Abstract
In this paper I detail three major mathematical developments that led to the emergence of Yang–Mills theories as the foundation for the standard model of particle physics. In less than 10 years, work on renormalizability, the renormalization group, and lattice quantum field theory highlighted the utility of Yang–Mills type models of quantum field theory by connecting poorly understood candidate dynamical models to emerging experimental results. I use this historical case study to provide lessons for theory construction in physics, and touch on issues related to renormalization group realism from a more historical perspective. In particular, I highlight the fact that much of the hard work in theory construction comes when trying to understand the consequences and representational capacities of a theoretical framework.
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Notes
By “representational capacity” I mean the ability of a model to properly capture the relevant phenomena for its domain. In the case of QCD, for example, it is important that it exhibit asymptotic freedom and quark confinement in order to match the experimental results from deep inelastic scattering and the absence of observed free quarks, respectively. I do not mean representation in some deeper sense, implying that we should therefore be realists about the model. That, I think, requires an extra step, and is outside the scope of this paper.
These terms will be defined in Sect. 1.1.
This is independent of the way in which dynamical models are interpreted. Dynamical models do not require a realist or mechanistic underlying interpretation. The dynamical models in the standard model—quantum chromodynamics and the electroweak model—are still the subject of heavy interpretive controversy, and many physicists involved in its construction take a clear instrumentalist view of the standard model. Nevertheless, the standard model is a clear case of a collection of dynamical models.
I use the example of general relativity here because it fits particularly well with model theory. Quantum field theory, on the other hand, is nowhere near as clearly or rigorously defined, and specifying models of quantum field theory in this sense is extremely difficult.
The data-phenomena distinction was first explicated by Bogen and Woodward (1988). This distinction is of some importance to my view, as phenomenological models, though closely connected to experiment, can only be used to describe experimental phenomena, not the data. In what follows I largely gloss over this distinction, and refer to the comparison of experimental data with phenomenological models. These statements should be understood to be shorthand for the conversion of experimental data into phenomena, followed by the comparison with phenomenological models.
The term “gauge” to describe the local symmetry operations comes from Weyl (1918), who sought to expand general relativity by allowing the metric field to include local variations to the length scale.
Divergences were also rampant in the classical relativistic theory. Unlike for nonrelativistic models of the atom—for which quantization introduced stability—quantization did not solve the divergence problem. Quantum electromagnetism suffered from logarithmic divergences instead of the steeper linear divergences of the classical theory, but the divergences remained. It wasn’t until the advent of QED that one had a (perturbatively) divergence free formulation of relativistic electrodynamics—quantum or classical.
In more modern terms, renormalization is a process which occurs after a regularization procedure. Regularization is a process by which divergent quantities are replaced by finite quantities depending on some arbitrary regularization parameter. Renormalization, on the other hand, is a process in which one takes the regularized theory and determines the “physical” form of the relevant parameters in the theory—usually masses and coupling constants—in such a way that they do not depend on the value of the regularization parameter. If this can be done, then the theory is renormalizable. A straightforward removal of the ground state energy value is therefore not a renormalization in this modern sense, but earlier views regarding “renormalization methods” were closer to “removing divergences from a theory.” In this older sense of the term, subtracting the ground state energy was a renormalization of the Hamiltonian.
For a more comprehensive account of the history of the development of QED, see Schweber (1994).
The resulting integrals, however, are not guaranteed to be amenable to analytic solution. It is still the case that diagrams with a higher order in the coupling constant \(\alpha \) lead to integrals that are enormously difficult to compute. For example, one major empirical success of QED is the high degree of precision match between the experimentally determined anomalous magnetic moment of the electron and the value predicted as a perturbative expansion of \(\alpha \) in QED. The current state of the art calculation cannot be carried out analytically, and provides a prediction to fifth order in \(\alpha \), or tenth order in e (Kinoshita 2013).
The heuristic argument for the connection between range of interaction and mass relies on a limit based on Heisenberg’s uncertainty principle. The energy-time version of the uncertainty relation is \(\varDelta E \varDelta t \ge 1/2 \hbar \). The timescale on which a boson can exist is related to the rest mass as \(t \approx \hbar /(2mc^2)\). So a particle traveling near the speed of light would have a range \(R \approx \hbar /(2mc)\). This argument is initially due to Wick (1938), explicating the Yukawa (1935) model of nuclear forces.
Even today, the mass gap problem in Yang–Mills theories is a topic of interest among mathematically inclined physicists. The Clay Institute has offered up $1 million as a reward for solving the mass-gap problem as one of their seven millennium problems.
