submissions are due May 31 and will be evaluated by the members of our steering board: John Baez, Bob Coecke, Kathryn Hess, Steve Lack, and Valeria de Paiva.

https://tinyurl.com/call-for-editors

We will announce a call for submissions in mid-June.

We’re looking forward to your ideas and submissions!

Best regards,

Brendan Fong, Nina Otter, and Joshua Tan

## April 30, 2018

### Tommaso Dorigo - Scientificblogging

### Tommaso Dorigo - Scientificblogging

*Frank D. Smith (Tony Smith for his friends) has been following this blog since the beginning. He is an independent researcher who is very interested in phenomena connected with the top quark and the Higgs boson. He has a theory of his own and he has been trying to check whether LHC data is compatible or not with it. His ideas are reported here as a guest post, as a tribute to his faithfulness to this site. Of course the views expressed below are his own, as I retain a healthy dose of scepticism to any bit of new physics apparent in today's data... Also, I will comment in the thread below to inform the reader of what my ideas are on his interpretation of public LHC results.*

## April 26, 2018

### Tommaso Dorigo - Scientificblogging

## April 25, 2018

### Clifford V. Johnson - Asymptotia

Turns out that Frank Buckley, the news anchor at KTLA 5, is not just a really great guy (evident from his manner on TV), but also a really excellent interviewer with a sharp curiosity that gives me hope that great journalism is still alive, well, and in good hands. I showed up at the station expecting to just have a pleasant chat around the book and be done with it, but I walked into the room and he'd done all his research and was sitting with extensive notes and so forth about lots of physics ideas he'd read in the book that he wanted to talk about! So we have a blast talking about the physics of our universe and the world around us in some in-depth detail. It was fantastic, and just the kind f conversation I hope that the book celebrates and inspires people to have!

Check out our interview here (embed below), and be sure to tune in to his [...] Click to continue reading this post

The post Frank Buckley Interviews… appeared first on Asymptotia.

## April 24, 2018

### Lubos Motl - string vacua and pheno

**And Keating's proposed Nobel prize reforms are left-wing lunacy**

Nick has asked whether Brian Keating, the designer of BICEP1 and the author of "Losing the Nobel Prize" (which will be released today), was conservative. At least according to some methodologies, the answer is Yes.

His 50-minute interview in Whiskey Politics, a right-wing podcast, has shown that he had the courage to hang the picture of George W. Bush in his University of California office – where most of his colleagues would prefer to hang Bush himself. Well, he didn't support Trump throughout most of his campaign, however.

He deplored the Che Café at UCSD where lots of taxpayer money is being spent to renovate the business and celebrate the mass killer by drinking coffee (which is a carcinogenic substance according to the Californian law but I guess that Che's café may get an exemption). And Keating has also followed me on Twitter so he can't be

*too*left-wing. ;-)

The interview is sort of amusing – about the group think in the Academia, about Keating's idiosyncratic claims that the Nobel prize will be boycotted and killed (he hates the nomination process, I don't quite get how he wants to pick the candidates instead), against tenure (which he says to greatly contribute to the amount of rubbish published by the soft, social scientists). He also gives an introduction to the Cosmic Microwave Background and its polarization and his feelings about his ex-boss and father-like figure Andrew Lang's suicide.

One of the comments he made was that just like the climatological community is pushed in a direction by the left-wing bias (Will Happer talked at the podcast in January), the left-wing group think also penetrates to cosmology – and it manifests itself as the support for the multiverse.

Well, it's not the first time I heard about this identification. I can see some justifications of this identification. But I think that the identification is oversimplified and exaggerated.

Eight years ago, I was invited to the French Riviera for a week. The scholars did things that were considered heretical according to the Academia's group think. So most of the folks were top defenders of the Intelligent Design. Richard Lindzen was there as a leading climate skeptic. And I was there because I was known to be politically incorrect. But it was assumed that I had to have such "right-wing" opinions about cosmology – which means to be against the Universe.

I didn't really meet those expectations. While I think that the anthropic principle is partly tautological and partly wrong (and lots of papers written to promote it have a very poor quality) – so that it's not useful to say true things about the Universe, at least at this moment – the very existence of the multiverse is a different thing. It seems rather likely – and probably more likely than 50% – that the multiverse is needed to properly understand the initial conditions at the Big Bang in our visible Universe, the vacuum selection, and other things.

Why do I think so? Well, inflation works and explains lots of things. And there are good reasons why a good inflationary theory may be automatically assumed to be eternal, and therefore produce the multiverse. It's a likely additional consequence of a theory in cosmology that seems to pass some tests to be believed to be correct. How could a rational person think that it doesn't matter? On top of that, string theory also has very good reasons to be the correct quantum theory of gravity and all other forces. And string theory seems to imply the landscape as well as the processes needed to change the vacuum of one type into another. An honest, competent, rational person just can't overlook these powerful arguments.

One can discuss the quasi-technical issues of whether or not the evidence for inflationary cosmology itself (or the string theory landscape) is strong or sufficient, whether the theory is natural, whether the most natural types of inflation are eternal, whether one should trust the eternal inflation in other parts of the multiverse that they seem to envision, and other things.

But the experience with the French Riviera and Brian Keating suggests that something more powerful than the rational arguments is deciding inside many folks. Many people apparently decide what to think about the multiverse by identifying the multiverse with some politics – usually left-wing politics. And if they like the left-wing politics, they decide to become the multiverse supporters; if they're not left-wing, they become the critics of the multiverse.

Needless to say, this rule isn't universally valid. There are lots of very left-wing people who are critics of the multiverse; and I am a right-wing example that is "mostly" a supporter of the multiverse. (Well, maybe the correlation between one's being religious and one's being a critic of the multiverse is stronger but it is surely not perfect, either.) But some people on both sides think that it "should be" valid. Why?

I think that the reasoning is just silly.

Whether the multiverse "exists" is a question about the world at the longest possible distance scales and time scales. But at the end, it's really just a question about the "size of the whole world". The multiverse research needs "more advanced, modern insights" but it's not "that different" from the question whether the Earth is flat, whether the Sun is the only star, whether the Milky Way is the only galaxy, or whether the Earth is the only inhabited planet. Even if you care about God's existence or in His holy absence, it's just a technical detail of a sort.

If God could have created (the laws that produced) a round Earth, small planets and large planets, one galaxy and billions of other galaxies, He could have created laws that produce a single patch of the visible Universe, a trillion of patches, googol to the 5th power of patches, or infinitely many patches. What is the problem? I think that you must imagine a very weak, anthropomorphic God if such things are a problem for you.

Years ago, Leonard Susskind promoted the multiverse as a weapon to kill God. Susskind believes that there is no God which is why it's so important to kill Him. ;-) His argument is that God has a good taste and creates pretty, ordered things. To prove that God is dead, just show that the Universe is maximally messy and the multiverse seems šitty enough for that – so that all the šit really looks beautiful to a staunch atheist. OK, Susskind stood on the opposite side than Keating but the underlying logic is equally unscientific and both of them "politicize" a topic that shouldn't be political.

If you look at the structural character of the argumentation, you could reasonably argue that the right identification is the other one: the multiverse and especially the anthropic principle often build on the kind of arguments that are similar to those by the Christian apologists. The anthropic principle differs from Christianity but both of them look like "some forms of faith". The evidence is really lacking and the belief in the importance of "the size of God" or "the number of intelligent observers' souls" seem to trump any "finite" empirical argument. So maybe this could be a better simplification: the most ambitious versions of the multiverse are on par with religion.

But my primary point is that none of these simplifications is the right starting point to discuss the existence of the multiverse and/or the existence of the multiverse or the validity of an inflationary theory. When things are simplified or politicized according to any of these vague templates, the discussion simply invites too many superficial people whose arguments are shallow and who will support

*any claim*whose apparent goal is to strengthen the "politically correct" side of the argument, independently of the quality of the claim. And that's just wrong.

The existence of the multiverse is a deep question but it's still a scientific, in some sense technical question, and no one should be assumed to defend one side of this debate or another just because it's claimed to be correlated with some (known) political or religious opinions of the person. It's the pressure arising from such expectations that is wrong for science; and it's the numerous people's inability to resist the pressure that also hurts proper objective science.

**Back to lawsuits against the Nobel committee**

At 33:40 of the interview, he discusses a website he founded that is meant to pressure the Nobel committee to reform the prize in some incomprehensible ways, in order to avoid the lawsuits and/or lost of allure, and also to help women and minorities. Holy cow. What does he exactly want, what is the justification, and how is this desire compatible with his being conservative?

Alfred Nobel wrote a will and some folks in his foundation tried to fulfill it. I think it would be very hard to fulfill it literally because Nobel didn't have a terribly good idea about the number of scientists who would exist in 2018, about the size of the relevant teams, and about their complex relationships with each other, with the organizers and sponsors of the scientific enterprises, and about the timescales it takes to complete an experiment or decide about the validity of a theory. If Nobel got familiar with all these things, he could very well agree that what is being done with his Nobel prize in physics is mostly reasonable. Or not.

Can Alfred Nobel sue the Nobel committee? He cannot because he's dead. Can someone else sue the committee on behalf of Alfred Nobel? I don't see how someone else could claim to better understand his intents than the committee that was specifically picked to do such decisions. But even if someone convinced the whole world that the committee deviates from the will in important aspects, what would it be good for? Does Keating really have a system for a better prize? It doesn't seem to be the case. That's an example of a situation that shows why it's so wise for the legal systems to demand the plaintiffs to have some standing. It seems clear that Keating has no standing in a hypothetical lawsuit about the "right way to interpret and fulfill Alfred Nobel's will".

After 42:00, he criticizes people's will to win the Olympic medals – some athletes would agree to die at age of 35 if they won one. Well, that's extreme but it's surely a reflection of a legitimate list of priorities that some people may have. A life that ends at this modest age but includes an Olympic victory may be considered a "better life" than a longer (just twice longer), more ordinary life, by some people. Some people simply are ambitious, some aren't. I think that the ambitions themselves are important for the progress of the mankind. So I don't share Keating's "horror" about it.

He says that the same extreme ambitions also exist in cosmology. Well, he has only provided us with some evidence from sports. But even if similar things exist in cosmology, and they may exist, I don't see anything unacceptable about it, either. Some people want to do great things (and even though the Nobel prize is just an honor, not the "real thing", as Feynman puts it, it's still a great enough thing for many people). This ambition exists independently of the Nobel prize. I think that Keating's logic is defective when he wants to sue the Nobel committee for the fact that some humans have ambitions. The ambitions are a universal constant of the humanity. In between the lines, I think that he is a great example of ambitious people himself.

Also, I understood some of his comments as urging the committee to give the Nobel prizes to everyone who wants it so that they're satisfied (Keating says that too many people fail to get the Nobel LOL). OK, that's a terrible idea (and the comment that "too many people are shut out" sounds like a joke; I literally cannot tell whether he's serious; of course that most people should be "shut out", it's a prestigious prize given at most to 3 physicists a year in a world that has over 7 billion people). I can't believe he's serious. They could bury meritocracy in this straightforward way. That would probably kill the people's interest in the Nobel prize, indeed. This move would

*actually*kill the prize, unlike the real world events that Keating

*incorrectly*predicts to lead to the death of the prize.

But the death of the Nobel prize wouldn't be enough to kill the people's ambitions. These people would naturally set other, more or less equivalent goals (when it comes to their will to shorten their lives), in front of themselves and these goals would arguably be less noble than a Nobel when it comes to the character of the activities that the people would do. And that would be bad for the mankind. One reason why Nobel's will is so useful for the mankind is that it is

*one of the motivations*that makes people do great things such as top science. If you kill that prize, you will reduce the motivation of the average people to do this great stuff – and that's bad! Nobel knew about that effect of a prize and he wanted to encourage people to do great things – one reason was that he felt guilty that the dynamite was going to do some bad things that he needed to compensate.

