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Mechanics as a general-relativistic gauge field theory, and relational quantization

J. T. François1,2,* and L. Ravera3,4,5,†

  • *Contact author: jordan.francois@uni-graz.at
  • †Contact author: lucrezia.ravera@polito.it
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APS Open Sci. 1, 000126 – Published 8 September, 2026

DOI: https://doi.org/10.1103/bttl-7rdd

Abstract

We treat the mechanics of point particles as a one-dimensional general-relativistic gauge field theory, which may be referred to as mechanical field theory (MFT), exploiting the bundle geometry of mechanical field space (MFS). 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 “spacetime” boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analog 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 degree of freedom—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.

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  84. The argument that it could be ascertained via the fact that when it holds the equation of motion ma=F of x reduces to the familiar mẍ=F—resembling the above discussion of “inertial clocks”—does not hold since ẍ is not an observable either.
  85. For the philosophically minded reader, we remark that this has unmistakable ontic structural realist undertones, and is especially strongly reminiscent of Eddington’s group theoretic “moderate” structural realism [146, 147]—see also Secs. 4.5 and 4.6 of Ref. [148]—according to which physical objects and their network of relations come as a (metaphysical) package, analogously to the elements of a group and their relations: A group element is only defined by its “position” relative to others with respect to the abstract group structure (i.e., the composition law) relating them. Analogously, the precise position of a physical object within the (lawful) relational network is constitutive of its “identity” and properties. Objects are then defined as nodes within the network and undetachable from it.
  86. As is, it holds, in particular, for ad hoc dressing fields—e.g., edge modes—but as discussed in Sec.  4b3, those typically lead to a wrongheaded physical picture.
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  89. See also Ref. [24] for the case of general-relativistic theories, i.e., only Diff(M), and applications of the bundle geometry of their field space to their covariant phase-space formalism.
  90. These were essential steps toward the application of the DFM, as explained in Sec.  4a, leading to an invariant relational formulation of gRGFTs, sketched below.
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APS Open Sci. 1, 000126– Published 8 September, 2026
Vol. 1
  • Received 13 June 2026
  • Accepted 19 August 2026
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