James Fraser

This paper examines the development of causal perturbation theory, a reformulation of perturbative quantum theory (QFT) starting from a causality condition rather than a time-evolution equation. We situate this program alongside other causality-based reformulations of relativistic quantum theory which flourished in the post-war period, contrasting it in particular with axiomatic QFT. Whereas the axiomatic QFT tradition tried to move beyond the perturbative expansion, causal perturbation theory can be thought of as a foundational investigation of this approximation method itself. Our reconstruction of this forgotten research program offers new perspectives on contemporary debates about relativistic causality conditions and the problem of ultraviolet divergences.

The presence of ultraviolet divergences in quantum field theory (QFT) perturbation series has long been a source of foundational worry about the theory. In the 1930s, the appearance of divergent integrals in the perturbative expansion led many physicists to conclude that QFT was an inconsistent framework. The invention of renormalization in the 1940s allowed finite predictions to be extracted for the perturbative expansion. However, confusion remained about the physical significance of the ultraviolet divergences, with many still taking them to signal the breakdown of QFT at very high energies. This talk examines a mathematical treatment of the ultraviolet divergences problem which developed in the 1950s-1970s but remains surprisingly little known. On this account ultraviolet divergences are seen as arising from the multiplication of distributions in the series coefficients. The notion of a distribution generalises the ordinary notion of a function to include more singular objects like the Dirac delta “function”. Crucially, unlike ordinary functions, the multiplication of distributions is not generally well-defined. Mathematical physicists showed that it is possible to treat the products of singular distributions that appear in QFT perturbation theory without manipulating divergent integrals. Thus, rather than indicating a physical problem, on this reconstruction the infinities that appear in conventional perturbative QFT are regarded as artefacts of an inappropriate mathematical formalization of the problem. They do not signal that QFT breaks down at very high energies.

The perturbative approach to quantum field theory has long been viewed with suspicion by philosophers of science. This article offers a diagnosis of its conceptual problems. Drawing on Norton’s discussion of the notion of approximation I argue that perturbative QFT ought to be understood as producing approximations without specifying an underlying QFT model. This analysis leads to a reassessment of common worries about perturbative QFT. What ends up being the key issue with the approach on this picture is not mathematical rigour, or the threat of inconsistency, but the need for a physical explanation of its empirical success. 1Three Worries about Perturbative Quantum Field Theory2The Perturbative Formalism 2.1Expanding the S-matrix2.2Perturbative renormalization3Approximations and Models4Perturbative Quantum Field Theory Produces Approximations5The Real Problem

Physicists use different theories to describe the world on different scales. In particular, they use the standard model of particle physics at very high energies, but move to various effective field theories, such as quantum electrodynamics, when modelling lower energy scattering processes. One way to explain this methodological fact is pragmatic in spirit. According to this view, physicists move to an effective field theory at lower energies in order to extract predictions and qualitative understanding which would be difficult or impossible to extract directly from a more fundamental theory. By contrast, another way of accounting for the methodological data is metaphysical in spirit. On this view, the reason that physicists use different theories at different scales is that the world actually exhibits different nomic structure on different scales. These two positions have recently been thrown into sharp relief in a debate between Woodward (2016) and Batterman (2021), with Woodward taking the pragmatist line and Batterman the metaphysical line. One difficulty with the metaphysical answer is that its content is unclear: in what sense exactly could an effective field theory provide a better representation of the nomic structure of the world in its domain of applicability than our world’s fundamental theory? This talk attempts to provide an answer to this question by developing a simple metaphysical model. I start by assuming a set of possible worlds which are nomically possible according to some fundamental theory (I suggest that one is free to adopt either a Humean or Non-Humean view of these fundamental nomic necessities). I then introduce an equivalence relation between possible worlds: two worlds are scale-E equivalent if they assign the same values to physical quantities below energy scale E (up to some finite level of precision P). The key question now is whether this relation induces equivalence classes on the set of nomically possible worlds. I argue that in the case of current quantum field theories this does indeed occur, as evidenced by the possibility of “integrating out” high energy degrees of freedom in the renormalisation group framework. The formation of equivalence classes in the fundamental theory's possibility space indicates that some of its high energy features do not make a difference to its low energy dynamical behaviour. This suggests the following reading of the claim that an effective field theory provides a better representation of the nomic structure of the world in its domain of applicability: the solutions of the effective field theory stand in one-to-one correspondence with equivalence classes induced by the scale-E equivalence relation, whereas the fundamental theory overcounts the physical possibilities which are relevant at scale E. In this sense, the effective field theory provides a more natural representation of low energy physics than the fundamental theory. I conclude by explaining how this metaphysical model offers a reduction compatible understanding of multiple realisability, once again pointing to renormalisation group results in quantum field theory to support this.

The orthodox characterization of spontaneous symmetry breaking in statistical mechanics appeals to novel properties of systems with infinite degrees of freedom, namely, the existence of multiple equilibrium states. This raises the same puzzles about the status of the thermodynamic limit fueling recent debates about phase transitions. I argue that there are prospects of explaining the success of the standard approach to SSB in terms of the properties of large finite systems. Consequently, despite initial appearances, the need to account for SSB phenomena does not offer decisive support to claims about the explanatory and representational indispensability of the thermodynamic limit.

In this article, we offer a detailed study of two important episodes in the early history of high-energy physics, namely the development of the Chew and the Nambu-Jona-Lasinio models. Our study reveals that both models resulted from the combination of an old Hamiltonian, which had been introduced by earlier researchers, and two new approximation methods developed by Chew and by Nambu and Jona-Lasinio. These new approximation methods, furthermore, were the key component behind the models’ success. We take this historical investigation to support two philosophical theses about the manner in which scientific modelling operates in high-energy physics. Both of these theses run counter to a view that is commonly accepted among philosophers of science: the view that all approximations can be embedded within an equivalent idealized system, and that whatever role the former might play in scientific modelling is therefore parasitic on the much more substantial work performed by the latter. Our first thesis, which we call “Distinctness,” states that approximation methods constitute an independent category of theoretical output from idealized systems. We thus believe that approximations and idealized systems constitute two independent types of objects, both of which are essential to the practice of modelling. Our second, more radical thesis is called “Content Determination.” Our claim here is that approximation methods can in fact be essential to assigning determinate physical content to the idealized systems with which they jointly operate. As we show, this is due to the fact that quantum field theory allows for a very thin characterization of idealized systems only, making the use of approximations necessary to supply additional content. We conclude the paper with a few reflections about the manner in which our two theses can be used to articulate David Kaiser’s views on the “vanishing of scientific theory” in physics after WWII.