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

In summary: What is the relation?In summary, Poincare invariance does not know anything about gauge symmetries, but two methods that use that symmetry (1) and (2) both arrive at the same result. This suggests that the connection between the two is deeper than initially thought.
  • #36
@hch71: you are right, there is again this strange relation between Poincare and gauge symmetry; for consistency reasons only specific dynamical models are allowed; quantum gauge symmetry is more restrictive than Poincare invariance.

Is this another coincidence? It is interesting that such a consistency condition shows up during regularization at the level of Slavnos-Taylor identities. 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.

Both structrures are related via fiber bundels; Poincare acts on the base space, gauge acts on the fibers. But gauge and Poincare do commute, so the overall symmetry structure is a direct product w/o and relation at the level of Lie groups. Perhaps some unifying principle (SUGRA? strings?) can tell us more about such a hidden principle.
 
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  • #39
thanks a lot for the references!
 
  • #40
You are welcome!
If I understand correctly, the situation described in the book and the paper is analogous to the transition from the Lorentz Group to the homogeneous Galilei group. Mathematically this is an Inonu-Wigner contraction for the limit c to infinity. In that limit there also arises a gauge transformation, namely for the phase of the wavefunction whose generator is mass in Galilean relativity.
What is strange is that the massless representations appear as limits of the massive representations and gauge transformations as ghosts of rotations/ boosts around axes perpendicular to p. Somehow one would expect massless representations to play a fundamental role.
 
  • #41
tom.stoer said:
@hch71: you are right, there is again this strange relation between Poincare and gauge symmetry; for consistency reasons only specific dynamical models are allowed; quantum gauge symmetry is more restrictive than Poincare invariance.

Is this another coincidence? It is interesting that such a consistency condition shows up during regularization at the level of Slavnos-Taylor identities. 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.

Both structrures are related via fiber bundels; Poincare acts on the base space, gauge acts on the fibers. But gauge and Poincare do commute, so the overall symmetry structure is a direct product w/o and relation at the level of Lie groups. Perhaps some unifying principle (SUGRA? strings?) can tell us more about such a hidden principle.
Last time I tried to make a connection between a spacetime symmetry and gauge symmetries (https://www.physicsforums.com/showthread.php?t=622084), I was reminded of the Colema-Mandula theorem, doesn't it apply here?
 
  • #42
Well, it does apply, that's what Tom was saying that "gauge and Poincaré do commute, so the overall symmetry structure is a direct product w/o and relation at the level of Lie groups". That's the Coleman-Mandula theorem in a nutshell.
 
  • #43
Yes.

And it's strange that Poincare and gauge symmetry are unrelated as Lie groups (and cannot be related due to the Coleman-Mandula theorem), but that they are somehow related w.r.t. their representations; that's the starting point of this thread.

But I still have to check the references
 
  • #44
And it's strange that Poincare and gauge symmetry are unrelated as Lie groups (and cannot be related due to the Coleman-Mandula theorem), but that they are somehow related w.r.t. their representations; that's the starting point of this thread.
The wiki page http://en.wikipedia.org/wiki/Coleman-Mandula_theorem says the theorem only applies to Lie algebras, not to the Lie groups. So it would apply to the Poincare algebra and the local gauge symmetry, not to the Lie groups. This is consistent with the sentence dextercioby quotes from you but not with this quote from your last post.
 
  • #45
It applies to Lie groups as well b/c they are nothing else but exponentiated Lie algebras.

Suppose you have two Lie algebras X and Y with two sets of generators

[tex]\{X^a\}, \{X^m\}[/tex]

and defining commutators and structure constants

[tex][X^a, X^b] = if^{abc}\,X^c[/tex]

[tex][Y^m, Y^n] = ig^{mnk}\,Y^k[/tex]

The two algebras commute, i.e.

[tex][X^a, Y^m] = 0;\;\forall a,m[/tex]

This is what is required by the Coleman-Mandula theorem for a gauge Lie algebra X and the Poisson algebra Y.

Now define the Lie group elements as

[tex]U[\theta] = \exp(i\,X^a\,\theta^a)[/tex]

[tex]V[\phi] = \exp(i\,Y^m\,\phi^m)[/tex]

b/c the generators of X and Y do commute, the group elements do commute as well, i.e.

[tex]U[\theta]\,V[\phi] = V[\phi]\,U[\theta];\;\forall \theta^a,\phi^m [/tex]
 
  • #46
Going from Lie algebras to Lie groups goes in this way only for symmetry groups (such as the universal cover of ISO(1,3) or the gauge groups SU(2) and SU(3)) which are simply connected.
 
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  • #47
dextercioby said:
Going from Lie algebras to Lie groups goes in this way only for symmetry groups (such as the universal cover of ISO(1,3) and or the gauge groups SU(2) and SU(3)) which are simply connected.

That' is my understanding too, that the exponentiation procedure to go from Lie algebra to Lie group doesn't apply in general, only in special cases due to the symmetry or in the flat case where one can identify the tangent space of a manifold with the manifold itself.
 