At 43:30, Keating starts to sound like a generic extreme left-wing fruitcake again. Rosalind Franklin wasn't given the prize for DNA just because of some petty details – she died before they made the decision. How can such an unimportant thing that the candidate is dead affect whether she wins? Honestly, Keating must be joking. Implicitly, he thinks that he's just like Rosalind Franklin which is why he launched this jihad against the Nobel prize. Holy cow.

These are real sour grapes, a textbook example of what they mean. There are very good meritocratic reasons (not just the death) why Franklin hasn't won the prize; and why Keating hasn't won one, either. Even if someone is the deepest thinker in the world, and it could very well be Edward Witten (or late Stephen Hawking) or someone else, there isn't any law of Nature that saying the Nobel prize is a necessary condition for him to be the world's deepest thinker. Unlike Keating and despite his modesty, Edward Witten knows that he may be the world's smartest man even without the "confirmation" from Stockholm. The Nobel prize is just an important prize with its own rules; the rules can't be precisely equivalent to everyone's definition of greatness. Keating seems to blame his colleagues that they have distorted definitions of greatness but it seems to me that Keating is one of the best examples that deserve that criticism of his.

While he's right-wing in some respects, I found his calls to "give the Nobel prize to women, minorities, and everyone who wants it so badly" to be examples of the generic, currently omnipresent, "progressive" insanity. Nobel wanted the price to go to one physicist a year and the cap was tripled soon. But the cap shouldn't be lifted or loosened (especially not substantially) because the prize would cease to play the positive role it plays.

by Luboš Motl (noreply@blogger.com) at April 24, 2018 07:34 AM

## April 22, 2018

### Lubos Motl - string vacua and pheno

Backreation wrote a review and Keating responded.

I used to think that the title was just a trick to emphasize the importance of Keating's work: He has done work that could have led to a Nobel prize but Nature wasn't generous enough, it has seemed for some 3 years. But the two articles linked to in the previous paragraph suggest that Keating is much more obsessed with the Nobel prize. That's ironic because the book seems to say that Keating is

*not*obsessed, and he doesn't even want such a lame prize, but it's his colleagues, the spherical bastards, who are obsessed. ;-)

OK, let me start to react to basic statements by Keating and Hossenfelder. First, Keating designed BICEP1 and lots of us were very excited about BICEP2, an upgraded version of that gadget. It could have seen the primordial gravitational waves. Even though I had

*theoretical prejudices*leading me to believe that those waves should be weak enough so that they shouldn't have been seen, I was impressed by the actual graphs and claims by the BICEP2 collaboration and willing to believe that they really found the waves and proved us wrong (by "us", I mean people around the Weak Gravity Conjecture and related schools of thought).

Keating has clearly designed a nice gadget and he deserves to be considered a top professional in his field. Because that gadget hasn't made a breakthrough that we would still believe to be real and solid, Keating hasn't won any major prize that also requires some collaboration of Mother Nature. He's still a top professional who rightfully earns a regular salary for that work and skills but his big lottery ticket hasn't won so he wasn't given a Nobel prize, an extraordinary donation.

During the excitement about BICEP2, if you told me that the Keating was this obsessed with the Nobel prize, I would have probably been more skeptical about the claims than I was. From my perspective, this obsession looks like a warning. If you really want a Nobel prize, it's natural to think that you make the arguments in favor of your discovery look a little bit clearer than what follows from your cold hard data. I don't really claim that Keating has committed such an "improvement" but I do claim that the expectation value of the "improvement" that I would have believed if I had known about his Nobel prize obsession would be positive and significant.

Keating seems to combine comments about his particular work with some more general criticism of the Nobel prize. Only 1/4 of the Nobel prize winners in physics are theorists; the rest are experimenters and observational people. Keating says that the fraction of theorists should be higher. I agree. He also says that experimenters shouldn't be getting Nobel prizes for things that some theorists outlined before them. I have mixed feelings about that claim – on some days, I would subscribe to that, on others, I wouldn't.

Hossenfelder seems upset about that very statement:

You read that right. No Nobel for the Higgs, no Nobel for B-modes, and no Nobel for a direct discovery of dark matter (should it ever happen), because someone predicted that.Ms Hossenfelder must have missed it but one of these experimental discoveries has been made, that of the Higgs boson, and the experimenters indeed

*didn't get any Nobel prize*. The 2013 Nobel prize went to Higgs and Englert, two of the theorists who discovered the mechanism and (perhaps) the particle theoretically. There have been several reasons why the experimenters haven't received the award (yet?): the CERN teams are too large, too many people could be said to deserve it (Alfred Nobel's limit is 3 – well, his will actually said 1 but soon afterwards, the number was tripled and another change would seem too radical now). But I think that Keating's thinking has also played a role. CERN has really done something straightforward. They knew what they should see. In my opinion, this makes the contribution by the experimenters less groundbreaking.

In 2017, Weiss, Thorn, and Barish got their experimental Nobel prize for something that was predicted by the theorists – such as Albert Einstein – namely the gravitational waves. But if you look at the justification, they got the prize both for LIGO and the discovery of the gravitational waves. So they were the "first men" who created LIGO and/or made it very powerful. It seems to me that no one who has done something this groundbreaking in particle physics experimentation was a visible member of the teams that discovered the Higgs boson. That discovery was made by a gradual improvement of the collider technology – by a large collective of people.

I think that if the primordial gravitational waves were discovered by BICEP2 and the discovery were confirmed and withstood the tests, Keating would both deserve the Nobel prize and he would get the Nobel prize. Now, some theorists have predicted strong enough primordial gravitational waves. But these waves may also be weak or non-existent. The difference from the Higgs boson is that the Higgs boson was really agreed to be necessarily there by good particle physicists, it was the unique player that makes the \(W_L W_L\to W_L W_L\) scattering unitary. On the other hand, there's no such uniqueness in the case of the primordial gravitational waves and their strength (and similarly in the problem of the identity of the dark matter). When the answers aren't agreed to be unique by the theorists, the experimenters play a much bigger role and they arguably deserve the Nobel prize.

Some people are very upset when Keating (or I) point out that the confirmation of a theory by an experiment – when the experimenter already knows what to look for – is less spectacular. For example:

The attitude may look condescending but there are very good reasons for this "condescension". The Nobel prize simplynaivetheorist said:Keating writes: "I am advocating that more theorists should win it, and experimentalists should not win it if they/we merely confirm a theory". Merely? that's an incredibly condescending attitude. Keating's rather lame response' affirms my decision to cancel my order for his book.

*is*meant to reward the original contribution and when someone is just confirming the work (theory) by someone else, this work is more derivative even if the first guy is a theorist and the second guy is an experimenter. It's great that experimenters are confirming or refuting hypotheses formulated by the theorists. But that's merely the scientific "business as usual". Prizes such as the Nobel prize are given for something extraordinary that isn't just "business as usual". One needs to be the really first person to do something – and luck or Nature's cooperation is often needed.

Keating seems to propose some boycotts of the Nobel prize or lawsuits against the Nobel prize. I don't get these comments. The inventor of some explosives got rich and created a system in which his money is invested and some fraction is paid to some people who are chosen as worthy the award by a committee that Nobel envisioned in his Nobel. It's a private activity. Well, one that has become globally famous, but the global fame is a consequence of the fame of Nobel himself and the winners (plus the money that attracts human eyes), not something that defines the award. Just because the award is followed by many people in the world doesn't mean that these people have the right to change the rules. After all, it's not their money.

As I said, the Nobel prize could be "better" according to many of us – and a higher percentage of the theorists could be a part of this "improvement". But this discussion is detached from reality. The Nobel prize is whatever it is. Alfred Nobel was a very practical person – explosives are rather practical compounds – and I believe that if he knew the whole list of the winners of his physics prize, he would be surprised by the high percentage of nerds and pure theorists. And maybe he would find it OK. And maybe he would want to increase the number of theorists, too. We don't know. But the prize has some traditional rules and expectations. Theorists only get their prizes for theories that have been experimentally verified – like Higgs and Englert.

The original BICEP2 claims about the very strong gravitational waves seem largely discredited now. This simple fact seems much more important for the question whether BICEP2 should be awarded a Nobel prize or not than some proposals to increase the number of theorists or reduce the number of experimental winners who just confirm predictions by theorists.

Concerning the obsession by the Nobel prizes, well, I think it's normal for the people who get close enough to be eligible to think about the prize. Some of the fathers of QCD knew that they deserved the prize and they were patiently waiting for some 30 years. The winners get some money directly, some extra money indirectly, and they may enjoy the life more than they did previously.

I think that the people who work on hep-th and ambitious hep-ph – like string theory and particle physics beyond the Standard Model – must know that according to the current scheme of things, the Nobel prize for their work is unlikely. But that doesn't mean that their work isn't the most valuable thing done in science. The best things in hep-th almost certainly

*are*the most valuable part of science. But things are just arranged in such a way that authors of such ground-breaking theoretical papers haven't gotten a Nobel prize and they're expected not to get it soon, either.

Is that such a problem? I don't think so. The Nobel prize is a distinguished award and – with the exception of the Nobel prize in peace and perhaps literature – it keeps on rewarding people who have done something important and who are usually very smart, too. But the precise criteria that decide who is rewarded are a bit more subtle – the physics prize isn't meant to reward people who are smart and/or made a deep contribution, without additional adjectives. The contributions must be confirmed experimentally because that's how "physics" is defined in the Nobel prize context. So there are rather good reasons why even Stephen Hawking hasn't ever received a Nobel prize although most quantum gravity theorists – and most formal theoretical particle physicists – would agree that his contributions to physics have been greater than those of the average Nobel prize winner. But the Hawking radiation hasn't really been seen. For me, the observation

*is*a formality – I have no real doubts about the existence of the Hawking radiation and other things – but I have no trouble to respect the rules of the game in which these

*formalities*decide about the prize. These are just the rules of the Nobel prize – and those ultimately reflect the rules of the scientific method.

By the way, I think that many people who have been doing similar things as your humble correspondent are often reminded that "they wanted a Nobel prize". It's possible that as a kid, I have independently talked about such things as well but at the end, I think that the obsession with the Nobel prize has primarily been widespread in my (or our) environment, not in my own thinking. The real excitement that underlined some of my important ideas – and even the hopes that one can get much further with these ideas – have had virtually nothing to do with the Nobel prize for over 20 years.

If you look rationally, the Nobel prize is just an honor. I actually think that my opinions about these matters – including the importance of the Nobel prize – were largely shaped by Feynman's view above since the moment when I read "Surely You're Joking Mr Feynman" for the first time. And I was 17. Well, the Nobel prize is still a

*better honor*than almost all others. After all, e.g. Richard Feynman who didn't like honors was one of those who got that particular honor. ;-) But it's unwise to be obsessed with the selection process and generic winners of that prize. At the end, the decision is one made by a smart but imperfect committee, and the prize primarily affects the winners only.

by Luboš Motl (noreply@blogger.com) at April 22, 2018 04:31 PM

## April 17, 2018

### Lubos Motl - string vacua and pheno

*Thanks to Willie Soon, Paul Halpern.*

**St Petersburg Times, Sunday, November 11th, 1928**

*Guest blog by John Nations, 3141 Twenty-sixth avenue South, City (St. Petersburg), Nov. 9, 1928*

*Mr Nations played with glimpses of string theory in 1928 and in that year, Lonnie Johnson recorded "Playing with the strings" about that achievement.*

Open forum (on the right side from the picture)

**UNDERSTANDING EINSTEIN**

Editor The Times:

A lot of people believe that Einstein is as transparent as boiler iron, one able authority estimating roughly that at least eight people in the world understand him.

