# Are Photons Goldstone Bosons?

I'm not interested in more speculative ideas, like the one of Bjorken et. al. that photons are Goldstones of broken Lorentz symmetry.

Instead, I want to understand if photons are simply the Goldstones of the spontaneously broken large gauge symmetry? (Large gauge symmetry here simply means those gauge transformations that do not become trivial at infinity.)

I recently read Stromingers "Lectures on the Infrared Structure of Gravity and Gauge Theory", where he argues that

large gauge symmetry is spontaneously broken, resulting in an infinite vacuum degeneracy with soft photons as the Goldstone bosons.

Moreover I discovered that already in the 70s there were quite a few papers that argued that photons are simply the Goldstones of the broken asymptotic gauge symmetry.

For example:

• Gauge invariance and mass by Richard A. Brandt and Ng Wing-Chiu Phys. Rev. D 10, 4198; who argued that "the physical photon can be interpreted as a Goldstone boson arising from the spontaneous breakdown of the R -transformation invariance." (R-transformations is simply another name for gauge transformations that are non-trivial at infinity)
• Spontaneous breakdown in quantum electrodynamics by R.Ferrari L.E.Picasso Nuclear Physics B Volume 31, Issue 2, 1 September 1971, Pages 316-330: "In the context of quantum electrodynamics we discuss the spontaneous breakdown of the symmetry associated to gauge transformations of the second kind, with gauge functions linear in the coordinates. We show that the photons (both physical and unphysical) can be considered as the Goldstone particles of this symmetry, and that the Ward identity and, in general, all self-photon theorems, are dynamical consequences of the spontaneous breakdown of the symmetry considered."

This seems like a well established fact. For example, in a recent paper by Yuta Hamada, Sotaro Sugishita they noted in passing:

The statement that photons and gravitons are NG bosons is not new and it is discussed in [20, 21, 22, 23].

I really like this perspective. However, I am a bit confused, because no textbook and almost no paper mentions this although the papers quoted above are 40+ years old.

• 1. The notion of "large gauge transformation" meaning "gauge transformation non-trivial at infinity" is non-standard, and you shouldn't redefine terminology that's already hard to use because people have rather different definitions and/or understanding of it. Nov 6 '17 at 12:19
• 2. What's the question here? You essentially seem to have answered your own question by already citing peer-reviewed articles saying "Yes.". What could any answer here do better than that? Recall that physics.SE is not a location to request peer-review. If there is a specific part in these papers you have questions about, please clarify that in your question. Nov 6 '17 at 12:19
• @ACuriousMind This use of the term "large gauge transformation" is due to Strominger, see e.g. 1703.05448. Therefore, this is apparently the correct notion for people currently working in this field. (I know that people working on non-perturbative QCD effects use the term differently). There are almost as many definitions as there are authors, so I don't see any point in arguing about what is standard and what isn't. The name doesn't matter as long as you define it - which I did. What, according to you is the standard name for these gauge transformations that are non-trivial at infinity?
– jak
Nov 6 '17 at 12:38
• @ACuriousMind My question is if this is a well established fact. Only because the papers are peer reviewed, doesn't mean that there aren't flaws in them. Or maybe we understand things nowadays differently. These papers are from the 70s (!). There is no textbook that I know of that mentions this. Therefore, for some reason this is not "textbook knowledge". I'm curious if anyone has insights why this is the case, or if this is well-known and I have only overlooked it until recently.
– jak
Nov 6 '17 at 12:42
• @JakobH I think it is indeed an established fact, the discussion in Picasso is fantastic. Two much more recent papers are arxiv.org/abs/1405.5532 and arxiv.org/abs/1412.6098. It is just a question of interpretation though, qed is still qed as you knew it, just the way you derive the results is different, but the results themselves are the same. Nov 7 '17 at 21:18

## 1 Answer

Whenever we have massless degrees of freedom, there is usually some underlying reason. For massless scalar fields, Goldstone’s theorem typically provides the reason. But we can also invoke Goldstone’s theorem to explain why the photon is gapless: we just need to extend its validity to higher form symmetries.

In Maxwell theory, there are two 2-forms which are conserved. Each can be thought of as the current for a global 1-form symmetry: $$\text{Electric 1-form symmetry}: J_e \propto *F$$ $$\text{Magnetic 1-form symmetry}: J_m \propto F$$ The electric 1-form symmetry shifts the gauge field by a flat connection: $$A → A+d\alpha$$. In contrast, the action of the magnetic 1-form symmetry is difficult to see in the electric description; instead, it shifts the magnetic gauge field $$\tilde{A}$$ by a flat connection. Relatedly, the electric 1-form symmetry acts on Wilson lines $$W$$; the magnetic 1-form symmetry acts on ’t Hooft lines $$T$$.

In the Coulomb phase we have $$\langle W\rangle \neq 0$$ and $$\langle T\rangle \neq 0$$, so that both symmetries are spontaneously broken. But a broken global symmetry should give rise to an associated massless Goldstone boson. This is nothing but the photon itself.

See section 3.6.2 in David Tong: Lectures on Gauge Theory.

I will be glad to discuss it, and extend the answer, if you have some questions:)

• Should we think of the photon as the Goldstone mode of the electric symmetry or the magnetic symmetry? Or does that question even make sense? Apr 1 at 17:03
• Yes. One can think that photon is the Goldstone mode of the electric symmetry and the magnetic symmetry. Apr 1 at 21:59
• How can it be the Goldstone mode of both? I'm imagining that we could explicitly break the magnetic symmetry by adding electric matter, in which case the photon would remain massless. But if we added magnetic matter, the photon would become gapped, right? Apr 4 at 17:31
• @NoethersOneRing , There are some differences in Goldstone theorem for higher form symetries. Here for breaking of 2 symmetries we have 1 goldstone mode. If we added magnetic matter, the photon would not become gapped. Apr 4 at 19:31
• @NoethersOneRing, situation is more complicated. See for example arxiv.org/abs/1203.1494 Apr 5 at 7:38