Theoretical vs Physical Equivalence

Updated 6 August 2025

Theoretical equivalence is a formal mapping between theories that preserves structure and deductive content via methods like definitional or categorical frameworks.
Physical equivalence requires that theories share identical empirical predictions and ontological claims through aligned interpretations and measurement protocols.
The topic highlights that formal equivalence alone is insufficient, as true physical equivalence demands rigorous treatment of boundary conditions and invariant empirical content.

Theoretical equivalence and physical equivalence are two central but distinct notions used to analyze when different mathematical representations or formulations describe “the same” physical theory. Their technical criteria, operational content, and implications for the structure of physical theories span a wide range of foundational debates and formal developments in physics, mathematics, and philosophy of science.

1. Operational and Formal Definitions

Theoretical equivalence refers to the existence of a mapping or correspondence—typically at the level of mathematical structure—between two theories that preserves (or translates in a well-defined way) their assertions, models, and deductive structure. Common formalizations include definitional equivalence, Morita equivalence, and categorical equivalence, each with varying degrees of strictness regarding the allowed translations or functors between theories (Barrett et al., 2015, Weatherall, 2018).

Physical equivalence, by contrast, is a more stringent requirement: two theories are physically equivalent if, beyond mere mathematical or syntactic correspondence, they make identical claims about the physical world—including empirical predictions and physical ontology—when supplied with appropriate interpretations, initial/boundary conditions, and connections to measurement (Haro, 2017, Butterfield, 2018, Weatherall, 2018). This requires that interpretations, measurement protocols, and empirical significance line up across the candidate theories.

Table: Key Criteria for Equivalence

Notion	Core Condition	Empirical Alignment
Definitional equivalence	Explicit definability (no new sorts)	Formal translation only
Morita equivalence	Explicit definability (allowing new sorts)	Formal translation only
Categorical equivalence	Functorial (category) equivalence	Must ensure arrows preserve empirical content
Physical equivalence	Interpretation and empirical prediction coincide	Physical ontology and operational content must be identical

2. Mathematical and Structural Notions of Equivalence

The literature details several formal frameworks:

Definitional and Morita Equivalence: The existence of explicit definitions translating the vocabulary of one theory into the other (definitional), and further, admitting new sorts (types of objects) as bridge definitions (Morita). The latter broadens the scope to cases where, for example, one formulation uses points and another uses both points and lines (as in geometry). Morita equivalence implies categorical equivalence, but the reverse does not generally hold (Barrett et al., 2015).
Categorical Equivalence: Two theories are categorically equivalent if their categories of models are equivalent via functors that are full, faithful, and essentially surjective. This concept is important in algebraic and geometric contexts (e.g., comparing vector potential and field strength formulations of electromagnetism), but its sufficiency as a criterion for physical equivalence is debated. Weatherall argues that categories can have the same structure “up to functors,” but still encode different physical information due to differences in internal constitution or interpretation (Weatherall, 2018, Weatherall, 2018).
Schema-Based Equivalences: Some frameworks treat the “bare theory” as a triple (state space, quantities, dynamics), and models as representations furnishing these structures with mathematical or physical specificity (Haro, 2019, Haro, 2017). Dualities (such as position-momentum or gauge/gravity duality) provide isomorphisms at the level of these bare theories, forming the basis for claims of theoretical equivalence, but require careful analysis of interpretative constraints to ensure physical equivalence (Butterfield, 2018).

3. Interpretational and Empirical Considerations

A recurring theme in the literature is that formal structural equivalence does not automatically entail interpretational or physical equivalence. Several scenarios are highlighted:

Two dual theories may be isomorphic at the level of mathematical structure yet disagree in their physical claims—either via contrary assertions about a common subject (Contr), or by describing isomorphic but distinct subject-matters (Diff) (Butterfield, 2018).
Empirical equivalence—the agreement on all observable predictions—is necessary, and sometimes assumed sufficient, for theoretical equivalence. However, two theories might be empirically equivalent while differing in their ontological or structural assumptions (e.g., Bohmian mechanics vs. standard quantum theory) (Weatherall, 2018).
Interpretation maps (which assign physical meaning to mathematical structure) are crucial. Two formally equivalent formulations are only physically equivalent if, under “internal interpretations” (those confined to the core of the theory), they refer to the same physical entities or properties (Haro, 2019, Haro, 2017). In the schema of (Haro, 2019), theoretical equivalence (formal isomorphism of model roots) must be supplemented with matching interpretations ("internal" and "unextendable") so that domains of application coincide.
Boundary conditions, not only dynamics, may encode crucial empirical content. Claims that two formulations are equivalent based solely on their dynamical equations (or equivalence up to total divergences in the action) can fail if one theory specifies distinct possibilities at the boundary (e.g., unique surface charge determined in the vector potential form of electromagnetism, which is ambiguous in the Faraday tensor formalism) (Wolf et al., 2023).