This should not be considered a disparagement. Those who understand Einstein can easily vindicate themselves by explaining him in "street" terms to those who avoid the subject for the sake of two things, honesty and delicacy. Those who admit that they understand Einstein might choose to tell just what would happen if an Antares should derail and go through the curve where space "curves around." It is beyond the small comprehensive powers of a large group, just what would happen to that great orb should it become entangled in a void of nothingness that isn't even space. When Mr. Einstein declares that space is not infinite but curves around, that settles it for those with broad vision, but not for the great masses who insist upon speculating upon what exists just outside the "curve" where space is claimed to stop.

**And as to time being the fourth dimension, a lot of ignorant folks might say that it is as good name for it as anything, but they might also ask about the ninth dimension or the tenth—not yet being reconciled to the fact that there has to be a fourth dimension tucked away somewhere in time, space or music, and figuring that since there is bound to be a fourth dimension there is bound to be a sixteenth dimension, since one is quite as reasonable as the other to their small conception.**

[Bold face added by LM for emphasis.]

A concise explanation of Einstein's theory of relativity would doubtless be appreciated by thousands of people, but anyone attempting an explanation should refrain from Einsteinian phraseology—the big crowd doesn't understand that. For instance, in attempting to explain the location and predicament of Antares should that orb break jail and plunge through Einstein's "curve 'around'" it would not be advisable to say: Function measured in speed, amplitude, frequency, infrequency etc.; nor that Antares bumped into fourth dimension and rebounded like a hailstone off a greenhouse.

*Maurice Ravel's "Bolero" premiered in 1928. Bolero is from Spanish "bula" (ball, whirling motion).*

That might all be to the point but so many could not understand. It would be more tenable, less abstruse, if explained in terms indigenous of the ignorant. Many of the ignorant persist in the belief that time and space are brothers and infinite, and when they are told that either space or time is limited they are sure to ask about what is outside of space or, after time ceases how long such a condition can prevail—it is very difficult to explain those simple little details so that the average man can grasp your meaning.

It is easy to state that Einstein is simple and clear and unerring, and not so difficult to explain him in terms that you do not understand yourself—that is the usual way it is done. It sometimes scares the crowd and makes them envious of your deep insight, but when a poor, dumb fellow who has been too weak to grasp the impossibility of attaching a meaning to your baffling claims, asks you some of these simple questions about what happens after time ends or outside the domains of space, it is comfortable to have a long list of scrambled, incoherent words already prepared to smother him or he will cause you trouble.

3141 Twenty-sixth avenue South, City, Nov. 9, 1928

\[

\left(\beta mc^2 + c\left(\sum_{n \mathop =1}^{3}\alpha_n p_n\right)\right) \psi (x,t) = i \hbar \frac{\partial\psi(x,t) }{\partial t}

\] The (original) Dirac equation above was also published in 1928. Too bad, Mr Dirac didn't cooperate with Mr Nations. They could have obtained string theory (or the superstring) 40 or 45 years earlier.

LM: Thanks for the nice contribution, John, and sorry for the delay before I published this guest blog. I guess that you are already dead by now and your house seems to be replaced by a highway. But if I understood you well, you recommend popularizers of relativity to start with plain English but switch to the fancy technical language as soon as the audience starts to ask something, even if the speaker doesn't understand the meaning of these fancy words, just to reduce the annoying questions. Clever. ;-)

Concerning the 9 spatial or 10 spacetime dimensions of string theory, it seems that you (or the people who annoyed you with obvious questions) found them as straightforward and as valid as the curved and possibly compact topology of the spacetime according to general relativity. It was a great guess. Indeed, when combined with the entanglement of quantum mechanics, when music of string theory and perhaps Antares are allowed, and when the Woit of nothingness is eliminated, Einstein's general relativity implies string theory with its 10-dimensional spacetime. Did you have a proof or did you guess? I am asking because even now, almost 90 years after your letter, I only have a partial proof of your statement.

Thanks, I will probably need a truly compact spacetime with closed time-like curves to get the answers from you.

by Luboš Motl (noreply@blogger.com) at April 17, 2018 04:24 PM

## April 01, 2018

### Lubos Motl - string vacua and pheno

Stephen Hawking just posted a new paper to the arXiv:

Imaginary time as a path to resurrection (screenshot)It's just five pages long but it's using some very hard mathematics so I haven't had the time to fully comprehend it yet.

The abstract looks simple and intriguing, however:

We exploit the machinery of imaginary time to circumvent any particular point on the temporal real axis. The methodology may also be considered a refined realization of the process called "the resurrection" by the laymen. We describe a successful experiment in which a lifetime started on the anniversary of Galileo Galilei's death was interrupted on Albert Einstein's birthday and, through the complex plane, continued on the anniversary of the resurrection of Jesus Christ.Is there a reader who understands the contours in the complex plane well enough?

I have some doubts about the applicability of the method. He could have easily continued himself through the contour. After all, the same trick was already made by Hawking's Jewish colleague 1985 or 1988 years ago. And no one wants to live forever, anyway. But a more important question is: May it be applied to objects such as food? If you like a particular fried chicken from Kentucky, can you make sure that you eat it as many times as you want?

OK, Hawking managed to be resurrected and write a paper again. But can he walk again? Only if he came again, we could admit that his derivation allows the second coming of Stephen Hawking.

by Luboš Motl (noreply@blogger.com) at April 01, 2018 05:52 AM

## March 29, 2018

### Robert Helling - atdotde

I got the impression that for many physicists that have not yet spent too much time with this, deep learning and in particular deep neural networks are expected to be some kind of silver bullet that can answer all kinds of questions that humans have not been able to answer despite some effort. I think this hope is at best premature and looking at the (admittedly impressive) examples where it works (playing Go, classifying images, speech recognition, event filtering at LHC) these seem to be more like those problems where humans have at least a rough idea how to solve them (if it is not something that humans do everyday like understanding text) and also roughly how one would code it but that are too messy or vague to be treated by a traditional program.

So, during some of the less entertaining talks I sat down and thought about problems where I would expect neural networks to perform badly. And then, if this approach fails even in simpler cases that are fully under control one should maybe curb the expectations for the more complex cases that one would love to have the answer for. In the case of the workshop that would be guessing some topological (discrete) data (that depends very discontinuously on the model parameters). Here a simple problem would be a 2-torus wrapped by two 1-branes. And the computer is supposed to compute the number of matter generations arising from open strings at the intersections, i.e. given two branes (in terms of their slope w.r.t. the cycles of the torus) how often do they intersect? Of course these numbers depend sensitively on the slope (as a real number) as for rational slopes [latex]p/q[/latex] and [latex]m/n[/latex] the intersection number is the absolute value of [latex]pn-qm[/latex]. My guess would be that this is almost impossible to get right for a neural network, let alone the much more complicated variants of this simple problem.

Related but with the possibility for nicer pictures is the following: Can a neural network learn the shape of the Mandelbrot set? Let me remind those of you who cannot remember the 80ies anymore, for a complex number c you recursively apply the function

[latex]f_c(z)= z^2 +c[/latex]

starting from 0 and ask if this stays bounded (a quick check shows that once you are outside [latex]|z| < 2[/latex] you cannot avoid running to infinity). You color the point c in the complex plane according to the number of times you have to apply f_c to 0 to leave this circle. I decided to do this for complex numbers x+iy in the rectangle -0.74

by Robert Helling (noreply@blogger.com) at March 29, 2018 07:35 PM

### Axel Maas - Looking Inside the Standard Model

One is that I am currently teaching, together with somebody from the philosophy department, a course on science philosophy of physics. It cam to me as a surprise that one thing the students of philosophy are interested in is, how I think. What are the objects, or subjects, and how I connect them when doing research. Or even when I just think about a physics theory. The other is the review I have have recently written. Both topics may seem unrelated at first. But there is deep connection. It is less about what I have written in the review, but rather what led me up to this point. This requires some historical digression in my own research.

In the very beginning, I started out with doing research on the strong interactions. One of the features of the strong interactions is that the supposed elementary particles, quarks and gluons, are never seen separately, but only in combinations as hadrons. This is a phenomenon which is called confinement. It always somehow presented as a mystery. And as such, it is interesting. Thus, one question in my early research was how to understand this phenomenon.

Doing that I came across an interesting result from the 1970ies. It appears that a, at first sight completely unrelated, effect is very intimately related to confinement. At least in some theories. This is the Brout-Englert-Higgs effect. However, we seem to observe the particles responsible for and affected by the Higgs effect. And indeed, at that time, I was still thinking that the particles affected by the Brout-Englert-Higgs effect, especially the Higgs and the W and Z bosons, are just ordinary, observable particles. When one reads my first paper of this time on the Higgs, this is quite obvious. But then there was the results of the 1970ies. It stated that, on a very formal level, there should be no difference between confinement and the Brout-Englert-Higgs effect, in a very definite way.

Now the implications of that serious sparked my interest. But I thought this would help me to understand confinement, as it was still very ingrained into me that confinement is a particular feature of the strong interactions. The mathematical connection I just took as a curiosity. And so I started to do extensive numerical simulations of the situation.

But while trying to do so, things which did not add up started to accumulate. This is probably most evident in a conference proceeding where I tried to put sense into something which, with hindsight, could never be interpreted in the way I did there. I still tried to press the result into the scheme of thinking that the Higgs and the W/Z are physical particles, which we observe in experiment, as this is the standard lore. But the data would not fit this picture, and the more and better data I gathered, the more conflicted the results became. At some point, it was clear that something was amiss.

At that point, I had two options. Either keep with the concepts of confinement and the Brout-Englert-Higgs effect as they have been since the 1960ies. Or to take the data seriously, assuming that these conceptions were wrong. It is probably signifying my difficulties that it took me more than a year to come to terms with the results. In the end, the decisive point was that, as a theoretician, I needed to take my theory seriously, no matter the results. There is no way around it. And it gave a prediction which did not fit my view of the experiments than necessarily either my view was incorrect or the theory. The latter seemed more improbable than the first, as it fits experiment very well. So, finally, I found an explanation, which was consistent. And this explanation accepted the curious mathematical statement from the 1970ies that confinement and the Brout-Englert-Higgs effect are qualitatively the same, but not quantitatively. And thus the conclusion was what we observe are not really the Higgs and the W/Z bosons, but rather some interesting composite objects, just like hadrons, which due to a quirk of the theory just behave almost as if they are the elementary particles.

This was still a very challenging thought to me. After all, this was quite contradictory to usual notions. Thus, it came as a very great relief to me that during a trip a couple months later someone pointed me to a few, almost forgotten by most, papers from the early 1980ies, which gave, for a completely different reason, the same answer. Together with my own observation, this made click, and everything started to fit together - the 1970ies curiosity, the standard notions, my data. That I published in the mid of 2012, even though this still lacked some more systematic stuff. But it required still to shift my thinking from agreement to really understanding. That came then in the years to follow.

The important click was to recognize that confinement and the Brout-Englert-Higgs effect are, just as pointed out in the 1970ies mathematically, really just two faces to the same underlying phenomena. On a very abstract level, essentially all particles which make up the standard model, are really just a means to an end. What we observe are objects which are described by them, but which they are not themselves. They emerge, just like hadrons emerge in the strong interaction, but with very different technical details. This is actually very deeply connected with the concept of gauge symmetry, but this becomes quickly technical. Of course, since this is fundamentally different from the usual way, this required confirmation. So we went, made predictions which could distinguish between the standard way of thinking and this way of thinking, and tested them. And it came out as we predicted. So, seems we are on the right track. And all details, all the if, how, and why, and all the technicalities and math you can find in the review.

