Abstract
The paper investigates the spontaneous breaking of gauge symmetries in gauge theories from a philosophical angle, taking into account the fact that the notion of a spontaneously broken local gauge symmetry, though widely employed in textbook expositions of the Higgs mechanism, is not supported by our leading theoretical frameworks of gauge quantum theories. In the context of lattice gauge theory, the statement that local gauge symmetry cannot be spontaneously broken can even be made rigorous in the form of Elitzur’s theorem. Nevertheless, gauge symmetry breaking does occur in gauge quantum field theories in the form of the breaking of remnant subgroups of the original local gauge group under which the theories typically remain invariant after gauge fixing. The paper discusses the relation between these instances of symmetry breaking and phase transitions and draws some more general conclusions for the philosophical interpretation of gauge symmetries and their breaking.
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Notes
See Healey (2007), Chapter 6, for an in-depth philosophical exposition of the distinction between empirical and purely formal symmetries that defends the standard account of gauge symmetries as purely formal.
See ’t Hooft (2007), p. 697.
See Earman (2004), p. 1239.
See Liu and Emch (2005), p. 153.
See Strocchi (2008) for a rigorous textbook account of SSB that avoids both unnecessary technicalities and misleading simplifications. Roughly speaking, for a state to exhibit SSB in the rigorous sense specified in that work, it needs to take an infinite amount of energy to transform the system from one asymmetric state into another. This is the reason why realistic systems (that is, systems without any potential barriers of infinite height) need to have infinitely many degrees of freedom to exhibit SSB. Furthermore, it does not suffice for a non-symmetric state to differ only slightly from a symmetric one (in the sense in which, say, a single particle state differs from a zero-particle, fully symmetric, vacuum state) to qualify as symmetry breaking. Cases like these are automatically excluded by the criterion in terms of the algebraic approach to quantum theories stated in the next paragraph.
Leggett (2006) provides an illuminating discussion of Bose-Einstein condensation in the absence of the thermodynamic limit that does not operate with the notion of a spontaneously broken gauge symmetry. The assumptions underlying Leggett’s approach are more realistic than those of the discussion made here, since the number of atoms in physically realized examples of BEC is not exceedingly large (between roughly 103 and 105) and there are important inter-particle interactions in these systems. Leggett’s “order parameter” (see Leggett 2006, Eq. 2.2.1), in terms of which he defines Bose-Einstein condensation, is not a symmetry breaking order parameter in the sense of Eq. 2.
See Liu and Emch (2005), p. 153. Liu and Emch focus on quantum spontaneous symmetry breaking, specifically, but the characterization of symmetry breaking as a natural phenomenon does not seem to be based on any particular quantum (as opposed to classical) aspects.
See Liu and Emch (2005), p. 153, fn. 14.
Note that to accept the characterization of SSB on the level of observables in quantum theories as a natural phenomenon, it does not seem necessary to endorse the standard ontic view of quantum states as states quantum systems “are in”. The main reason for adopting the alternative, epistemic, conception of quantum states is that it elegantly avoids the paradoxes of measurement and nonlocality. (See Friederich 2011, for more details and an exploration of how the view might be spelled out in detail.) According to the epistemic conception of quantum states, quantum states reflect the state assigning agents’ epistemic relations to these systems, so no such thing as the “true” quantum state of a quantum system is acknowledged, and SSB cannot be characterized in terms of quantum systems’ “being in” quantum states that break some symmetry of the algebra of observables. Nevertheless, proponents of the epistemic conception of states can hold that SSB is a natural phenomenon in that an observable called a “witness” of a symmetry of the observables may have a value that, if known, requires the assignment of a state that breaks that symmetry. (For an explanation of the notion of an observable being a “witness” for SSB, see Liu and Emch 2005, p. 145.) The characterization of SSB in quantum theories as a natural phenomenon seems therefore independent of the question of whether quantum states are conceived of as ontic or non-ontic.
Alternatively, one may reserve the notion of a phase transition for the physical process of crossing a phase boundary. This is the sense in which, for instance, cosmologists speak of phase transitions in the early universe. For a detailed and rigorous account of phase transitions in the sense of phase boundaries, see Sewell (1986), Chapter 4. Here I gloss over the difficulties of giving a rigorous account of thermodynamic notions such as the Gibbs potential in the relativistic, quantum field theoretical, context, assuming that at least for all practical purposes these difficulties can be met.
Equivalently, one could have arrived at a Lagrangean of exactly the same form by imposing the unitary gauge θ = 0.
See Struyve (2011) for a detailed discussion of these questions. If one chooses to use only gauge invariant fields, however, one has to pay careful attention to the constraints for the variable η occurring in Eq. 12, see Struyve (2011), Section 4, and Strocchi (2008), p. 194. The analysis in terms of the reduced phase space approach given in Struyve (2011), Section 7, avoids this problem.
