Your Week in Seminars: Conformal Edition

Another week has gone by here in Cornell. The last leaves are turning red, a hint of snow passed us on the weekend, and the undergrads have hit the streets and parties in minimal clothing, then did the same again next day wearing a set of cat ears. And in the physics department, we had the usual three talks.



On Monday, the colloquium speaker was Holger Müller from UC Berkeley, talking about Gravitational Redshift, Equivalence Principle, and Matter Waves.

The center of the talk was Muller's experimental device, an atom interferometer. Many of you will remember the Michelson-Morley interferometer, the device used to disprove the existence of the ether. A light-interferometer essentially takes a beam of light, splits it in two and then merges it back again, using the result of the interference between the two parts to learn something about the relative difference between the optical paths taken by the two.

The atom interferometer, then, performs a similar function with atom wavefunctions. An atom is shot up into the air and a laser is directed towards it and calibrated to interact with the atom half the time. The atom's wavefunction is split two trajectories, at the end of which another laser is calibrated to bring the two paths back together. The detector can then measure the path difference between the two trajectories, and as we have excellent ways of measuring time and the mass and energy of the atom, this amounts to a very accurate measurement of g, the free-fall constant.

Muller went on to show how his team has been using the interferometer to perform very accurate measurements of General Relativity, from its isotropy to the universality of free fall motion for objects of different masses. There were some neat tricks described, and they mentioned the ability to measure those minute differences in gravity experienced by moving the system one meter upwards.

It's always a little difficult to get excited about tests that confirm an accepted theory, especially one like General Relativity, but I think this is important work. To paraphrase the words of fellow Virtuos Jared, GR is always going to be right up until we find where it breaks.


On Wednesday, David Kaplan talked about Conformality Lost.

This talk was about QCD, but not about QCD.

One of the features of QCD, or really field theories in general, is the running of the coupling constants. Where in classical theories the strength of the interaction between two particles is constant and depends only on the distance between them, field theory shows us how the strength of the interaction changes with the energy of the participating particles. This is crucial, for instance, for theories of grand unification that posit that the known forces are all the same at very high energies.

In QCD, in particular, the running of the constant also has to with confinement and asymptotic freedom. Confinement is the notion that quarks can never break free of each other, and so we never observe them alone in nature, only within particles such as protons, neutrons, baryons and mesons. Asymptotic freedom is the notion that at high energies, if we collide another particle with a quark, it behaves as if it was free of other influence. If we associate long distances with low energy and short distances with high energy, we can see how the coupling must flow from very small at one end to very large at the other end.

One of the interesting things about the running of the coupling is that it defines a scale for the theory. If the coupling is different for particles of energy E₁ and E₂, then we can choose some value of the coupling and describe our energy in relation to the energy relative to this scale. Theories without running coupling are called conformal and have no natural scale. QCD, it seems, behave this way if you take it all the way to asymptotic freedom.

Kaplan talked about the investigation of this conformal stage of the theory, its existence and inexistence. As an analog he showed a quantum-mechanical system of a particle in a Coulombic, potential. The minimum energy of this system is given by solution of a quadratic equation, which can have either two solutions, one or none, depending on the relation between the mass of the particle and the strength of the potential. A scale exists in this case only if there are two solutions: a single energy is meaningless, of course, because we can always add a constant, but if there's two of them then the difference defines a scale.

This toy model, it turns out, can be analogous to a QCD with the equivalent parameter being the relative number of flavors (kind of particles) and colors (different charges in the theory, red, blue and green in our regular QCD).

There were a number of interesting results from this model, the most exciting one, perhaps, being the possible existence of a "mirror" QCD theory beyond the conformal point of QCD, a sort of theory with a different number of colors and different gauge groups. Kaplan ended his talk by talking of at least one possible candidate for this mirror theory that they had recently found.


Finally, on Friday, we had Ami Katz from Boston University talk about CFT/AdS.

AdS/CFT has been a big buzzword for the last decade or so. The CFT here stands for conformal field theory of the kind mentioned in the previous summary, and AdS stands for Anti-de Sitter space, a geometry of spacetime possible in general relativity. The slash in between stands for a duality that allows results from one theory to be interpreted in the other and vice versa. This has some exciting implications since it allows us to use each theory in the regime where we can solve it.

Particle theorists are, in general, trying to use the CFT to solve for high-energy theories that behave like AdS. Katz had apparently rewritten the duality as CFT/AdS, to signal that he was asking the opposite question, starting with a CFT and asking whether it is a good fit for the duality.

A large part of the talk was dedicated to making an analogy from CFTs into conventional field theories. We know pretty well when a field theory is a good description of reality and when it tends to break down. This has to do, usually, with some cutoff energy, a scale at which new physics comes into play. As long as we stay at energies far below that cutoff, the effects of the unknown physics will be a small correction to the calculations we make with our known physics.

In CFTs, we had just said, there is no energy scale, and so the question must be different. The relevant question, apparently, is the dimensionality of operators - not what their energy scale is, but how they scale with a change of energy. For instance, a derivative behaves like inverse distance, and distance behaves like inverse energy, so a single derivative scales linearly in energy, while a double derivative scale quadratically.

I didn't understand much past the half-point of this lecture, but the bottom line appeared to be that a well-behaved CFT has a gap in its operators dimensionality, allowing us to focus on one operator and plenty of its derivatives before coming to the scaling of the next operator. This kind of gap allows our perturbative corrections to remain perturbative when we go to the AdS side.


That's it for last week, with its conformal ups and down. As usual, we're past the first seminar of the new week, which was non-wimpy talk about WIMPs. Still ahead this week are superconductors (and more AdS/CFT, presumably) and some non-thermal histories of the universe.

(that is, of course, if I don't freeze first - temperatures have dropped below zero already. It's so much colder when you work in Celsius)