4. Physical Equivalence in Fundamental Physics

Physical equivalence is examined in multiple contexts:

General Probabilistic Theories (GPTs): Physical equivalence of pure states is operationally defined by the existence of unit-preserving affine bijections on the state space that map one pure state to any other, enforcing symmetry (transitivity) among pure states. This induces a symmetric structure on state space, which, combined with a decomposability principle (every mixed state is a convex combination of perfectly distinguishable pure states), leads to strong constraints: in low dimensions, either a simplex (classical probability) or the Bloch ball (qubit) (Kimura et al., 2010).
Thermodynamics: Thermodynamic equivalence is defined via reversible transformations that relate the thermodynamic potentials (such as the grand potential) of two physical systems up to proportionality or more general affine relations. It is a transitive relation, dividing systems into equivalence classes with shared macroscopic characteristics (e.g., ideal quantum Fermi gases, classical gases, 1D hard rod gas), even if microdynamics differ (Ciccariello, 2014).
Gauge/Gravity Dualities: In dualities such as AdS/CFT, formal (bare theory) equivalence exists at the level of isomorphic triples (states, quantities, dynamics), but physical equivalence requires that only the invariant “common core” (e.g., conformal boundary data) is interpreted as physically real; model-dependent structures (such as the bulk geometry) are auxiliary (Haro, 2017).
Contextuality Frameworks: Equivalence between different frameworks of contextuality (e.g., sheaf-theoretic and equivalence-based) can be established categorically under assumptions such as factorizability, such that operational and sheaf-theoretic non-contextuality are shown to be (categorically) equivalent. This reflects a strong form of theoretical and physical equivalence regarding the phenomena encoded by contextuality (Wester, 2017).

5. Breakdown and Limitations of Equivalence

Physical equivalence is nontrivially sensitive to:

Quantum Anomalies and Regularization: At the quantum level, field redefinitions that yield equivalent classical theories (e.g., between Jordan and Einstein frames of scalar-tensor gravity) can become inequivalent due to anomalous symmetries, different counterterm structures, or non-covariant behavior under quantization. This results in distinct S-matrix elements and necessitates different UV completions (Herrero-Valea, 2016).
Canonical and Quantum Formulations: Even when classical field equations are related by a conformal transformation (e.g., in F(R) gravity), differences in Noether symmetries, canonical structures, and quantum Hamiltonians result in nonequivalent quantum theories—classical field equation equivalence does not extend to mathematical or physical equivalence at the quantum level (Sk. et al., 2016).
Category Structure Limitations: Categorical equivalence cannot capture all relevant theory structure. The existence of autoequivalences that are not naturally isomorphic to the identity (the failure of the 'G' property) exposes the inability of pure categorical structure to distinguish physically relevant properties like the detailed constitution of spacetime in GR (Weatherall, 2018).

6. Extensions: Invariant Content and Foundational Principles

The notion of equivalence is sharpened by appeal to invariance under transformations or under change of presentation:

Univalent Foundations: In mathematics, the “equivalence principle” is rigorously enforced in univalent foundations, where equality of objects is identified with equivalence (e.g., isomorphism of sets or categories), and all properties and constructions are thereby invariant under equivalence. This ensures that theoretical equivalence is realized in the most robust structural sense; nevertheless, it does not assert physical equivalence since the connection to empirical content is absent (Ahrens et al., 2022).
Physical Equivalence of Nonadditive Quantities: In the thermodynamic formalism for dynamical systems, asymptotically additive and additive families of potentials are physically equivalent in the sense that their long-term averages coincide, permitting the transfer of results from additive to nonadditive frameworks, provided regularity conditions are satisfied (Holanda, 2022).

7. Philosophical and Practical Consequences

Assessments of equivalence have significant repercussions:

Interpretation-Dependence: Structural or formal equivalence is only meaningful when operationalized with explicit attention to interpretation, measurement protocols, and physical content.
Boundary Content: Model pluralism and pragmatics in theory choice require enrichment of the formal (semantic) structure of a theory to include boundary possible models (BPMs) as well as dynamical and kinematical content, especially for the adjudication of empirical equivalence (Wolf et al., 2023).
Limits of Formal Criteria: Duality maps or categorical equivalences cannot by themselves ensure physical equivalence. A full account necessitates the matching of interpretations, the agreement of physical predictions across all conditions (including boundaries and quantum corrections), and in some cases the explicit construction of invariants under the equivalence.

In summary, theoretical equivalence concerns the formal possibility of translating the entire framework of one theory into another, while physical equivalence requires this translation to preserve empirical predictions and physical ontology under all physically meaningful constructions and interpretations. The literature reveals that only by supplementing formal equivalence with careful interpretational and empirical considerations can one reliably adjudicate when two theories are “the same” in the physical sense.