Canonical path-integral measures for Holst and Plebanski gravity: II. Gauge invariance and physical inner product

Muxin Han (1, 2)

1 AEI - Max-Planck-Institut für Gravitationsphysik ( Albert-Einstein-Institut )
2 Institut f. Theoretische Physik III

This article serves as a continuation for the discussion in [ 1 ], we analyze the invariance properties of the gravity pathintegral measure derived from canonical framework, and discuss which path-integral formula may be employed in the concrete computation e.g. constructing a spin-foam model, so that the final model can be interpreted as a physical inner product in the canonical theory. The present article is divided into two parts, the first part is concerning the gauge invariance of the canonical path-integral measure for gravity from the reduced phase space quantization. We show that the path-integral measure is invariant under all the gauge transformations generated by all the constraints. These gauge transformations are the local symmetries of the gravity action, which is implemented without anomaly at the quantum level by the invariant path-integral measure. However, these gauge transformations coincide with the spacetime diffeomorphisms only when the equations of motion is imposed. But the path-integral measure is not invariant under spacetime diffeomorphisms, i.e. the local symmetry of spacetime diffeomorphisms become anomalous in the reduced phase space path-integral quantization. In the second part, we present a path-integral formula, which formally solves all the quantum constraint equations of gravity, and further results in a rigging map in the sense of refined algebraic quantization (RAQ). Then we give a formal path-integral expression of the physical inner product in loop quantum gravity (LQG). This path-integral expression is simpler than the one from reduced phase space quantization, since all the gauge fixing conditions are removed except the time-gauge. The resulting path-integral measure is different from the product Lebesgue measure up to a local measure factor containing both the spacetime volume element and the spatial volume element. This formal path-integral expression of the physical inner product can be a starting point for constructing a spin-foam model. Contents 1 Introduction 3 2 A general dynamical system with both first-class and second-class constraints 5 2.1 Reduce phase space quantizations of the constraint system. 3 The gauge invariance of the path-integral measure of gravity 14 3.1 The path-integral measure in ADM formalism. 4 Path-integral and refined algebraic quantization 23 4.1 Refined algebraic quantization.

Mots clés: Physical Sciences

Muxin Han. Canonical path-integral measures for Holst and Plebanski gravity: II. Gauge invariance and physical inner product. Classical and Quantum Gravity, 2010, 27 (24), pp.245015. ⟨10.1088/0264-9381/27/24/245015⟩. ⟨hal-00654154⟩