Are Photons Goldstone Bosons?

Question

I'm not interested in more speculative ideas, like the one of [Bjorken et. al.][1] 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"][2], 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][3] 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][4] 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][5] 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]: https://arxiv.org/abs/hep-th/0111196
  [2]: https://arxiv.org/abs/1703.05448
  [3]: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.10.4198
  [4]: http://www.sciencedirect.com/science/article/pii/0550321371902355?via%3Dihub
  [5]: https://arxiv.org/abs/1709.05018

Comments

@Vladimir Kalitvianski I asked, because I was not sure if this is an established fact or not. (I'm pretty sure that most theoretical physicists do not know this, for whatever reason.) Is there any textbook that discusses this?

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I belong to an old generation of physicists and in my opinion the masslessness of photons follows from the Maxwell equations for the propagating to infinity electric and magnetic fields (strengths). Any other explanation is a secondary - like some property of a massless theory which cannot be a true reason, but just a property. Besides, photons are always coupled to charges, either to the "sourcing charges" or to the "sucking ones". In the latter case the photons are just external fields for probe charges, which scatter or absorb external photons, depending on their being free or coupled to some other charges. For some reason we forget it and think of photons as of a gas/flux of massless particles existing per se.

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@VladimirKalitvianski: Hard photons (produced by a laser, say) are not coupled to charges, in the sense that they are not external fields of any charge.

The photon field is the Goldstone field of  the asymptotic gauge symmetry, independent of how or whether it is interpreted in a particle language.

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@ArnoldNeumaier: I am surprised to see that you call the near field (for example, that with the strength $E_{{\rm{near}}}\propto 1/R^2$) a "cloud" of "coherent states of photons", and the propagating to infinity field ($E_{{\rm{prop}}}\propto 1/R$) are "hard" ones as they are "detectable". All frequencies may propagate to infinity, so photons are of all frequencies and are detectable (long radio waves is an example). All frequencies may be present in the near field too and the sum of them gets into the equations of motion of detecting device charges. Being sourced means being coupled to the sourcing charges. Being detected means being coupled to the probe charges ("absorbers"). There is no other meaning of fields/photons and their properties. Otherwise (i.e., when non detectable) they are void of physical meaning.

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@ArnoldNeumaier: No, the Coulomb, or more generally the retarded near field is not detectable at infinity: it is just too weak (zero). It has no effect at all at infinity. The propagating photons, on the contrary, may be felt at any distance: they may cause excitations, chemical reaction, heating, etc.

After production, the laser pulse has no meaning if it does not get into a measuring device. You even cannot judge about your laser efficiency without measuring the distribution the radiated energy and the lost energy in the total energy supplied to the laser. Measuring is always implied explicitly or implicitly.

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@ArnoldNeumaier: Thank you for your clarifying answer. But your statement that EMF does not have any equation of motion contradicts the Maxwell achievements.

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@VladimirKalitvianski: Maxwell achieved nothing for quantum fields. Moreover, I didn't claim that the electromagnetic field has no equations of motions - only that virtual particles (and hence the photon cloud) don't.

The interacting electron/positron and electromagnetic fields are not asymptotic quantities and obey operator product expansions, which are the relativistic QFT analogue of the equation of motion. And the coherent state factor in the asymptotic electron wave function is represented in the interaction picture, and hence moves freely with the center of mass of the electron. 

Answer 1

Check out this paper: https://arxiv.org/abs/1412.5148 It is more or less a simple application of the Goldstone/Noether procedure. If you apply the charge operator ($\star F$) to a Coulomb ground state you produce a photon.

By the way it is not good to call these "large gauge transformations" because it is really a global symmetry, but with connections as parameters.

Comments

Thanks a lot! I'll read it

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In my opinion "large gauge transformations" is simply the name given to global symmetries with connections as parameters.

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"Large-scale" gauge transformations are nearly "global" ones, in contrast to the local ones.

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A gauge transformation is a redundancy in the mathematical description of the physics. It is not physical. A 1-form symmetry is however a physical property of the system, and must be preserved under dual descriptions of the theory, unlike the gauge group.

