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arXiv:1602.07160

The role of symmetry in the interpretation of quantum mechanics

Olímpia Lombardi, Sebastián Fortín

Published Feb 23, 2016• 6 citations
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Abstract

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.

DOI
10.48550/arxiv.1602.07160

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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.
In this paper it will be argued that any realist interpretation of quantum mechanics intending to preserve the objectivity of the set of the definite-valued observables should require such a set to be invariant under the symmetry group of the theory. In particular, it will be shown that the natural way to reach this goal is to appeal to the Casimir operators of the Galilean group. Additionally, this idea will be generalized in two ways: (i) by selecting the definite-valued observables of a system in quantum field theory as those represented by the Casimir operators of the Poincaré group, and (ii) by extending the strategy to gauge symmetries.
Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.
Let me begin by briefly explaining the concept of ‘interpretation’ relevant to this discussion. Certain physical theories postulate abstract structural constraints which events are held to satisfy. Such theories are termed ‘principle theories’. Interpretations of principle theories aim to explain their relation to the theories they replace. Interpretations are therefore concerned with the nature of the transitions between theories. Theories of space-time structure provide the most accessible illustration of principle theories. For example, Newtonian mechanics in the absence of gravitation represents the 4-dimensional geometry of space-time by the inhomogeneous Galilean group, which acts transitively in the class of free motions, i.e. the inhomogeneous Galilean group is the symmetry group of the free motions: it is a subgroup of the symmetry group of every mechanical system, and the largest such subgroup. Einstein’s special principle of relativity is the hypothesis that the symmetry group of the free motions is the Poincaré group.
Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.
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.