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Philosophy of Science

Relativistic Invariance and Modal Interpretations

John Earman, Laura Ruetsche

Published Oct 1, 2005• 50 citations
Publisher

Abstract

A number of arguments have been given to show that the modal interpretation of ordinary nonrelativistic quantum mechanics cannot be consistently extended to the relativistic setting. We find these arguments inconclusive. However, there is a prima facie reason to think that a tension exists between the modal interpretation and relativistic invariance; namely, the best candidate for a modal interpretation adapted to relativistic quantum field theory, a prescription due to Rob Clifton (2000), comes out trivial when applied to a number of systems of physical interest. However, it is far from clear whether this difficulty for the modal interpretation is traceable to relativistic invariance per se or to the infinite number of degrees of freedom involved. In any case, the proponents of the modal interpretation have work to do.

DOI
10.1086/505448

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A number of arguments have been given to show that the modal interpretation of ordinary nonrelativistic quantum mechanics cannot be consistently extended to the relativistic setting. We find these arguments inconclusive. However, there is a prima facie reason to think that a tension exists between the modal interpretation and relativistic invariance; namely, the best candidate for a modal interpretation adapted to relativistic quantum field theory, a prescription due to Rob Clifton (2000), comes out trivial when applied to a number of systems of physical interest. However, it is far from clear whether this difficulty for the modal interpretation is traceable to relativistic invariance per se or to the infinite number of degrees of freedom involved. In any case, the proponents of the modal interpretation have work to do.
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Nonrelativistic quantum mechanics is conceptually and mathematically simpler, and does, after all, work very well in a low-energy regime. Any solution of the measurement problem, or any account of the ontology of quantum theory, must yield sensible results in this regime, and so nonrelativistic quantum mechanics can serve as a testing ground for the viability of interpretations. On any approach, empirical adequacy requires recovery of at least an approximation to the quantum probabilities for outcomes of experiments, insofar as these have been verified by experiment. A quantum-mechanical theory is what results when one subjects a classical theory of a system with finitely many degrees of freedom to quantization. The task of extending additional beables theories to the context of quantum field theories remains a work in progress. A dynamical collapse theory modifies the evolution of the quantum state. Standard quantum theory predicts correlations between outcomes of spacelike separated experiments that violate Bell inequalities.
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Abstract The modal interpretation is actually a family of interpretations sharing the feature that a system’s density operator constrains the possibilities for assigning definite values to its observables (Healey 1989, Dieks & Vermaas 1998). We start with a brief review of the basic idea behind these interpretations from an algebraic point of view (Zimba & Clifton 1998, Halvorson & Clifton 1999). Consider a ‘universe’ comprised of a quantum system U, with finitely many degrees of freedom, represented by a Hilbert space Hu = ®i Hi. At any given time, U will occupy a pure vector state x E Hu that determines a reduced density operator Ds on the Hilbert space Hi of any subsystem S U. Let B(Hs) de­