To make now full circle to the starting point: That what happened during this decade in my mind was that the way I thought about how the physical theory I tried to describe, the standard model, changed. In the beginning I was thinking in terms of particles and their interactions. Now, very much motivated by gauge symmetry, and, not incidental, by its more deeper conceptual challenges, I think differently. I think no longer in terms of the elementary particles as entities themselves, but rather as auxiliary building blocks of actually experimentally accessible quantities. The standard 'small-ball' analogy went fully away, and there formed, well, hard to say, a new class of entities, which does not necessarily has any analogy. Perhaps the best analogy is that of, no, I really do not know how to phrase it. Perhaps at a later time I will come across something. Right now, it is more math than words.

This also transformed the way how I think about the original problem, confinement. I am curious, where this, and all the rest, will lead to. For now, the next step will be to go ahead from simulations, and see whether we can find some way how to test this actually in experiment. We have some ideas, but in the end, it may be that present experiments will not be sensitive enough. Stay tuned.

by Axel Maas (noreply@blogger.com) at March 29, 2018 01:09 PM

## March 28, 2018

### Marco Frasca - The Gauge Connection

Recently, I wrote a paper together with Masud Chaichian (see here) containing a mathematical proof of confinement of a non-Abelian gauge theory based on Kugo-Ojima criterion. This paper underwent an extended review by several colleagues well before its submission. One of them has been Taichiro Kugo, one of the discoverers of the confinement criterion, that helped a lot to improve the paper and clarify some points. Then, after a review round of about two months, the paper has been accepted in Physics Letters B, one of the most important journals in particle physics.

This paper contains the exact beta function of a Yang-Mills theory. This confirms that confinement arises by the combination of the running coupling and the propagator. This idea was around in some papers in these latter years. It emerged as soon as people realized that the propagator by itself was not enough to grant confinement, after extended studies on the lattice.

It is interesting to point out that confinement is rooted in the BRST invariance and asymptotic freedom. The Kugo-Ojima confinement criterion permits to close the argument in a rigorous way yielding the exact beta funtion of the theory.

## March 20, 2018

### Marco Frasca - The Gauge Connection

Some days ago, Rencontres of Moriond 2018 ended with the CERN presenting a wealth of results also about the Higgs particle. The direction that the two great experiments, ATLAS and CMS, took is that of improving the measurements on the Standard Model as no evidence has been seen so far of possible new particles. Also, the studies of the properties of the Higgs particle have been refined as promised and the news are really striking.

In a communicate to the public (see here), CERN finally acknowledge, for the first time, a significant discrepancy between data from CMS and Standard Model for the signal strengths in the Higgs decay channels. They claim a 17% difference. This is what I advocated for some years and I have published in reputable journals. I will discuss this below. I would like only to show you the CMS results in the figure below.

ATLAS, by its side, is seeing significant discrepancy in the ZZ channel () and a compatibility for the WW channel. Here are their results.

On the left the WW channel is shown and on the right there are the combined and ZZ channels.

The reason of the discrepancy is due, as I have shown in some papers (see here, here and here), to the improper use of perturbation theory to evaluate the Higgs sector. The true propagator of the theory is a sum of Yukawa-like propagators with a harmonic oscillator spectrum. I solved exactly this sector of the Standard Model. So, when the full propagator is taken into account, the discrepancy is toward an increase of the signal strength. Is it worth a try?

This means that this is not physics beyond the Standard Model but, rather, the Standard Model in its full glory that is teaching something new to us about quantum field theory. Now, we are eager to see the improvements in the data to come with the new run of LHC starting now. In the summer conferences we will have reasons to be excited.

## March 17, 2018

### Cormac O’Raifeartaigh - Antimatter (Life in a puzzling universe)

Like many physicists, I woke to some sad news early last Wednesday morning, and to a phoneful of requests from journalists for a soundbyte. In fact, although I bumped into Stephen at various conferences, I only had one significant meeting with him – he was intrigued by my research group’s discovery that Einstein once attempted a steady-state model of the universe. It was a slightly scary but very funny meeting during which his famous sense of humour was fully at play.

*Yours truly talking steady-state cosmology with Stephen Hawking*

I recalled the incident in a radio interview with RTE Radio 1 on Wednesday. As I say in the piece, the first words that appeared on Stephen’s screen were *“I knew..”* My heart sank as I assumed he was about to say *“I knew about that manuscript*“. But when I had recovered sufficiently to look again, what Stephen was actually saying was *“I knew ..your father”*. Phew! You can find the podcast here.

*Hawking in conversation with my late father (LHS) and with Ernest Walton (RHS)*

RTE TV had a very nice obituary on the Six One News, I have a cameo appearence a few minutes into the piece here.

In my view, few could question Hawking’s brilliant contributions to physics, or his outstanding contribution to the public awareness of science. His legacy also includes the presence of many brilliant young physicists at the University of Cambridge today. However, as I point out in a letter in today’s Irish Times, had Hawking lived in Ireland, he probably would have found it very difficult to acquire government funding for his work. Indeed, he would have found that research into the workings of the universe does not qualify as one of the “strategic research areas” identified by our national funding body, Science Foundation Ireland. I suspect the letter will provoke an angry from certain quarters, but it is tragically true.

**Update**

The above notwithstanding, it’s important not to overstate the importance of one scientist. Indeed, today’s **Sunday Time**s contains a good example of the dangers of science history being written by journalists. Discussing Stephen’s 1974 work on black holes, Bryan Appleyard states *“The paper in effect launched the next four decades of cutting edge physics. Odd flowers with odd names bloomed in the garden of cosmic speculation – branes, worldsheets , supersymmetry …. and, strangest of all, the colossal tree of string theory”.*

What? String theory, supersymmetry and brane theory are all modern theories of particle physics (the study of the world of the very small). While these theories were used to some extent by Stephen in his research in cosmology (the study of the very large), it is ludicrous to suggest that they were launched by his work.

## March 16, 2018

### Sean Carroll - Preposterous Universe

Stephen Hawking died Wednesday morning, age 76. Plenty of memories and tributes have been written, including these by me:

- “Stephen Hawking’s Most Profound Gift to Physics,” in
*The New York Times*— a piece concentrating on black hole evaporation and the information-loss puzzle. - “Stephen Hawking Was Very Particular About His Tea,” in
*The Atlantic*— more focused on our personal interactions and Hawking’s human side.

I can also point to my Story Collider story from a few years ago, about how I turned down a job offer from Hawking, and eventually took lessons from his way of dealing with the world.

Of course Hawking has been mentioned on this blog many times.

When I started writing the above pieces (mostly yesterday, in a bit of a rush), I stumbled across this article I had written several years ago about Hawking’s scientific legacy. It was solicited by a magazine at a time when Hawking was very ill and people thought he would die relatively quickly — it wasn’t the only time people thought that, only to be proven wrong. I’m pretty sure the article was never printed, and I never got paid for it; so here it is!

(If you’re interested in a much better description of Hawking’s scientific legacy by someone who should know, see this article in *The Guardian* by Roger Penrose.)

**Stephen Hawking’s Scientific Legacy**

Stephen Hawking is the rare scientist who is also a celebrity and cultural phenomenon. But he is also the rare cultural phenomenon whose celebrity is entirely deserved. His contributions can be characterized very simply: Hawking contributed more to our understanding of gravity than any physicist since Albert Einstein.

“Gravity” is an important word here. For much of Hawking’s career, theoretical physicists as a community were more interested in particle physics and the other forces of nature — electromagnetism and the strong and weak nuclear forces. “Classical” gravity (ignoring the complications of quantum mechanics) had been figured out by Einstein in his theory of general relativity, and “quantum” gravity (creating a quantum version of general relativity) seemed too hard. By applying his prodigious intellect to the most well-known force of nature, Hawking was able to come up with several results that took the wider community completely by surprise.

By acclimation, Hawking’s most important result is the realization that black holes are not completely black — they give off radiation, just like ordinary objects. Before that famous paper, he proved important theorems about black holes and singularities, and afterward studied the universe as a whole. In each phase of his career, his contributions were central.

**The Classical Period**

While working on his Ph.D. thesis in Cambridge in the mid-1960’s, Hawking became interested in the question of the origin and ultimate fate of the universe. The right tool for investigating this problem is general relativity, Einstein’s theory of space, time, and gravity. According to general relativity, what we perceive as “gravity” is a reflection of the curvature of spacetime. By understanding how that curvature is created by matter and energy, we can predict how the universe evolves. This may be thought of as Hawking’s “classical” period, to contrast classical general relativity with his later investigations in quantum field theory and quantum gravity.

Around the same time, Roger Penrose at Oxford had proven a remarkable result: that according to general relativity, under very broad circumstances, space and time would crash in on themselves to form a * singularity*. If gravity is the curvature of spacetime, a singularity is a moment in time when that curvature becomes infinitely big. This theorem showed that singularities weren’t just curiosities; they are an important feature of general relativity.

Penrose’s result applied to black holes — regions of spacetime where the gravitational field is so strong that even light cannot escape. Inside a black hole, the singularity lurks in the future. Hawking took Penrose’s idea and turned it around, aiming at the past of our universe. He showed that, under similarly general circumstances, space must have come into existence at a singularity: the Big Bang. Modern cosmologists talk (confusingly) about both the Big Bang “model,” which is the very successful theory that describes the evolution of an expanding universe over billions of years, and also the Big Bang “singularity,” which we still don’t claim to understand.

Hawking then turned his own attention to black holes. Another interesting result by Penrose had shown that it’s possible to extract energy from a rotating black hole, essentially by bleeding off its spin until it’s no longer rotating. Hawking was able to demonstrate that, although you can extract energy, the area of the event horizon surrounding the black hole will always increase in any physical process. This “area theorem” was both important in its own right, and also evocative of a completely separate area of physics: thermodynamics, the study of heat.

Thermodynamics obeys a set of famous laws. For example, the first law tells us that energy is conserved, while the second law tells us that entropy — a measure of the disorderliness of the universe — never decreases for an isolated system. Working with James Bardeen and Brandon Carter, Hawking proposed a set of laws for “black hole mechanics,” in close analogy with thermodynamics. Just as in thermodynamics, the first law of black hole mechanics ensures that energy is conserved. The second law is Hawking’s area theorem, that the area of the event horizon never decreases. In other words, the area of the event horizon of a black hole is very analogous to the entropy of a thermodynamic system — they both tend to increase over time.

**Black Hole Evaporation**

Hawking and his collaborators were justly proud of the laws of black hole mechanics, but they viewed them as simply a formal analogy, not a literal connection between gravity and thermodynamics. In 1972, a graduate student at Princeton University named Jacob Bekenstein suggested that there was more to it than that. Bekenstein, on the basis of some ingenious thought experiments, suggested that the behavior of black holes isn’t simply * like* thermodynamics, it actually

*thermodynamics. In particular, black holes have entropy.*

*is*Like many bold ideas, this one was met with resistance from experts — and at this point, Stephen Hawking was the world’s expert on black holes. Hawking was certainly skeptical, and for good reason. If black hole mechanics is really just a form of thermodynamics, that means black holes have a temperature. And objects that have a temperature emit radiation — the famous “black body radiation” that played a central role in the development of quantum mechanics. So if Bekenstein were right, it would seemingly imply that black holes weren’t really black (although Bekenstein himself didn’t quite go that far).

To address this problem seriously, you need to look beyond general relativity itself, since Einstein’s theory is purely “classical” — it doesn’t incorporate the insights of quantum mechanics. Hawking knew that Russian physicists Alexander Starobinsky and Yakov Zel’dovich had investigated quantum effects in the vicinity of black holes, and had predicted a phenomenon called “superradiance.” Just as Penrose had showed that you could extract energy from a spinning black hole, Starobinsky and Zel’dovich showed that rotating black holes could emit radiation spontaneously via quantum mechanics. Hawking himself was not an expert in the techniques of quantum field theory, which at the time were the province of particle physicists rather than general relativists. But he was a quick study, and threw himself into the difficult task of understanding the quantum aspects of black holes, so that he could find Bekenstein’s mistake.

