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arXiv:quant-ph/0008030

Are Rindler Quanta Real? Inequivalent particle concepts in quantum field theory

Rob Clifton, Hans Halvorson

Published Aug 5, 2000• 0 citations
arXivHTMLPDF

Abstract

Philosophical reflection on quantum field theory has tended to focus on how it revises our conception of what a particle is. However, there has been relatively little discussion of the threat to the "reality" of particles posed by the possibility of inequivalent quantizations of a classical field theory, i.e., inequivalent representations of the algebra of observables of the field in terms of operators on a Hilbert space. The threat is that each representation embodies its own distinctive conception of what a particle is, and how a "particle" will respond to a suitably operated detector. Our main goal is to clarify the subtle relationship between inequivalent representations of a field theory and their associated particle concepts. We also have a particular interest in the Minkowski versus Rindler quantizations of a free Boson field, because they respectively entail two radically different descriptions of the particle content of the field in the very same region of spacetime. We shall defend the idea that these representations provide complementary descriptions of the same state of the field against the claim that they embody completely incommensurable theories of the field.

DOI
10.48550/arxiv.quant-ph/0008030

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Philosophical reflection on quantum field theory has tended to focus on how it revises our conception of what a particle is. However, there has been relatively little discussion of the threat to the "reality" of particles posed by the possibility of inequivalent quantizations of a classical field theory, i.e., inequivalent representations of the algebra of observables of the field in terms of operators on a Hilbert space. The threat is that each representation embodies its own distinctive conception of what a particle is, and how a "particle" will respond to a suitably operated detector. Our main goal is to clarify the subtle relationship between inequivalent representations of a field theory and their associated particle concepts. We also have a particular interest in the Minkowski versus Rindler quantizations of a free Boson field, because they respectively entail two radically different descriptions of the particle content of the field in the very same region of spacetime. We shall defend the idea that these representations provide complementary descriptions of the same state of the field against the claim that they embody completely incommensurable theories of the field.
Philosophical reflection on quantum field theory has tended to focus on how it revises our conception of what a particle is. However, there has been relatively little discussion of the threat to the ‘reality’ of particles posed by the possibility of inequivalent quantizations of a classical field theory, i.e. inequivalent representations of the algebra of observables of the field in terms of operators on a Hilbert space. The threat is that each representation embodies its own distinctive conception of what a particle is, and how a ‘particle’ will respond to a suitably operated detector. Our main goal is to clarify the subtle relationship between inequivalent representations of a field theory and their associated particle concepts. We also have a particular interest in the Minkowski versus Rindler quantizations of a free Boson field, because they respectively entail two radically different descriptions of the particle content of the field in the very same region of spacetime. We shall defend the idea that these representations provide complementary descriptions of the same state of the field against the claim that they embody completely incommensurable theories of the field.
Despite its name, Quantum Field Theory (QFT) has been built to describe interactions between localizable particles. For this reason the actual formalism of QFT is partly based on a suitable generalization of the one already used for systems of point particles. This circumstance gives rise to a number of conceptual problems, stemming essentially from the fact that the existence within QFT of non-equivalent representations implies the existence of field theories allowing, within the same theory, different, inequivalent, descriptions of particles. This led some authors to claim that in QFT the concept itself of particle should be abandoned. In this paper we will shortly discuss the validity of this claim, as well as the possibilities, so far existing, of building alternative versions of QFT, not designed in advance to allow some kind of particle representation. We will also spend some words about the generalizations of the concept itself of particle which could grant for a better cohabitation of particles and fields within a wider formulation of QFT. The latter is indispensable if we want to extend the range of application of QFT from particle physics or simple condensed matter physics to other domains of scientific research.
I examine some problems standing in the way of a successful `field interpretation' of quantum field theory. The most popular extant proposal depends on the Hilbert space of `wavefunctionals.' But since wavefunctional space is unitarily equivalent to many-particle Fock space, two of the most powerful arguments against particle interpretations also undermine this form of field interpretation. 1. Introduction 2. Field Interpretations and Field Operators 3. The Wavefunctional Interpretation 4. Fields and Inequivalent Representations 4.1. The Rindler representation 4.2. Spontaneous symmetry breaking 4.3. Coherent representations 5. The Fate of Fields in Interacting QFT 6. Conclusions
Abstract Sagredo: Do we not see here another example of that all­pervading principle of complementarity which excludes the simultaneous applicability of concepts to the real objects of our world? Is it not so that, rather than being frustrated by this limitation of our conceptual grasp of the reality, we see in this unification of opposites the deepest and most satisfactory result of the dialectical process in our struggle for understanding? ­ Josef Jauch, Are Quanta Real? A Galilean Dialogue (1973) We are extremely grateful to John Earman and Laura Ruetsche for many stimulating discussions which provided the impetus for writing this paper, and to Aristidis Arageorgis for writing a provocative and inspiring dissertation. We would also like to thank Klaas Landsman and Rainer Verch for help with the proof of Proposition 6.
This is an opinionated survey of some interpretive puzzles in quantum field theory. The problem of inequivalent representations is sketched, including its connections with competing accounts of physical equivalence. The controversy between variant formulations of the theory, algebraic versus Lagrangian, is given a conciliatory resolution. Arguments against particles are addressed, demarcating clearly between different forms of particle interpretation. Field interpretations are then considered, including wavefunctional, spacetime state realist and Heisenberg operator realist interpretations. Ruetsche’s coalesced structure interpretation is presented and juxtaposed with an alternative, more traditional view of the theory’s laws and state space. Finally, the CPT theorem is discussed, together with its implications about the nature of spacetime.
The availability of unitarily inequivalent representations of the canonical commutation relations constituting a quantization of a classical field theory raises questions about how to formulate and pursue quantum field theory. In a minimally technical way, I explain how these questions arise and how advocates of the Hilbert space and of the algebraic approaches to quantum theory might answer them. Where these answers differ, I sketch considerations for and against each approach, as well as considerations which might temper their apparent rivalry.