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Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

How to take particle physics seriously: A further defence of axiomatic quantum field theory

Doreen Fraser

Published Apr 14, 2011• 80 citations
Publisher
DOI
10.1016/j.shpsb.2011.02.002

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The debate between Fraser and Wallace (2011) over the foundations of quantum field theory (QFT) has spawned increased focus on both the axiomatic and conventional formalisms. The debate has set the tone for future foundational analysis, and has forced philosophers to “pick a side”. The two are seen as competing research programs, and the major divide between the two manifests in how each handles renormalization. In this paper I argue that the terms set by the Fraser-Wallace debate are misleading. AQFT and CQFT should be viewed as complementary formalisms that start from the same physical basis. Further, the focus on cutoffs as demarcating the two approaches is also highly misleading. Though their methods differ, both axiomatic and conventional QFT seek to use the same physical principles to explain the same domain of phenomena.
Quantum field theory (QFT) is the physical framework that integrates quantum mechanics and the special theory of relativity; it is the basis of many of our best physical theories. QFT's for interacting systems have yielded extraordinarily accurate predictions. Yet, in spite of unquestionable empirical success, the treatment of interactions in QFT raises serious issues for the foundations and interpretation of the theory. This dissertation takes Haag's theorem as a starting point for investigatingthese issues. It begins with a detailed exposition and analysis of different versions ofHaag's theorem. The theorem is cast as a reductio ad absurdum of canonical QFT prior to renormalization. It is possible to adopt different strategies in response to this reductio: (1) renormalizing the canonical framework; (2) introducing a volume i.e., long-distance) cutoff into the canonical framework; or (3) abandoning another assumption common to the canonical framework and Haag's theorem, which is the approach adopted by axiomatic and constructive field theorists. Haag's theorem doesnot entail that it is impossible to formulate a mathematically well-defined Hilbert space model for an interacting system on infinite, continuous space. Furthermore, Haag's theorem does not undermine the predictions of renormalized canonical QFT; canonical QFT with cutoffs and existing mathematically rigorous models for interactions are empirically equivalent to renormalized canonical QFT. The final two chapters explore the consequences of Haag's theorem for the interpretation of QFT with interactions. I argue that no mathematically rigorous model of QFT on infinite, continuous space admits an interpretation in terms of quanta (i.e., quantumparticles). Furthermore, I contend that extant mathematically rigorous models for physically unrealistic interactions serve as a better guide to the ontology of QFT than either of the other two formulations of QFT. Consequently, according to QFT, quanta do not belong in our ontology of fundamental entities.
In his book `Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most naturally have the structure of ontological equivalence classes, we show that they cannot be seriously introduced into the quantum formalism without rendering it incoherent, as under some circumstances the formalism permits an equality between members of distinct equivalence classes. This is labeled the `equality of inequivalents problem'. Three possible reactions are discussed: First, one could deny the problem by challenging some of the assumptions that underlie its formulation; second, one could accept the problem and take it to be grounds for dismissing Heisenberg's or similar distinctions altogether; third, one could accept the problem but take its source to be not Heisenberg's distinction but the quantum formalism. A plausibility argument in support of the third reaction is given by analogy: it is shown that axiomatic probability suffers from a similar problem if one wishes to seriously distinguish between things which are possibilities and things which are not. An enriched axiomatization of probability is proposed which captures the concept of probability as a measure over possibilities and thereby overcomes the equality of inequivalents problem in axiomatic probability.