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Mechanics as a general-relativistic gauge field theory, and Relational Quantization

J. François, L. Ravera

TL;DR

This work reframes Mechanics as a 1D general-relativistic gauge theory (MFT) built on the Mechanical Field Space (MFS) and uses the Dressing Field Method (DFM) to extract invariant relational degrees of freedom. It shows that the standard path integral on MFS differs from the conventional QM path integral, which naturally emerges when one quantizes relationally on the moduli space ${\\mathcal M}$ of d.o.f.; this provides a concrete instance of Relational Quantization (RQ). The paper then extends these ideas to general-relativistic gauge theories (gRGFT), demonstrating automatic anomaly cancellation via cocyclic dressings and outlining a seesaw mechanism for residual second-kind transformations. The results establish QM as a special case of relational quantization and offer a principled framework toward Relational Quantum Gravity, with NR mechanics serving as a proof-of-concept for the broader program. Overall, the Relational Quantization program promises a covariant, clock-invariant path to quantum gravity and related gauge theories by treating physical d.o.f. as relational, dressable structures on field space.

Abstract

We treat Mechanics as a 1-dimensional general-relativistic gauge field theory, Mechanical Field Theory (MFT), introducing what we call the Mechanical Field Space (MFS) and exploiting its bundle geometry. The diffeomorphism covariance of MFT encodes its relational character, arising - as in all general-relativistic physics - via the conjunction of a hole and a point-coincidence argument. Any putative "boundary problem", meaning the claim that boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analogue of the PI used in gauge field theory, is conceptually and technically distinct from the standard PI of Quantum Mechanics. We then use the Dressing Field Method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining Relational Quantization - i.e. the quantization of invariant relational d.o.f. - is shown to reproduce the standard PI of Quantum Mechanics. This establishes the soundness of Relational Quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.

Mechanics as a general-relativistic gauge field theory, and Relational Quantization

TL;DR

This work reframes Mechanics as a 1D general-relativistic gauge theory (MFT) built on the Mechanical Field Space (MFS) and uses the Dressing Field Method (DFM) to extract invariant relational degrees of freedom. It shows that the standard path integral on MFS differs from the conventional QM path integral, which naturally emerges when one quantizes relationally on the moduli space of d.o.f.; this provides a concrete instance of Relational Quantization (RQ). The paper then extends these ideas to general-relativistic gauge theories (gRGFT), demonstrating automatic anomaly cancellation via cocyclic dressings and outlining a seesaw mechanism for residual second-kind transformations. The results establish QM as a special case of relational quantization and offer a principled framework toward Relational Quantum Gravity, with NR mechanics serving as a proof-of-concept for the broader program. Overall, the Relational Quantization program promises a covariant, clock-invariant path to quantum gravity and related gauge theories by treating physical d.o.f. as relational, dressable structures on field space.

Abstract

We treat Mechanics as a 1-dimensional general-relativistic gauge field theory, Mechanical Field Theory (MFT), introducing what we call the Mechanical Field Space (MFS) and exploiting its bundle geometry. The diffeomorphism covariance of MFT encodes its relational character, arising - as in all general-relativistic physics - via the conjunction of a hole and a point-coincidence argument. Any putative "boundary problem", meaning the claim that boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analogue of the PI used in gauge field theory, is conceptually and technically distinct from the standard PI of Quantum Mechanics. We then use the Dressing Field Method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining Relational Quantization - i.e. the quantization of invariant relational d.o.f. - is shown to reproduce the standard PI of Quantum Mechanics. This establishes the soundness of Relational Quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.
Paper Structure (35 sections, 171 equations, 1 figure, 1 table)

This paper contains 35 sections, 171 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Geometry of Mechanical Field Space $\Phi$: Kinematics
  3. The Mechanical Field Space and its natural transformation groups
  4. Differential structures
  5. Tangent bundle and subbundles
  6. Differential forms
  7. Gauge transformations
  8. Connections on Mechanical Field Space
  9. Associated bundle of regions and integration
  10. Integration map
  11. Mechanical Field Theory on $\Phi$: Dynamics
  12. The variational principle for MFT, and its covariance group
  13. The relationality of MFT: the Hole and Point-Coincidence arguments
  14. The Problem of Time and the Boundary Problem
  15. Model of MFT: Non-relativistic Mechanics
  16. ...and 20 more sections

Figures (1)

  • Figure 1: The configuration space-time bundle $\mathscr{C}$vs the standard configuration space $Q$.