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Quantum mechanics: symmetry and interpretation

Sebastián Fortín, Olímpia Lombardi

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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 relativistic quantum field theory the notion of a local operation is regarded as basic: each open space-time region is associated with an algebra of observables representing possible measurements performed within this region. It is much more difficult to accommodate the notions of events taking place in such regions or of localized objects. But how can the notion of a local operation be basic in the theory if this same theory would not be able to represent localized measuring devices and localized events? After briefly reviewing these difficulties we discuss a strategy for eliminating the tension, namely by interpreting quantum theory in a realist way. To implement this strategy we use the ideas of the modal interpretation of quantum mechanics. We then consider the question of whether the resulting scheme can be made Lorentz invariant.
The best mathematical arguments against a realistic interpretation of quantum mechanics - that gives definite but partially unknown values to all observables - are analysed and shown to be based on reasoning that is not compelling. This opens the door for an interpretation that, while respecting the indeterministic nature of quantum mechanics, allows to speak of definite values for all observables at any time that are, however, only partially measurable. The analysis also suggests new ways to test the foundations of quantum theory.
The quantum-mechanical formalism developed in a previous article and based on the use of a rigged Hilbert space Φ⊂H⊂Φ′ is here enlarged by taking into account the symmetry properties of the system. First, the compatibility of a particular symmetry with this structure is obtained by requiring Φ to be invariant under the corresponding representation U of the symmetry group in H. The symmetry is then realized by the restriction of U to Φ and its contragradient representation Ǔ in Φ′. This double manifestation of the symmetry is related to the so-called active and passive points of view commonly used for interpreting symmetry operations. Next, a general procedure is given for constructing a suitable space Φ out of the labeled observables of the system and the representation U describing its symmetry properties. This general method is then applied to the case where U is a semidirect product G=T[squared times]Δ, with T Abelian. Finally, the examples of the Euclidean, the Galilei, and the Poincaré groups are briefly studied.
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.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.