[Paper Review] Theoretical Equivalence in Physics

James Owen Weatherall|arXiv (Cornell University)|Oct 18, 2018

Philosophy and History of ScienceArts and Humanities79 references3 citations

TL;DR

This paper examines theoretical equivalence in physics by analyzing multiple criteria—empirical equivalence, interpretational equivalence, definitional equivalence, categorical equivalence, and duality—arguing that formal structures like category theory and Morita equivalence offer robust frameworks for assessing when two physical theories are truly equivalent, while cautioning that duality does not always preserve physical interpretation.

ABSTRACT

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including (generalized) definitional equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality.

Motivation & Objective

• To clarify the philosophical and formal criteria for determining when two physical theories are theoretically equivalent.
• To examine the role of empirical equivalence as a necessary but not always sufficient condition for theoretical equivalence.
• To investigate how formal frameworks like definitional equivalence, Morita equivalence, and categorical equivalence capture theoretical content across different physical theories.
• To analyze the relationship between duality in high-energy physics and philosophical notions of theoretical equivalence.
• To assess whether dual theories preserve physical interpretation, challenging the assumption that duality implies full theoretical equivalence.

Proposed method

• Surveying philosophical literature on theoretical equivalence, focusing on empirical, interpretational, and formal criteria.
• Applying formal tools such as generalized definitional equivalence and categorical equivalence to compare physical theories.
• Using category theory to assess whether different formulations of a theory yield equivalent model categories.
• Analyzing duality in physics (e.g., T-duality, S-duality) through the lens of isomorphisms between different formulations of a single theory.
• Evaluating whether dual theories preserve physical meaning under translation, using De Haro’s schema of bare theories and dual realizations.
• Contrasting philosophical notions of equivalence with physical examples, especially in quantum mechanics and general relativity.

Experimental results

Research questions

• RQ1 What conditions must be met for two physical theories to be considered theoretically equivalent?
• RQ2 To what extent is empirical equivalence sufficient for theoretical equivalence, and when might it fail?
• RQ3 How do formal equivalence relations like categorical equivalence or Morita equivalence capture the theoretical content of physical theories?
• RQ4 Do dualities in high-energy physics represent genuine theoretical equivalences, or do they fail to preserve physical interpretation?
• RQ5 In what ways can a duality between theories be understood as an isomorphism between different formulations of a single bare theory?

Key findings

• Empirical equivalence is necessary but not sufficient for theoretical equivalence, as theories may agree on predictions yet differ in interpretation or structure.
• Generalized definitional equivalence and categorical equivalence provide formal criteria that better capture theoretical content than empirical agreement alone.
• Morita equivalence offers a robust framework for comparing theories with different mathematical formulations but is distinct from categorical equivalence in its scope and implications.
• Duality does not always imply theoretical equivalence, as dual formulations may assign different physical meanings to the same formal structure.
• De Haro’s schema reveals that dual theories can be isomorphic at the formal level but fail to preserve physical interpretation, challenging the idea that duality implies full equivalence.
• Butterfield’s critique suggests that some dualities may not reflect equivalent physical theories if they do not preserve physical meaning under translation, even when formally isomorphic.