History state formalism for scalar particles

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 Wooters formalism, are discussed. A related consistent second quantization formulation is also introduced.

Figure 1

Pictorial representation of the two equivalent characterizations of the quantity $⟨y|{\mathrm{\Pi }}_{m^{\prime }}^{+}{\mathrm{\Pi }}_m^{+}|x⟩$. Each line represents an amplitude $D(y-x)$.

Figure 2

Integration region in the variables $m^2$ and $\mathbf{p}$. Here $|\mathbf{p}|$ is the modulus of the three-momentum $\mathbf{p}$.