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HHU Journals

The Methodological Character of Symmetry Principles

Otávio Bueno

Published Jun 28, 2021• 1 citation
Publisher

Abstract

In this paper, I argue that symmetry principles in physics (in particular, in quantum mechanics) have a methodological character, rather than an ontological or an epistemological one. First, I provide a framework to address three related issues regarding the notion of symmetry: (i) how the notion can be characterized; (ii) one way of discussing the nature of symmetry principles, and (iii) a tentative account of some types of symmetry in physics. To illustrate how the framework functions, I then consider the case of the early formulation of quantum mechanics, examining the different roles played by symmetry in this context. Finally, I raise difficulties for ontological and purely epistemological interpretations of symmetry principles, and offer a methodological alternative.

DOI
10.24338/abs-2006.321

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In this paper, I argue that symmetry principles in physics (in particular, in quantum mechanics) have a methodological character, rather than an ontological or an epistemological one. First, I provide a framework to address three related issues regarding the notion of symmetry: (i) how the notion can be characterized; (ii) one way of discussing the nature of symmetry principles, and (iii) a tentative account of some types of symmetry in physics. To illustrate how the framework functions, I then consider the case of the early formulation of quantum mechanics, examining the different roles played by symmetry in this context. Finally, I raise difficulties for ontological and purely epistemological interpretations of symmetry principles, and offer a methodological alternative.
In the non-reflexive formulation of quantum physics, attention is given to a metaphysical view of quantum objects (q-objects) as non-individuals, characterised as entities to which the standard notion of identity lacks sense. The reference to the nature of q-objects is usually omitted in other interpretations and the most we can find is some vague reference to “to each [quantum] system we associate a Hilbert space”, and nothing more is said about them; we need to infer from the theory their characteristics, and it gives us several alternatives. One of them, to be justified here in broad aspects, is the non-individual view. It is important to enlighten that we are not trying to just find a way to deal with quantum entities but to defend a possible ontology of non-individuals, which can be applied also to ‘particles’ arising from quantum fields. Our account is based on two assumptions: (i) the ‘classical theory of identity’ does not apply to q-objects, and (ii) their non-individuality must be attributed from the start, that is, as a primitive notion and not ‘made a posteriori’ say by assuming symmetry principles. The underlying logic is called non-reflexive because the reflexive law of identity ∀x(x=x)\forall{x{({x = x})}}∀x(x=x) is supposed not to hold for q-objects. Since all the details demand a lot of space, here we only sketch and justify our view. The references provide a more detailed account.
Many scientific theories purport to describe empirically inaccessible aspects of the world. The agenda of foundational physics is more ambitious still: to characterize the world at the level of 'primitive ontology'. In this pursuit it often posits new and peculiar physical features, a proclivity aided by the abstract, mathematical way in which foundational theories are framed. But how do we decipher their physical content, and in particular the accounts of primitive ontology they offer, and why think that content is true? This dissertation explores the relationship between these questions -- between the problem of ontological interpretation in foundational physics and the nature of scientific realism. I do not argue for a unified account. Rather, I examine three contexts in which their interactions are particularly interesting and philosophically significant. First, I consider the role scientific methodology plays in a prominent naturalistic defense of scientific realism: the no-miracles argument. The most sophisticated form of this argument rests on the broad claim that successful scientific methodology is irredeemably theory dependent. I examine the nature of this theory dependence within the context of Maxwell's development of electromagnetism, and argue that the presence of competing ontological interpretations undermines the realist's attempt to draw sweeping epistemic conclusions from methodological success. I next consider the concept of theoretical reformulation and how it applies to the claim that Lagrangian dynamics is a reformulation of (part of) Newtonian dynamics. How should this claim be understood in light of the fact that the world is non-classical? What can this tell us about the concept of theory reformulation itself, which plays a central role in arguments against scientific realism? I provide an analysis of theory reformulation in terms of counterfactual interpretative judgments, and then cast doubt on its justification in the classical case by developing a non-Newtonian interpretation of Lagrangian dynamics. Finally, I consider whether competing ontological interpretations raise underdetermination problems for realism about foundational physics. After first re-formulating the underdetermination argument to avoid recent objections that it fails to pose any distinctive threat, I suggest a formulation of realism that vitiates the underdetermination threats posed by competing ontological interpretations.
