Revisiting the Fradkin–Vilkovisky theorem

Abstract

The status of the usual statement of the Fradkin-Vilkovisky theorem, claiming complete independence of the Batalin-Fradkin-Vilkovisky path integral on the gauge fixing "fermion" even within a nonperturbative context, is critically reassessed. Basic, but subtle reasons why this statement cannot apply as such in a nonperturbative quantisation of gauge invariant theories are clearly identified. A criterion for admissibility within a general class of gauge fixing conditions is provided for a large ensemble of simple gauge invariant systems. This criterion confirms the conclusions of previous counter-examples to the usual statement of the Fradkin-Vilkovisky theorem.

The Cosmological Constant of One-Dimensional Matter Coupled Quantum Gravity Is Quantised

“…l An alternative approach which maintains explicit BRST invariance at all stages is presented in Ref. 19, confirming once again the discussion developed here.…”

“…In fact, it is possible to circumvent the ill defined singular products 0 · δ(0) stemming from the non-normalisable character of thep λ eigenstates in still another manner, that also explicitly preserves manifest BRST invariance at all stages and does not require to considerÛ BRST (t 2 , t 1 ) matrix elements for non-BRST invariant external states, and still reach exactly the identical conclusion. 19 Any gauge fixing leads to a dynamics defined over the space of gauge orbits of the system, hence to a gauge invariant dynamics, but which of these gauge orbits are included and with which multiplicity is dependent on the choice of gauge fixing procedure. The correct physics and dynamics is represented only for an admissible gauge fixing procedure, namely one which selects, up to a common weight factor, once and only once each of the gauge orbits and thus induces an admissible covering of the space of gauge orbits.…”

“…First, given initial values q n i and p n,i at t = t i for the phase space degrees of freedom, let us construct the general solutions to the equations of motion (17) whatever the choice for the Lagrange multiplier function λ(t). For this purpose, consider the function τ (t), with τ i = t i , such that…”

“…It then follows that in terms of this variable τ , the equations of motion (17) reduce to those of the uncoupled matter system,…”

“…In fact, it is possible to circumvent the ill defined singular products 0 • δ(0) stemming from the non-normalisable character of the pλ eigenstates in still another manner, that also explicitly preserves manifest BRST invariance at all stages and does not require to consider ÛBRST (t 2 , t 1 ) matrix elements for non-BRST invariant external states, and still reach exactly the identical conclusion. 19 Any gauge fixing leads to a dynamics defined over the space of gauge orbits of the system, hence to a gauge invariant dynamics, but…”