Taking particle physics seriously: A critique of the algebraic approach to quantum field theory

doi:10.1016/j.shpsb.2010.12.001

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

Volume 42, Issue 2, May 2011, Pages 116-125

https://doi.org/10.1016/j.shpsb.2010.12.001 Get rights and content

Abstract

I argue against the currently prevalent view that algebraic quantum field theory (AQFT) is the correct framework for philosophy of quantum field theory and that “conventional” quantum field theory (CQFT), of the sort used in mainstream particle physics, is not suitable for foundational study. In doing so, I defend that position that AQFT and CQFT should be understood as rival programs to resolve the mathematical and physical pathologies of renormalization theory, and that CQFT has succeeded in this task and AQFT has failed. I also defend CQFT from recent criticisms made by Doreen Fraser.

Section snippets

Prelude: a fable

Once upon a time there was a community of physicists. This community believed, and had good reason to believe, that quantum mechanics, not classical mechanics, was the right framework in which to do physics. They also had a good understanding, at the classical level, of the dynamics of solid bodies (vibrations in crystals, for instance): they knew, for example, that some such bodies could be analysed using Lagrangians like $L = \frac{1}{2} {\dot{ϕ}}^{2} - \frac{1}{2} (\nabla ϕ)^{2} + (higher terms)$ where $ϕ (x)$ is the displacement of the part

AQFT and CQFT as rival programs

As is widely known,³ the original attempts—by Dirac, Jordan, Heisenberg and others—to develop quantum field theory foundered on the problem of infinities: attempts to calculate physical quantities

Understanding renormalization in CQFT

One of the revolutions in post-1960s quantum field theory has been the application of QFT methods to condensed-matter physics, and the flow of ideas in both directions between condensed-matter and particle physics. In particular, ideas from condensed-matter physics strongly influenced Wilson and Kogut in their analysis of renormalization, and the condensed-matter approach to renormalization will be a useful starting point for our purposes.

In (real, not fabled!) condensed-matter physics,⁵

Understanding renormalization in AQFT

As I have already noted, algebraic quantum field theory deals with the paradoxes of renormalization by going back to first principles. Instead of writing down concrete examples of quantum field theories (using the admittedly deeply unattractive method of starting with a classical field and then “quantizing” it), which associated field operators to every spacetime point, and then trying to modify those theories to get rid of the infinities, the AQFT theorist writes down a formal mathematical

Comparing research programs: CQFT versus AQFT

I argued in Section 3 that CQFT and AQFT, because they take different and conceptually incompatible approaches to the problem of renormalization, should be understood as rival research programs and evaluated on that basis. We have now reviewed each program; how do they compare to one another?

To begin with the “conventional” or “cutoff” approach: CQFT makes a very large number of novel empirical predictions. Chief amongst these are the hundreds of cross-sections, decay rates, mass ratios,

Another problem for AQFT: why trust claims about the arbitrarily small?

For the sake of argument, let us put aside the lack of positive reasons to expect that an empirically adequate AQFT could be constructed. Is there anything about the success of CQFT that actually rules them out?

One argument might run as follows¹¹:

1.
We have a very well

Fraser's defence of AQFT

By and large, philosophical explorations of AQFT have proceeded with little or no explanation as to why that framework is appropriate for the philosophy of quantum field theory. An honorable—and refreshing—exception is Fraser (2009), who explicitly argues that “an interpretation of QFT should be based on a rigourous axiomatic variant of QFT rather than any of the other variants.” Fraser engages directly with CQFT (notably in the form which I defended in Wallace, 2006) and tries to show why it

Other objections to CQFT

Although I am not aware of any defences of AQFT (as the basis for philosophy of QFT) in print other than Fraser's, I have come across various other objections in conversation with colleagues; here I will try to set out, and respond to, the more common of them.

Objection: CQFT is not Poincaré-covariant.

Response: It is true that those cutoff schemes we can actually concretely implement (notably, putting the theory on a lattice) violate Poincarè convariance (and, indeed, translation and rotation

Conclusion

It has not been my purpose in this paper to disparage the mathematical physicists who continue to search for physically realistic interacting algebraic quantum field theories. Leaving aside the mathematical interest of the task, it is impossible to know in advance just which mathematical highways and byways may prove to be a useful source of insights for the future progress of physics. Let a thousand flowers bloom—so long as it is understood that there is no requirement in quantum field theory

Acknowledgements

I am grateful for comments and constructive criticism from David Baker, Gordon Fleming, Hilary Greaves, Nick Huggett, Wayne Myrvold, Laura Ruetsche, Simon Saunders, Chris Timpson, and especially Jeremy Butterfield and Doreen Fraser.

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Taking particle physics seriously: A critique of the algebraic approach to quantum field theory

Abstract

Section snippets

Prelude: a fable

AQFT and CQFT as rival programs

Understanding renormalization in CQFT

Understanding renormalization in AQFT

Comparing research programs: CQFT versus AQFT

Another problem for AQFT: why trust claims about the arbitrarily small?

Fraser's defence of AQFT

Other objections to CQFT

Conclusion

Acknowledgements

Studies in History and Philosophy of Modern Physics

Studies in the History and Philosophy of Modern Physics

Against field interpretations of quantum field theory

British Journal for the Philosophy of Science

Antimatter

British Journal for the Philosophy of Science

The theory of critical phenomena: An introduction to the renormalization group

Conceptual developments of 20th century field theories

Gauge theory of elementary particle physics

Infrared divergence in quantum electrodynamics

Physical Review

Are Rindler quanta real? Inequivalent particle concepts in quantum field theory

British Journal for the Philosophy of Science

A Dirac sea pilot-wave model for quantum field theory

Journal of Physics A

The principles of quantum mechanics

Curie's principle and spontaneous symmetry breaking

International Studies in the Philosophy of Science

Haag's theorem and its implications for the foundations of quantum field theory

Erkenntnis

Quantum field theory: Underdetermination, inconsistency, and idealization

Philosophy of Science

Generalized functions

A λϕ4 quantum field without cutoffs. I

Physical Review (2)

A $λ ϕ^{4}$ quantum field without cutoffs. I