The fate of ‘particles’ in quantum field theories with interactions

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Abstract

Most philosophical discussion of the particle concept that is afforded by quantum field theory has focused on free systems. This paper is devoted to a systematic investigation of whether the particle concept for free systems can be extended to interacting systems. The possible methods of accomplishing this are considered and all are found unsatisfactory. Therefore, an interacting system cannot be interpreted in terms of particles. As a consequence, quantum field theory does not support the inclusion of particles in our ontology. In contrast to much of the recent discussion on the particle concept derived from quantum field theory, this argument does not rely on the assumption that a particulate entity be localizable.

Introduction

Quantum field theory (QFT) is the basis of the branch of physics known as ‘particle physics.’ However, the philosophical question of whether quantum field theories genuinely describe particles is not straightforward to answer. Since QFTs are formulated in terms of fields (i.e., mathematical expressions that associate quantities with points of spacetime), the issue is whether the formalism can be interpreted in terms of a particle notion. What is at stake is whether QFT, one of our current best physical theories, supports the inclusion of particles in our ontology. This paper advances an argument that, because systems which interact cannot be given a particle interpretation, QFT does not describe particles.

Even proponents of a particle interpretation of QFT acknowledge that the particle concept inherent in a QFT would differ from the classical particle concept in many ways. To distinguish the QFT concept from the classical one, the former has been dubbed the ‘quanta’ concept (e.g., Teller, 1995, p. 29). Redhead and Teller (1992) argue that one way in which quanta differ from classical particles is that quanta are not capable of bearing labels.1 That is, they lack a property that is variously termed ‘haecceity,’ ‘primitive thisness,’ or ‘transcendental individuality.’ However, Teller argues that the quanta notion should still be considered a particlelike notion because quanta are aggregable (Teller, 1995, p. 30). There are, for example, states in which we definitely have two quanta and states in which we definitely have three quanta, and these can be combined to yield a state in which we definitely have five quanta. Quanta are also particlelike insofar as they possess the same energies as classical, relativistic, non-interacting particles.

This minimal notion of quanta as entities satisfying a countability condition and a relativistic energy condition will be employed in the following investigation. These quanta may significantly differ from classical particles in other respects. For example, the question of whether there is an appropriate sense in which quanta are localized has been the subject of recent debate (see Fleming, 2001; Halvorson & Clifton, 2002; Malament, 1996). However, for our purposes, this debate can be set aside. The particlelikeness of the quanta notion will not challenged; instead, the arguments presented below aim to show that the domain of application of the quanta concept is so strictly limited that quanta cannot be admitted into our ontology.

The quanta interpretation of QFT is based on special properties of the mathematical representation for free systems in QFT. Free fields describe the world in the absence of interactions. But in the real world there are always interactions. This raises a crucial question: Can the quanta interpretation be extended to interacting systems? The analysis presented here aims to supply a comprehensive answer to this question. Ultimately, the answer is ‘no;’ an interacting system cannot be described in terms of quanta. This inquiry is in the same spirit as recent discussions (which are restricted to free systems) of whether a unique quanta notion is available for accelerating observers or in more general spacetime settings (i.e., non-stationary spacetimes) (Arageorgis et al., 2002, Arageorgis et al., 2003; Clifton & Halvorson, 2001). The present investigation adopts the opposite approach: the restriction to free systems will be dropped, and the restriction to inertial observers on flat Minkowski spacetime will be retained. The commonality is that the interpretive conclusions rest on the employment of unitarily inequivalent representations of the canonical commutation relations. In this case, the fact that the representations for free and interacting systems are necessarily unitarily inequivalent is invoked in the first stage of the argument. However, the structure of the argument diverges from the above-mentioned discussions after this first stage; further work is required to establish that the unitarily inequivalent representation for the interacting field cannot possess the relevant formal properties.

After a brief review of the Fock representation for a free system and the standard argument that it supports a quanta interpretation, three methods for obtaining a quanta interpretation for an interacting system will be evaluated. The first method is simply to use the Fock representation for a free system to represent an interacting system. Since this method proves unsuccessful, it is necessary to generalize the definition of Fock representation so that it is applicable to interacting systems. In order to distinguish these definitions, I will reserve the term ‘Fock representation’ for free systems and refer to the results of attempts to formulate analogous representations for interacting systems as ‘ΦOK representations.’2 In principle, there are two methods for extending the definition of a Fock representation which are allied with the two approaches to defining a Hilbert space representation in QFT: the ‘constructive’ method of applying to an interacting field the same quantization procedure that generates a Fock representation from a classical free field and the ‘axiomatic’ method of specifying a Hilbert space representation by stipulating formal conditions. The former method will be investigated in Section 4 and the latter in Section 5. Following the failure of both methods, a final, last-ditch attempt to retain a quanta interpretation for interacting systems will be critiqued in Section 6. The implications of the conclusion that interacting systems cannot be described in terms of quanta for metaphysics and for the foundations of QFT will be assessed in Section 7.

Section snippets

The Fock representation for a free field

Every introductory QFT textbook contains a discussion of how to construct a Fock space representation of the equal-time canonical commutation relations (ETCCRs) for a free field. The construction proceeds by effecting a positive–negative Fourier decomposition of a classical free field and then promoting the coefficients to operators. The details of this construction will be discussed in Section 4. In this section, the properties of the final product of the construction—a Fock representation for

Method #1: using the Fock representation for a free field

The simplest strategy for obtaining a quanta interpretation for an interacting field would be to use the Fock representation for a free system to represent a given interacting system, and then to try to extract a quanta interpretation. That is, the hope is that the Hilbert space spanned by n-particle states for the free field contains states that can be interpreted as containing n quanta in the presence of the interaction. As we shall see, this simple strategy fails because there is no state in

Method #2: application of the construction that generates a Fock representation

A Fock representation for a free system is generated from the classical free field by a quantization procedure. One approach to generalizing the definition of a Fock representation to interacting fields is to apply the same mathematical construction to a classical interacting field. The mathematical construction of a Fock representation proceeds by Fourier decomposing the free field satisfying the classical field equation into positive and negative frequency parts and then promoting the

Method #3: stipulation of formal conditions

The method for obtaining a ΦOK1 representation pursued in the previous section was to quantize a given classical interacting field in the same manner in which a given classical free field is quantized to produce a Fock representation. Following the failure of this method, a different method for extending the definition of ‘Fock representation’ to interactions will be pursued in this section. The strategy is to arrive at a Hilbert space representation of the ETCCRs by stipulating formal

Scattering theory does not support a quanta interpretation

To recapitulate the argument thus far, we set out to find a mathematical representation for an interacting system that sustains a quanta interpretation in the same manner that a Fock representation furnishes a quanta interpretation for a free system. All three methods for accomplishing this have proven unsuccessful. Generalizations of two of the strategies cannot be ruled out entirely, but success seems exceedingly unlikely. Since in QFT there is no known alternative for establishing that a

Conclusion

A Fock representation sustains a quanta interpretation for a free field. The goal of this paper was to determine whether an interacting field possesses a mathematical representation that sustains a quanta interpretation in the same manner. The simplest solution would have been to use the Fock representation for a free field to represent an interacting field. This option was ruled out by Haag's theorem. It then became necessary to find another Hilbert space representation for an interacting

Acknowledgments

For helpful comments, thanks to an anonymous referee, Gordon Belot, Tony Duncan, Nick Huggett, Wayne Myrvold, Laura Ruetsche, Andrew Wayne, and especially John Earman. I would also like to thank the Social Sciences and Humanities Research Council of Canada for financial support in the form of a Doctoral Fellowship.

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