A large gauge transformation is simply parametrized by a function to the group which cannot be globally written as the exponential of a map to the Lie algebra.

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See also this paper https://arxiv.org/abs/1309.4721 , where we applied these ideas to studying different confining phases of Yang-Mills theory.

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@RyanThorngren Do you know an explicit example of a large gauge transformation? I've read this definition before, but at least all the large gauge transformations that are considered when one studies the QCD vacuum can be written as an exponential function. For example, the usual example $U^{\left( 1\right) }\left( \vec{x}\right) = \frac{1}{|x|} (1+ \vec {x} \cdot \vec \sigma)$ with winding number $1$, can also be written as $U^{\left( 1\right) }\left( \vec{x}\right) =\exp\left( \frac{i\pi x^{a}\tau^{a}}{\sqrt{x^{2}+c^{2}}}\right)$. (See e.g. page 91 in Quarks, Leptons and Gauge Fields by K. Huang). As far as I know, the difference is that large gauge transformations are parametrized by functions that do not go to $0$ as $|x| \to \infty$, but to multiples of $2 \pi$. Moreover, the class of gauge transformations that aren't necessarily constant, i.e. which still depend on the angles $\phi, \theta$, at infinity, are what we call asymptotic gauge transformations. The "large gauge transformations", defined this way are a subset of the asymptotic gauge transformations. Analogously, the global gauge transformations are those that are constant elements of the Lie group $G$ at infinity, not necessarily the identity. Again, these global transformations are a subset of the asymptotic transformations.

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@RyanThorngren One more thing: you wrote "gauge transformation is a redundancy in the mathematical description of the physics. It is not physical." I think this is not correct, unless you use a different definition of gauge transformation than I do. (Which I'm pretty sure you do). In my understanding, only local gauge transformations, i.e. only those which are non-trivial within a finite volume are non-physical and mere redundancies. The other gauge symmetries I quoted above are physical:

• The global gauge symmetry is responsible e.g. for the conservation of electrical charge.
• The spontaneously broken asymptotic gauge symmetry is responsible for the massless excitations we call photons. 
• Large gauge transformations are responsible for the non-trivial vacuum of QFTs like QCD. ($\theta$-term, instantons, etc.)

I'm pretty sure that you know and agree that these kind of "gauge" transformations are physical. I'm really interested in how you call these transformations, i.e. a map from my terminology to yours. (Especially, because I tried to read the papers you quoted, but unfortunately had problems understanding the terminology. In particular the physical motivation of these higher q-form symmetries is unclear to me. Although now I suspect these are simply what I call asymptotic gauge symmetries etc. above. ) 

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I would like to say that any reasonable variable change (not only a gauge one) leaves the final (or exact ) results the same physically. However, in the perturbative treatment, it is important to start from an initial approximation as close as possible to the exact solution - in order to catch correctly physics and to have small perturbative (numerical) corrections.

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Hi @JakobS I think the best terminology is to reserve the term "gauge transformation" for a redundancy in our description of the system. Whereas symmetries correspond to conservation laws by Noether, gauge symmetries correspond to constraints on the Hilbert space.

In field theory in Minkowski space, we only consider transformations parmetrized by a map $\phi:\mathbb{R}^4 \to G$ which tends to the identity at infinity. The reason we don't want to consider nontrivial gauge transformations at infinity is that this would preclude us from preparing asymptotic charged particle states in the Hilbert space, which we need to compute S matrix elements.

A better definition of "large" is "not connected to the identity transformation", which is precisely what winding number tells you. The SU(2) transformation you wrote down acts as the identity at infinity and is therefore a legit gauge transformation.

Hope this helps!

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@JakobS If you're interested in applying these ideas to QCD also check out https://arxiv.org/abs/1703.00501 .

