Representations of the Poincare group, spin and gauge invariance

Whatever deserves the name ''particle'' must move like a single, indivisible object. The Poincare group must act on the description of this single object; so the state space of the object carries a unitary representation of the Poincare group. This splits into a direct sum or direct integral of irreducible reps. But splitting means divisibility; so in the indivisible case, we have an irreducible representation. Thus particles are described by irreducible unitary reps of the Poincare group. Additional parameters characterizing the irreducible representation of an internal symmetry group = gauge

On the other hand, not all irreducible unitary reps of the Poincare group qualify. Associated with the rep must be a consistent and causal free field theory. As explained in Volume 1 of Weinberg's book on quantum field theory, this restricts the rep further to those with positive mass, or massless reps with quantized helicity.

Weinberg's book on QFT argues for gauge invariance from causality + masslessness. He discusses massless fields in Chapter 5, and observes (probably there, or in the beginning of Chapter 8 on quantum electrodynamics) roughly the following:

Since massless spin 1 fields have only two degrees of freedom, the 4-vector one can make from them does not transform correctly but only up to a gauge transformation making up for the missing longitudinal degree of freedom. Since sufficiently long range elementary fields (less than exponential decay) are necessarily massless, they must either have spin <1 or have gauge behavior.

To couple such gauge fields to matter currents, the latter must be conserved, which means (given the known conservation laws) that the gauge fields either have spin 1 (coupling to a conserved vector current), or spin 2 (coupling to the energy-momentum tensor). [Actually, he does not discuss this for Fermion fields, so spin 3/2 (gravitinos) is perhaps another special case.]

Spin 1 leads to standard gauge theories, while spin 2 leads to general covariance (and gravitons) which, in this context, is best viewed also as a kind of gauge invariance.

There are some assumptions in the derivation, which one can find out by reading Weinberg's papers

• Phys.Rev. 133 (1964), B1318-B1322 any spin (massive)
• Phys.Rev. 134 (1964), B882-B896 any spin II (massless)
• Phys.Rev. 135 (1964), B1049-B1056 grav. mass = inertial mass
• Phys.Rev. 138 (1965), B988-B1002 derivation of Einstein
• Phys.Rev. 140 (1965), B516-B524 infrared gravitons
• Phys.Rev. 181 (1969), 1893-1899 any spin III (general reps.)
on 'Feynman rules for any spin' and some related questions, which contain a lot of important information about applying the irreducible representations of the Poincare group for higher spin to field theories, and their relation to gauge theories and general relativity. A perhaps more understandable version of part of the material is in
Note that there are plenty of interactions that can be constructed using the representation theory of the Lorentz group (and Weinberg's constructions), and there are plenty of (compound) particles with spin >2. See the tables of the particle data group, e.g., Delta(2950) (randomly chosen from the table).
constructs covariant propagators and complete vertices for spin J bosons with conserved currents for all J. See also

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