Instead, he surprised himself, and in the process turned theoretical physics on its head. What Hawking eventually discovered was that Bekenstein was right — black holes * do* have entropy — and that the extraordinary implications of this idea were actually true — black holes are not completely black. These days we refer to the “Bekenstein-Hawking entropy” of black holes, which emit “Hawking radiation” at their “Hawking temperature.”

There is a nice hand-waving way of understanding Hawking radiation. Quantum mechanics says (among other things) that you can’t pin a system down to a definite classical state; there is always some intrinsic uncertainty in what you will see when you look at it. This is even true for empty space itself — when you look closely enough, what you thought was empty space is really alive with “virtual particles,” constantly popping in and out of existence. Hawking showed that, in the vicinity of a black hole, a pair of virtual particles can be split apart, one falling into the hole and the other escaping as radiation. Amazingly, the infalling particle has a negative energy as measured by an observer outside. The result is that the radiation gradually takes mass away from the black hole — it evaporates.

Hawking’s result had obvious and profound implications for how we think about black holes. Instead of being a cosmic dead end, where matter and energy disappear forever, they are dynamical objects that will eventually evaporate completely. But more importantly for theoretical physics, this discovery raised a question to which we still don’t know the answer: when matter falls into a black hole, and then the black hole radiates away, where does the information go?

If you take an encyclopedia and toss it into a fire, you might think the information contained inside is lost forever. But according to the laws of quantum mechanics, it isn’t really lost at all; if you were able to capture every bit of light and ash that emerged from the fire, in principle you could exactly reconstruct everything that went into it, even the print on the book pages. But black holes, if Hawking’s result is taken at face value, seem to destroy information, at least from the perspective of the outside world. This conundrum is the “black hole information loss puzzle,” and has been nagging at physicists for decades.

In recent years, progress in understanding quantum gravity (at a purely thought-experiment level) has convinced more people that the information really is preserved. In 1997 Hawking made a bet with American physicists Kip Thorne and John Preskill; Hawking and Thorne said that information was destroyed, Preskill said that somehow it was preserved. In 2007 Hawking conceded his end of the bet, admitting that black holes don’t destroy information. However, Thorne has not conceded for his part, and Preskill himself thinks the concession was premature. Black hole radiation and entropy continue to be central guiding principles in our search for a better understanding of quantum gravity.

**Quantum Cosmology**

Hawking’s work on black hole radiation relied on a mixture of quantum and classical ideas. In his model, the black hole itself was treated classically, according to the rules of general relativity; meanwhile, the virtual particles near the black hole were treated using the rules of quantum mechanics. The ultimate goal of many theoretical physicists is to construct a true theory of quantum gravity, in which spacetime itself would be part of the quantum system.

If there is one place where quantum mechanics and gravity both play a central role, it’s at the origin of the universe itself. And it’s to this question, unsurprisingly, that Hawking devoted the latter part of his career. In doing so, he established the agenda for physicists’ ambitious project of understanding where our universe came from.

In quantum mechanics, a system doesn’t have a position or velocity; its state is described by a “wave function,” which tells us the probability that we would measure a particular position or velocity if we were to observe the system. In 1983, Hawking and James Hartle published a paper entitled simply “Wave Function of the Universe.” They proposed a simple procedure from which — in principle! — the state of the entire universe could be calculated. We don’t know whether the Hartle-Hawking wave function is actually the correct description of the universe. Indeed, because we don’t actually have a full theory of quantum gravity, we don’t even know whether their procedure is sensible. But their paper showed that we could talk about the very beginning of the universe in a scientific way.

Studying the origin of the universe offers the prospect of connecting quantum gravity to observable features of the universe. Cosmologists believe that tiny variations in the density of matter from very early times gradually grew into the distribution of stars and galaxies we observe today. A complete theory of the origin of the universe might be able to predict these variations, and carrying out this program is a major occupation of physicists today. Hawking made a number of contributions to this program, both from his wave function of the universe and in the context of the “inflationary universe” model proposed by Alan Guth.

Simply talking about the origin of the universe is a provocative step. It raises the prospect that science might be able to provide a complete and self-contained description of reality — a prospect that stretches beyond science, into the realms of philosophy and theology. Hawking, always provocative, never shied away from these implications. He was fond of recalling a cosmology conference hosted by the Vatican, at which Pope John Paul II allegedly told the assembled scientists not to inquire into the origin of the universe, “because that was the moment of creation and therefore the work of God.” Admonitions of this sort didn’t slow Hawking down; he lived his life in a tireless pursuit of the most fundamental questions science could tackle.

### Ben Still - Neutrino Blog

It has been available in the UK since September 2017 and you can buy it from Foyles / Waterstones / Blackwell's / AmazonUK where it is receiving ★★★★★ reviews

It is released in the US this Wednesday 21st March 2018 and you can buy it from all good book stores and Amazon.com

I just wanted to share a few reviews of the book as well because it makes me happy!

*Spend a few hours perusing these pages and you'll be in a much better frame of mind to understand your place in the cosmos... The astronomically large objects of the universe are no easier to grasp than the atomically small particles of matter. That's where Ben Still comes in, carrying a box of Legos. A British physicist with a knack for explaining abstract concepts... He starts by matching the weird properties and interactions described by the Standard Model of particle physics with the perfectly ordinary blocks of a collection of Legos. Quarks and leptons, gluons and charms are assigned to various colors and combinations of plastic bricks. Once you've got that system in mind, hang on: Still races off to illustrate the Big Bang, the birth of stars, electromagnetism and all matter of fantastical-sounding phenomenon, like mesons and beta decay. "Given enough plastic bricks, the rules in this book and enough time," Still concludes, "one might imagine that a plastic Universe could be built by us, brick by brick." Remember that the next time you accidentally step on one barefoot.*--Ron Charles, The Washington Post

Complex topics explained simply

*An excellent book. I am Head of Physics at a school and have just ordered 60 copies of this for our L6th students for summer reading before studying the topic on particle physics early next year. Highly recommended.*- Ben ★★★★★ AmazonUK

It's beautifully illustrated and very eloquently explains the fundamentals of particle ...

This is a gem of a pop science book. It's beautifully illustrated and very eloquently explains the fundamentals of particle physics without hitting you over the head with quantum field theory and Lagrangian dynamics. The author has done an exceptional job. This is a must have for all students and academics of both physics and applied maths! - Jamie ★★★★★ AmazonUK

## March 02, 2018

### Cormac O’Raifeartaigh - Antimatter (Life in a puzzling universe)

Like most people in Ireland, I am working at home today. We got quite a dump of snow in the last two days, and there is no question of going anywhere until the roads clear. Worse, our college closed quite abruptly and I was caught on the hop – there are a lot of things (flash drives, books and papers) sitting smugly in my office that I need for my usual research.

*The college on Monday evening*

That said, I must admit I’m finding it all quite refreshing. For the first time in years, I have time to read interesting things in my daily email; all those postings from academic listings that I never seem to get time to read normally. I’m enjoying it so much, I wonder how much stuff I miss the rest of the time.

*The view from my window as I write this*

This morning, I thoroughly enjoyed a paper by Nicholas Campion on the representation of astronomy and cosmology in the works of William Shakespeare. I’ve often wondered about this as Shakespeare lived long enough to know of Galileo’s ground-breaking astronomical observations. However, anyone expecting coded references to new ideas about the universe in Shakespeare’s sonnets and plays will be disappointed; apparently he mainly sticks to classical ideas, with a few vague references to the changing order.

I’m also reading about early attempts to measure the parallax of light from a comet, especially by the great Danish astronomer Tycho de Brahe. This paper comes courtesy of the History of Astronomy Discussion Group listings, a really useful resource for anyone interested in the history of astronomy.

While I’m reading all this, I’m also trying to keep abreast of a thoroughly modern debate taking place worldwide, concerning the veracity of an exciting new result in cosmology on the formation of the first stars. It seems a group studying the cosmic microwave background think they have found evidence of a signal representing the absorption of radiation from the first stars. This is exciting enough if correct, but the dramatic part is that the signal is much larger than expected, and one explanation is that this effect may be due to the presence of Dark Matter.

If true, the result would be a major step in our understanding of the formation of stars, plus a major step in the demonstration of the existence of Dark Matter. However, it’s early days – there are many possible sources of a spurious signal and signals that are larger than expected have a poor history in modern physics! There is a nice article on this in The Guardian, and you can see some of the debate on Peter Coles’s blog In the Dark. Right or wrong, it’s a good example of how scientific discovery works – if the team can show they have taken all possible spurious results into account, and if other groups find the same result, skepticism will soon be converted into excited acceptance.

All in all, a great day so far. My only concern is that this is the way academia should be – with our day-to-day commitments in teaching and research, it’s easy to forget there is a larger academic world out there.

**Update**

Of course, the best part is the walk into the village when it finally stops chucking down. can’t believe my local pub is open!

*Dunmore East in the snow today*

## March 01, 2018

### Sean Carroll - Preposterous Universe

So here’s something intriguing: an observational signature from the very first stars in the universe, which formed about 180 million years after the Big Bang (a little over one percent of the current age of the universe). This is exciting all by itself, and well worthy of our attention; getting data about the earliest generation of stars is notoriously difficult, and any morsel of information we can scrounge up is very helpful in putting together a picture of how the universe evolved from a relatively smooth plasma to the lumpy riot of stars and galaxies we see today. (Pop-level writeups at *The Guardian* and *Science News*, plus a helpful Twitter thread from Emma Chapman.)

But the intrigue gets kicked up a notch by an additional feature of the new results: the data imply that the cosmic gas surrounding these early stars is quite a bit cooler than we expected. What’s more, there’s a provocative explanation for why this might be the case: the gas might be cooled by interacting with dark matter. That’s quite a bit more speculative, of course, but sensible enough (and grounded in data) that it’s worth taking the possibility seriously.

[**Update:** skepticism has already been raised about the result. See this comment by Tim Brandt below.]

Let’s think about the stars first. We’re not seeing them directly; what we’re actually looking at is the cosmic microwave background (CMB) radiation, from about 380,000 years after the Big Bang. That radiation passes through the cosmic gas spread throughout the universe, occasionally getting absorbed. But when stars first start shining, they can very gently excite the gas around them (the 21cm hyperfine transition, for you experts), which in turn can affect the wavelength of radiation that gets absorbed. This shows up as a tiny distortion in the spectrum of the CMB itself. It’s that distortion which has now been observed, and the exact wavelength at which the distortion appears lets us work out the time at which those earliest stars began to shine.

Two cool things about this. First, it’s a *tour de force* bit of observational cosmology by Judd Bowman and collaborators. Not that collecting the data is hard by modern standards (observing the CMB is something we’re good at), but that the researchers were able to account for all of the different ways such a distortion could be produced *other* than by the first stars. (Contamination by such “foregrounds” is a notoriously tricky problem in CMB observations…) Second, the experiment itself is totally charming. EDGES (Experiment to Detect Global EoR [Epoch of Reionization] Signature) is a small-table-sized gizmo surrounded by a metal mesh, plopped down in a desert in Western Australia. Three cheers for small science!

But we all knew that the first stars had to be somewhen, it was just a matter of when. The surprise is that the spectral distortion is larger than expected (at 3.8 sigma), a sign that the cosmic gas surrounding the stars is colder than expected (and can therefore absorb more radiation). Why would that be the case? It’s not easy to come up with explanations — there are plenty of ways to heat up gas, but it’s not easy to cool it down.

