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International Journal of Quantum Chemistry

Selected stable states in large quantum systems

Dorian M. Hatch

Published Jul 1, 1975• 1 citation
Publisher

Abstract

Abstract The utility of an infinite number of degrees of freedom is examined in the context of quantum systems. Such systems are known to allow inequivalent representations of the commutation relations, qualitative differentiation (spontaneously broken symmetries) arising from cooperative effects, and superselected stable states. This freedom is discussed and then used in an exactly soluble model of the measurement process.

DOI
10.1002/qua.560090407

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Abstract The utility of an infinite number of degrees of freedom is examined in the context of quantum systems. Such systems are known to allow inequivalent representations of the commutation relations, qualitative differentiation (spontaneously broken symmetries) arising from cooperative effects, and superselected stable states. This freedom is discussed and then used in an exactly soluble model of the measurement process.
Philosophical debate on the measurement problem of quantum mechanics has, for the most part, been confined to the non-relativistic version of the theory. Quantizing quantum field theory, or making quantum mechanics relativistic, yields a conceptual framework capable of dealing with the creation and annihilation of an indefinite number of particles in interaction with fields, i.e. quantum systems with an infinite number of degrees of freedom. I want to show that a solution to the standard measurement problem is available if we exploit the properties of the infinite quantum models available in this broader conceptual framework. There is a qualitative difference between quantum systems with an infinite number of degrees of freedom and quantum systems with a finite number of degrees of freedom. In the infinite case, there exist many unitarily inequivalent irreducible Hilbert space representations of the algebra of observables of the system. In the finite case, there is only one such representation.
Quantum systems with infinite degrees of freedom exhibit properties, like relaxation to equilibrium, phase-transitions and the existence of inequivalent representations of the CAR and CCR , that can satisfactorily be dealt with by means of the methods and techniques of algebraic quantum statistical mechanics [80, 81, 130]. The point of departure from standard quantum mechanics, is that in an infinite dimensional context one is usually provided with the algebraic properties of the relevant observables, but, in general, not with an a priori given representation on a Hilbert space; the latter rather depends on to the physical properties of the systems under consideration [324, 342, 343].
There has been much debate about how the semantic content of quantum theories with infinitely many degrees of freedom ought to be understood. This article focuses instead on the epistemological status of the limit of infinite degrees of freedom. Do we have strong grounds to believe that the actual world is an infinite quantum system? I consider two types of argument for thinking that we do: indispensability arguments, which appeal to the role that unitarily inequivalent Hilbert space representations play in accounting for physical phenomena like spontaneous symmetry breaking, and what I call extrapolation arguments, which support the physical significance of the limit of infinite degrees of freedom by pointing to infinite volume cosmological models. Neither of these strategies turns out to be conclusive as they stand, leaving the relevance of novel features of infinite quantum systems to real-world physics murky. I go on to discuss ways of motivating foundational work on infinite quantum systems that do not turn on the representational veracity of these features, and point out that when it comes to the question of the epistemological status of unitarily inequivalent Hilbert space representations, other sources of non-uniqueness than the limit of infinite degrees of freedom must also be considered.
Since the canonical commutation relations for a finite number of degrees of freedom have many inequivalent irreducible representations, the states of a physical system may span more than one such represen-tation. Superseparability is defined to be the case in which no meaning can be attached to a superposition of vectors belonging to inequivalent representations. In this report, which is basically nonmathematical, we trace the origins of superseparability and suggest two experiments that may establish the existence of the phenomenon. We then dis-cuss which-path experiments and interaction-free measurements in the light of superseparability, and conclude with stating some open prob-lems. A mathematical appendix provides the essential mathematical background. ∗Preliminary version 1
When quantum theory is applied to systems with an infinite number of degrees of freedom, wholly new interpretative issues arise. This is due to the fact that such theories have physically inequivalent instantiations. Here the interpretative issues of quantum field theory and of the quantum theory of systems at the thermodynamic limit are discussed. In quantum mechanics the so-called commutation operators on the basic descriptive operators fix the descriptive realm of the theory. This chapter addresses the question of how we are to understand this theory that seems to allow for distinct, incompatible ways of representing the basic features of the world.