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International Journal of Modern Physics A

A time operator in the simulations of the Dirac equation

M. Bauer

Published Aug 7, 2019• 4 citations
Publisher

Abstract

Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.

DOI
10.1142/s0217751x19501148

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It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.

•Jul 24, 2019•0 citations
Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
The Dirac equation admits a natural time-invariant algebra Q of unbounded operators of the Hilbert space H = L 2(ℝ3, ℂ4). Time-invariance means that eiHtQe−iHt=Q,{{e}^{{iHt}}}Q{{e}^{ - }}^{{iHt}} = Q, eiHtQe−iHt=Q, with the unitary eiHt:H→H.{e^{iHt}}:{\mathbf{H}} \to {\mathbf{H}}. eiHt:H→H. This is not a specific feature of the Dirac equation; any 1-st order strictly hyperbolic system (under mild additional assumptions) admits such an algebra. We propose to introduce Q as algebra of “really predictable observables” into v. Neumann’s theory of physical states and observables: Only the self-adjoint operators A ∈ Q can be really predicted. For other observables — especially for most dynamical observables — one must find an observable A0 ∈ Q “close” to A, perhaps only for a given physical state ψ. Energy H belongs to Q, but location (angular) momentum, spin,…, do not. This seems to explain at a glance the triumphs and difficulties Dirac’s theory encountered.
We present a classical optics simulation of the one-dimensional Dirac equation for a free particle. Positive- and negative-energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle. Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator---known as Zitterbewegung. We are also able to measure the so-called mean-position operator which is a single-particle observable and presents no oscillations. Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.
A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.
Abstract It is first shown that the Dirac’s equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4 × 4 relativistic time operator for spin-1/2 particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but non-zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli’s objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found .
It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.