Symmetries and representations in Group Field Theory

Tuesday, January 24th
Alexander Kegeles, Albert Einstein Institute
Title: Field theoretical aspects of GFT: symmetries and representations 
PDF of the talk (1M)
Audio+Slides [.mp4 11MB]


by Jorge Pullin, Louisiana State University


In loop quantum gravity the quantum states are labeled by loops, more precisely by graphs formed by lines that intersect at vertices and that are “colored”, meaning each line is associated with an integer. They are known as "spin networks". As the states evolve in time these graphs "sweep" surfaces in four dimensional space-time constituting what is known as a “spin foam”. This is a representation of a quantum space-time in loop quantum gravity. The spin foams connect an initial spin network with a final one and the formalism gives a probability for such “transition” from a given spatial geometry to a future spatial geometry to occur. The picture that emerges has some parallel with ordinary particle physics in which particles transition from initial to final states, but also some differences.

However, it was found that one could construct ordinary quantum field theories such that the transition probabilities of them coincided with those stemming from spin foams connecting initial to final spatial geometries in loop quantum gravity. This talk concerns itself with such quantum field theories, known generically as Group Field Theories (GFTs). The talk covered two main aspects of them: symmetries and representations.

Symmetries are important in that they may provide mathematical tools to solve the equations of the theory and identify conserved quantities in it. There is a lot of experience with symmetries in local field theories, but GFT’s are non-local, which adds challenges. Ordinary quantum field theories are formulated starting by a quantity known as the action, which is an integral on a domain. A symmetry is defined as a map of the points of such domain and of the fields that leaves the integral invariant. In GFTs the action is a sum of integrals on different domains. A symmetry is defined as a collections of maps acting on the domains and fields that leave invariant each integral in the sum. An important theorem of great generality stretching from classical mechanics to quantum field theory is Noether’s theorem, that connects symmetries with conserved quantities. The above notion of symmetry for GFTs allows to introduce a Noether’s theorem for them. The theorem could find applicability in a variety of situations, in particular certain relations that were noted between GFTs and recoupling theory and better understand various models based on GFTs.

In a quantum theory like GFTs the quantum states structure themselves into a mathematical set known as Hilbert space. The observable quantities of the theory are represented as operators acting on such space. Hilbert spaces are generically infinite dimensional and this introduces a series of technicalities both in their own definition and in the definition of observables for quantum theories. In particular one can find different families of inequivalent operators related to the same physical observables. This is what is known as different representations of the algebra of observables. Algebra in this context means that one can compose observables to form either new observables or linear combinations of known observables. An important type of representation in quantum field theory is known as Fock representation. It is the representation on which ordinary particles are based. Another type of representations is the condensate representation which, instead of particles, describes their collective (excitations) behaviour and is very convenient for systems with large (infinite) number of particles. A discussion of Fock and condensate like representations in the context of GFTs was presented and the issue of when representations are equivalent or not was also addressed.

Future work looks at generalizing the notion of symmetries presented to find further non-standard symmetries of GFTs. Also investigating “anomalies”. This is when one has a symmetry in the classical theory that may not survive upon quantization. The notion of symmetry can also be used to define an idea of “ground state” or fundamental state of the theory. In ordinary quantum field theory in flat space-time this is done by seeking the state with lower energy. In the context of GFTs one will invoke more complicated notions of symmetries to define the ground state. Several other results of ordinary field theories, like the spin statistics theorem, may be generalizable to the GFT context using the ideas presented in this talk.