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The modal interpretation of algebraic quantum field theory

Rob Clifton

Published Sep 23, 2009• 0 citations
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Abstract

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

DOI
10.1093/oso/9780199270156.003.0005

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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
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.
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.
This book is about how to understand quantum mechanics by means of a modal interpretation. Modal interpretations provide a general framework within which quantum mechanics can be considered as a theory that describes reality in terms of physical systems possessing definite properties. Quantum mechanics is standardly understood to be a theory about probabilities with which measurements have outcomes. Modal interpretations are relatively new attempts to present quantum mechanics as a theory which, like other physical theories, describes an observer-independent reality. In this book, Pieter Vermaas summarises the results of this work. The book will be of great value to undergraduates, graduate students and researchers in philosophy of science, and physics departments with an interest in learning about modal interpretations of quantum mechanics.
Earman and Ruetsche ([2005]) have cast their gaze upon existing no-go theorems for relativistic modal interpretations, and have found them inconclusive. They suggest that it would be more fruitful to investigate modal interpretations proposed for ‘really relativistic theories,’ that is, algebraic relativistic quantum field theories. They investigate the proposal of (Clifton [2000]), and extend Clifton's result that, for a host of states, his proposal yields no definite observables other than multiples of the identity. This leads Earman and Ruetsche to a suspicion that troubles for modal interpretations of such relativistic theories ‘are due less to the Poincaré invariance of relativistic QFT vs. the Galilean invariance of ordinary nonrelativistic QM than to the infinite number of degrees of freedom of former vs. the finite number of degrees of freedom of the latter’ (pp. 577–8). I am skeptical of this suggestion. Though there are troubles for modal interpretations of relativistic quantum field theory that are due to its being a field theory—that is, due to infinitude of the degrees of freedom—they are not the only troubles faced by modal interpretations of quantum theories set in relativistic spacetime; there are also troubles traceable to relativistic causal structure. 1. Introduction 2. Relativity and Causal Structure 3. The Theorems Purified 4. Evolving States in a Relativistic Context
Modal interpretations are non-collapse interpretations, where the quantum state of a system describes its possible properties rather than the properties that it actually possesses. Among them, the atomic modal interpretation (AMI) assumes the existence of a special set of disjoint systems that fixes the preferred factorization of the Hilbert space. The aim of this paper is to analyze the relationship between the AMI and our recently presented modal-hamiltonian interpretation (MHI), by showing that the MHI can be viewed as a kind of “atomic” interpretation in two different senses. On the one hand, the MHI provides a precise criterion for the preferred factorization of the Hilbert space into factors representing elemental systems. On the other hand, the MHI identifies the atomic systems that represent elemental particles on the basis of the Galilei group. Finally, we will show that the MHI also introduces a decomposition of the Hilbert space of any elemental system, which determines with precision what observables acquire definite actual values.