Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation

TL;DR

This paper derives the correct path integral measure for Palatini and Holst formulations of gravity using reduced phase space techniques, clarifying differences from previous measures and connecting canonical LQG with spin foam models.

Contribution

It provides the first derivation of the path integral measure for these gravity formulations and resolves discrepancies with prior results, advancing understanding of quantum gravity approaches.

Findings

Derived the path integral measure for Holst and Palatini gravity.

Identified differences from previous measures and explained their origin.

Connected canonical LQG with spin foam models through measure analysis.

Abstract

An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic "Lebesgue measure" usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed. In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element. From this path integral for the Holst formulation of GR we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical LQG and the spin foam approach.