History state formalism for scalar particles

DIAZ, NAHUEL L.; MATERA, JUAN MAURICIO; ROSSIGNOLI, RAUL D.

Physical Review D

Americal Physical Society

Año: 2019 vol. 100

2470-0010

We present a covariant quantum formalism for scalar particles based on anenlarged Hilbert space. The particular physical theory can be introducedthrough a timeless Wheeler DeWitt-like equation, whose projection ontofour-dimensional coordinates leads to the Klein Gordon equation. The standardquantum mechanical product in the enlarged space, which is invariant andpositive definite, implies the usual Klein Gordon product when applied to itseigenstates. Moreover, the standard three-dimensional invariant measure emergesnaturally from the flat measure in four dimensions when mass eigenstates areconsidered, allowing a rigorous identification between definite mass historystates and the standard Wigner representation. Connections with the freepropagator of scalar field theory and localized states are subsequentlyderived. The formalism also allows the superposition of different theories andremains valid in the presence of a fixed external field, revealing specialorthogonality relations. Other details such as extended identities for thecurrent density, the quantization of parameterized theories and thenonrelativistic limit, with its connection to the Page and Wootters formalism,are discussed. A related consistent second quantization formulation is alsointroduced.