Solving general gauge theories on inner product spaces

Abstract

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form |ph = e [Q,ψ] |ph 0 where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge fixing operator, and |ph 0 BRST invariant states determined by a hermitian set of BRST doublets in involution. |ph 0 does not belong to an inner product space although |ph does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for |ph 0 and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that |ph 0 are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set |ph 0 are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.

A note on BRST quantization of SU(2) Yang-Mills mechanics

“…We discuss next the possibility for these states to have finite norm (see also [16,1]). The spectrum of the hamiltonian in such a basis would be guaranteed to be physical.…”

Quantum fluids in the Kähler parametrization

“…where χ is a "gauge fixing" fermion with gh(χ) = −1 and |φ phys 0 is a trivial BRST state determined by a complete irreducible set of BRST doublets in involution [27]. If the BRST operator is nilpotent and the theory has a Lie group gauge symmetry, the state |φ phys is a BRST singlet.…”

“…In general, these transformations change the representation of δ -operator in terms of operators fromΣ and, consequently, transform the operators Λ A . Since we have obtained our states in the representations given by the relations (27) and (28), respectively, we are interested in those morphisms ofΣ that leave the BRST operator invariant in these representations. The transformations that satisfy this property are endomorphisms that form the U (1) 4 group that have a two scale and a two rotation actions, respectively,…”

Generalized BRST Quantization and Massive Vector Fields

“…To see this one may observe that Q is invariant under the scale transformations (γ 1 and γ 2 are real parameters and U 1,2 unitary operators) 15) and under the rotations (θ 1 and θ 2 are real parameters and R 1,2 unitary operators)…”

“…where ψ is a gauge fixing fermion, and where |φ is determined by a complete irreducible set of BRST doublets in involution [15]. Since the BRST charge is no longer nilpotent in this generalized BRST case, operators and states are decomposed into higher multiplets than doublets.…”