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

Jonathan Engle, Muxin Han, Thomas Thiemann

TL;DR

This work derives the correct path-integral measure for Holst and Plebanski–Holst gravity using a reduced phase-space approach, showing that the measure contains local factors proportional to spacetime and spatial volumes rather than a naive Lebesgue form. By transforming to configuration variables, the authors obtain a Holst path integral in terms of a spacetime connection $oldsymbol{ ext{ω}}_mu^{IJ}$ and a tetrad $oldsymbol{e}_mu^I$, with the Palatini limit recovered as $eta o ext{infty}$. They then extend the construction to the Plebanski–Holst theory, obtaining a constrained BF representation summed over the $(II ext{±})$ sectors and establishing a concrete link between Holst gravity and the Plebanski BF framework. The results clarify inconsistencies with earlier BHNR work, discuss implications for spin-foam models, and outline a discretization program that preserves the nontrivial measure, aiming to bridge canonical LQG and spin foams in a gauge-consistent, background-independent setting.

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.

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

TL;DR

This work derives the correct path-integral measure for Holst and Plebanski–Holst gravity using a reduced phase-space approach, showing that the measure contains local factors proportional to spacetime and spatial volumes rather than a naive Lebesgue form. By transforming to configuration variables, the authors obtain a Holst path integral in terms of a spacetime connection and a tetrad , with the Palatini limit recovered as . They then extend the construction to the Plebanski–Holst theory, obtaining a constrained BF representation summed over the sectors and establishing a concrete link between Holst gravity and the Plebanski BF framework. The results clarify inconsistencies with earlier BHNR work, discuss implications for spin-foam models, and outline a discretization program that preserves the nontrivial measure, aiming to bridge canonical LQG and spin foams in a gauge-consistent, background-independent setting.

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.

Paper Structure

This paper contains 17 sections, 4 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. The path-integral measure for the Holst action
  3. Reduced Phase space path integral
  4. Configuration path integral in terms of spacetime $so(\eta)$-connection and tetrad
  5. Basic relations between variables
  6. Proof of bijectivity
  7. Rewriting the measure
  8. Final path integral
  9. The construction of path-integral measure for Plebanski-Holst, by way of Holst
  10. Basic strategy and some definitions
  11. The coordinate transformation and its bijectivity
  12. The change of measure
  13. Final path integral
  14. An alternative way to construct Plebanski-Holst path-integral from Holst
  15. Consistency with Buffenoir, Henneaux, Noui and Roche
  16. ...and 2 more sections

Key Result

Lemma 3.1

$(\det e^i_a)(e^0_t - e^0_h f^h_k e^k_t)$ is equal to the 4-volume element $\mathcal{V} = \det e^I_\alpha$.

Theorems & Definitions (4)

  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4