BRST

BRST stands for Becchi-Rouet-Stora-Tyutin symmetry, a quantum gauge symmetry used in the quantization of gauge theories. To avoid gauge dependence after fixing a gauge, BRST introduces ghost and anti-ghost fields and a BRST operator Q_B, a fermionic, nilpotent charge with Q_B^2 = 0. The BRST transformations mix gauge, ghost, and anti-ghost fields in a way that leaves the gauge-fixed action invariant.

The physical state space in BRST quantization is defined as the cohomology of Q_B: states for which Q_B|psi> = 0 modulo states of the form Q_B|chi>. Observables must commute with Q_B up to BRST-exact terms, i.e., be BRST-closed modulo BRST-exact. In the path integral, gauge fixing is implemented with the Faddeev-Popov determinant expressed through ghost fields, and BRST symmetry yields Slavnov-Taylor identities, which generalize Ward identities and guarantee gauge-independence of physical amplitudes and perturbative renormalizability.

BRST symmetry, discovered by Becchi, Rouet and Stora, with later refinement by Tyutin, provides a universal framework for quantizing non-Abelian gauge theories, including Yang-Mills theories, and has applications in string theory and topological field theories. Related constructions include anti-BRST symmetry; BRST cohomology underpins the classification of physical states and observables in gauge theories.