Physical Equivalence in Theory and Models

• Physical equivalence is the condition under which different mathematical representations yield indistinguishable empirical predictions, ensuring consistency across various physical theories.
• It is applied in fields such as random dynamical systems, thermodynamic formalism, and gravitational theories to relate symmetry, gauge choice, and model parameterizations.
• The concept is crucial for evaluating models through empirical, interpretational, and definitional criteria, thereby affirming theoretical consistency and guiding equivalence proofs.

Physical equivalence delineates when distinct mathematical or formal representations—whether measures, state spaces, theories, or dynamical models—encode the same physical content, i.e., yield indistinguishable empirical predictions or describe empirically indistinguishable physical situations. While often evoked to justify identifications between different model presentations (e.g., gauges, frames, parameterizations, or dual theories), technical and philosophical analyses reveal that such equivalence is contingent on both formal symmetries and detailed empirical or interpretive criteria. The notion thus plays a central role in the foundations of statistical mechanics, dynamical systems, quantum theory, classical and quantum gravity, and the theory of physical theories themselves.

1. Physical Equivalence in Random Dynamical Systems

In random dynamical systems (RDS), the principle of physical equivalence manifests in the identification between "physical measures"—empirical time-averaged measures describing the statistical behavior of Lebesgue-almost every initial point—and Sinai-Ruelle-Bowen (SRB) measures, which are mathematically characterized by absolute continuity along unstable manifolds. Blumenthal–Young established that for i.i.d. sequences of $C^2$ random diffeomorphisms $\{f_\omega\}_{\omega\in\Omega}$ on a compact manifold $M$, under mild integrability and nonuniform hyperbolicity conditions, the limit points $\mu_\omega$ of forward images of a stationary measure $\mu$ are random SRB measures: they satisfy absolute continuity on the relevant local unstable manifolds and possess positive Lyapunov exponents. Crucially, for random perturbations, the sample (or "physical") measures coincide exactly with random SRB measures, and the entropy formula for the system is given by summing positive Lyapunov exponents weighted by their multiplicities:

$$h_\mu(\{f^n\}) = \sum_{i:\lambda_i>0} m_i \lambda_i$$

This equivalence fails generically in the deterministic case, where bifurcations or sinks may preclude alignment of forward iterates with the unstable foliation, as seen in the Newhouse phenomenon and the figure-eight attractor (Blumenthal et al., 2018).

2. Physical Equivalence in Thermodynamic Formalism of Flows

Thermodynamic formalism extends the notion of physical equivalence to additive and nonadditive families of continuous potentials associated with flows. For a compact metric space $(Y,d)$ with continuous flow $\phi_t$, an asymptotically additive family $\Phi = (\varphi_t)_{t \ge 0}$ is physically equivalent to an additive family $(S_t b)_{t \ge 0}$ if

$$\lim_{t\to\infty} \frac{1}{t} \|\varphi_t - S_t b\|_\infty = 0$$

Thus, additive and asymptotically additive potentials yield the same thermodynamic pressure, multifractal spectra, and share equilibrium measures. In hyperbolic flows, this equivalence enables the reduction of the nonadditive formalism to the classical additive case, preserving all key statistical mechanical quantities up to sublinear corrections (Holanda, 2022).

3. Gauge and Frame Transformations: Physical vs. Mathematical Equivalence

A central locus for physical equivalence is the question of whether theories or models related by field redefinitions, gauge choices, or conformal (Weyl) rescalings are truly physically equivalent. In gravitational theories, especially scalar-tensor and $f(R)$ theories:

• The Einstein and Jordan frames, related by conformal transformations $g_{\mu\nu} \to \Omega^2(x) g_{\mu\nu}$, are physically equivalent only when all physical observables are formulated in terms of dimensionless, frame-invariant quantities. Once all fields and couplings are expressed in units of local Planck mass (or another invariant scale), the two frames yield identical predictions for cosmological and inflationary observables (Postma et al., 2014).
• However, field redefinition equivalence does not universally translate to physical equivalence. For instance, Palatini $f(R)$ theories and Brans–Dicke models with $\omega = -3/2$, though related mathematically at the level of actions, are not physically equivalent because they yield different free-fall geodesics and fail in distinct ways against observational tests such as Mercury’s perihelion precession. The specification of which metric couples to matter determines empirical predictions, and thus only one model may pass a given solar-system test (Fatibene et al., 2013). Similarly, Noether symmetries, quantum constraints, and Hamiltonians do not generically carry over from the Jordan to Einstein frames, leading to demonstrable inequivalence at both the classical and quantum levels unless the theory reduces to General Relativity (Sk. et al., 2016).

In quantum field theory, Weyl transformations induce contact operators in the effective action that exactly match the local terms arising from graviton exchange diagrams in the Jordan frame. S-matrix elements, both at tree and one-loop level, are invariant between frames if all such local contributions are properly included, establishing physical equivalence in this context (Hill et al., 2020).