If one can call any formalism for quantum field theory “dominant” at this time. Trust in the reliability of quantum field theory was at an all time low, but the formalism was still used on a heuristic basis in order to arrive at the S-matrix. The S-matrix was then thought to contain all of the physical content involved in particle physics.
For quantum field theories involving fermions, the field theory has to be somewhat artificial in that Grassmann fields are used in place of classical real-valued fields. This is to ensure the appropriate anticommutation relations upon quantization.
Note that canonical transformations are distinct from gauge transformations. A canonical transformation is a change of field variables, leading to (anti)commutation relations involving different field operators. Though this leads to the problem of unitarily inequivalent representations in quantum field theory, fields related by canonical transformations are generally thought to represent the same physical situation.
In the context of the above quote, Veltman was actually working on renormalizing explicitly massive Yang–Mills theory, which was ultimately a failure. What Veltman accomplished was to renormalize massive Yang–Mills theory up to one loop. This was an important feat, especially in light of the more modern view of the quantum field theories as being effective theories of matter; a theory that is renormalizable to one loop can be used to generate low-energy predictions on scattering. Fermi’s theory of the weak force was one-loop renormalizable, and Veltman further showed that Yang–Mills theory with an explicit mass term was equally useful. The steps Veltman took here can also apply to the massless Yang–Mills case, or the case where mass is obtained for the vector bosons through spontaneous symmetry breaking.
In the course of proving renormalizability of Yang–Mills theories, this was a minor step. However, this second paper was hugely influential in the development of the Standard Model, as it proved at least one-loop renormalizability of the Glashow–Salam–Weinberg electroweak model. The model was therefore proven to be usable for first order predictions, which were later tested and confirmed the adequacy of the model. By this time, ’t Hooft and Veltman (1972) had demonstrated full renormalizability using the dimensional regularization procedure (see below), and the electroweak model was accepted as the appropriate description of the newly unified electromagnetic and weak interactions.
This is assuming an infinitely extended ferromagnet. In general, true critical phenomena in statistical mechanics require the thermodynamic limit be taken as an idealization: the volume and number of particles both go to infinity such that the density N / V of particles remains constant.
This work on operator product expansions would eventually be published as Wilson (1969), after Wilson resolved some issues with the expected behaviour of expansion coefficients in the strong coupling domain.
Wilson uses an analogy with a simple classical mechanical system—a ball at the crest of a hill—to argue that the singularities inherent in a particular form of differential equation may be an artifact of the variables chosen to represent the equation. This is also familiar in the context of solutions to Einstein’s field equations, where coordinate singularities can arise, and an appropriate transformation of coordinates must be done to remove the false singularity.
’t Hooft presented the final equation for the scaling of the beta function at a conference in 1972, but never published the results. Gross, Wilczek, and Politzer would eventually win the 2004 Nobel prize for the theoretical prediction of asymptotic freedom.
This change is central to the physical disanalogies between models in quantum field theory and condensed matter physics (cf. Fraser and Koberinski 2016). Causal and modal structures change dramatically when time is converted to a spatial dimension.
Mass generation—in the form of spontaneous symmetry breaking—was not discussed in this paper. For a detailed analysis of the formal analogy between spontaneous symmetry breaking in the Higgs mechanism and in superconductivity, see Fraser and Koberinski (2016).
There is not a consensus that an effective field theory view of the standard model is the best way to interpret the utility of quantum field theoretic models. Many people working in axiomatic and/or algebraic quantum field theory, for example, aim to provide an exact model for realistic interactions, to which the standard perturbative methods of conventional quantum field theory create an asymptotic expansion (e.g., Streater and Wightman (1964), Buchholz and Verch (1995), Halvorson and Müger (2007), Feintzeig (2017)). These may still be effective theories in the sense that they have a limited domain of applicability, but they would then be candidates for a more standard philosophical interpretation. Others have criticized the limited utility of a realist interpretation of effective field theories based on the renormalization group (Fraser 2018; Ruetsche 2018). Though these are important philosophical issues, they are orthogonal to the discussion here. For the purposes of the main discussion, I will uncritically accept the effective field theory view, and attempt to explain why a proof of renormalizability is still epistemically important in HEP.
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Acknowledgements
I would like to thank Doreen Fraser, Wayne Myrvold, and Marie Gueguen for helpful comments on earlier drafts of this paper. I am also grateful to two anonymous referees, whose suggestions helped to strengthen this paper.
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Koberinski, A. Mathematical developments in the rise of Yang–Mills gauge theories. Synthese 198 (Suppl 16), 3747–3777 (2021). https://doi.org/10.1007/s11229-018-02070-z
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DOI: https://doi.org/10.1007/s11229-018-02070-z