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Perturbative expansions and the foundations of quantum field theory

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Abstract

Perturbative expansions have played a peculiarly central role in quantum field theory, not only in extracting empirical predictions but also in investigations of the theory’s mathematical and conceptual foundations. This paper brings the special status of QFT perturbative expansions into focus by tracing the history of mathematical physics work on perturbative QFT and situating a contemporary approach, perturbative algebraic QFT, within this historical context. Highlighting the role that perturbative expansions have played in foundational investigations helps to clarify the relationships between the formulations of QFT developed in mathematical physics and high-energy phenomenology.

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Notes

  1. The reception of the ultraviolet divergences problem in the 30 s and 40 s is a complex matter which is treated only very briefly here. See Rueger [55] and Schweber [59], chapters 1 and 2, for more detailed accounts.

  2. Rudolf Haag cites the 1957 conference on the “On Mathematical Aspects Of Quantum Field Theory” in Lille as the “baptism of mathematical physics as a discipline distinguished from theoretical physics” [37, 274]. It is interesting to note that Julian Schwinger attended this conference, however, evidencing how porous the boundaries between these sub-fields were in this period.

  3. To give some more details of Dyson’s programme in this period, the goal was to use the renormalised expansion to construct Heisenberg field operators, rather than just the asymptotic S-matrix [23]. In order to achieve this he introduced the so-called intermediate representation, a representation of the time evolution of the theory which separated the high and low momentum field modes, with the hope being that the high momentum part of the theory could be constructed via a convergent renormalised series expansion [24, 27]. Interestingly, while Dyson’s intermediate representation was not taken up in the 50 s, Kenneth Wilson mentions it as an inspiration for his approach to the renormalisation group [76].

  4. When it came to models of the strong nuclear interaction, the expansion parameter was expected to be much larger than QED’s, and consequently, the fact that the expansion was likely only asymptotic had a more direct phenomenological relevance—see [17] for some discussion. The large-order behaviour of the series would also become important once more with the rise of perturbative quantum chromodynamics (QCD) in the 70 s and 80 s, leading to a new wave of results on the large-order behaviour of QFT perturbative expansions.

  5. The impression that the perturbative expansion was not powerful enough to resolve the foundational problems of QFT was likely further bolstered by new putatively non-perturbative results which emerged in the 1950 s. Källén and Landau put forward new arguments for thinking that QED broke down in the ultraviolet regime which seemed to be non-perturbative in nature and potentially more severe than the [46, 48]. These developments, and the broader context of debates about the inconsistency of QFT in the 1950 s, are discussed in Blum [4].

  6. As is evidenced by the curious 1961 volume “Dispersion Relations and the Abstract Approach to Field Theory” which gathered together early axiomatic QFT papers with more phenomenologically orientated works by Mandelstram, Pomeranchuk and others as if they were all contributions to a common programme [47]. Fleshing out these connections is a thus far unexplored historical problem.

  7. The fact that infinite degrees of freedom led to inequivalent Hilbert space quantisations was noted already by Von Neumann [73]. Besides Friedrichs, another early proponent of the significance of this fact was Irving Segal, who advocated an algebraic approach to quantum theory long before Haag and Kastler; see [60, 61].

  8. The idea that a causality condition of some kind needed to be added to Poincare covariance was prevalent across a wide range of theoretical approaches to relativistic quantum theory in the 50 s and 60 s, as evidenced by Bogoliubov’s causality condition discussed in Sect. 2.4. This broader context is discussed in Blum and Fraser [5].

  9. Later on, the Osterwalder–Schrader axioms for QFT on Euclidean space would become the primary focus of constructive field theory, with analytic continuation properties being used to establish the connection with Minkowski spacetime. See [44] for a discussion of the development of constructive field theory which covers the rise of Euclidean methods.

  10. Thanks to Kseniia Mohelsky for translating this Russian language source.

  11. For a detailed reconstruction of this problem, see [5].

  12. A topological *-algebra is an algebra equipped with an involution (a linear operation *, with the properties that \(A^{**}=A\), \((AB)^*=B^*A^*\) and \(1^*=1\)) such that both the involution and the algebra operations are continuous in the given topology.

  13. Note that the historical origins of pAQFT have not been systematically studied and these brief comments about the role of QFT on curved spacetimes should be read as preliminaries to such an investigation. Interestingly, the idea of connecting the causal perturbation theory and algebraic approach to QFT was suggested much earlier by Il’in and Slavnov [42]. While this does not seem to have been taken up at the time, it suggests that the pieces were in place to connect causal perturbation theory and AQFT much earlier. Whether the focus on QFT on curved spacetimes at the turn of the century was merely a catalyst or played a more fundamental role in the development of pAQFT is a matter for further study.

  14. In fact, a major focus of this workshop was on how general covariance could be implemented in the context of renormalised perturbative QFT.

  15. Furthermore, the \(f(x)\rightarrow 1\) limit may not even make sense in a curved spacetime setting so there may actually be physical reasons not to take this limit.

  16. This operator is in fact closely related to the path integral. Formally, we have that

    $$\begin{aligned} (\mathcal {T}F)(0)=\int F(\varphi ) d\mu _{\Delta ^F}(\varphi ), \end{aligned}$$

    where \(d\mu _{\Delta ^F}(\varphi )\) is the Gaussian measure with covariance \(i\Delta ^F\) and the integral is the standard path integral. This connection to more conventional path integral approaches to QFT provides a powerful starting point for formulating general conjectures starting from pAQFT.

  17. There remains the question whether the renormalised \(\mathcal {T}_n\)s can be seen as n-fold products coming from some binary product. This question was answered affirmatively by Fredenhagen and Rejzner [31]. We will keep denoting this, now renormalised, time-ordered product by \(\cdot _{\mathcal {T}}\), with the corresponding \(\mathcal {T}\). The idea to treat the renormalised time-ordered product as a binary product opened up many new possibilities since one can drastically simplify the algebraic structure of the theory.

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Acknowledgements

We would like to thank two anonymous referees and participants of the History of Quantum Field Theory and Rotman Institute Philosophy of Physics reading groups for useful comments on earlier versions of this paper. James Fraser would like to acknowledge funding from the Epistemology of the Large Hadron Collider project (234743567/FOR2063, Deutsche Forschungsgemeinschaft) and the ASYMPTOPHYS project (ANR-22-CE54-0002, hosted by IHPST, CNRS, UMR8590, France).

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  1. IHPST, CNRS and Paris 1 Panthéon-Sorbonne University, Paris, France

    James D. Fraser

  2. University of York, York, UK

    Kasia Rejzner

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Fraser, J.D., Rejzner, K. Perturbative expansions and the foundations of quantum field theory. EPJ H 49, 10 (2024). https://doi.org/10.1140/epjh/s13129-024-00075-6

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