One bold hypothesis is put forward by Rennan Barkana in a companion paper. One way to cool down gas is to have it interact with something even colder. So maybe — cold dark matter? Barkana runs the numbers, given what we know about the density of dark matter, and finds that we could get the requisite amount of cooling with a relatively light dark-matter particle — less than five times the mass of the proton, well less than expected in typical models of Weakly Interacting Massive Particles. But not completely crazy. And not really constrained by current detection limits from underground experiments, which are generally sensitive to higher masses.

The tricky part is figuring out how the dark matter could interact with the ordinary matter to cool it down. Barkana doesn’t propose any specific model, but looks at interactions that depend sharply on the relative velocity of the particles, as . You might get that, for example, if there was an extremely light (perhaps massless) boson mediating the interaction between dark and ordinary matter. There are already tight limits on such things, but not enough to completely squelch the idea.

This is all extraordinarily speculative, but worth keeping an eye on. It will be full employment for particle-physics model-builders, who will be tasked with coming up with full theories that predict the right relic abundance of dark matter, have the right velocity-dependent force between dark and ordinary matter, and are compatible with all other known experimental constraints. It’s worth doing, as currently all of our information about dark matter comes from its gravitational interactions, not its interactions directly with ordinary matter. Any tiny hint of that is worth taking very seriously.

But of course it might all go away. More work will be necessary to verify the observations, and to work out the possible theoretical implications. Such is life at the cutting edge of science!

## February 25, 2018

### Jon Butterworth - Life and Physics

Of which I wrote on average about 1/8 of a character…

See here.

## February 20, 2018

## February 08, 2018

### Sean Carroll - Preposterous Universe

Or *is* it?

I’ve talked before about the issue of why the universe exists at all (1, 2), but now I’ve had the opportunity to do a relatively careful job with it, courtesy of Eleanor Knox and Alastair Wilson. They are editing an upcoming volume, the Routledge Companion to the Philosophy of Physics, and asked me to contribute a chapter on this topic. Final edits aren’t done yet, but I’ve decided to put the draft on the arxiv:

Why Is There Something, Rather Than Nothing?

Sean M. CarrollIt seems natural to ask why the universe exists at all. Modern physics suggests that the universe can exist all by itself as a self-contained system, without anything external to create or sustain it. But there might not be an absolute answer to why it exists. I argue that any attempt to account for the existence of something rather than nothing must ultimately bottom out in a set of brute facts; the universe simply is, without ultimate cause or explanation.

As you can see, my basic tack hasn’t changed: this kind of question might be the kind of thing that doesn’t have a sensible answer. In our everyday lives, it makes sense to ask “why” this or that event occurs, but such questions have answers only because they are embedded in a larger explanatory context. In particular, because the world of our everyday experience is an emergent approximation with an extremely strong arrow of time, such that we can safely associate “causes” with subsequent “effects.” The universe, considered as all of reality (i.e. let’s include the multiverse, if any), isn’t like that. The right question to ask isn’t “Why did this happen?”, but “Could this have happened in accordance with the laws of physics?” As far as the universe and our current knowledge of the laws of physics is concerned, the answer is a resounding “Yes.” The demand for something more — a *reason why* the universe exists at all — is a relic piece of metaphysical baggage we would be better off to discard.

This perspective gets pushback from two different sides. On the one hand we have theists, who believe that they can answer why the universe exists, and the answer is God. As we all know, this raises the question of why God exists; but aha, say the theists, that’s different, because God *necessarily* exists, unlike the universe which could plausibly have not. The problem with that is that nothing exists necessarily, so the move is pretty obviously a cheat. I didn’t have a lot of room in the paper to discuss this in detail (in what after all was meant as a contribution to a volume on the philosophy of physics, not the philosophy of religion), but the basic idea is there. Whether or not you want to invoke God, you will be left with certain features of reality that have to be explained by “and that’s just the way it is.” (Theism could possibly offer a better account of the nature of reality than naturalism — that’s a different question — but it doesn’t let you wiggle out of positing some brute facts about what exists.)

The other side are those scientists who think that modern physics explains why the universe exists. It doesn’t! One purported answer — “because Nothing is unstable” — was never even supposed to explain why the universe exists; it was suggested by Frank Wilczek as a way of explaining why there is more matter than antimatter. But any such line of reasoning has to start by assuming a certain set of laws of physics in the first place. Why is there even a universe that obeys those laws? This, I argue, is not a question to which science is ever going to provide a snappy and convincing answer. The right response is “that’s just the way things are.” It’s up to us as a species to cultivate the intellectual maturity to accept that some questions don’t have the kinds of answers that are designed to make us feel satisfied.

## February 07, 2018

### Axel Maas - Looking Inside the Standard Model

It is best to answer these questions in reverse order.

So, do elementary particles have a size at all? Well, elementary particles are called elementary as they are the most basic constituents. In our theories today, they start out as pointlike. Only particles made from other particles, so-called bound states like a nucleus or a hadron, have a size. And now comes the but.

First of all, we do not yet know whether our elementary particles are really elementary. They may also be bound states of even more elementary particles. But in experiments we can only determine upper bounds to the size. Making better experiments will reduce this upper bound. Eventually, we may see that a particle previously thought of as point-like has a size. This has happened quite frequently over time. It always opened up a new level of elementary particle theories. Therefore measuring the size is important. But for us, as theoreticians, this type of question is only important if we have an idea about what could be the more elementary particles. And while some of our research is going into this direction, this project is not.

The other issue is that quantum effects give all elementary particles an 'apparent' size. This comes about by how we measure the size of a particle. We do this by shooting some other particle at it, and measure how strongly it becomes deflected. A truly pointlike particle has a very characteristic reflection profile. But quantum effects allow for additional particles to be created and destroyed in the vicinity of any particle. Especially, they allow for the existence of another particle of the same type, at least briefly. We cannot distinguish whether we hit the original particle or one of these. Since they are not at the same place as the original particle, their average distance looks like a size. This gives even a pointlike particle an apparent size, which we can measure. In this sense even an elementary particle has a size.

So, how can we then distinguish this size from an actual size of a bound state? We can do this by calculations. We determine the apparent size due to the quantum fluctuations and compare it to the measurement. Deviations indicate an actual size. This is because for a real bound state we can scatter somewhere in its structure, and not only in its core. This difference looks pictorially like this:

So, do we know the size already? Well, as said, we can only determine upper limits. Searching for them is difficult, and often goes via detours. One of such detours are so-called anomalous couplings. Measuring how they depend on energy provides indirect information on the size. There is an active program at CERN underway to do this experimentally. The results are so far say that the size of the W is below 0.0000000000000001 meter. This seems tiny, but in the world of particle physics this is not that strong a limit.

And now the interesting question: Why do we do this? As written, we do not want to make the W a bound state of something new. But one of our main research topics is driven by an interesting theoretical structure. If the standard model is taken seriously, the particle which we observe in an experiment and call the W is actually not the W of the underlying theory. Rather, it is a bound state, which is very, very similar to the elementary particle, but actually build from the elementary particles. The difference has been so small that identifying one with the other was a very good approximation up to today. But with better and better experiments may change. Thus, we need to test this.

Because then the thing we measure is a bound state it should have a, probably tiny, size. This would be a hallmark of this theoretical structure. And that we understood it. If the size is such that it could be actually measured at CERN, then this would be an important test of our theoretical understanding of the standard model.

However, this is not a simple quantity to calculate. Bound states are intrinsically complicated. Thus, we use simulations for this purpose. In fact, we actually go over the same detour as the experiments, and will determine an anomalous coupling. From this we then infer the size indirectly. In addition, the need to perform efficient simulations forces us to simplify the problem substantially. Hence, we will not get the perfect number. But we may get the order of magnitude, or be perhaps within a factor of two, or so. And this is all we need to currently say whether a measurement is possible, or whether this will have to wait for the next generation of experiments. And thus whether we will know whether we understood the theory within a few years or within a few decades.

by Axel Maas (noreply@blogger.com) at February 07, 2018 11:18 AM

## February 05, 2018

### Matt Strassler - Of Particular Significance

This week, the community of high-energy physicists — of those of us fascinated by particles, fields, strings, black holes, and the universe at large — is mourning the loss of one of the great theoretical physicists of our time, Joe Polchinski. It pains me deeply to write these words.

Everyone who knew him personally will miss his special qualities — his boyish grin, his slightly wicked sense of humor, his charming way of stopping mid-sentence to think deeply, his athleticism and friendly competitiveness. Everyone who knew his research will feel the absence of his particular form of genius, his exceptional insight, his unique combination of abilities, which I’ll try to sketch for you below. Those of us who were lucky enough to know him both personally and scientifically — well, we lose twice.

Polchinski — Joe, to all his colleagues — had one of those brains that works magic, and works magically. Scientific minds are as individual as personalities. Each physicist has a unique combination of talents and skills (and weaknesses); in modern lingo, each of us has a superpower or two. Rarely do you find two scientists who have the same ones.

Joe had several superpowers, and they were really strong. He had a tremendous knack for looking at old problems and seeing them in a new light, often overturning conventional wisdom or restating that wisdom in a new, clearer way. And he had prodigious technical ability, which allowed him to follow difficult calculations all the way to the end, on paths that would have deterred most of us.

One of the greatest privileges of my life was to work with Joe, not once but four times. I think I can best tell you a little about him, and about some of his greatest achievements, through the lens of that unforgettable experience.

*[To my colleagues: this post was obviously written in trying circumstances, and it is certainly possible that my memory of distant events is foggy and in error. I welcome any corrections that you might wish to suggest.]*

Our papers between 1999 and 2006 were a sequence of sorts, aimed at understanding more fully the profound connection between quantum field theory — the language of particle physics — and string theory — best-known today as a candidate for a quantum theory of gravity. In each of those papers, as in many thousands of others written after 1995, Joe’s most influential contribution to physics played a central role. This was the discovery of objects known as “D-branes”, which he found in the context of string theory. (The term is a generalization of the word `membrane’.)

I can already hear the polemical haters of string theory screaming at me. ‘*A discovery in string theory,’* some will shout, pounding the table, ‘

*an untested and untestable theory that’s not even wrong, should not be called a discovery in*.’ Pay them no mind; they’re not even close, as you’ll see by the end of my remarks.

**physics****The Great D-scovery**

In 1989, Joe, working with two young scientists, Jin Dai and Rob Leigh, was exploring some details of string theory, and carrying out a little mathematical exercise. Normally, in string theory, strings are little lines or loops that are free to move around anywhere they like, much like particles moving around in this room. But in some cases, particles aren’t in fact free to move around; you could, for instance, study particles that are trapped on the surface of a liquid, or trapped in a very thin whisker of metal. With strings, there can be a new type of trapping that particles can’t have — you could perhaps trap one end, or both ends, of the string within a surface, while allowing the middle of the string to move freely. The place **where**** a string’s end may be trapped** — whether a point, a line, a surface, or something more exotic in higher dimensions — is what we now call a “D-brane”.

*[The `D’ arises for uninteresting technical reasons.]*

Joe and his co-workers hit the jackpot, but they didn’t realize it yet. What they discovered, in retrospect, was that D-branes are an *automatic* feature of string theory. They’re not optional; you can’t choose to study string theories that don’t have them. And they aren’t just surfaces or lines that sit still. They’re physical objects that can roam the world. They have mass and create gravitational effects. They can move around and scatter off each other. They’re just as real, and just as important, as the strings themselves!

It was as though Joe and his collaborators started off trying to understand why the chicken crossed the road, and ended up discovering the existence of bicycles, cars, trucks, buses, and jet aircraft. It was that unexpected, and that rich.

