My impression for how Poincare and Lorentz representations are linked in $3+1$ dimensions is:

  • Assuming positive mass for simplicity, irreducible representations of the Poincare group are indexed by their mass $M$ and spin $s$.
  • Irreducible representations of the Lorentz group are indexed by two numbers $(s_1, s_2)$ and when restricted to the rotation subgroup contains spins from $|s_1 - s_2|$ up to $s_1 + s_2$ in integer steps.
  • Particles transform in representations of the Poincare group, while fields transform in representations of the Lorentz group. They are related by defining $$|\mathbf{p}, \lambda \rangle = a^\dagger(\mathbf{p}, \lambda) |0 \rangle$$ where $\lambda$ is the helicity, and the field is defined in terms of the creation and annihilation operators in the usual way.
  • The only requirement is that the Lorentz representation used for the field contains the spin $s$ used in the Poincare irrep. A relativistic wave equation is imposed to eliminate the unwanted degrees of freedom.

For example, a massive spin $1$ particle may be embedded in a vector field, which is in the $(1/2, 1/2)$ irrep of the Lorentz group. This vector field has one extra degree of freedom that is eliminated by the equation $\partial_\mu A^\mu = 0$.

Assuming this is right, I'm wondering why I've never heard of "alternative" embeddings. For instance, why can't one embed a spin $1$ particle inside the $(1, 0)$ irrep, which doesn't have any extra degrees of freedom at all, or inside the $(1, 1)$ irrep? For massive particles, I've only ever seen spin $0$ in $(0, 0)$, spin $1/2$ in $(1/2, 0) + (0, 1/2)$, and spin $1$ in $(1/2, 1/2)$. Are other embeddings very clunky, or not allowed, or are they actually used somewhere?


1 Answer 1


A field in the $(A,B)$ representation of the Lorentz Group has a propagator that scales as $|p|^{2(s-1)}$ for $|p|\to\infty$, where $s=A+B$ is the "spin" of the field (Ref.1 §12.1). Therefore, the propagator is a decaying (or constant) function of $p$ if and only if $s=0,\,1/2,\,1$. Otherwise, the propagator grows in the UV and the theory is non-renormalisable (unless we have SUSY1, which in principle may allow you to go up to $s=3/2,2$). If fields of that type are actually realised in Nature, their effect is invisible in the IR (essentially, by dimensional analysis; more formally, by the standard classification of irrelevant interactions, cf. Ref.1 §12.3). That is why they have not been detected so far.

This leaves as the only options $(0,0),\,(\frac12,0),\,(0,\frac12),\,(1,0),\,(\frac12,\frac12),\,(0,1)$. All of these are used in the Standard Model but for $(1,0),(0,1)$ (these are the self-dual and anti-self dual anti-symmetric second rank representations, respectively). There is nothing intrinsically wrong about these representations; they just happen to be irrelevant for the Standard Model: no known particle is described by such a field. They are indeed sometimes used in toy models. Let me in fact quote a paragraph from Ref.1 §5.9:

Although there is no ordinary four-vector for massless particles of helicity $h=\pm1$, there is no problem in constructing an antisymmetric tensor $F_{\mu\nu}$ for such particles. [...] Why should we want to use a [gauge dependent vector field $A^\mu$] in constructing theories of massless particles of spin one, rather than being content with fields like $F_{\mu\nu}$ [which is gauge-independent]? The presence of derivatives in eq. 5.9.34 means that an interaction density constructed solely from $F_{\mu\nu}$ and its derivatives will have matrix elements that vanish more rapidly for small massless particle energy and momentum than one that uses the vector field $A^\mu$. Interactions in such a theory will have a correspondingly rapid fall-off at large distances, faster than the usual inverse-square law. This is perfectly possible, but gauge-invariant theories that use vector fields for massless spin one particles represent a more general class of theories, including those that are actually realized in nature.

Parallel remarks apply to gravitons, massless particles of helicity $\pm2$. [...] in order to incorporate the usual inverse-square gravitational interactions we need to introduce a field $h_{\mu\nu}$ that transforms as a symmetric tensor, up to gauge transformations of the sort associated in general relativity with general coordinate transformations. Thus, in order to construct a theory of massless particles of helicity $\pm2$ that incorporate long-range interactions, it is necessary for it to have a symmetry something like general covariance. As in the case of electromagnetic gauge invariance, this is archived by coupling the field to a conserved "current" $\theta^{\mu\nu}$, now with two spacetime indices, satisfying $\partial_\mu\theta^{\mu\nu}=0$. The only such conserved tensor is the energy-momentum tensor, aside from possible total derivative terms that do not affect the long-range behaviour of the force produced. The fields of massless particles of spin $j\ge3$ would have to couple to conserved tensors with three or more spacetime indices, but aside from total derivatives there are none, so highs-spin massless cannot produce long-range forces.

In short: most "non-standard" representations are fine but phenomenologically useless. The only non-trivial cases are $(1,0),\,(0,1)$, but they don't seem to be realised in Nature. A possible reason is that they mediate short-ranged interactions (but non-confining: it can be proven that confinement arises only if you have non-abelian gauge interactions) and are therefore not seen in actual experiments. If any such particle existed, we would need much larger accelerators.


  1. Weinberg's QFT, Vol.1.

1: Fields of higher spin are always of the gauge type, because of the usual mismatch between field components and particle degrees of freedom. If you consider a bosonic field of arbitrary spin, you can always fix the gauge in such a way that its propagator is $\mathcal O(k^{-2})$ in the UV; similarly, a fermionic field can be gauge-fixed so that its propagator scales as $\mathcal O(k^{-1})$. Thus, it appears that any field is power-counting renormalisable. The catch is that the theory is gauge invariant if and only if you have a Ward-Takahashi-Slavnov-Taylor identity to control the unphysical degrees of freedom. You therefore need a conserved current which, as per Coleman-Mandula-Haag–Łopuszański–Sohnius-etc., is at most of the vector type if bosonic, or has spin $3/2$ if fermionic. In other words, you may only introduce $s=1$ fields if the symmetries form a regular algebra, or $s=2$ if you allow superalgebras. You cannot introduce any higher spin, simply because of the lack of a conserved current (cf. Weinberg's quote above).

Alternatively, if you don't want to fix the gauge (say, by working with massive particles and introducing Proca-like auxiliary conditions), the propagators always grow like $|p|^{2(s-1)}$, and the problematic large-$k$ behaviour is only cancelled if you have conserved currents at the vertices (so that the terms proportional to $k^\mu$ vanish, cf. [this PSE post](https://physics.stackexchange.com/a/383961/84967)). But, as in the previous paragraph, such currents can only be, at most, of the supersymmetric type, so no particle of spin higher than $s=2$ is allowed. See also [Weinberg–Witten theorem](https://en.wikipedia.org/wiki/Weinberg–Witten_theorem).
  • $\begingroup$ With respect to the argument from Weinberg, isn't it more important that the coefficient functions in the mode expansion of eq. 5.9.1 are shown to be proportional to momenta in eq. 5.9.33? It means that if we were to construct theories with $f^{\mu\nu}(x)$ as the dynamical field (or indeed just the $(0,1)$ or $(1,0)$ part), we would yield a different class of propagators than when we construct gauge invariant theories with a vector field. $\endgroup$ Commented Dec 12, 2021 at 15:25

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