BRST symmetry

In theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters BRST stand for Becchi, Rouet, Stora, and (independently) Tuytin 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.

For the special case of gauge theories with a connection, a BRST charge (sometimes also a BRS charge) is an operator usually denoted <math>Q<math>. The operator <math>Q<math> is defined as

<math>Q = c^i (L_i-\frac 12 {f_{ij}}^k b_j c^k)<math>

where <math>c^i,b_i<math> are the Faddeev-Popov ghosts and antighosts, respectively, <math>L_i<math> are the infinitesimal generators of the Lie group, and <math>f_{ij}{}^k<math> are its structure constants. The most important general property of <math>Q<math> is nilpotency: <math>Q^2=0<math>.

The physical states are identified as elements of cohomology of the operator <math>Q<math>, i.e. as states that are BRST-closed (they satisfy <math>Q|\psi\rangle=0<math>) but not BRST-exact (they cannot be written as <math>|\psi\rangle=Q|\phi\rangle<math>. The BRST theory is in fact linked to the standard resolution in Lie algebra cohomology. 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. This means Q acts as an antiderivation. See algebra representation of a Lie superalgebra.

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