The ongoing transformation of quantum field theory

Quantum field theory (QFT) is the basic framework of particle physics. Particles arise from the quantized energy levels of field oscillations; Feynman diagrams are the simple tool for approximating their interactions. The “standard model”, the success of which is capped by the recent observation of a Higgs boson lookalike, is a quantum field theory.

But just like everything mathematical, quantum field theory has hidden depths. For the past decade, new pictures of the quantum scattering process (in which particles come together, interact, and then fly apart) have incrementally been developed, and they presage a transformation in the understanding of what a QFT describes.

At the center of this evolution is “N=4 super-Yang-Mills theory”, the maximally supersymmetric QFT in four dimensions. I want to emphasize that from a standard QFT perspective, this theory contains nothing but scalar particles (like the Higgs), spin-1/​2 fermions (like electrons or quarks), and spin-1 “gauge fields” (like photons and gluons). The ingredients aren’t something alien to real physics. What distinguishes an N=4 theory is that the particle spectrum and the interactions are arranged so as to produce a highly extended form of supersymmetry, in which particles have multiple partners (so many LWers should be comfortable with the notion).

In 1997, Juan Maldacena discovered that the N=4 theory is equivalent to a type of string theory in a particular higher-dimensional space. In 2003, Edward Witten discovered that it is also equivalent to a different type of string theory in a supersymmetric version of Roger Penrose’s twistor space. Those insights didn’t come from nowhere, they explained algebraic facts that had been known for many years; and they have led to a still-accumulating stockpile of discoveries about the properties of N=4 field theory.

What we can say is that the physical processes appearing in the theory can be understood as taking place in either of two dual space-time descriptions. Each space-time has its own version of a particular large symmetry, “superconformal symmetry”, and the superconformal symmetry of one space-time is invisible in the other. And now it is becoming apparent that there is a third description, which does not involve space-time at all, in which both superconformal symmetries are manifest, but in which space-time locality and quantum unitarity are not “visible”—that is, they are not manifest in the equations that define the theory in this third picture.

I cannot provide an authoritative account of how the new picture works. But here is my impression. In the third picture, the scattering processes of the space-time picture become a complex of polytopes—higher-dimensional polyhedra, joined at their faces—and the quantum measure becomes the volume of these polyhedra. Where you previously had particles, you now just have the dimensions of the polytopes; and the fact that in general, an n-dimensional space doesn’t have n special directions suggests to me that multi-particle entanglements can be something more fundamental than the separate particles that we resolve them into.

It will be especially interesting to see whether this polytope combinatorics, that can give back the scattering probabilities calculated with Feynman diagrams in the usual picture, can work solely with ordinary probabilities. That was Penrose’s objective, almost fifty years ago, when he developed the theory of “spin networks” as a new language for the angular momentum calculations of quantum theory, and which was a step towards the twistor variables now playing an essential role in these new developments. If the probability calculus of quantum mechanics can be obtained from conventional probability theory applied to these “structures” that may underlie familiar space-time, then that would mean that superposition does not need to be regarded as ontological.

I’m talking about this now because a group of researchers around Nima Arkani-Hamed, who are among the leaders in this area, released their first paper in a year this week. It’s very new, and so arcane that, among physics bloggers, only Lubos Motl has talked about it.

This is still just one step in a journey. Not only does the paper focus on the N=4 theory—which is not the theory of the real world—but the results only apply to part of the N=4 theory, the so-called “planar” part, described by Feynman diagrams with a planar topology. (For an impressionistic glimpse of what might lie ahead, you could try this paper, whose author has been shouting from the wilderness for years that categorical knot theory is the missing piece of the puzzle.)

The N=4 theory is not reality, but the new perspective should generalize. Present-day calculations in QCD already employ truncated versions of the N=4 theory; and Arkani-Hamed et al specifically mention another supersymmetric field theory (known as ABJM after the initials of its authors), a deformation of which is holographically dual to a theory-of-everything candidate from 1983.

When it comes to seeing reality in this new way, we still only have, at best, a fruitful chaos of ideas and possibilities. But the solid results—the mathematical equivalences—will continue to pile up, and the end product really ought to be nothing less than a new conception of how physics works.