This is the first posting of a planned series that will discuss the BRST method for handling gauge symmetries and related mathematical topics. I’ve been writing a more formal paper about this, but given the substantial amount of not-well-known background material involved, it seems like a good idea to first put together a few expository accounts of some of these topics. And what better place for this than a blog?
Many readers who are used to my usual attempts to be newsworthy and entertaining, often by scandal-mongering or stirring up trouble of one kind or another, may be very disappointed in these posts. They’re quite technical, hard to follow, and of low-to-negative entertainment value. You probably would do best to skip them and wait for more of the usual fare, which should continue to appear from time to time.
Quantum Mechanics and Representation Theory
A quantum mechanical physical system is given by the following mathematical structure:
If a physical system has a symmetry group , there is a unitary representation of on . This means that for each we get a unitary operator satisfying
i.e. the map from group elements to unitary operators is a homomorphism. The act on by taking an operator to its conjugate .
When is a Lie group with Lie algebra , differentiating gives a unitary representation of on . This means that for each we get a skew-Hermitian operator on , satisfying
i.e. the map taking Lie algebra elements (with the Lie bracket in ) to skew-Hermitian operators (with commutator of operators) is a homomorphism. On , acts by the differential of the conjugation action of , this action is just that of taking the commutator with .
The Lie bracket is not associative, but to any Lie algebra , one can construct an associative algebra called the universal enveloping algebra for . If one identifies with left-invariant vector fields on , which are first-order differential operators on functions on , then is the algebra of left-invariant differential operators on of all orders, with product the composition of differential operators. A Lie algebra representation is precisely a module over , i.e. a vector space with an action of .
So, the state space of a quantum system with symmetry group carries not only a unitary representation of , but also a unitary representation of , or equivalently, an action of the algebra . acts by the operator . In this way a representation gives a sub-algebra of the algebra of observables. Most of the important observables that show up in practice come from a symmetry in this way. An interesting philosophical question is whether the quantum system that governs the real world is purely determined by symmetry, i.e. such that ALL its observables come from symmetries in this manner.
Much of the structure of common quantum mechanical systems is governed by the fact that they carry space-time symmetries. In our 3-space, 1-time dimensional world, these include:
For each basis element one gets a momentum operator
Another example is the symmetry of phase transformations of the state space . Here , and one gets an operator that can be normalized to have integral eigenvalues.
This last example also comes in a local version, where we make independent phase transformations at different points in space-time. This is an example of a “gauge symmetry”, and the question of how it gets represented on the space of states is what will lead us into the BRST story. Next posting in the series will be about gauge symmetry, then on to BRST.
If you want some idea of where this is headed, you can take a look at slides from a colloquium talk I gave recently at the Dartmouth math department. They’re very sketchy, the postings in this series should add some detail.