And yet, nobody, not even Joe and his colleagues, quite realized what they’d done. Rob Leigh, Joe’s co-author, had the office next to mine for a couple of years, and we wrote five papers together between 1993 and 1995. Yet I think Rob mentioned his work on D-branes to me just once or twice, in passing, and never explained it to me in detail. Their paper had less than twenty citations as 1995 began.

In 1995 the understanding of string theory took a huge leap forward. That was the moment when it was realized that all five known types of string theory are different sides of the same die — that there’s really only one string theory. A flood of papers appeared in which certain black holes, and generalizations of black holes — black strings, black surfaces, and the like — played a central role. The relations among these were fascinating, but often confusing.

And then, on October 5, 1995, a paper appeared that changed the whole discussion, forever. It was Joe, explaining D-branes to those of us who’d barely heard of his earlier work, and showing that many of these black holes, black strings and black surfaces were actually D-branes in disguise. His paper made everything clearer, simpler, and easier to calculate; it was an immediate hit. By the beginning of 1996 it had 50 citations; twelve months later, the citation count was approaching 300.

*So what? Great for string theorists, but without any connection to experiment and the real world. What good is it to the rest of us?* Patience. I’m just getting to that.

**What’s it Got to Do With Nature?**

Our current understanding of the make-up and workings of the universe is in terms of particles. Material objects are made from atoms, themselves made from electrons orbiting a nucleus; and the nucleus is made from neutrons and protons. We learned in the 1970s that protons and neutrons are themselves made from particles called quarks and antiquarks and gluons — specifically, from a “sea” of gluons and a few quark/anti-quark pairs, within which sit three additional quarks with no anti-quark partner… often called the `valence quarks’. We call protons and neutrons, and all other particles with three valence quarks, `baryons”. (Note that there are no particles with just one valence quark, or two, or four — all you get is baryons, with three.)

In the 1950s and 1960s, physicists discovered short-lived particles much like protons and neutrons, with a similar sea, but which contain one valence quark and one valence anti-quark. Particles of this type are referred to as “mesons”. I’ve sketched a typical meson and a typical baryon in Figure 2. (The simplest meson is called a “pion”; it’s the most common particle produced in the proton-proton collisions at the Large Hadron Collider.)

But the quark/gluon picture of mesons and baryons, back in the late 1960s, was just an idea, and it was in competition with a proposal that mesons are little strings. These are not, I hasten to add, the “theory of everything” strings that you learn about in Brian Greene’s books, which are a billion billion times smaller than a proton. In a “theory of everything” string theory, often *all* the types of particles of nature, including electrons, photons and Higgs bosons, are tiny tiny strings. What I’m talking about is a “theory of mesons” string theory, a much less ambitious idea, in which only the mesons are strings. They’re much larger: just about as long as a proton is wide. That’s small by human standards, but immense compared to theory-of-everything strings.

Why did people think mesons were strings? ** Because there was experimental evidence for it! **(Here’s another example.) And that evidence didn’t go away after quarks were discovered. Instead, theoretical physicists gradually understood

*why*quarks and gluons might produce mesons that behave a bit like strings. If you spin a meson fast enough (and this can happen by accident in experiments), its valence quark and anti-quark may separate, and the sea of objects between them forms what is called a “flux tube.” See Figure 3.

*[In certain superconductors, somewhat similar flux tubes can trap magnetic fields.]*It’s kind of a thick string rather than a thin one, but still, it shares enough properties with a string in string theory that it can produce experimental results that are similar to string theory’s predictions.

And so, from the mid-1970s onward, people were confident that quantum field theories like the one that describes quarks and gluons can create objects with stringy behavior. A number of physicists — including some of the most famous and respected ones — made a bolder, more ambitious claim: *that quantum field theory and string theory are profoundly related, in some fundamental way.* But they weren’t able to be precise about it; they had strong evidence, but it wasn’t ever entirely clear or convincing.

In particular, there was an important unresolved puzzle. If mesons are strings, then what are baryons? What are protons and neutrons, with their three valence quarks? What do they look like if you spin them quickly? The sketches people drew looked something like Figure 3. A baryon would perhaps become three joined flux tubes (with one possibly much longer than the other two), each with its own valence quark at the end. In a stringy cartoon, that baryon would be three strings, each with a free end, with the strings attached to some sort of junction. This junction of three strings was called a “baryon vertex.” If mesons are little strings, the fundamental objects in a string theory, what is the baryon vertex from the string theory point of view?! Where is it hiding — what is it made of — in the mathematics of string theory?

*[Experts: Notice that the vertex has nothing to do with the quarks. It’s a property of the sea — specifically, of the gluons. Thus, in a world with only gluons — a world whose strings naively form loops without ends — it must still be possible, with sufficient energy, to create a vertex-antivertex pair. Thus field theory predicts that these vertices must exist in closed string theories, though they are linearly confined.]*

No one knew. But isn’t it interesting that the most prominent feature of this vertex is that it is a location **where a string’s end can be trapped**?

Everything changed in the period 1997-2000. Following insights from many other physicists, and using D-branes as the essential tool, Juan Maldacena finally made the connection between quantum field theory and string theory precise. He was able to relate strings with gravity and extra dimensions, which you can read about in Brian Greene’s books, with the physics of particles in just three spatial dimensions, similar to those of the real world, with only non-gravitational forces. It was soon clear that the most ambitious and radical thinking of the ’70s was correct — that almost every quantum field theory, with its particles and forces, can alternatively be viewed as a string theory. It’s a bit analogous to the way that a painting can be described in English or in Japanese — **fields/particles and strings/gravity are, in this context, two very different languages for talking about exactly the same thing.**

The saga of the baryon vertex took a turn in May 1998, when Ed Witten showed how a similar vertex appears in Maldacena’s examples. *[Note added: I had forgotten that two days after Witten’s paper, David Gross and Hirosi Ooguri submitted a beautiful, wide-ranging paper, whose section on baryons contains many of the same ideas.]* Not surprisingly, this vertex was a D-brane — specifically a D-particle, an object on which the strings extending from freely-moving quarks could end. It wasn’t yet quite satisfactory, because the gluons and quarks in Maldacena’s examples roam free and don’t form mesons or baryons. Correspondingly the baryon vertex isn’t really a physical object; if you make one, it quickly diffuses away into nothing. Nevertheless, Witten’s paper made it obvious what was going on. *To the extent real-world mesons can be viewed as strings, real-world protons and neutrons can be viewed as strings attached to a D-brane.*

**Working with Joe**

That project arose during my September 1999 visit to the KITP (Kavli Institute for Theoretical Physics) in Santa Barbara, where Joe was a faculty member. Some time before that I happened to have studied a field theory (called *N=1**) that differed from Maldacena’s examples only slightly, but in which meson-like objects do form. One of the first talks I heard when I arrived at KITP was by Rob Myers, about a weird property of D-branes that he’d discovered. During that talk I made a connection between Myers’ observation and a feature of the N=1* field theory, and I had one of those “aha” moments that physicists live for. I suddenly knew what the string theory that describes the N=1* field theory must look like.

But for me, the answer was bad news. To work out the details was clearly going to require a very difficult set of calculations, using aspects of string theory about which I knew almost nothing *[non-holomorphic curved branes in high-dimensional curved geometry.]* The best I could hope to do, if I worked alone, would be to write a conceptual paper with lots of pictures, and far more conjectures than demonstrable facts.

But I was at KITP. Joe and I had had a good personal rapport for some years, and I knew that we found similar questions exciting. And Joe was the brane-master; he knew everything about D-branes. So I decided my best hope was to persuade Joe to join me. I engaged in a bit of persistent cajoling. Very fortunately for me, it paid off.

I went back to the east coast, and Joe and I went to work. Every week or two Joe would email some research notes with some preliminary calculations in string theory. They had such a high level of technical sophistication, and so few pedagogical details, that I felt like a child; I could barely understand anything he was doing. We made slow progress. Joe did an important warm-up calculation, but I found it really hard to follow. If the warm-up string theory calculation was so complex, had we any hope of solving the full problem? Even Joe was a little concerned.

And then one day, I received a message that resounded with a triumphant cackle — a sort of “we got ’em!” that anyone who knew Joe will recognize. Through a spectacular trick, he’d figured out how use his warm-up example to make the full problem easy! Instead of months of work ahead of us, we were essentially done.

From then on, it was great fun! Almost every week had the same pattern. I’d be thinking about a quantum field theory phenomenon that I knew about, one that should be visible from the string viewpoint — such as the baryon vertex. I knew enough about D-branes to develop a heuristic argument about how it should show up. I’d call Joe and tell him about it, and maybe send him a sketch. A few days later, a set of notes would arrive by email, containing a complete calculation verifying the phenomenon. Each calculation was unique, a little gem, involving a distinctive investigation of exotically-shaped D-branes sitting in a curved space. It was breathtaking to witness the speed with which Joe worked, the breadth and depth of his mathematical talent, and his unmatched understanding of these branes.

*[Experts: It’s not instantly obvious that the N=1* theory has physical baryons, but it does; you have to choose the right vacuum, where the theory is partially Higgsed and partially confining. Then to infer, from Witten’s work, what the baryon vertex is, you have to understand brane crossings (which I knew about from Hanany-Witten days): Witten’s D5-brane baryon vertex operator creates a physical baryon vertex in the form of a D3-brane 3-ball, whose boundary is an NS 5-brane 2-sphere located at a point in the usual three dimensions. And finally, a physical baryon is a vertex with n strings that are connected to nearby D5-brane 2-spheres. See chapter VI, sections B, C, and E, of our paper from 2000.]*

Throughout our years of collaboration, it was always that way when we needed to go head-first into the equations; Joe inevitably left me in the dust, shaking my head in disbelief. That’s partly my weakness… I’m pretty average (for a physicist) when it comes to calculation. But a lot of it was Joe being so incredibly good at it.

Fortunately for me, the collaboration was still enjoyable, because I was almost always able to keep pace with Joe on the conceptual issues, sometimes running ahead of him. Among my favorite memories as a scientist are moments when I taught Joe something he didn’t know; he’d be silent for a few seconds, nodding rapidly, with an intent look — his eyes narrow and his mouth slightly open — as he absorbed the point. “Uh-huh… uh-huh…”, he’d say.

But another side of Joe came out in our second paper. As we stood chatting in the KITP hallway, before we’d even decided exactly which question we were going to work on, Joe suddenly guessed the answer! And I couldn’t get him to explain which problem he’d solved, much less the solution, for several days!! It was quite disorienting.

This was another classic feature of Joe. Often he *knew* he’d found the answer to a puzzle (and he was almost always right), but he couldn’t say anything comprehensible about it until he’d had a few days to think and to turn his ideas into equations. During our collaboration, this happened several times. (I never said “Use your words, Joe…”, but perhaps I should have.) Somehow his mind was working in places that language doesn’t go, in ways that none of us outside his brain will ever understand. In him, there was something of an oracle.

**Looking Toward The Horizon**

Our interests gradually diverged after 2006; I focused on the Large Hadron Collider *[also known as the Large D-brane Collider]*, while Joe, after some other explorations, ended up thinking about black hole horizons and the information paradox. But I enjoyed his work from afar, especially when, in 2012, Joe and three colleagues (Ahmed Almheiri, Don Marolf, and James Sully) blew apart the idea of ** black hole complementarity**, widely hoped to be the solution to the paradox. [I explained this subject here, and also mentioned a talk Joe gave about it here.] The wreckage is still smoldering, and the paradox remains.

Then Joe fell ill, and we began to lose him, at far too young an age. One of his last gifts to us was his memoirs, which taught each of us something about him that we didn’t know. Finally, on Friday last, he crossed the horizon of no return. If there’s no firewall there, he knows it now.

