QFT and Representation Theory - Part 2.htm
Let’s continue from Part 1. We are discussing, critically, a paper "Quantum Field Theory and Representation Theory: A Sketch" by Peter Woit. The paper is also available from his website. This must be not a very popular paper as the Citebase reveals only 0 citations and 16 downloads of the paper. For a comparison, the PhD Thesis of Lubos Motl, "Nonperturbative Formulations of Superstring Theory", available from http://arxiv.org/abs/hep-th/0109149, shows 0 citations and 53 downloads, while "Higher Yang-Mills Theory" by John Baez shows 19 citations and 29 downloads (these downloads numbers are, as it seems, only from uk.arXiv.org ).
We continue commenting, sentence by sentence, the introductory part of the paper. Peter Woit writes:
The relation between quantum mechanics and representation theory has been formalized as the subject of \lq\lq geometric quantization" which ideally associates to a classical mechanical phase space (a symplectic manifold M) a complex vector space V in a functorial manner. This functor takes at least some subgroup $G$ of the symplectomorphisms (canonical transformations) of M to unitary transformations of V, making V a unitary $G$-representation.
While the above is true, it is not the whole truth, nor even an essential part of the truth. Every symplectic manifold M is naturally endowed with the canonical volume form, and therefore also with a natural measure, let’s call it m. Symplectomorphism are automatically measure preserving transformations. We can always build the Hilbert space of square integrable functions on M. Then all the symplectomorphisms are represented by unitary transformations, and the construction is certainly functorial. In geometric quantization we aim in "cutting this Hilbert space in half", otherwise the task described in the quoted paragraph above would be trivial!
The theory of geometric quantization has never been very popular among physicists for at least
two reasons (in addition to the fact that the mathematical apparatus required is rather
extensive and mostly unfamiliar to physicists). The first is that it seems to have very little
to say about quantum field theory. The quantum field theory of the standard model of particle
physics is built upon the geometrical concepts of gauge fields and the Dirac operator on
spinors, concepts which have no obvious relation to those used in geometric quantization.
Again, that is not exactly correct. The problem of the Dirac operator and spinors arises in relativistic quantum mechanics, without the need of going to quantum field theory. There exist generalizations of geometric quantization that unable us to deal with spinning (free) particles. Studying the Poincare groups, its Lie algebra, and using an appropriately adjusted coadjoint orbit methods one gets a sort of relativistic quantum mechanics. The problems arise when we want to have a consistent physical interpretation, but these problems arise already in a non-relativistic case.
The second problem is that the most well-developed formalism for doing calculations within
the standard model is the path integral formalism. While no rigorous version of this exists,
all evidence is that consistent calculations can be performed using this formalism, at least
within perturbation theory or outside of perturbation theory with a lattice cut-off. Even in the
simple case of quantum mechanics the relationship between the path integral quantization and
geometric quantization has been quite unclear making it impossible to see how the ideas of
geometric quantization can be useful in the much more complex situation of standard model
quantum field theory.
I do not think that the fact that there is no evident relation between path integral formalism and geometric quantization constitutes a problem. There is no evident relation between path integral formalism and calculating of eigenvalues and eigenstates of operators, and yet physicists calculate these eigenvalues and eigenvectors for ages for a simple reason that these calculations seem to be useful. If physicists would find geometric quantization "useful" - they would use it disregarding the lack of its relationship to the path integral methods.
To be continued
Chronostalker
