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arXiv:1806.01472 · Physical review. D/Physical review. D.

History state formalism for Dirac’s theory

N. L. Diaz, R. Rossignoli

Published Feb 15, 2019• 36 citations
arXivHTMLPDF

Abstract

We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $\ensuremath{\tau}$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of a hypothetical superposition of different masses. Analytical expressions for both the space-time density and electron-time entanglement are provided for two particular families of electron states, the former including Pryce localized particles.

DOI
10.1103/physrevd.99.045008

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Progress of Theoretical Physics•Dec 1, 1969•15 citations
We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter \ensuremathτ\ensuremath{\tau}\ensuremathτ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of a hypothetical superposition of different masses. Analytical expressions for both the space-time density and electron-time entanglement are provided for two particular families of electron states, the former including Pryce localized 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.
We analyze the application of the history state formalism to quantum walks. The formalism allows one to describe the whole walk through a pure quantum history state, which can be derived from a timeless eigenvalue equation. It naturally leads to the notion of system-time entanglement of the walk, which can be considered as a measure of the number of orthogonal states visited in the walk. We then focus on one-dimensional discrete quantum walks, where it is shown that such entanglement is independent of the initial spin orientation for real Hadamard-type coin operators and real initial states (in the standard basis) with definite site parity. Moreover, in the case of an initially localized particle it can be identified with the entanglement of the unitary global operator that generates the whole history state, which is related to its entangling power and can be analytically evaluated. Besides, it is shown that the evolution of the spin subsystem can also be described through a spin history state with an extended clock. A connection between its average entanglement (over all initial states) and that of the operator generating this state is also derived. A quantum circuit for generating the quantum walk history state is provided as well.
We discuss some fundamental properties of discrete system-time history states. Such states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We consider the general case where system-clock pairs can interact, analyzing first their different representations and showing there is always a special clock basis for which the evolution for a given initial state can be described by a constant Hamiltonian HHH. It is also shown, however, that when the evolution operators form a complete orthogonal set, the history state is maximally entangled for any initial state, as opposed to the case of a constant HHH, and can be generated through a simple double-clock setting. We then examine the quadratic system-time entanglement entropy, providing an analytic evaluation and showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable one to efficiently determine overlaps between system states and also evolution operators at any two times.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.
A satisfactory proper time operator is found for Dirac's quantum relativistic electron. A self-consistent scheme is developed in which this operator and Beck's proper time derivative have the properties they should have. As an application of the proper time formalism as a research tool, properties of three position operators (which are generally considered as covariant) are studied in regard to their behaviour under the inhomogeneous Lorentz group and under ordinary and proper time evolution. They are Pryce's class d, Bacry's and R. J. Finkelstein's operators. It results that all three have some unreasonable properties for Dirac's electron. The reason for the difficulty in the case of Finkelstein's operator can be traced back to an incorrect definition of it. By making some changes a satisfactory position operator is obtained but it turns out to be equivalent to Bunge's one (up to an arbitrary and unimportant constant of motion). The results of this paper reinforce those of previous work which show that (for Dirac's electron) Bunge's position operator seems to be in a privileged place as regards the solution of the localization problem. Incidentaly, the interpretation of some relations previously used as auxiliary formulas for computations, allows us to obtain a manifestly covariant expression for Dirac's Hamiltonian and a second manifestly covariant form of Dirac's equation.