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On the Structure of Charge in Classical and Quantum Physics

Benjamin H. Feintzeig

Published Apr 21, 2026• 0 citations
OpenAlex

Abstract

This paper takes up a suggestion that the structure associated with physical charge in quantum theories is best captured mathematically by the collection of charge representations of the quantum theory. In order to assess the evidence for this view, I investigate whether this charge structure has been preserved across theory change from classical to quantum physics. I show that one can analogously take charge structure in classical physics to be captured by the collection of charge representations of a classical theory. But one faces the potential problem that the collections of classical and quantum charge representations appear to not be isomorphic. I solve the problem by showing that if one attends carefully to more sophisticated category-theoretic tools for comparing structure, then one finds a precise and substantive sense in which charge structure is preserved from classical to quantum theories. I argue these results lead to greater understanding of inequivalent representations in quantum physics.

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This paper takes up a suggestion that the structure associated with physical charge in quantum theories is best captured mathematically by the collection of charge representations of the quantum theory. In order to assess the evidence for this view, I investigate whether this charge structure has been preserved across theory change from classical to quantum physics. I show that one can analogously take charge structure in classical physics to be captured by the collection of charge representations of a classical theory. But one faces the potential problem that the collections of classical and quantum charge representations appear to not be isomorphic. I solve the problem by showing that if one attends carefully to more sophisticated category-theoretic tools for comparing structure, then one finds a precise and substantive sense in which charge structure is preserved from classical to quantum theories. I argue these results lead to greater understanding of inequivalent representations in quantum physics.
Abstract This article presents a novel sense in which theoretical structure has been preserved across the transition from classical to quantum physics. I import mathematical tools from category theory that have been used for structural comparisons in the context of theoretical equivalence and apply these tools to new situations involving theory change. The structural preservation takes the form of a categorical equivalence between categories of models of classical and quantum physics. I situate the significance of this structural preservation in terms of prospects for theory construction in quantum physics.
It is well known that classical and quantum theories carry distinct types of representations, each type of representation corresponding to possible values of generalized charges in the classical or quantum context. This paper demonstrates a sense in the structure of these representation theories is preserved from classical to quantum physics. To show this, I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts. Specifically, I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context. These morphisms are significant because they give rise to induced representations of classical and quantum theories, respectively. I consider quantization and the classical limit as determining functors between these categories. I treat quantization via the strict deformation quantization of a Poisson algebra and the classical limit via the extension of a uniformly continuous bundle of C*-algebras. With these tools, I prove that the quantization and classical limit functors are "almost-inverse" to each other, thus establishing a categorical equivalence.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.
The availability of unitarily inequivalent representations of the canonical commutation relations constituting a quantization of a classical field theory raises questions about how to formulate and pursue quantum field theory. In a minimally technical way, I explain how these questions arise and how advocates of the Hilbert space and of the algebraic approaches to quantum theory might answer them. Where these answers differ, I sketch considerations for and against each approach, as well as considerations which might temper their apparent rivalry.
This dissertation gives a general account of the properties of unitarily inequivalent representations (UIRs) in both canonical quantum field theory and algebraic quantum field theory. A simple model is constructed and then used to show how to build a broad spectrum of UIRs including a version of Haag’s theorem. Haag and Kastler,P, two of the founding fathers of algebraic quantum field theory, argue that the problems posed by UIRs are solved by adopting a notion of equivalence that is weaker than unitary equivalence, which they refer to as physical equivalence. In the dissertation, it is shown that their notion does not provide a suitable classificatory schema. Some of the most important physical representations fail to satisfy the mathematical conditions of their notion. However, Haag and Kastler's notion has an unexpected connection with classical observables. A theorem is proven in which two representations make the same predictions with respect to all classical observables if and only if they satisfy their notion of physical equivalence. Following Haag and Kastler's lead, it was claimed by most proponents of algebraic quantum field theory that all physical content resides in a specific class of observables. It is shown in the dissertation that such claims are exaggerated and misleading. UIRs are used to elucidate the nature of quantum field theory by showing that UIRs have different expectation values for some classical observables of the system, such as temperature and chemical potential, which are not in Haag and Kastler’s specific class. It is shown how UIRs may be used to construct classical observables. To capture the physical content of quantum field theory it is shown that a much larger algebra than that of Haag and Kastler is necessary. Finally, the arguments that UIRs are incommensurable theories are shown to be flawed. The lesson of UIRs is that the mathematical structures in both canonical quantum field theory and Haag and Kastler’s version of algebraic quantum field theory are not sufficient to capture all of the physical content that UIRs represent. A suitable algebraic structure for quantum field theory is provided in the dissertation.