What is the correct possibility space for a given particle of a quantum field? The firm orthodoxy, initiated by Wigner in 1939, is that we should look to the irreducible representations of the Poincaré group. But the justifications for this are murky, and offer two rational reconstructions. According to one, the justification is that the Poincaré group comprises all the actions that can be performed on a simple system; but this is false, made particularly relevant in the case of interactions. According to another, conflicting reconstruction, Poincaré invariance is a reality condition; but this applies only to the entire system of fields, not to individual particles. Therefore it ought not constrain the possibility spaces of individual particles.
Consequently, I argue, we have reason to consider “larger” state spaces than are normally considered. On one such approach, which I will advocate, the elements of these state spaces represent entire four-dimensional histories rather than instantaneous states—most of which are “off mass-shell”, in that the relativistic mass condition is violated. I conclude by arguing that off-mass-shell particles, far from being mere formal inventions for the sake of S-matrix calculations, are just as “real” as their free counterparts—indeed, they are to be found as much in classical relativistic theories as in quantum relativistic theories.