The coset construction for particles of arbitrary spin

Abstract

When a Poincaré-invariant system spontaneously breaks continuous internal symmetries, Goldstones's theorem demands the existence of massless, spin-zero excitations in a one-to-one correspondence with the broken symmetry generators. When a system spontaneously breaks Poincaré symmetry, however, the kinds of excitations that satisfy Goldstone's theorem can be quite unusual. In particular, they may have any spin and need not be particles or even quasiparticles. The standard coset construction used to formulate effective actions of Goldstones, however, is rather restrictive and is incapable of generating the full spectrum of possibilities allowed by Goldstone's theorem. We propose a (partial) remedy to this problem by postulating a novel coset construction for systems that spontaneously break Poincaré symmetry. This new construction is capable of generating effective actions with a wide range of Goldstone excitations---including fermionic degrees of freedom---even when all symmetries are bosonic. To demonstrate it's utility, we focus on constructing effective actions for point particles of various spins. We recover the known result that a particle of spin $s$ requires an $\mathcal N=2s$ supersymmetric worldline reparameterization gauge symmetry, which we implement at the level of the coset construction. In the process, we discover that massless particles require a novel kind of inverse Higgs constraint that bears some resemblance to the dynamical inverse Higgs constraints that appear in certain fermi liquid effective field theories. We then consider particles that, in addition to quantum spin, have finite spatial extent and are free to rotate. We derive a novel action for such particles and find a `spin-orbital' coupling between the intrinsic quantum spin and the physical-rotation degrees of freedom.