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@RyanThorngren Thanks a lot for your reply! Hmm, yeah... there seems to be a problem with terminology. Have a look e.g. at Eq. (2.10.1) in Strominger's lecture notes. He talks about trivial, and non-trivial gauge transformations. Modding the trivial gauge transformations out from all allowed transformations, leaves us with the "asymptotic symmetry group". Are these "q-form global symmetries" anyhow connected to Strominger's "asymptotic symmetry group"? There seems to be a connection, because in the lecture notes quoted above, the photons are Goldstones of this broken asymptotic symmetry group. In contrast, in the paper you linked to the photons are Goldstones of "these higher-form global symmetries". The thing is that Strominger's concept of the "asymptotic symmetric group" as given in Eq. (2.10.1) makes perfect sense to me, while I have trouble grasping the "higher-form global symmetry" concept. Any hint how these are related would be awesome!

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@JacobS you are right, there is some care required with the definitions, and some of the popular but blunt statements about the nature of gauge transformations need further qualification.

Standard accounts, e.g. Henneaux-Teitelboim's "Quantization of gauge systems" give the definition of gauge transformations as parameterized variations that leave the Lagrangin invariant up to a totdal spacetime derivative. For compactly supported choice of gauge parameters this happens to be examples of what in other texts are called "gauge transformations", namely compactly supported symmetries of the Lagrangian, up to total derivatives.

But for non-compactly supported gauge parameters, we still get something that needs to be called "gauge transformation".

In my A first Idea of Quantum Field Theory I speak of "gauge symmetry" for the compactly supported symmetries, and of "gauge-parameterized gauge symmetry" for the other case. (Keeping in mind that, despite the terminology, neigher definition in general subsumes the other, but that's just how it is).

Another point to notice is that, with either of the two definition, one has to be careful with the ever popular slogan "gauge symmetry is just a redundancy". This is in general too naive, namely when there are more that one gauge symmetries connecting any two gauge equivalent configuration.

If one strictly belives that all gauge symmetry is redundancy, then for instance locality of field theory breaks (exposition pdf)

There is related discussion on PhysicsForums right now, around here

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@UrsSchreiber: It seems to me that you concentrate on "some classifications", according to some criteria, but you do not discuss the question whether physically reasonable solutions exist or not for a given nice Lagrangian. Meanwhile, even in Classical Filed Theory the exact physically reasonable solutions do not exist. We content ourself with approximate solutions of some other equations, and this is the problem to be solved, in my opinion.

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@VladimirKalitvianski I have to admit that I am at a loss seeing how your comment is at all related to anything I wrote in my comment, or, for that matter, to anything at all.

Your first sentence has the words "some classifications according to some criteria": apart from me not having mentioned any classification of anything, what could be more vague of you than speaking of "some classification according to some criteria" without specifying any context? I have not the least of a clue what you are referring to here.

Next you say "even in Classical Filed Theory the exact physically reasonable solutions do not exist". This seems content-free to me I just can't see what you mean to be saying. What is "the" even referring to?

Before getting to the point that I could agree or disagree with you on anything, I have trouble parsing your message to anything of semantic content.

Why is that? It's not the first time, I have this problem with many of your messages. I am wondering if there is a way that you could improve your communication method, for it would help this forums here, given your high rate of activity.

My suspicion is that you are internally captured in some old arguments that you have once been involved in, with some other people, and are providing counter-arguments here in reply to statements that once were made elsewhere, by other people. Could this be?

Maybe @ArnoldNeumaier can help? In other places he seems to have been able to understand what you have in mind.

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@UrsSchreiber: You have just mentioned transformations with "compactly and non-compactly supported gauge parameters". It is a specific classification, isn't it? And in my other comment I clearly said that any transformation is good if reasonable (reversible, for example). New equations/Lagrangian may have quite a different form, but it does not invalidate such transformations since the exact solutions, if they exist, return the right physics. And if there is no physically reasonable exact solution, what is the use of any transformation? Why do you pretend that I am vague?

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@VladimirKalitvianski, you ask "Why do you pretend that I am vague?"

Please note that I am not pretending anything, I am just giving you frank feedback on the trouble I have with reading your comments. Since both you and me are frequent contributors here, it becomes awkward if we don't openly voice such stark problems.