Abstract Symmetry principles play an important part in theoretical physics, both as a means of understanding conservation laws and as computational devices providing quick and elegant solutions to problems that would otherwise be tedious to solve. In quantum theory, their role is especially significant since they not only clarify basic concepts and simplify the formal development of the subject, but they also provide detailed information about physical systems for which no explicit dynamical models exist. However, the methods used to investigate symmetry in quantum systems are often based on abstract group-theoretical techniques, which bypass the canonical formalism. They include no general rules for going over to the classical limit and for directly relating the symmetries of the quantum system to those of a corresponding classical system.
We consider symmetry as a foundational concept in quantum mechanics and rewrite quantum mechanics and measurement axioms in this description. We argue that issues related to measurements and physical reality of states can be better understood in this view. In particular, the abstract concept of symmetry provides a basis-independent definition for observables. Moreover, we show that the apparent projection/collapse of the state as the final step of measurement or decoherence is the result of breaking of symmetries. This phenomenon is comparable with a phase transition by spontaneous symmetry breaking, and makes the process of decoherence and classicality a natural fate of complex systems consisting of many interacting subsystems. Additionally, we demonstrate that the property of state space as a vector space representing symmetries is more fundamental than being an abstract Hilbert space, and its L2L2L2 integrability can be obtained from the imposed condition of being a representation of a symmetry group and general properties of probability distributions.
Standard quantum mechanics poses serious challenges to some principles deeply entrenched in traditional metaphysics. In particular, the theory defies the ontological category of individual object through three of its features: contextuality, non-separability, and indistinguishability. The difficulties have prompted the development of non-traditional ontological frameworks intended to account for the quantum domain. Some approaches advocate replacing the classical conception of a realm of objects endowed with properties by an ontology that either dispenses with the category of object altogether or treats it as a derivative notion. The Modal-Hamiltonian Interpretation (MHI) of quantum mechanics and Ontic Structural Realism (OSR) exemplify this position. MHI proposes an ontology in which quantum systems are non-objectual bundles of properties linked by definite physical relations. The OSR defends the ontological primacy of structures, within which objects are construed as nodes of relations. The aim of this article is to compare these two ontological approaches, highlighting both their points of convergence and their divergences, particularly in light of the ontological challenges posed by quantum mechanics. With this aim, the article is structured as follows. Section 2 outlines the ontology of MHI, introducing the distinction between type-properties and case-properties, characterizing quantum systems as non-objectual bundles, and analyzing the role of symmetry groups in providing physical content to the theory and governing actualization. Section 3 turns to OSR, examining its motivations, its distinction between physical and mathematical structures, and the modal character of structures. Section 4 offers a systematic comparison of the two approaches, considering their respective motivations and scope, their shared commitment to an object-free ontology, the role played by symmetry groups within each framework, and their contrasting interpretations of modality. Finally, Section 5 presents some concluding remarks.
Despite the impressive amount of literature on the foundations of quantum mechanics, the relevance of symmetry in interpretation is not properly acknowledged. In fact, although it is usually said that quantum mechanics is invariant under the Galilean group, the invariance is usually not discussed in the case of the interpretation of the theory. This situation is particularly pressing for realist positions, which want to know how reality would be if quantum mechanics were true. From a realist perspective, it seems reasonable to require that an interpretation of quantum mechanics, to be admissible, preserves the same symmetries of the theory. In this paper we will take a realist standpoint in order to study what physical constraints the Galilean group imposes on interpretation, and we will extrapolate the conclusions drawn for quantum mechanics to the case of quantum field theory.