History state formalism for scalar particles

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 non relativistic limit, with its connection to the Page and Wooters formalism, are discussed. A related consistent second quantization formulation is also introduced.

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

“…On the one hand, this will permit us to prove full equivalence of the so-obtained relational quantum dynamics with the manifestly gauge-invariant formulation in terms of relational Dirac observables on H phys of Dynamics (i). As an aside, this will also resolve the normalization issue of physical states appearing in Diaz et al [50], where the kinematical rather than physical inner product was used to normalize physical states, thus yielding a divergence (when used for equal mass states). On the other hand, the covariant clock POVM will allow us, in section 6, to address the observation by Kuchař [1] that using the Minkowski time observable leads to incorrect localization probabilities for relativistic particles in the Page-Wootters formalism.…”

“…(i) a Dirac quantization scheme, wherein relational observables are constructed that encode correlations between evolving and clock degrees of freedom [1,2,4,, (ii) the Page-Wootters formalism, which defines a relational dynamics in terms of conditional probabilities for clock and evolving degrees of freedom [7,25,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], and (iii) classical or quantum deparametrizations, which result in a reduced quantum theory that only treats the evolving degrees of freedom as quantum [1,2,7,10,30,31,[58][59][60][61][62][63][64][65].…”

“…This degeneracy is not covered by our previous construction. While quadratic clock Hamiltonians are standard in the literature on relational observables [approach (i)] and deparametrizations [approach (iii)], see e.g., [4,10,11,19,29,31], relativistic particle models have only recently been studied in the Page-Wootters formalism [approach (ii)] [45,[49][50][51]. However, Kuchař's criticism (a) that the Page-Wootters approach yields incorrect localization probabilities in relativistic settings has yet to be addressed.…”

Time Observables in a Timeless Universe

“…Perhaps a PaW formulation of spacetime may represent a first step towards removing this asymmetry. This might help to develop relativistic generalizations of the PaW formalism, see also [31,32,33,34].…”

Peaceful coexistence of thermal equilibrium and the emergence of time

“…Back in 1983 Don N. Page and William K. Wootters (PaW) suggested that time can be an emergent property of entanglement between subsystems in a globally static Universe [4,5], a proposal that has recently attracted much attention [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] as a viable route for a new description of space-time, including a new perspective for merging quantum clocks and gravity [24,25] (see Appendix A for a brief summary of PaW).…”