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

[DrDu]

Although the mass zero representations involve some gauge transformations, I got the impression that the gauge transformations are path independent with base space being labeled by the parameters of the Poincare group. So in fiber space, the section is still trivial.
Hence also the Coleman-Mandula theorem is not violated.

 

[strangerep]

Weinberg states that the continuous spin states for m=0 can be excluded for topological reasons,

Precisely where does he say that? I'm thinking maybe this is an example of how the term "continuous spin" can be misleading. See Weinberg vol1, bottom of p71 and top of p72. The particular representations being discussed here are excluded for phenomenological reasons: "Massless particles are not observed to have any continuous degree of freedom like $\theta$; [...]" -- top of p72.

The topological thing is a distinct issue...

ie. a 4π rotation can be transformed along a continuous path into a 0 rotation whence only integer or non-integer spins are possible.

Having effectively excluded any nontrivial action by the two translation-like generators of ISO(2), we are left with $J_3$. (See Weinberg's eq(2.5.39) and discussion following.) To recover the usual restriction to integer or half-integer spins for massless particles, when we have only $J_3$ to work with (and the usual algebraic proof does not go through), he uses a topological argument later in ch2 -- see top of p90.

 

[strangerep]

[...] and gauge transformations as ghosts of rotations/ boosts around axes perpendicular to p. [...]

Y. S. Kim and collaborators also advocated that line, but never (afaik) explained how those transformations applied to the massless $A_\mu$ field are also required (by Poincare analysis alone) to act as local phase factors on the massive electron field. I think this is a serious flaw/omission.

 

[strangerep]

And it's strange that Poincare and gauge symmetry are unrelated as Lie groups (and cannot be related due to the Coleman-Mandula theorem), [...]

Since Poincare transformations are physically implementable, but gauge transformations are not, I actually find it comforting that the C-M thm says what it does. :-)

 

[strangerep]

[...] and why Casimir operators of Poincare symmetry tell something about the construction of gauge symmetries.

Poincare symmetry is really only constraining the form of the interaction. Cf. the usual "instant form" of dynamics -- we wish to construct interacting representations of the Poincare group, but later find that in general they are unitarily inequivalent to the free representations. Perhaps this is not surprising, since the interacting theory typically admits a different/larger dynamical group than applies to the free theory. Also, the C-M thm does not exclude transformations which map between inequivalent representations, but afaik this possibility has not been explored by anyone, at least not to the point of constructing a satisfactory theory.

 

[DrDu]

Y. S. Kim and collaborators also advocated that line, but never explain how those transformations applied to the massless $A_\mu$ field also are also required (by Poincare analysis alone) to act a local phase factors of the massive electron field. I think this is a serious flaw, or at least a notable omission.

Yes, Kim has published a lot of articles on that topic. But maybe it is noteworthy that one of his collaborators has been Wigner himself, e.g.:
http://www.ysfine.com/yspapers/stg.pdf
As far as the transformation properties of the electron are concerned, I don't think there is a problem. Evidently the electronic wave function may be multiplied by an arbitrary phase factor as long as it is path independent, i.e. only depends on the parameters of the total Poincare transform.

 

[DrDu]

Since Poincare transformations are physically implementable, but gauge transformations are not, I actually find it comforting that the C-M thm says what it does. :-)

Btw this is not the case in Galilean relativity. There is a combination of translations, boosts time reversal which is equivalent to identity in Galilean group but generates a phase factor proportional to mass in the extended Galilean group. This leads to mass superselection in Galilean relativity.

 

[strangerep]

[...] maybe it is noteworthy that one of his collaborators has been Wigner himself, e.g.: http://www.ysfine.com/yspapers/stg.pdf

I know, but it still doesn't address my objection, no matter who the authors are. :-)

As far as the transformation properties of the electron are concerned, I don't think there is a problem. Evidently the electronic wave function may be multiplied by an arbitrary phase factor as long as it is path independent, i.e. only depends on the parameters of the total Poincare transform.

My point is that although the electronic wave function may be multiplied by such an arbitrary phase factor, this remains an arbitrary manual injection into the theory. The Poincare irrep analysis alone does not compel it, nor do the EM gauge transformations necessarily induce such a transformation on the electron -- one must write down the interacting Lagrangian to find it.

 

[strangerep]

Btw this is not the case in Galilean relativity. There is a combination of translations, boosts time reversal which is equivalent to identity in Galilean group but generates a phase factor proportional to mass in the extended Galilean group. This leads to mass superselection in Galilean relativity.

OK, I'm not familiar with the details of that. Could you give me a reference, pls?

(BTW, are you familiar with Gerry Kaiser's approach in which the mass term in the centrally extended Galilean group arises by contraction from the Poincare group -- if one remains in an irreducible representation of the latter, i.e., preserving the $M^2$ Casimir (hence remaining on-shell) during the contraction. What you said above sounds somewhat different, though.)

 

[DrDu]

My point is that although the electronic wave function may be multiplied by such an arbitrary phase factor, this remains an arbitrary manual injection into the theory. The Poincare irrep analysis alone does not compel it,

Of course, nevertheless I would not consider this a flaw of the theory.
How should a massive particle alone know that you are considering other particles to be present?

nor do the EM gauge transformations necessarily induce such a transformation on the electron -- one must write down the interacting Lagrangian to find it.

How should a massive particle alone know that you are considering other particles to be present?
I think you have to analyze the transformation of representations containing both an electron and a photon.

 

[DrDu]

OK, I'm not familiar with the details of that. Could you give me a reference, pls?

http://jmp.aip.org/resource/1/jmapaq/v4/i6/p776_s1
especially formula V.7

 

[TrickyDicky]

One question, certainly the Poincare group and the gauge symmetry group commute, does the little group-SE(2) commute with the gauge symmetry?

 

[tom.stoer]

yes, why not?

 

[A. Neumaier]

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)

The problem in the nonabelian case is that a massive nonabelian gauge theory (without a symmetry-breaking mechanism) is not renormalizable. If nonrenormalizable theories were better understood, the massless limit would probably work out well.

 

[A. Neumaier]

I would expect an underlying principle (unfortunately unknown) which explains why gauge symmetry adds constraints to Poincare multiplets and why Casimir operators of Poincare symmetry tell something about the construction of gauge symmetries.

Gauge symmetry adds constraints to Poincare multiplets because an interaction with a gauge field can be gauge invariant only if the coupling is to a conserved current. A Fourier transform gives the transversality condition. But transversality is Poincare invariant only in massless representations.

Weinberg's paper Phys.Rev. 134 (1964), B882-B896 should explain everything to your satisfaction.

 

[time601]

the paper on arxiv(arxiv:1403.2698) maybe explain it.

 

[tom.stoer]

thanks for the hint; seems to be a new explanation based on spacetime-symmetry, but still unrelated to local gauge invariance

 

[strangerep]

the paper on arxiv(arxiv:1403.2698) maybe explain it.

I think that paper does not "explain" it any better than Weinberg. Banishment of the extra degrees of freedom (so-called "continuous spin") is still quite arbitrary. See top right of their p3 and also the last paragraph of their conclusion.

Weinberg's justification is that particles with such continuous spin degrees of freedom are not observed experimentally. Hence we arbitrarily ignore that possibility when constructing photonic quantum fields.

The justification of Chang-Li and Feng-Jun in the above paper is just as arbitrary: they seem to think that "just admitting" that certain states are unphysical is somehow different from Weinberg. They seem to think that Weinberg is using an extra "experimental hypothesis", whereas in fact he is just referring to (lack of) experimental evidence as a way to make the arbitrary exclusion reasonable.