  • #48
TrickyDicky said:
That' is my understanding too, that the exponentiation procedure to go from Lie algebra to Lie group doesn't apply in general, only in special cases due to the symmetry or in the flat case where one can identify the tangent space of a manifold with the manifold itself.
Afaik it applies to each component.

What I wanted to show is that two commuting Lie algebras introduce two commuting Lie groups; this holds for all components of the Poincare group b/c you can write all group elements as matrices

[tex]X^a_i = S_i\,X^a\,T_i[/tex]

or something like that; here i labels the connected components with i=0 containing the identity
 
  • #49
tom.stoer said:
What I wanted to show is that two commuting Lie algebras introduce two commuting Lie groups; this holds for all components of the Poincare group b/c you can write all group elements as matrices

[tex]X^a_i = S_i\,X^a\,T_i[/tex]

or something like that; here i labels the connected components with i=0 containing the identity

In this case it does seem to work out.
 
  • #50
What I wanted to say is that there is no known way to betray Coleman and Manula except via introducing SUSY
 
  • #51
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.
 
  • #52
DrDu said:
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.
 
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  • #53
DrDu said:
[...] 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.
 
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  • #54
tom.stoer said:
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. :-)
 
  • #55
tom.stoer said:
[...] 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.
 
  • #56
strangerep said:
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.
 
  • #57
strangerep said:
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.
 
  • #58
DrDu said:
[...] 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.
 
  • #59
DrDu said:
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.)
 
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  • #60
strangerep said:
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?

strangerep said:
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.
 
  • #61
strangerep said:
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 [Broken]
especially formula V.7
 
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  • #62
One question, certainly the Poincare group and the gauge symmetry group commute, does the little group-SE(2) commute with the gauge symmetry?
 
  • #63
yes, why not?
 
  • #64
tom.stoer said:
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.
 
  • #65
tom.stoer said:
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.
 
  • #66
the paper on arxiv(arxiv:1403.2698) maybe explain it.
 
  • #67
thanks for the hint; seems to be a new explanation based on spacetime-symmetry, but still unrelated to local gauge invariance
 
  • #68
time601 said:
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.
 
<h2>1. What is photon helicity?</h2><p>Photon helicity refers to the intrinsic angular momentum of a photon, which can be either left-handed or right-handed. It is related to the direction of the photon's spin, and plays an important role in understanding the behavior of photons in various physical processes.</p><h2>2. What is Wigner's unitary representation of the Poincare group?</h2><p>Wigner's unitary representation is a mathematical framework that describes the behavior of particles, including photons, under the symmetries of the Poincare group. This representation is used in quantum field theory to understand the properties and interactions of particles.</p><h2>3. How does gauge symmetry relate to photon helicity?</h2><p>Gauge symmetry is a fundamental symmetry in physics that describes the invariance of physical laws under certain transformations. In the case of photon helicity, gauge symmetry is related to the conservation of angular momentum and the behavior of photons in different reference frames.</p><h2>4. What is the significance of the Poincare group in understanding photon helicity?</h2><p>The Poincare group is a mathematical structure that describes the symmetries of spacetime, including translations, rotations, and boosts. Understanding the behavior of photons, including their helicity, requires a deep understanding of these symmetries and how they relate to the properties of particles.</p><h2>5. How is Wigner's unitary representation used in practical applications?</h2><p>Wigner's unitary representation is used in many practical applications, including particle physics experiments, quantum computing, and the development of new technologies. It provides a powerful mathematical framework for understanding the behavior of particles and their interactions, and has led to many important discoveries in the field of physics.</p>

1. What is photon helicity?

Photon helicity refers to the intrinsic angular momentum of a photon, which can be either left-handed or right-handed. It is related to the direction of the photon's spin, and plays an important role in understanding the behavior of photons in various physical processes.

2. What is Wigner's unitary representation of the Poincare group?

Wigner's unitary representation is a mathematical framework that describes the behavior of particles, including photons, under the symmetries of the Poincare group. This representation is used in quantum field theory to understand the properties and interactions of particles.

3. How does gauge symmetry relate to photon helicity?

Gauge symmetry is a fundamental symmetry in physics that describes the invariance of physical laws under certain transformations. In the case of photon helicity, gauge symmetry is related to the conservation of angular momentum and the behavior of photons in different reference frames.

4. What is the significance of the Poincare group in understanding photon helicity?

The Poincare group is a mathematical structure that describes the symmetries of spacetime, including translations, rotations, and boosts. Understanding the behavior of photons, including their helicity, requires a deep understanding of these symmetries and how they relate to the properties of particles.

5. How is Wigner's unitary representation used in practical applications?

Wigner's unitary representation is used in many practical applications, including particle physics experiments, quantum computing, and the development of new technologies. It provides a powerful mathematical framework for understanding the behavior of particles and their interactions, and has led to many important discoveries in the field of physics.

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