Quantization without gauge fixing: avoiding Gribov ambiguities through the physical projector

Abstract

The quantisation of gauge invariant systems usually proceeds through some gauge fixing procedure of one type or another. Typically for most cases, such gauge fixings are plagued by Gribov ambiguities, while it is only for an admissible gauge fixing that the correct dynamical description of the system is represented, especially with regards to non perturbative phenomena. However, any gauge fixing procedure whatsoever may be avoided altogether, by using rather a recently proposed new approach based on the projection operator onto physical gauge invariant states only, which is necessarily free on any such issues. These different aspects of gauge invariant systems are explicitely analysed within a solvable U(1) gauge invariant quantum mechanical model related to the dimensional reduction of Yang-Mills theory.

Revisiting the Fradkin–Vilkovisky theorem

“…all define gauge fixing choices which are not admissible [3,4,5,8]. For instance when F (λ) = aλ 3 , the modular domain D[F ] is finite and given by the interval [− 2∆t/a, 2∆t/a] in modular space.…”

The Quantum Geometer's Universe: Particles, Interactions and Topology

“…[30] So far, the physical projector approach has been applied to some well-known integrable systems, as well as to some of the issues surrounding the problems of quantum gravity. [31] For example, it has been applied [23,32] to the gauge invariant mechanical models in 0+1 dimensions described previously, with Lagrangian…”

The physical projector and topological quantum field theories:U(1) Chern-Simons theory in 2 + 1 dimensions

“…The physical projector approach is thus quite an efficient approach to the quantisation of gauge invariant systems, which does not require any gauge fixing procedure whatsoever and thus avoids the potential Gribov problems inherent to such procedures. Some of its advantages have been illustrated here in the instance of the U(1) Chern-Simons theory, as well as for some simple gauge invariant quantum mechanical systems elsewhere [9,10]. Hence, it appears timely now to start exploring the application [22] of this alternative method to the quantisation of gauge invariant systems of more direct physical interest, within the context of the recent developments surrounding M-theory compactified to low dimensions, and aiming beyond that towards the gauge invariant theories of the fundamental interactions among the elementary quantum excitations in the natural Universe.…”

“…Nonetheless, the correct representation of the true quantum dynamics of the system is achieved in this new approach, which uses in an essential way the projection operator [7] onto the subspace of gauge invariant physical states of a given gauge invariant system. Some of the advantages of the physical projector approach have already been explored and demonstrated in a few simple quantum mechanical gauge invariant systems [9,10]. In the present work, we wish to illustrate how the same methods are capable to also deal with the intricacies of topological quantum field theories which, even though possessing only a finite set of physical states, require an infinite number of degrees of freedom and states for their formulation.…”