Abstract
This article has two aims. First, I undertake an extensive review of the Higgs mechanism and its connections with spontaneous symmetry breaking and the Goldstone theorem. I take the opportunity to expound and discuss a certain number of philosophical issues, amongst them surplus structure and redundancies. Second, I offer a defence of the metaphor according to which ‘gauge fields eat Goldstone bosons to gain a mass’ as sensible rather than merely misleading. It is sensible because there is a direct physical correspondence between the longitudinal polarization of massive gauge fields and Goldstone bosons, which is not merely set by a gauge-fixing procedure. In these terms, I wish to argue that the mechanism which allows for the discovery of the Higgs boson has more than merely heuristic and methodological virtue.
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Notes
For further references, see Earman (2004a).
The term “heuristic” refers to interacting theories that have not yet been mathematically well defined.
An important distinction exists between global and local symmetries: both are variational, but the former pertain to finite dimensional symmetry groups and the latter to infinite ones. In the Higgs mechanism, the symmetries are local, which is particularly problematic (see Sect. 5.1).
It is worth pointing that this imaginary mass poses serious difficulties (see Lyre 2008).
We used two short-cuts here for simplification purposes: (1) we normally deal with vacuum expectation values of fields \(\varPhi _{0}=\langle 0\vert \varPhi \vert 0\rangle \) in the quantum case; (2) we assimilate the vacuum state and vacuum expectation value (VEV) of a field since the eigenvalues \(\varPhi _{0}\) in the vacuum are distinct, and each is respectively associated with one vacuum state.
In the case where \(\mu ^2=-m^2<0\), \(\varPhi _{0}=0\) and the vacuum state conserves the full \(O(N)\) symmetry.
An infinite wavelength oscillation is not necessarily an unphysical object, it may correspond to the fact that some underlying medium experiences a uniform phase.
An intuitive solution is also produced by approximate symmetries (Weinberg 1996).
Note that this is another argument against the analogy with the ferromagnet, cf. Sect. 2.
By ‘nomological sufficiency’, I mean theoretical arguments that are sufficient but not necessary to give a consistent answer to a law-like problem.
By ‘intrinsic’, I mean that the mass is the ‘own’ property of a particle, and not dependent on the environment.
Massive vector fields have four polarization states: one time-like, two transverse and one longitudinal.
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Acknowledgments
I am particularly grateful to Nazim Bouatta and Jeremy Butterfield for detailed comments and helpful discussions on the present essay. I would like to thank Koray Karaca and two anonymous referees for their helpful advice and comments.
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Rivat, S. On the Heuristics of the Higgs Mechanism. J Gen Philos Sci 45, 351–367 (2014). https://doi.org/10.1007/s10838-014-9258-4
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DOI: https://doi.org/10.1007/s10838-014-9258-4