The inverse free propagator of the gauge fields can be thought of as the coefficient of the part in the Lagrangean \(\mathcal L\) which is quadratic in the gauge fields. For the Abelian case this part of the Lagrangean is given by \(-\frac{1}{4}\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)^2\), and the resulting inverse free propagator for the gauge field is, in momentum representation, given by \(\eta_{\mu\nu}k^2-k_\mu k_\nu\) (where η μν corresponds to the matrix diag[1, − 1, − 1, − 1]). As remarked in the main text, the operator \(\eta_{\mu\nu}k^2-k_\mu k_\nu\) is not invertible, which can be seen from the fact that it has k ν as an eigenvector with eigenvalue zero.
For an alternative approach that does not use gauge fixing see the version of continuum perturbation theory used in Buchmüller et al. (1994) to study the electroweak phase transition in gauge invariant terms.
Elitzur proved the theorem for the case of a Higgs field with fixed modulus, see Elitzur (1975). The result was generalized to the case of a Higgs field with variable modulus by de Angelis, de Falco and Guerra, see De Angelis et al. (1978). See Itzykson and Drouffe (1989), Chapter 6.1.3, for a useful textbook version of the proof.
See Fröhlich et al. (1981).
See Smeenk (2006), p. 498.
Detailed calculations (see, for instance, Kajantie et al. 1996) have shown that for values of the Higgs mass not excluded by experiment the electroweak phase transition is actually not a real phase transition (in the sense of an abrupt change in thermodynamic quantities) but rather a steep crossover between two qualitatively different regimes of electroweak theory, meaning that at least some expectation values of observables vary very strongly (yet analytically) from one regime to the other. In the context of the present paper, however, the question of whether, for realistic values of the Higgs mass, the electroweak phase transition is a genuine phase transition or rather a continuous crossover is not important since we are concerned here with the more general question of whether the notion of a spontaneously broken local gauge symmetry is needed to give meaning to the distinction between the high and low temperature phases of the electroweak theory, which are sharply separated for some values of the Higgs mass.
See, for instance, Buchmüller et al. (1994), pp. 134–6.
This can be seen, for instance, from the fact that ghost fields formally correspond to spinless fermion fields the physical existence of which is excluded by the spin-statistics theorem.
The example taken is Eq. 1.1 in Caudy and Greensite (2008).
There are other symmetries besides local gauge symmetries and their global subgroups which can be broken in quantized gauge theories such as, for instance, chiral symmetry in QCD or centre symmetry in non-Abelian gauge theories (the centre of a group is the set of elements which commutes with all other elements), which seems to be linked to the confinement-deconfinement phase transition, see Greensite (2011). The present paper is not concerned with the breaking of these symmetries but only with that of local gauge symmetries and their global subgroups.
See Caudy and Greensite (2008). More precisely, their results are for a model with a fixed-modulus Higgs field in the fundamental colour representation. Their results clearly show that in a certain range of parameters the system exhibits the typical features of a “Higgs phase” such as, for instance, the appearance of a massive spectrum associated with the gauge field degrees of freedom, even though there is no “Mexican hat potential” (which makes sense only for a Higgs field with a variable modulus).
See Greensite (2011) for an introduction to the problem of confinement that includes an in-depth discussion of how confinement should be defined in the first place.
An interesting line can be drawn from the Fradkin-Shenker result to more recent developments involving duality in supersymmetric gauge theories and their ramifications for string theory. See Intriligator and Seiberg (1996) for an introduction to supersymmetric gauge theories that makes this connection.
See Kosso (2000), p. 359.
See Morrison (2003), p. 356.
See Morrison (2003), p. 357.
See Morrison (2003), p. 359.
See Morrison (2003), p. 361.
See loc. cit.
See loc. cit.
See Weinberg (1974), p. 3359.
Or, equivalently, on the choice of gauge transformation g(x;A) as in Eq. 15, used to define a remnant subgroup of the original local gauge group.
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Acknowledgements
I would like to thank Kerry McKenzie, Robert Harlander, Dennis Lehmkuhl, Holger Lyre, Michael Kobel, Michael Krämer, Michael Stöltzner, Ward Struyve and anonymous referees who reviewed this article for many helpful comments. Furthermore, I am grateful to Jeff Greensite, Gernot Münster and Franco Strocchi for useful answers to questions I raised.
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Friederich, S. Gauge symmetry breaking in gauge theories—in search of clarification. Euro Jnl Phil Sci 3, 157–182 (2013). https://doi.org/10.1007/s13194-012-0061-y
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DOI: https://doi.org/10.1007/s13194-012-0061-y