Abstract
Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR 0,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: (p 2,p), (p 2,p +,p T ), (p 2,p −,p T ), (p 2,π), and (p 2,π +,π T ) where\(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \). We find that, only in the case ofR 0 with (p 2,p) coordinates,R + with (p 2,p +,p T ) coordinates, andD with (p 2, π) or (p 2,π +,π T ) 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.
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Almond, D.J. The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle. Z. Phys. C - Particles and Fields 15, 71–92 (1982). https://doi.org/10.1007/BF01573428
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DOI: https://doi.org/10.1007/BF01573428