Joseph P. Day on Holographic Quantum Chromodynamics

What do we really know about the world we live in? Are we merely prisoners of our primitive notions; like the captives of Plato’s cave deceived by the illusion that we live in a lower dimensional world when the truth of nature is actually much richer? More than 40 years since the theoretical grounds have been laid for the commonly accepted theory of the strong interaction, namely quantum chromodynamics (QCD), we still presently struggle with many unanswered questions.

My name is Joseph Day. I am originally from Southern California. I did my undergraduate studies at the University of California Santa Barbara and I’ve received my master’s degree at California State University Long Beach. Furthermore, I have worked at Los Alamos National Laboratory and have been a visiting scholar at Stanford University. I arrived in Austria in September of 2010 after accepting an offer from my present PhD advisor, Willibald Plessas, to attend the University of Graz where I am a currently studying.

In the current version of the Standard Model of particle physics we have three known forces. One force governs electrodynamic interactions; this interaction tells us about common everyday experiences like light, electricity, magnetism, how our cell phone works, and is the force we would experience if a tram hit us. Another force governs the weak interactions; this helps us to understand things like radioactive decay of nuclei. The force that interests us today is the strong force. This force involves the interactions of the constituents of the proton and neutron known as quarks and gluons. This is extremely interesting because although we have seen much indirect evidence, from numerous experiments, beginning with deep inelastic experiments at SLAC in the 1960’s, these particles cannot ever directly be seen due to a peculiar phenomenon known as confinement (The astute reader may notice that we haven’t included gravity. This force is absent from the Standard Model at the present time. While on large scales we understand gravity quite well due to Einstein’s general relativity, on very short length scales we don’t understand at all how gravity can be assimilated into current models of particle physics).

The modern accepted theory of the strong force, QCD, was put forth in 1972 by Gell-Mann, Fritzsch, and Leutwyler. A consequence of this theory, known as asymptotic freedom, was revealed in 1973 and subsequently Gross, Wilczek, and Politzer were awarded the 2004 Nobel Prize for their discovery. What asymptotic freedom tells us is that depending upon at which energy scale (how closely) we look at our theory, the physical interaction between the particles is different. A parameter of QCD known as the strong coupling constant actually varies. This coupling tells us how strong the interactions with each particle are. At high energies the coupling becomes quite small (much less than one), this allows the particles to move around freely and unimpeded (hence asymptotic freedom). Another consequence of this high-energy interaction is it allows theoretical physicists to use one of our favorite tools known as perturbation theory to calculate certain properties of the system. The problem arises at lower energies, in the regime of proton and neutron masses. Here the coupling constant grows quite large and the particles interact with each other strongly. Additionally we can no longer use our beloved perturbation theory as this only works in weakly coupled theories. Contrary to popular belief physicists are not very smart; once we have lost of our tool of perturbation theory we don’t really know how to solve anything. This is now the state of the art for 40 years. Given the state of affairs, we have resorted to either attacking the problem via brute force with ever more powerful computers, or to model building. Model building means we make a guess about how the world should approximately behave and then see if the calculations from our model make any sense. The model I will present in this article is known as AdS/QCD. Many predictions of the model seem to make a lot of sense; however, at a fundamental level we don’t understand why. These deeper questions are the beauty of research in physics and allow us plenty of interesting work to do in the future.

The model of AdS/QCD is motivated by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence first put forth by Maldacena in 1997. What Maldacena conjectured was that there is a mapping between a higher dimensional gravitational theory and a conformal field theory at the boundary. The gravitational theory lives in a geometrical space known as anti-de Sitter space; this is a maximally symmetric spacetime with negative curvature. At the boundary of this geometric space we have a (d+1)-dimensional gravitational theory that is mapped to a d-dimensional conformal field theory. A way to think of this is to imagine yourself in a movie theater. Three-dimensional information that was recorded on the film is projected onto a flat two-dimensional screen. Since our brains have evolved to process the world in three-dimensions we have no problem reconstructing this two-dimensional information back to three dimensions in our mind. Another example is that of a hologram. If you have ever played with a hologram sticker you undoubtedly noticed, that although your sticker was a flat two-dimensional object, it had three-dimensional information encoded into it. For this reason, this property in physics has come to be known as the holographic principle.

Holographic projection of the theory in AdS space

Figure 1: Holographic projection of the theory in AdS space represented by the inner part of the sphere, to the boundary where we live represented by the outer blue half-sphere. (Figure courtesy of Stan Brodsky)

The model of AdS/QCD I have worked on, also known as light-front holography, was developed by Stan Brodsky and Guy de Téramond. We investigate QCD using a method known as light-front quantization. This is essentially a rewriting of QCD in different coordinates. Dirac first proposed the forms of relativistic dynamics in 1949; figure 2 schematically illustrates the front form. The usefulness of this is that it becomes immediately apparent the correspondence between variables in (3+1)-dimensional (three space + one time) light-front QCD and equations in the (4+1)-dimensional AdS space. Mapping between these equations allows us to find simple closed form solutions for previously very complicated problems. Given this light-front holographic mapping we are able to calculate many things that physicists are interested in with regard to low-energy phenomena. Some important examples are things like the masses of particles, properties of the inner structure of composite particles known as form factors, as well as cross sections. A short note for the more technical reader (others may skip to the next paragraph); a priori QCD is not a conformal field theory, in practice there are several ways to mitigate this in. One is breaking the conformal symmetry via introducing a confining mechanism by hand, most often a soft-wall dilaton profile. Another is if we consider a semi-classical approximation to QCD in the chiral limit, and neglect particle creation and absorption, the coupling becomes constant, we have a zero beta function, and this approximate theory is conformally invariant.

The light-like invariant hyper surface of the front form

Figure 2: The light-like invariant hyper-surface of the front form, represented by the plane, intersecting with the light cone.

The model of AdS/QCD supposes that in fact the physics that we experience in our day-to-day life are a result of a higher dimensional reality, which we only experience the projection of. Much like the prisoners of Plato’s cave we interpret the observations and experiences in our world as real objects, but perhaps these are only reflections of reality. While this statement is meant to be provocative and thought inducing, one should be careful not to oversell such a viewpoint. While indeed, this is a very interesting line of research, we are far from knowing the truth. Additionally, attempts to interpret what higher dimensional mathematics actually mean quickly leave the realm of science and delve into philosophy. Many leading physicists suspect that the AdS/CFT correspondence will eventually allow us to solve many very deep fundamental questions of nature; two such examples would be the problem of confinement and how gravity and the Standard Model are related. However, we are not there yet! Future experimental results from the Large Hadron Collider (LHC) as well as a more rigorous understanding of this correspondence will help lead us into the future. As physicists we find ourselves at a very interesting time in our history. With the discovery of the Higgs, the final piece of the 40-year-old standard model finally experimentally fell into place. As physicists we know for many reasons that we still lack understanding of vast aspects of nature; questions of dark matter, dark energy, neutrino oscillations, and various other irreconcilable experimental facts make this clear. The next decade and beyond will be very illuminating for particle theorists.