Abstract
We present a covariant quantum formalism for scalar particles based on an enlarged Hilbert space. The particular physical theory can be introduced through a timeless Wheeler DeWitt-like equation, whose projection onto four-dimensional coordinates leads to the Klein Gordon equation. The standard quantum mechanical product in the enlarged space, which is invariant and positive definite, implies the usual Klein Gordon product when applied to its eigenstates. Moreover, the standard three-dimensional invariant measure emerges naturally from the flat measure in four dimensions when mass eigenstates are considered, allowing a rigorous identification between definite mass history states and the standard Wigner representation. Connections with the free propagator of scalar field theory and localized states are subsequently derived. The formalism also allows the superposition of different theories and remains valid in the presence of a fixed external field, revealing special orthogonality relations. Other details such as extended identities for the current density, the quantization of parameterized theories and the nonrelativistic limit, with its connection to the Page and Wootters formalism, are discussed. A related consistent second quantization formulation is also introduced.
Highlights
The introduction of the concept of time in a quantum mechanical framework [1,2,3] has recently attracted renewed attention [4,5,6,7,8,9,10,11,12,13]
One of the main results is the definition of a consistent Hilbert space for the Klein-Gordon equation [29,30], in both the free case and in the presence of an external field, where the inner product is the canonical product in four dimensions
While corresponding results for the free case were previously obtained in the context of quantum gravity [15,16,17,18], the fourdimensional (4d) space was there considered as an auxiliary Hilbert space
Summary
The introduction of the concept of time in a quantum mechanical framework [1,2,3] has recently attracted renewed attention [4,5,6,7,8,9,10,11,12,13]. One of the main results is the definition of a consistent Hilbert space for the Klein-Gordon equation [29,30], in both the free case and in the presence of an external field, where the inner product is the canonical product in four dimensions This construction, and the subsequent proper normalization of fixed mass states, which are eigenstates of a Wheeler DeWitt-like equation [31], ensure the usual three-dimensional (3d) norm. The present treatment of interactions reveals that such general states are already implied when expressing the corresponding solutions in terms of the free states, in analogy with the off-shell contributions in perturbative treatments for interacting many particle systems These results provide a new perspective which could be suitable to deal with the Hilbert space problem of the Wheeler DeWitt framework of quantum gravity [21,22,25,31].
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