Quantum deformations of 4d spin foam models

by Hanno Sahlmann, Asia Pacific Center for Theoretical Physics and Physics Department, Pohang University of Science and Technology, Korea.

• Winston Fairbairn, Hamburg University

Title: Quantum deformation of 4d spin foam models
PDF of the slides (300k)
Audio [.wav 36MB], Audio [.aif 3MB].

The work Winston Fairbairn talked about is very intriguing because it brings together a theory of quantum gravity, and some very interesting mathematical objects called quantum groups, in a way that may be related to the non-zero cosmological constant that is observed in nature! Let me try to explain what these things are, and how they fit together.

Quantum groups

A group is a set of things that you can multiply with each other to obtain yet another element of the group. So there is a product. Then there also needs to be a special element, the unit, that when multiplied with any element of the group, just gives back the same element. And finally there needs to be an inverse to every element, such that if one multiplies an element with its inverse, one gets the unit. For example, the integers are a group under addition, and the rotations in space are a group under composition of rotations. Groups are one of the most important mathematical ingredients in physical theories because they describe the symmetries of a physical system. A group can act in different ways on physical systems. Each such way is called a representation.

Groups have been studied in mathematics for hundreds of years, and so a great deal is known about them. Imagine the excitement when it was discovered that there exists a more general (and complicated) class of objects that nevertheless have many of the same properties as groups, in particular with respect to their mathematical representations. These objects are called quantum groups. Very roughly speaking, one can get a quantum group by thinking about the set of functions on a group. Functions can be added and multiplied in a natural way. And additionally, the group product, the inversion and the unit of the group itself, induce further structures on the set of functions in the group.

The product of functions is commutative - fg and gf are the same thing. But one can now consider set of functions that have all the required extra structure to make them set of functions acting over a group - except for the fact that the product is now not commutative anymore. Then the elements cannot be functions on a group anymore -- in fact they can't be functions at all. But one can still pretend that they are “functions” on some funny type of set: A quantum group. 

Particular examples of quantum groups can be found by deforming the structures one finds for ordinary groups. In these examples, there is a parameter q that measures how big the deformations are. q=1 corresponds to the structure without deformation. If q is a complex number with q^n=1 for some integer n (i.e., q is a root of unity), the quantum groups have particular properties. Another special class of deformations is obtained for q a real number. Both of these cases seem to be relevant in quantum gravity.

Quantum gravity

Finding a quantum theory of gravity is an important goal of modern physics, and it is what loop quantum gravity is all about. Since gravity is also a theory of space, time, and how they fit together in a space-time geometry, quantum gravity is believed to be a very unusual theory, one in which quantities like time and distance come in discrete bits, (atoms of space-time, if you like) and are not all simultaneously measureable.

One way to think about quantum theory in general is in terms of what is known as "path integrals". Such calculations answer the question of how probable it is that a given event (for example two electrons scattering off each other) will happen. To compute the path integral, one must sum up complex numbers (amplitudes), one for each way that the thing under study can happen. The probability is then given in terms of this sum. Most of the time this involves infinitely may possible ways, the electrons for example can scatter by exchanging one photon, or two, or three, or..., the first photon can be emitted in infinitely many different places, and have different energies etc. Therefore computing path integrals is very subtle, needs approximations, and can lead to infinite values. Path integrals were introduced into physics by Feynman. Not only did he suggest to think about quantum theory in terms of these integrals, he also introduced an ingenious device useful for their approximate calculation. To each term in an approximate calculation of some particular process in quantum field theory, he associated what we now call its Feynman diagram. The nice thing about Feynman diagrams is that they not only have a technical meaning. They can also be read as one particular way in which a process can happen. This makes working with them very intuitive.

(image from Wikipedia)

It turns out that  loop quantum gravity can also be formulated using some sort of path integrals. This is often called spin foam gravity. The spin foams in the name are actually very nice analogs to Feynman diagrams in ordinary quantum theory: They are also a technical device in an approximation of the full integral - but as for Feynman diagrams, one can read them as a space-time history of a process - only now the process is how space-time itself changes!

Associating the amplitude to a given diagram usually involves integrals. In the case of quantum field theory there is an integral over the momentum of each particle involved in the process. In the case of spin foams, there are also integrals or infinite sums, but those are over the labels of group representations! This is the magic of loop quantum gravity: Properties of quantized space-time are encoded in group representations. The groups most relevant for gravity are known as technically as SL(2,C) -- a group containing all the Lorentz transformations, and SU(2), a subgroup related to spatial rotations.  

