The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle

Abstract

Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR0,R+, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p2,p), (p2,p+,pT), (p2,p−,pT), (p2,π), and (p2,π+,πT) where\(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \). We find that, only in the case ofR0 with (p2,p) coordinates,R+ with (p2,p+,pT) coordinates, andD with (p2, π) or (p2,π+,πT) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.