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  Why are continuous spin representations forbidden in a local field theory?

+ 6 like - 0 dislike

The question originates from this comment of Arnold. 

(A continuous spin representation is a massless representation of Poincare group which represents nontrivially the generators $J_2+K_1$ and $-J_1+K_2$.)

asked Nov 4, 2014 in Theoretical Physics by Jia Yiyang (2,630 points) [ revision history ]
edited Nov 4, 2014 by Arnold Neumaier
Most voted comments show all comments

BTW, I'm not sure if they are actually forbidden. I'm not well-versed in the reasoning myself, but I recently heard a talk on continuous spin particles by Philip Schuster. You might want to check out some recent work by Natalia Toro and him. The talk had some very interesting content.

@Siva, Thanks for the information!

The one I heard is not available online. Here's an older one available online: http://www.perimeterinstitute.ca/videos/spin-and-long-range-forces-unfinished-tale-last-massless-particle (since it's a nascent area, it's developing quite quickly and talks might get dated every few months). This article doesn't have much physics, but conveys their approach and surprise at the observations. I think the papers should be quite readable.

The relevant papers are arXiv:1302.1198 and arXiv:1302.1577.

@Siva, @JiaYiyang  @ValterMoretti: After having looked more closely at the work by Schuster & Toro, I still can't make sense of it. 

For example, their latest paper on the subject, arxiv:1404.0675, discusses the form of the wave function in momentum space in (4.12). But the equation doesn't make sense - the delta factor restricts the wave function to be defined on the manifold $\eta^2+1=0$, while the second factor contains the differential operator $k\cdot \partial_\eta$ that is not defined on this manifold. I have similar problems interpreting the action (1.1), and hence the whole paper.

Other recent papers on the continuous spin representation such as that by Mund et al. or by Schroer don't produce local fields (as required by the standard S-matrix arguments) but only string-localized fields, whose physical adequacy is still under discussion.

It seems clear that the authors of arxiv:1404.0675 have difficulties turning their construction into one with proper physical content. On p.28, they write: 

We do not yet know of any local matter sector that furnishes an appropriately conserved current that can serve as a source for (5.1) when ρ = 0. 6 This is the central open problem to solve in formulating a complete CSP-theory. 

Moreover, on p.30 they find themselves unable to construct field operators that are both local and gauge invariant, and hence would have physical content. Thus it seems that the analysis given in the references in my answer below remains unchallenged.

Note that the complaint quoted in this comment is inappropriate - the massless vector representation used in QED gives rise to a well-defined local and covariant field strength on the Fock space constructed from the space of physical photon wave functions (e.g., Silberstein vectors).

@ArnoldNeumaier, thanks for the update.

1 Answer

+ 8 like - 0 dislike

Elementary particles must satisfy the principles of relativistic quantum field theory. This implies that they are described by nontrivial irreducible unitary representations of the Poincare group, compatible with a vacuum state and causality.

Having a unitary representation of the Poincare group characterizes relativistic invariance. Irreducibility corresponds to the elementarity of the particle. The vacuum is excluded by forbidding the trivial representation.

Finally, causality requires the principle of locality, namely that commutators (or in case of fermions anticommutators) of the creation and annihilation fields at points with spacelike relative position must commute. Otherwise, the dynamics of distant points would be influenced in a superluminal way.

This rules out many of the irreducible unitary representations (completely classified by Wigner in 1939), leaving only those with nonnegative mass and finite spin. Of the other irreducible unitary representations, all of which were classified by Wigner in 1939, the massless continuous spin (also referred to as infinite spin) representations are those most difficult to dismiss of.

On page 71 of his QFT book, Weinberg simply says that massless particles are not observed to have a continuous degree of freedom. Weinberg uses an empirical fact (''are not observed to have'') to eliminate this case in his analysis. He says that there are such representation, but that they are irrelevant as they don't match observation. One can eliminate the continuous spin representation also by causality arguments; but these arguments are lengthy:

     L.F. Abbott,
     Massless particles with continuous spin indices,
     Phys. Rev. D 13 (1976), 2291-2294.

     K. Hirata,
     Quantization of massless fields with continuous spin,
     Prog. Theor. Phys. 58 (1977) 652-666.

But Weinberg doesn't want to do more representation theory than necessary. Since these representations do not lead to causal quantum fields, he refers to experience to be able to take a shortcut.