And I have to admit that I still don't really see what you are talking about.

Maybe somebody else here can explain in other words what you are concerned with. (If nobody can, that might highlight the problem! But maybe Arnold can help.)

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@UrsSchreiber: I am clearly speaking of existence or non existence of physical solutions of a Theory equations/Lagrangian you make some transformations of. I suppose you imply existence of such solutions. Can you confirm my supposition, please?

P.S. Arnold is fond of 2D QFT (contrary to me). It could be instructive to him to listen carefully to the talk by Arthur Jaffe, especially about playing with space-time dimensions.

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@UrsSchreiber Thanks for your comment. I've read your recent introductions (the PhysicsForums Insights, and in the nLab) and liked them. I think it's a real problem that the few people who do try to get these things like gauge symmetries, global symmetries etc. precise do not talk to each other and therefore do not seem to speak a common language. Thus it's no wonder that almost everyone seems extremely confused.  So, to come back to my original question: are you able to translate the approaches by Strominger or also the approach by Kapustin, Seiberg, et. al. into your language? Would you agree with the statement that gauge bosons are Goldstones? (And if yes, in your language, of which broken symmetry?)

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@VladimirKalitvianski you write " I am clearly speaking of existence or non existence of physical solutions of a Theory equations/Lagrangian you make some transformations of. I suppose you imply existence of such solutions. Can you confirm my supposition, please?"

Sure, we are talking about examples like the Yang-Mills Lagrangian and the Chern-Simons Lagrangian, whose equations of motion are well known and whose solution spaces are well studied.

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@JakobS I still have to study these articles to answer these questions. Maybe later.

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Dear Urs, it was you who taught me that the "gauge transformations between gauge transformations" are redundancy in the redundancy (and so on). One imagines something like a chain complex of gauge symmetries, and the global symmetries are its cohomology.

JakobS, I do not know the precise relationship between 1-form symmetries and asymptotic symmetries but one observation is that the conserved charges are the same, namely surface integrals of F and \star F. We can leverage this observation to propose a thought experiment. We know the 1-form symmetries are spontaneously broken. This implies a (rather large) ground state degeneracy. A natural question is "which of these ground states are coherent?" in the sense "which are classically observed?". Take spontaneous 0-form symmetry breaking as an example. Let Q be the charge operator (a volume integral of j^0 over all space). In a finite size system, there is an exponentially small splitting between energy eigenstates of definite Q, with the neutral Q as the lowest energy state. In the infinite size limit, this degeneracy goes quickly to zero and we see degenerate ground states that are characteristic of spontaneous symmetry breaking. However, these degenerate ground states are distinguished by a global operator Q which is impossible to measure. Instead, there are *local* order parameters f which [Q,f] not 0 which we can measure, and when we do, because of this noncommutation, we completely decohere the charge in the observed vacuum state. The states of definite Q are called "cat states" in the literature by a colorful analogy with Shroedinger's famous feline.

So in the case of our 1-form symmetries, Q is now a surface integral and f is something like a Wilson line. In the usual classical gauge theory basis (where we expect to see photons), the Wilson line is a diagonal operator e^i \int A and so our observed states have indefinite Q. However, suppose we only have experimental access to a finite region of spacetime (a causal diamond, say). Then only the Q and f which intersect can noncommute, and we can only measure f in our small region. Therefore, we can imagine a preparation of the system where we observe states with both gauge bosons and definite Q *at infinity*. I think these are the states Strominger et al are talking about. See also this paper https://arxiv.org/abs/1709.08632 which shows that indeed the asymptotic charges are precisely those which are inaccesible to our experiments.

Besides being incredibly impressionistic, this discussion still surffers from the problem that on a contractible space there are no nontrivial 1-form symmetry generators. Maybe one needs to think about a causal diamond inside a torus or something? I don't know. At best it seems like the two approaches are clearly related but not the same. I would guess that Strominger's approach is the shadow of the 1-form symmetry approach in contractible space. I'm happy to discuss more, over email, say.