- BRST formalism
(A draft of an alternate exposition has been added at

BRST quantization .)In

theoretical physics , the**BRST formalism**is a method of implementingfirst class constraint s. 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 withgauge invariance . For example, the BRST methods are often applied to gauge theory and quantizedgeneral relativity .**Classical version**This is related to a supersymplectic

manifold where pure operators are graded by integralghost numbers and we have a BRSTcohomology .**Quantum version**The space of states is not a Hilbert space (see below). This

vector space is both**Z**_{2}-graded and**R**-graded. If you wish, you may think of it as a**Z**_{2}×**R**-graded vector space . The former grading is the parity, which can either be even or odd. The latter grading is theghost 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**Z**_{2}×**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 acochain complex describing acohomology .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 thestandard resolution inLie algebra cohomology .Recall that the space of states is

**Z**_{2}-graded. If A is a pure graded operator, then the BRST transformation maps A to[ Q,A) where[ ,) is thesupercommutator . 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 rightJacobian associated with constraints that gauge-fix the symmetry.The BRST is a

supersymmetry . It generates theLie 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 anantiderivation .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 aHilbert 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 anoperator 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 toGribov 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 theLie bracket , NOT thecommutator .:$QB=0$:$Qb=B$

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 ghost s and antighosts, respectively, $L\_i$ are the infinitesimal generators of theLie 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*]

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