However, the literature also discusses almost acceptable variations of traditional quantum fields involving continuous spin representations:

     J. Yngvason,
     Zero-mass infinite spin representations of the Poincare group and quantum field theory,
     Comm. Math. Phys. 18, 195-203 (1970)

     G.J. Iverson and G. Mack,
     Quantum fields and interactions of massless particles: The continuous spin case,
     Annals of Physics 64, 211-253 (1971)

     J. Mund, B. Schroer and J. Yngvason,
     String-Localized Quantum Fields and Modular Localization,
     Comm. Math. Phys 268 (2006), 621-672.

     X. Bekaert and J. Mourad,
     The continuous spin limit of higher spin field equations, 
     J. High Energy Phys. 2006 (2006), 115.

     R. Longo, V. Morinelli and K.-H. Rehren,
     Where infinite spin particles are localizable,

Note that some higher derivative string theories give rise to particles belonging to the continuous spin representation:

    G.K. Savvidy,
    Tensionless strings: physical fock space and higher spin fields,
    Int. J. Mod. Phys. A 19 (2004) 3171-3194. 

    J. Mourad,
    Continuous spin particles from a string theory,

Note also that irreducibility (while characterizing elementary particles) is not necessary for causality. A generalized free causal field theory carrying a reducible representation is described in

      R. F. Streater,
      Local fields with the wrong connection between spin and statistics,
      Comm. Math. Phys. Volume 5, Number 2 (1967), 88-96.

Recent work by Schuster & Toro does not contradict the findings reported in the above references, as the latter are not constructing local and gauge invariant fields, hence fail to give their constructions a proper physical meaning. See this and this comment in this thread. 

For possible relations to dark matter see these papers by Schroer: arxiv.org/abs1601.02477, 

answered Nov 4, 2014 by Arnold Neumaier (12,690 points) [ revision history ]
edited Mar 12 by Arnold Neumaier

Wow, this is packed, I need to chew on your answer. Thanks and +1.

Too bad I have no right to vote up. +1, Arnold!

But I have a question: as far as the real electron is always coupled to the electromagnetic degrees of freedom and has no "definite mass", the proof may fail for such a coupled electron. Those degrees of freedom (whose number is infinite) have also their own "spins" and as a result the "dressed" electron might have an arbitrary and uncertain spin, depending on the EM field state. How comes that those degrees of freedom do not influence the observed electron spin 1/2? Or do they?

All I mentioned is about free field representations, where there is no question of mass being definite. In a coupled theory, the physical electrons (being infraparticles) are states of an asymptotic generalized free electron field with an unsharp mass (i.e., the 2-point function in the Kallen-Lehmann representation is an integral over all states with mass at least the physical electron mass), multiplied with a coherent state of the electromagnetic field (defining the e/m field of the electron, a boosted Coulomb field). Nevertheless, the spin of a physical electron is (by definition) always definite, equal to 1/2, since spin is never renormalized. 

Note that during interaction, i.e., outside the asymptotic regime, only the quantum field representation is sensible, and the concept of an electron doesn't make sense anymore - the system would be in a superposition of an arbitrary (possibly even infinite) number of electrons, positrons, and photons, with arbitrary masses. One can point to individual electrons only when the asymptotics is valid.

Thank you for your answer. +1.

You say the spin is not renormalized, but I speak of its being a "non conserved quantity" in case of coupling. If "before" interaction it was definite and conserved, the interaction with the "proper" field may give another total spin, which we assign to the real electron. Probably there is a mechanism of its "conservation" or "interaction independence".

By the way, we may consider a case of non relativistic scattering to avoid extra electrons and positrons, and deal with soft photons solely where only one electron is present and meaningful.

Please open a new question if you want to have this discussed further, since it has nothing to do with continuous spin representations.


Great answer! In addition, one has to recall that causal fields are required to build a special-relativistic theory, in order to preserve the Lorentz invariance of the temporal ordering of fields. Thus, Lorentz invariance itself forbids continuous degrees of freedom connected with the helicity (at least in local field theories).

In the above very comprehensive answer I noticed incorrect journal references:

Geoffrey J. Iverson, and Gerhard Mack,
Quantum Fields and Interactions of Massless Particles: the Continuous Spin Case
Annals of Physics 64, 211-253 (1971)

Jakob Ynvason,
Zero-Mass Infinite Spin Representations of the Poincaré Group and Quantum Field Theory
Commun. math. Phys. 18, 195-203 (1970)

Thanks; corrected!

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