The Real Problem with Perturbative Quantum Field Theory

Abstract

The perturbative approach to quantum field theory (QFT) has long been viewed with suspicion by philosophers of science. This paper offers a diagnosis of its conceptual problems. Drawing on Norton's ([2012]) 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.

The Elementary Particles of Quantum Fields

“…However, as he has also pointed out, "the perturbative framework should be understood as a method for producing approximations without addressing the project of constructing interacting QFT models", for then "the mathematical sloppiness we do find in perturbative calculations in the physics literature becomes a less pressing foundational concern" and "Haag's theorem is not as disastrous for the perturbative framework as it initially seems. In brief, the result does not undermine standard perturbative calculations because they do not posit the existence of a model satisfying the relevant set of inconsistent assumptions" [25]. The idea is to consider weaker interactions, i.e., those for which the interaction strength g (for our KG model, this is λ) is sufficiently small, to understand what takes place during interactions.…”

I ain’t afraid of no ghost

“…So the divergences related to renormalization are distinct from those related to the quadratic terms. More recent work attends to the interpretation of the divergent series produced by the quantization procedure (Fraser, 2020;Miller, 2021). These divergences do arise in the finite-dimensional case; indeed, the special character of such divergent series was first identified by Poincaré (1892) in the context of celestial mechanics.…”

“…Fraser (2020) suggests that we understand these series as approximations with no underlying exact model, while Miller (2021) develops a novel semantics for theories that assign divergent perturbative expansions to observables.7 My exposition of the finite-dimensional model followsMnev (2019). Under appropriate conditions this approach can even reproduce the exact value of the integral(Johnson-Freyd, 2015).…”

Whence the Effectiveness of Effective Field Theories?

“…While the bare parameters, which are defined at the cutoff, may still be regarded as mere mathematics, this view allows that there are physically meaningful parameters defined at high energies below the high-energy cutoff, which sensitively relate to low-energy parameters. Thus, the high-energy theory ought not to be conceived of as a continuum model with an infinite cutoff; instead, following Fraser (2017), we should view the high-energy theory as having a finite cutoff, and the infinite limit as a way to approximate certain properties of the finite theory. This step avoids the conceptual problems which previously arose when renormalisation was viewed as a process for correcting infinite bare parameters, but allows there to be connections between parameter values defined at different length-scales.…”