Photon helicity: Wigner's unitary rep. of Poincare group and gauge symmetry

[tom.stoer]

1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |k^μ, h> with k_μk^μ = 0 and h = ±1 and we have two d.o.f.

2) For gauge theories with massless gauge bosons like QED and QCD it is well known that the 4-vector A_μ carries two unphysical d.o.f. which can be eliminated by gauge fixing (a la Dirac, Gupta-Bleuler, BRST, ...). An obvious way to see this is to
i) use the temporal gauge A° = 0 to eliminate one unphysical d.o.f. A° (∏° = 0 b/c there's no ∂°A° in the Lagrangian ~ F²)
ii) keep the corresponding Euler-Lagrange equation (Gauss law G) as constraint to define the physical Hilbert space as its kernel G|phys> = 0 which fixes the residual gauge symmetry of time-indep. gauge transformations ∂°θ = 0
b/c we have 4 components in A_μ and 2 gauge fixing conditions A° = 0 and G ~ 0 we arrive at 4-2 = 2 d.o.f.

The method 2) gives us exactly the two helicity states described in 1) But 1) is using Poincare invariance whereas 2) is using gauge invariance w/o ever looking at Poincare invariance. So it seems that it's sheer coincidence that 1) and 2) arrive at the same results.

Where's the relation?

 

[DrDu]

Hm, you are looking at the gauge transformation properties of a 4-vector for a massless particle.
"4-vector" and "massless" nicely specify the representation of the Poincare group. So I don't see that 2) is never looking at Poincare invariance.

 

[DrDu]

On the other hand, there are Weyl fermions which are also massless two component helicity eigenstates but transform differently under gauge transformations.

 

[tom.stoer]

The simple question is: why does gauge symmetry reduce exactly from 4 to 2 d.o.f. as required by the Poincare representation. Doing the math there is no obvious relation (of couzrse everything is Poincare invaraint, but Poincare invariance does not know anything about gauge symmetries)

 

[George Jones]

Isn't there a relationship between 1) and 2)? See VI of Weinberg's quantum field theory text, and also papers by Kim.

Caveat: I wrote the above without giving any real thought to the matter.

 

[tom.stoer]

I'll have a look at Weinberg's book

 

[TrickyDicky]

Tom, I'm not completely sure what you mean by "Poincare invariance does not know anything about gauge symmetries", can you elaborate?

 

[tom.stoer]

I mean that when you go through the math of 1) and 2) there is absolutely now relation between both approaches; however they both arrive at the same result.

 

[TrickyDicky]

The Poincare group is the group of isometries of Minkowski space, the 4-vector Aμ of QED is brought from the Minkowskian formulation of the Maxwell equations and these equations have that gauge symmetry also in the QM context (since we are in QFT), so I guess this is how the gauge symmetry knows about the Poincare invariance (rather than the other way around), does this make any sense?

 

[tom.stoer]

Yes, of course this guess makes sense. I think there is some deep connection, but I can't see it. That's why I am asking.

btw.: the same applies to linearized gravity as well; there are two graviton helicity states; and there is a gauge symmetry which reduces 10 components of the metric to two d.o.f.

 

[TrickyDicky]

Yes, of course this guess makes sense. I think there is some deep connection, but I can't see it. That's why I am asking.

btw.: the same applies to linearized gravity as well; there are two graviton helicity states; and there is a gauge symmetry which reduces 10 components of. the metric to two d.o.f.

I also think the connection between gauge symmetries and the Poincare group is worth invstigating.
Linearized gravity has as background Minkowski spacetime so it also seems the gauge symmetries d.o.f. reduction is related to that fact.

 

[tom.stoer]

No, it isn't. The same reduction i.e. the same number of d.o.f. holds for arbitrary curved spacetimes in GR

 

[strangerep]

The method 2) gives us exactly the two helicity states described in 1) But 1) is using Poincare invariance whereas 2) is using gauge invariance w/o ever looking at Poincare invariance. So it seems that it's sheer coincidence that 1) and 2) arrive at the same results.
Where's the relation?

The Wigner method for massless particles requires an additional input to be put in by hand: that there are no particle types in existence with (so-called) "continuous spin". (Personally, I find that term a bit misleading, but it's in wide usage even though Weinberg doesn't seem to use that phrase.)

In method 2, you presumably have a Lagrangian that respects Poincare invariance, and then you find unwanted degrees of freedom which must be handled/banished somehow, e.g., by gauge-fixing or a constraint approach.

So both methods can be thought of as "Poincare + extra arbitrary input".

IMHO, it is both puzzling and intriguing that the Poincare group does not give exactly the right set of answers for elementary particle classification without some extra empirical input.

 

[DrDu]

I don't think that gauge invariance is necessary in that context. If you take a massive A_mu and let m tend to 0, the longitudinal and time-like photons also decouple from the transversal ones. That's how e.g. Zee calculates the photon propagator.

 

[TrickyDicky]

No, it isn't. The same reduction i.e. the same number of d.o.f. holds for arbitrary curved spacetimes in GR

You are right, quite disturbing, isn't it?
Look at strangerep answer about the extra arbitrary info for both approaches, I think that is the key to the connection.

 

[tom.stoer]

I don't think that gauge invariance is necessary in that context. If you take a massive A_mu and let m tend to 0, the longitudinal and time-like photons also decouple from the transversal ones. That's how e.g. Zee calculates the photon propagator.

This approach fails in non-abelian gauge theories.

 

[TrickyDicky]

This approach fails in non-abelian gauge theories.