4. Symmetry Transformations, Decompositional and Diffeomorphism Equivalence

Physical equivalence is often rooted in the invariance of physical laws under certain relabelings, decompositions, or gauge transformations:

• The Principle of Decompositional Equivalence (PDE) asserts that observable predictions remain unchanged under arbitrary reassignments of system-environment decompositions in both classical and quantum mechanics. Formally, the global Hamiltonian $H_U$ and the universal state $|\Psi\rangle$ are invariant under any permutation of the degrees of freedom, i.e., for any $\Pi$,

$$\Pi H_U \Pi^{-1} = H_U\,, \quad \Pi |\Psi_{1,\ldots,N}\rangle = |\Psi_{\Pi(1),\ldots,\Pi(N)}\rangle$$

This symmetry implies that no experimental protocol can ever establish the absolute identity of a subsystem across measurements, rendering oracular completeness and the identification of persistent "objects" as extra-theoretical assumptions (Fields, 2010).

• In diffeomorphism-invariant (generally covariant) theories, two paradigms—Leibniz Equivalence and Newton Equivalence—articulate different conceptions of physical equivalence. Under Leibniz Equivalence, all models related by active diffeomorphisms represent the same physical situation; under Newton Equivalence, these models represent physically distinct but equally possible situations. These philosophies have significant implications for the substantivalism debate and the interpretation of symmetry in general relativity (Johns, 2019).
• In gauge theories like linearized massive gravity, actions differing by gauge fixing or mass term parameterization (e.g., Fierz–Pauli at the FP point versus generic mass) can be shown to define physically equivalent quantum theories if their generating functionals are related by a finite field-dependent BRST transformation, ensuring identical correlation functions and S-matrix elements (Raval et al., 2018).

5. Empirical and Interpretational Notions of Physical Equivalence

Contemporary philosophy of science delineates several formal criteria:

• Empirical equivalence: Two theories are empirically equivalent if they assign the same probability distribution to all experimental outcomes across all possible preparations.
• Interpretational equivalence: This stricter criterion requires that, given an interpretation mapping formalism to ontology, both theories make identical claims about the world.
• Definitional equivalence: Two first-order theories are definitionally equivalent if there exist intertranslatable extensions yielding logical equivalence after expansion.
• Categorical equivalence: Theories are categorically equivalent if their categories of models are equivalent (via full, faithful, essentially surjective functors) and their physical interpretations align across this equivalence.

While empirical and even categorical equivalence are necessary for physical equivalence, they are not sufficient. Interpretational equivalence is sensitive to how formal symbols are read: dual models may be formally isomorphic yet make contradictory claims (as in T-duality or AdS/CFT, where radii or even spacetime dimensions are inverted) (Weatherall, 2018, Butterfield, 2018, Haro, 2017). Thus, physical equivalence ultimately rests on both formal and interpretational correspondences.

6. Physical Equivalence of States and Measures: Statistical and Probabilistic Frameworks

In general probabilistic theories (GPTs), the physical equivalence of states formalizes the indistinguishability (under all operationally realizable observations) of two states $s_1, s_2$: there exists an affine bijection $\Phi$ on the space of effects such that $e(s_1) = \Phi(e)(s_2)$ for all effects $e$. When all pure states are physically equivalent—i.e., the affine symmetry group acts transitively—the convex state space inherits a high degree of symmetry. Combining symmetric structure with the decomposability postulate (every state decomposes into perfectly distinguishable pure states) singles out the classical simplex or the Bloch ball in low-dimensional systems, directly linking physical equivalence postulates to quantum state space geometry (Kimura et al., 2010). Similarly, in thermodynamic formalism, physically equivalent families of potentials produce identical equilibrium states and multifractal spectra (Holanda, 2022).

7. Physical Equivalence in the Presence of Boundary Conditions and Beyond

Empirical and theoretical equivalence criteria must include not only the equations of motion (dynamics) but also the structure of boundary conditions. For instance, general relativity and teleparallel gravity differ by a total divergence term in the action; only after integrating appropriate boundary data—e.g., Gibbons–Hawking–York boundary term—do they become physically equivalent for all physically relevant (bounded) models. Similar caveats arise in electromagnetic duals (Faraday-tensor vs. potential) and affect which empirical situations each formulation distinguishes. Agreement on both dynamics and boundary conditions is required for full empirical (and hence physical) equivalence. The inclusion of boundary data refines and sometimes rescues purported equivalences between formulations (Wolf et al., 2023).

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In summary, physical equivalence is a cluster concept which, depending on context, involves technical criteria—gauge or diffeomorphism invariance, symmetry and representation-theoretic properties, equivalence of S-matrix elements, or formal category-theoretic equivalence—as well as interpretational agreement on empirical content and ontology. Its rigorous analysis is indispensable for properly understanding gauge and gravitational dualities, the relation between different mathematical representations of physical theory, the status of physical observables under symmetries and transformations, and the foundations of probabilistic and statistical frameworks in physics.