BRST formalism

(A draft of an alternate exposition has been added at BRST quantization.)

In theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters BRST stand for Becchi, Rouet, Stora, and (independently) Tyutin who discovered this formalism. It is a sophisticated method to deal with quantum physical theories with gauge invariance. For example, the BRST methods are often applied to gauge theory and quantized general relativity.

Classical version

This is related to a supersymplectic manifold where pure operators are graded by integral ghost numbers and we have a BRST cohomology.

Quantum version

The space of states is not a Hilbert space (see below). This vector space is both Z₂-graded and R-graded. If you wish, you may think of it as a Z₂×R-graded vector space. The former grading is the parity, which can either be even or odd. The latter grading is the ghost number. Note that it is R and not Z because unlike the classical case, we can have nonintegral ghost numbers. Operators acting upon this space are also Z₂×R-graded in the obvious manner. In particular, "Q" is odd and has a ghost number of 1.

Let H_n be the subspace of all states with ghost number n. Then, "Q" restricted to H_n maps H_n to H_n+1. Since Q²=0, we have a cochain complex describing a cohomology.

The physical states are identified as elements of cohomology of the operator $Q$, i.e. as vectors in Ker Q_n+1/Im Q_n. The BRST theory is in fact linked to the standard resolution in Lie algebra cohomology.

Recall that the space of states is Z₂-graded. If A is a pure graded operator, then the BRST transformation maps A to [Q,A) where [,) is the supercommutator. BRST-invariant operators are operators for which [Q,A)=0. Since the operators are also graded by ghost numbers, this BRST transformation also forms a cohomology for the operators since [Q, [Q,A))=0.

Although the BRST formalism is more general than the Faddeev-Popov gauge fixing, in the special case where it is derived from it, the BRST operator is also useful to obtain the right Jacobian associated with constraints that gauge-fix the symmetry.

The BRST is a supersymmetry. It generates the Lie superalgebra with a zero-dimensional even part and a one dimensional odd part spanned by Q. [Q,Q)={Q,Q}=0 where [,) is the Lie superbracket (i.e. Q²=0). This means Q acts as an antiderivation.

Because Q is Hermitian and its square is zero but Q itself is nonzero, this means the vector space of all states prior to the cohomological reduction has an indefinite norm! This means it is not a Hilbert space!

For more general flows which can't be described by first class constraints, see Batalin-Vilkovisky

Example

For the special case of gauge theories (of the usual kind described by sections of a principal G-bundle) with a quantum connection form A, a BRST charge (sometimes also a BRS charge) is an operator usually denoted $Q$.

Let the $mathfrak\{g\}$-valued gauge fixing conditions be $G=xipartial^mu\; A\_mu$ where ξ is a positive number determining the gauge. There are many other possible gauge fixings, but they will not be covered here. The fields are the $mathfrak\{g\}$-valued connection form A, $mathfrak\{g\}$-valued scalar field with fermionic statistics, b and c and a $mathfrak\{g\}$-valued scalar field with bosonic statistics B. c deals with the gauge transformations wheareas b and B deals with the gauge fixings. There actually are some subtleties associated with the gauge fixing due to Gribov ambiguities but they will not be covered here.

:$QA=Dc$

where D is the covariant derivative.

:$Qc=\{iover\; 2\}\; [c,c]\; \_L$

where [,] _L is the Lie bracket, NOT the commutator.

:$QB=0$:$Qb=B$

Q is an antiderivation.

The BRST Lagrangian density

:$mathcal\{L\}=-\{1over\; 4g^2\}Tr\; [F^\{mu\; u\}F\_\{mu\; u\}]\; +\{1over\; 2g^2\}Tr\; [BB]\; -\{1over\; g^2\}Tr\; [BG]\; -\{xiover\; g^2\}Tr\; [partial^mu\; b\; D\_mu\; c]$

While the Lagrangian density isn't BRST invariant, its integral over all of spacetime, the action is.

The operator $Q$ is defined as

:$Q\; =\; c^i\; left(L\_i-frac\; 12\; \{f\_\{ij^k\; b\_j\; c^k\; ight)$

where $c^i,b\_i$ are the Faddeev-Popov ghosts and antighosts, respectively, $L\_i$ are the infinitesimal generators of the Lie group, and $f\_\{ij\}\{\}^k$ are its structure constants.

References

* [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=1&index1=2938340 brst cohomology on arxiv.org]