Generalization of Faddeev–Popov rules in Yang–Mills theories: N = 3,4 BRST symmetries

Abstract

The Faddeev–Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, [Formula: see text], with respect to [Formula: see text]-parametric Abelian SUSY transformations with odd-valued parameters [Formula: see text], [Formula: see text] and generators [Formula: see text]: [Formula: see text], for [Formula: see text], implying the substitution of an [Formula: see text]-plet of ghost fields, [Formula: see text], instead of the parameter, [Formula: see text], of infinitesimal gauge transformations: [Formula: see text]. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the [Formula: see text] and [Formula: see text] symmetric cases. The superspace of [Formula: see text] SUSY irreducible representation includes, in addition to Yang–Mills fields [Formula: see text], [Formula: see text] ghost odd-valued fields [Formula: see text], [Formula: see text] and [Formula: see text] even-valued [Formula: see text] for [Formula: see text], [Formula: see text]. To construct the quantum action, [Formula: see text], by adding to the classical action, [Formula: see text], of an [Formula: see text]-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires [Formula: see text] additional fields, [Formula: see text]: antighost [Formula: see text], [Formula: see text] even-valued [Formula: see text], 3 odd-valued [Formula: see text] and Nakanishi–Lautrup [Formula: see text] fields. The action of [Formula: see text] transformations on new fields as [Formula: see text]-irreducible representation space is realized. These transformations are the [Formula: see text] BRST symmetry transformations for the vacuum functional, [Formula: see text]. The space of all fields [Formula: see text] proves to be the space of an irreducible representation of the fields [Formula: see text] for [Formula: see text]-parametric SUSY transformations, which contains, in addition to [Formula: see text] the [Formula: see text] ghost–antighost, [Formula: see text], even-valued, [Formula: see text], odd-valued [Formula: see text] and [Formula: see text] fields. The quantum action is constructed by adding to [Formula: see text] an [Formula: see text]-exact gauge-fixing term with a gauge boson, [Formula: see text]. The [Formula: see text] SUSY transformations are by [Formula: see text] BRST transformations for the vacuum functional, [Formula: see text]. The procedures are valid for any admissible gauge. The equivalence with [Formula: see text] BRST-invariant quantization method is explicitly found. The finite [Formula: see text] BRST transformations are derived and the Jacobians for a change of variables related to them but with field-dependent parameters in the respective path integral are calculated. They imply the presence of a corresponding modified Ward identity related to a new form of the standard Ward identities and describe the problem of a gauge-dependence. An introduction into diagrammatic Feynman techniques for [Formula: see text] BRST invariant quantum actions for Yang–Mills theory is suggested.