Are Photons Goldstone Bosons?

Question

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.

Answer 1

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.

See also sections 1.3-1.5 in Xiao-Gang Wen: Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons

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