What, we may already wonder, will Joe’s scientific legacy be, decades from now? It’s difficult to foresee how a theorist’s work will be viewed a century hence; science changes in unexpected ways, and what seems unimportant now may become central in future… as was the path for D-branes themselves in the course of the 1990s. For those of us working today, D-branes in string theory are clearly Joe’s most important discovery — though his contributions to our understanding of black holes, cosmic strings, and aspects of field theory aren’t soon, if ever, to be forgotten. But who knows? By the year 2100, string theory may be the accepted theory of quantum gravity, or it may just be a little-known tool for the study of quantum fields.

Yet even if the latter were to be string theory’s fate, I still suspect it will be D-branes that Joe is remembered for. Because — as I’ve tried to make clear — they’re * real*. Really real. There’s one in every proton, one in every neutron. Our bodies contain them by the billion billion billions. For that insight, that elemental contribution to human knowledge, our descendants can blame Joseph Polchinski.

Thanks for everything, Joe. We’ll miss you terribly. You so often taught us new ways to look at the world — and even at ourselves.

## January 29, 2018

### Georg von Hippel - Life on the lattice

by Georg v. Hippel (noreply@blogger.com) at January 29, 2018 11:49 AM

## January 25, 2018

### Alexey Petrov - Symmetry factor

Some of you might remember my previous post about non-linear teaching, where I described a new teaching strategy that I came up with and was about to implement in teaching my undergraduate Classical Mechanics I class. Here I want to report on the outcomes of this experiment and share some of my impressions on teaching.

### Course description

Our Classical Mechanics class is a *gateway class* for our physics majors. It is the first class they take after they are done with general physics lectures. So the students are already familiar with the (simpler version of the) material they are going to be taught. The goal of this class is to start *molding physicists out of physics students*. It is a rather small class (max allowed enrollment is 20 students; I had 22 in my class), which makes professor-student interaction rather easy.

### Rapid-response (non-linear) teaching: generalities

To motivate the method that I proposed, I looked at some studies in experimental psychology, in particular in memory and learning studies. What I was curious about is how much is currently known about the process of learning and what suggestions I can take from the psychologists who know something about the way our brain works in retaining the knowledge we receive.

As it turns out, there are some studies on this subject (I have references, if you are interested). The earliest ones go back to 1880’s when German psychologist Hermann Ebbinghaus hypothesized the way our brain retains information over time. The “forgetting curve” that he introduced gives approximate representation of information retention as a function of time. His studies have been replicated with similar conclusions in recent experiments.

The upshot of these studies is that loss of learned information is pretty much exponential; as can be seen from the figure on the left, in about a day we only retain about 40% of what we learned.

Psychologists also learned that one of the ways to overcome the loss of information is to (meaningfully) retrieve it: this is how learning happens. Retrieval is critical for robust, durable, and long-term learning. It appears that every time we retrieve learned information, it becomes more accessible in the future. It is, however, important *how* we retrieve that stored information: simple re-reading of notes or looking through the examples will not be as effective as re-working the lecture material. It is also important *how often* we retrieve the stored info.

So, here is what I decided to change in the way I teach my class in light of the above-mentioned information (no pun intended).

### Rapid-response (non-linear) teaching: details

To counter the single-day information loss, I changed the way homework is assigned: instead of assigning homework sets with 3-4-5 problems per week, I introduced two types of homework assignments: *short homeworks* and *projects*.

Short homework assignments are *single-problem assignments* given after each class that must be done by the next class. They are designed such that a student needs to re-derive material that was discussed previously in class (with small new twist added). For example, if the block-down-to-incline problem was discussed in class, the short assignment asks to redo the problem with a different choice of coordinate axes. This way, instead of doing an assignment in the last minute at the end of the week, the students are forced to work out what they just learned in class every day (meaningful retrieval)!

The second type of assignments, *project homework assignments* are designed to develop understanding of how topics in a given chapter relate to each other. There are as many project assignments as there are chapters. Students get two weeks to complete them.

At the end, the students get to solve approximately the same number of problems over the course of the semester.

For a professor, the introduction of short homework assignments changes the way class material is presented. Depending on how students performed on the previous short homework, I adjusted the material (both speed and volume) that we discussed in class. I also designed examples for the future sections in such a way that I could repeat parts of the topic that posed some difficulties in comprehension. Overall, instead of a usual “linear” propagation of the course, we moved along something akin to helical motion, returning and spending more time on topics that students found more difficult (hence “rapid-response or non-linear” teaching).

Other things were easy to introduce: for instance, using Socrates’ method in doing examples. The lecture itself was an open discussion between the prof and students.

### Outcomes

So, I have implemented this method in teaching Classical Mechanics I class in Fall 2017 semester. It was not an easy exercise, mostly because it was the first time I was teaching this class and had no grader help. I would say the results confirmed my expectations: introduction of short homework assignments helps students to perform better on the exams. Now, my statistics is still limited: I only had 20 students in my class. Yet, among students there were several who decided to either largely ignore short homework assignments or did them irregularly. They were given zero points for each missed short assignment. All students generally did well on their project assignments, yet there appears some correlation (see graph above) between the total number of points acquired on short homework assignments and exam performance (measured by a total score on the Final and two midterms). This makes me thing that short assignments were beneficial for students. I plan to teach this course again next year, which will increase my statistics.

I was quite surprised that my students generally liked this way of teaching. In fact, they were disappointed that I decided not to apply this method for the Mechanics II class that I am teaching this semester. They also found that problems assigned in projects were considerably harder than the problems from the short assignments (this is how it was supposed to be).

For me, this was *not* an easy semester. I had to develop my set of lectures — so big thanks go to my colleagues Joern Putschke and Rob Harr who made their notes available. I spent a lot of time preparing this course, which, I think, affected my research outcome last semester. Yet, most difficulties are mainly Wayne State-specifics: Wayne State does not provide TAs for small classes, so I had to not only design all homework assignments, but also grade them (on top of developing the lectures from the ground up). During the semester, it was important to grade short assignments in the same day I received them to re-tune lectures, this did take a lot of my time. I would say TAs would certainly help to run this course — so I’ll be applying for some internal WSU educational grants to continue development of this method. I plan to employ it again next year to teach Classical Mechanics.

## January 22, 2018

### Axel Maas - Looking Inside the Standard Model

However, if such a GUT exists, then it must not only explain the forces, but also somehow why we see the numbers and kinds of elementary particles we observe in nature. And now things become complicated. As discussed in the last entry on GUTs there maybe a serious issue in how we determine which particles are actually described by such a theory.

To understand how this issue comes about, I need to put together many different things my research partners and I have worked on during the last couple of years. All of these issues are actually put into an expert language in the review of which I talked in the previous entry. It is now finished, and if your interested, you can get it free from here. But it is very technical.

So, let me explain it less technically.

Particle physics is actually superinvolved. If we would like to write down a theory which describes what we see, and only what we see, it would be terribly complicated. It is much more simple to introduce redundancies in the description, so-called gauge symmetries. This makes life much easier, though still not easy. However, the most prominent feature is that we add auxiliary particles to the game. Of course, they cannot be really seen, as they are just auxiliary. Some of them are very obviously unphysical, called therefore ghosts. They can be taken care of comparatively simply. For others, this is less simple.

Now, it turns out that the weak interaction is a very special beast. In this case, there is a unique one-to-one identification between a really observable particle and an auxiliary particle. Thus, it is almost correct to identify both. But this is due to the very special structure of this part of particle physics.

Thus, a natural question is whether, even if it is special, it is justified to do the same for other theories. Well, in some cases, this seems to be the case. But we suspected that this may not be the case in general. And especially not in GUTs.

Now, recently we were going about this much more systematically. You can again access the (very, very technical) result for free here. There, we looked at a very generic class of such GUTs. Well, we actually looked at the most relevant part of them, and still by far not all of them. We also ignored a lot of stuff, e.g. what would become quarks and leptons, and concentrated only on the generalization of the weak interaction and the Higgs.

We then checked, based on our earlier experiences and methods, whether a one-to-one identification of experimentally accessible and auxiliary particles works. And it does essentially never. Visually, this result looks like

On the left, it is seen that everything works nicely with a one-to-one identification in the standard model. On the right, if one-to-one identification would work in a GUT, everything would still be nice. But a our more precise calculation shows that the actually situation, which would be seen in an experiment, is different. There is non one-to-one identification possible. And thus the prediction of the GUT differs from what we already see inn experiments. Thus, a previously good GUT candidate is no longer good.

Though more checks are needed, as always, this is a baffling, and at the same time very discomforting, result.

Baffling as we did originally expect to have problems under very special circumstances. It now appears that actually the standard model of particles is the very special case, and having problems is the standard.

It is discomforting because in the powerful method of perturbation theory the one-to-one identification is essentially always made. As this tool is widely used, this seems to question the validity of many predictions on GUTs. That could have far-reaching consequences. Is this the case? Do we need to forget everything about GUTs we learned so far?

Well, not really, for two reasons. One is that we also showed that methods almost as easily handleable as perturbation theory can be used to fix the problems. This is good, because more powerful methods, like the simulations we used before, are much more cumbersome. However, this leaves us with the problem of having made so far wrong predictions. Well, this we cannot change. But this is just normal scientific progress. You try, you check, you fail, you improve, and then you try again.

And, in fact, this does not mean that GUTs are wrong. Just that we need to consider somewhat different GUTs, and make the predictions more carefully next time. Which GUTs we need to look at we still need to figure out, and that will not be simple. But, fortunately, the improved methods mentioned beforehand can use much of what has been done so far, so most technical results are still unbelievable useful. This will help enormously in finding GUTs which are applicable, and yield a consistent picture, without the one-to-one identification. GUTs are not dead. They likely just need a bit of changing.

This is indeed a dramatic development. But one which fits logically and technically to the improved understanding of the theoretical structures underlying particle physics, which were developed over the last decades. Thus, we are confident that this is just the next logical step in our understanding of how particle physics works.

by Axel Maas (noreply@blogger.com) at January 22, 2018 04:54 PM

## Subscriptions

- ATLAS Experiment
- Alexey Petrov - Symmetry factor
- Andrew Jaffe - Leaves on the Line
- Anton Zeilinger - Quantinger
- Axel Maas - Looking Inside the Standard Model
- Ben Still - Neutrino Blog
- CERN Bulletin
- Christian P. Robert - xi'an's og
- Clifford V. Johnson - Asymptotia
- Cormac O’Raifeartaigh - Antimatter (Life in a puzzling universe)
- Cosmic Variance
- David Berenstein, Moshe Rozali - Shores of the Dirac Sea
- Dmitry Podolsky - NEQNET: Non-equilibrium Phenomena
- Emily Lakdawalla - The Planetary Society Blog
- Georg von Hippel - Life on the lattice
- Geraint Lewis - Cosmic Horizons
- Imaginary Potential
- Jaques Distler - Musings
- Jester - Resonaances
- John Baez - Azimuth
- Jon Butterworth - Life and Physics
- Life as a Physicist
- Lubos Motl - string vacua and pheno
- Marco Frasca - The Gauge Connection
- Matt Strassler - Of Particular Significance
- Michael Schmitt - Collider Blog
- Peter Coles - In the Dark
- Peter Steinberg - Entropy Bound
- Phil Plait - Bad Astronomy
- Physicsworld blog
- Quantum Diaries
- Robert Helling - atdotde
- Sean Carroll - Preposterous Universe
- Sujit Datta - metadatta
- Symmetrybreaking - Fermilab/SLAC
- Teilchen blog
- The Great Beyond - Nature blog
- The n-Category Cafe
- Tommaso Dorigo - Scientificblogging
- US/LHC Blogs
- ZapperZ - Physics and Physicists
- arXiv blog
- astrobites - astro-ph reader's digest

## Feeds

**Last updated:**

May 24, 2018 01:51 PM

*All times are UTC.*

**Suggest a blog:**

planet@teilchen.at