Mechanical Field Theory Overview

• Mechanical Field Theory is a reformulation of classical mechanics as a gauge field theory using the mechanical field space and diffeomorphism invariance to emphasize relational observables.
• The framework employs the dressing field method to systematically remove gauge redundancies from the path integral, thereby resolving boundary issues while recovering standard quantum mechanics.
• Its unified geometric approach not only clarifies the quantization of 1D dynamical systems but also extends to higher-dimensional general-relativistic gauge field theories.

Mechanical Field Theory (MFT) is the formulation of classical mechanics, and by extension all one-dimensional (1D) dynamical systems, as a general-relativistic gauge field theory with a focus on the relational, diffeomorphism-invariant structure of the configuration space and its quantization. The central object is the “Mechanical Field Space” (MFS), a principal bundle with diffeomorphism group structure, from which quantum mechanics emerges via a manifestly relational path integral after elimination of gauge redundancy through the Dressing Field Method. MFT provides an explicit field-theoretic and geometric framework for addressing classical mechanics, highlighting the role of reparametrization invariance (“diffeomorphism covariance”) and resolving “boundary problems” via relational quantization procedures that generalize to broader general-relativistic gauge field theories (François et al., 20 Oct 2025).

1. Mechanical Field Space and Bundle Geometry

Mechanical Field Space (MFS), denoted as $\Phi$, consists of all histories (fields) $\varphi = (x, t)$, where $x$ represents the “spatial” degree of freedom and $t$ is the clock field, both defined on a one-dimensional parameter manifold $I$. This space is equipped with the structure of an infinite-dimensional principal fiber bundle:

• Base: the moduli space $\mathcal{M} = \Phi / \mathrm{Diff}(I)$ contains equivalence classes $[\varphi]$ under the action of the diffeomorphism group.
• Structure group: $\mathrm{Diff}(I)$, the group of reparametrizations of $I$, acts by right pullback: $R_{\psi} : \Phi \to \Phi$, $\varphi \mapsto \varphi^{\psi} = \psi^{*} \varphi$.
• Projection: $\pi : \Phi \to \mathcal{M}$, $\varphi \mapsto [\varphi]$.

This construction is fully analogous to the configuration space of general relativistic field theories, but in 1D, the parameter manifold $I$ plays the role of “worldline parameter.” The geometry naturally encodes both the standard physical histories (through the base) and the redundant clock parametrizations (through the fibers).

2. Diffeomorphism Covariance and Relational Structure

The covariant formulation ensures invariance under arbitrary reparametrizations of $I$ (“diffeomorphisms”), making the theory manifestly gauge-invariant:

• Gauge symmetry: If $\varphi$ is a solution, so is $\psi^* \varphi$ for any $\psi \in \mathrm{Diff}(I)$.
• Relational character: Only relational data—the values of $x$ as functions of the physical variable $t$—are considered physically meaningful. This corresponds to the “point-coincidence argument” in general relativity, resolving any “hole argument” or alleged boundary-induced symmetry breaking. Thus, the physically observable content is encoded in $[ \varphi ] \in \mathcal{M}$, not in the coordinate-dependent representatives.

3. Path Integral and Its Gauge-Redundancy

A conventional “bare” path integral for MFT is defined over the field space $\Phi$: $$Z(\varphi) = \int \mathcal{D}\varphi\, e^{\frac{i}{\hbar} S(\varphi)}$$ with $S(\varphi) = \int_{I} L(\varphi)$, e.g., $$L(\varphi) = \frac{1}{2}m\langle v, v \rangle - V(x)$$, where $v$ is the velocity as defined in terms of $x$ and $t$.

However, this path integral is over redundant degrees of freedom, rendering it conceptually and technically distinct from the standard quantum mechanical path integral, which is defined only over the physical (“relational”) degrees of freedom.

Traditional gauge-fixing (e.g., choosing a specific parametrization like $dt/d\tau = 1$) yields path integrals that only after further manipulation reproduce standard quantum mechanical results, and introduces complications such as the so-called boundary problem and the possibility of Gribov ambiguities.

4. Relational Quantization Principle

Relational Quantization is the procedure whereby only gauge-invariant (relational) degrees of freedom are quantized, inverting the usual gauge-fixing paradigm. In the MFT context, this amounts to defining the path integral directly on the moduli space $\mathcal{M}$, or equivalently on basic (dressed) fields $\widehat{\varphi}$: $$Z^{\wedge} = \int \mathcal{D} \widehat{\varphi}\, e^{\frac{i}{\hbar} S(\widehat{\varphi})}$$ If the clock field $t$ is chosen as a “dressing field,” then $$\widehat{\varphi} = (x \circ t^{-1}, t \circ t^{-1}) = (x, \mathrm{id}_t)$$ so the dynamics is entirely encoded in $x$ as a function of $t$, eliminating parametrization redundancy. In the case of standard non-relativistic mechanics, this procedure recovers the textbook Feynman path integral.

This principle is fully general and underpins relational quantization schemes for all generally covariant (gauge) field theories.

5. Dressing Field Method and Manifest Invariance

The Dressing Field Method (DFM) is the systematic procedure to map “bare” fields in the principal bundle $\Phi$ to “basic” (dressed, gauge-invariant) variables:

• Dressing field construction: One identifies a map $u: J \to I$ (e.g., the clock field $t$), and pulls back the fields with $u$ to define

$\widehat{\varphi} = u^* \varphi$

• Equivariance: The construction ensures $\widehat{\varphi}$ is fully horizontal and $\mathrm{Diff}(I)$-invariant.
• Dressed forms: All geometrical and quantum structures (differential forms, measures, actions) are constructed from their dressed counterparts (e.g., given a form $\alpha$, its basic version is $$\widehat{\alpha} = \alpha(d\widehat \varphi; \widehat \varphi)$$).

With the clock $t$ as the dressing, one expresses all quantities in terms of $x(t)$, manifestly relabeling invariant and eliminating all unphysical gauge freedom from the path integral.

6. Extension to General-Relativistic Gauge Field Theories

Although developed in the context of 1D mechanics, the MFT formalism generalizes seamlessly to any general-relativistic gauge field theory (“gRGFT”). For field theories on higher-dimensional manifolds, the principal bundle structure persists, now with structure group $\mathrm{Diff}(M) \ltimes H$ (where $H$ is any internal gauge group), and the DFM can be adapted to explicitly construct gauge-invariant (relational) field variables.

The relational quantization path integral then adopts the form: $$Z^{\text{basic}} = \int \mathcal{D}\phi^{\wedge}\, e^{\frac{i}{\hbar} S^{\wedge}(\phi^{\wedge})}$$ where $\phi^{\wedge}$ are the dressed, fully invariant field variables. This approach avoids gauge-fixing ambiguities and provides a generalized quantum field theory consistently quantizing only genuinely physical observables.

7. Conceptual and Technical Significance

The mechanical field theory framework recasts classical mechanics as a 1D general-relativistic gauge field theory embedded in the formal structure of principal fiber bundles. The relational resolution of the “boundary problem”—i.e., the supposed symmetry breaking due to boundary conditions—demonstrates that only relational observables are physical. The explicit separation between bare (gauge-dependent) and dressed (gauge-invariant) path integrals clarifies the foundational structure underlying quantization in systems with reparametrization invariance. Through the dressing field method, standard quantum mechanics is precisely recovered as the quantum relational dynamics of mechanical field theory, and a rigorous geometric foundation is established for extending quantization procedures to general covariant gauge field theories including quantized gravity (François et al., 20 Oct 2025).