Cosmological constant

In some theories of gravity, empty space “has weight”, and hence influences the dynamics of the universe. This influence is governed by a quantity called the cosmological constant. Until a bit more than ten years ago, the possibility of a non-zero cosmological constant was not considered very seriously, but to everybody’s surprise, astronomers then discovered strong evidence that there is a positive cosmological constant. Creating empty space creates energy! The effect of this is so large that it seems to dominate cosmological evolution at the present epoch (and there has been theoretical evidence for something like a cosmological constant in earlier epochs, too). Quantum field theory in fact predicts that there should be energy in empty space, but the observed cosmological constant is tremendously much smaller than what would be expected. So the explaining the observed value of the cosmological constant presents quite a mystery for physics.

Spin foam gravity with quantum groups

Now I can finally come to the talk. As I've said before, path integrals are complicated objects, and infinities do crop up quite frequently in their calculation. Often these infinities are a due to problems with the approximations one has made, and sometimes several can be canceled against each other, leaving a finite result. To analyze those cases, it is very useful to first consider modifications of the path integral that remove all infinities, for example by restricting the ranges of integrations and sums. This kind of modification is called "introducing a regulator", and it certainly changes the physical content of the path integral. But introducing a regulator can help to analyze the situation and rearrange the calculation in such a way that in the end the regulator can be removed, leaving a finite result. Or one may be able to show that the existence of the regulator is in fact irrelevant at least in certain regimes of the theory.

Now back to gravity: For the case of Euclidean (meaning a theory of pure space rather than a theory of space-time, this is unphysical but simplifies certain calculations) quantum gravity in three dimensions, there is a nice spinfoam formulation due to Ponzano and Regge, but as can be anticipated, it gives divergent answers in certain situations. Turaev and Viro then realized that replacing the group by its quantum group deformation at a root of unity furnishes a very nice regularization. First of all, it does what a regulator is supposed to do, namely render the amplitude finite. This happens because the quantum group in question, with q a root of unity, turns out to have only finitely many irreducible representations, so the infinite sums that were causing the problems are now replaced by finite sums. Moreover, as the original group was only deformed, and not completely broken, one expects that the regulated results stay reasonably close to the ones without regulator. In fact, something even nicer happens: It turned out (work by Mizoguchi and Tada) that the amplitudes in which the group is replaced by its deformation into a quantum group correspond to another physical theory -- quantum gravity with a (positive) cosmological constant! The deformation parameter q is directly related to the value of the constant. So this regulator is not just a technical tool to make the amplitudes finite. It has real physical meaning.

Winston's talk was not about three dimensional gravity, but on the four dimensional version - the real thing, if you like. He was considering what is called the EPRL vertex, a new way to associate amplitudes to spin foams, devised by Engle, Pereira, Rovelli and Livine, which has created a lot of excitement among people working on loop quantum gravity. The amplitudes obtained this way are finite in a surprising number of circumstances, but infinities are nevertheless encountered as well. Winston Fairbairn, together with Catherine Meusburger (and, independently, Muxin Han), were now able to write down the new vertex function in which the group is replaced by a deformation into a quantum group. In fact, they developed a nice graphical calculus to do so. What is more, they were able to show that it gives finite amplitudes. Thus the introduction of the quantum group does its job as a regulator.

As for the technical details, let me just say that they are fiercely complicated. To appreciate the intricacy of this work, you should know that the group SL(2,C) involved is what is known as non-compact, which makes its quantum group deformations very complicated and challenging structures (intuitively, compact sets have less chance of producing infinities than non-compact ones). Also, the EPRL vertex function relies on a subtle interplay between SU(2) and SL(2,C). One has to understand this interplay on a very abstract level to be able to translate it to quantum groups. The relevant type of deformation in this case has a real parameter q. In this case, there are still infinitely many irreducible representations – but it seems that it is the quantum version of the interplay between SU(2) and SL(2,C) that brings about the finiteness of the sums.

Thanks to this work, we now have a very interesting question on our hands: Is the quantum group deformation of the EPRL theory again related to gravity with a cosmological constant? Many people bet that this is the case, and the calculations to investigate this question have already begun, for example in a recent preprint by Ding and Han. Also, this begs the question of how fundamentally intertwined quantum gravity is with quantum groups. There were some interesting discussions about this already during and after the talk. At this point, the connection is still rather mysterious on a fundamental level.