Right, it is the non-abelian case that needs by-hand additions, not justified by the Poincare group. After all, the Poincare translations are abelian so why should it inform non-abelian gauge symmetries?

 

[tom.stoer]

Right, it is the non-abelian case that needs by-hand additions, not justified by the Poincare group.

What do you mean by "by-hand additions"?

... the Poincare translations are abelian so why should it inform non-abelian gauge symmetries?

The Poincare group always commutes with local gauge symmetries, even for the non-abelian case; they have nothing to do with each other. And it's not only about translations but about the full non-abelian Poincare group

 

[TrickyDicky]

What do you mean by "by-hand additions"?

I guess the same thing as strangerep in #13.

And it's not only about translations but about the full non-abelian Poincare group

How does that contradict the fact that the translation subgroup is abelian?
I must have misunderstood you, so what "fails in non-abelian gauge theories" in your opinion then?

 

[DrDu]

This approach fails in non-abelian gauge theories.

How?

 

[tom.stoer]

Starting with massive vector bosons with the limit m → 0 fails in non-abelian gauge theories (afaik it breaks the SlavnovTaylor identities which indicates an anomaly, i.e. the gauge symmetry is bot restored in the limit m → 0; but I am not sure about that)

 

[TrickyDicky]

I think the photon's helicity by itself is not the best example to clarify what you seem to be interested in, I mean how is it related to renormalization (Slavnov-Taylor) or non-abelian gauges?

 

[tom.stoer]

I think the photon's helicity by itself is not the best example to clarify what you seem to be interested in, I mean how is it related to renormalization (Slavnov-Taylor) or non-abelian gauges?

It isn't.

This was only to clarify that the above mentioned m² → 0 limit does not work for non-abelian gauge theories.

The approach mentioned in 2) in the post #1 is quite general and does work for both abelian and non-abelian gauge theories. In addition in the canonical approach using A°=0 gauge plus gauge fixing of residual symmetries generated by the Gauss law there are no Slavnov-Taylor identitites b/c the gauge symmetry is reduced to the identity in the physical Hilbert space.

So the question from post #1 still remains: what is the hidden relation between 1) two physical helicity states derived from Poincare representations and 2) two physical helicity states derived from gauge fixing. No answer so far :-(

 

[TrickyDicky]

So the question from post #1 still remains: what is the hidden relation between 1) two physical helicity states derived from Poincare representations and 2) two physical helicity states derived from gauge fixing. No answer so far :-(

Maybe not the answer you are looking for but there's been some answers ;-)
The hidden relation seems to lie on the unspoken aasumptions, in 1) it's not true that is derived only from the Poincare rep. You are also singling out one of the possibilities for photons in the Wigner classification, leaving out the continuous spin one.
And in 2) you are fixing gauges in such way that you obtain the same result(and using the EM gauge symmetry from Maxwell).
Is there additionally some meaningful connection we can't see? Maybe, but IMHO this particular case doesn't seem to need it.

 

[tom.stoer]

The hidden relation seems to lie on the unspoken aasumptions, in 1) it's not true that is derived only from the Poincare rep. You are also singling out one of the possibilities for photons in the Wigner classification, leaving out the continuous spin one.

I have to check this.

And in 2) you are fixing gauges in such way that you obtain the same result(and using the EM gauge symmetry from Maxwell).

I do not fix the gauge in such way that I obtain the same result. I simply fix the gauge! Gauge fixing always means eliminating unphysical d.o.f. But there is no choice. There are no different approaches to with more or less d.o.f.; the result is unique.

 

[DrDu]

Starting with massive vector bosons with the limit m → 0 fails in non-abelian gauge theories (afaik it breaks the SlavnovTaylor identities which indicates an anomaly, i.e. the gauge symmetry is bot restored in the limit m → 0; but I am not sure about that)

Does this hold also in classical field theory or does it only occur in connection with renormalization?

 

[tom.stoer]

Good question; I don't know. If it's really the Slavnov-Taylor identity then it's obviously only true in quantum field theory - which is strange b/c it means that a quantization a la Dirac (first solve the constraint - then quantize) should work.

 

[TrickyDicky]

http://physics.stackexchange.com/qu...ave-only-two-possible-eigenvalues-of-helicity

 

[tom.stoer]

rather interesting, but still w/o any explicit explanation regarding a relation of 1) and 2) I'll check Weinberg but as far as I remember he doesn't explain this

 

[strangerep]

So the question from post #1 still remains: what is the hidden relation between 1) two physical helicity states derived from Poincare representations and 2) two physical helicity states derived from gauge fixing. No answer so far :-(

Really? I'm hurt.

Just kidding. Here's a more expanded attempt at an answer...

In the massless Poincare irreps one encounters the little group ISO(2), aka E(2). The two translation-like generators of the latter suggest that particle types with continuous spin should exist. Since no such particles are known, one must postulate that only massless particles exist which transform trivially under those generators. Weinberg sect 2.5 discusses this, but I find the explanation in Maggiore sect 2.7 a helpful additional reference.

Later, one finds that the photon fields constructed in this way do not transform covariantly by themselves -- which is not surprising since the translation-like generators in ISO(2) are the contractions from 2 of the generators of the ordinary SO(3) rotation group.

Weinberg explains that to "solve" this dilemma, one couples this recalcitrant photon field to a conserved current in the Lagrangian in such a way that the latter compensates for the noncovariant behaviour of the photon field by itself. But within this (minimal) coupling recipe lurk extra gauge degrees of freedom. If there is indeed a "hidden relation" of the kind you wanted, I think this is it.