The Wigner function in the relativistic quantum mechanics

Kowalski, Krzysztof; Rembieliński, Jakub

doi:10.1016/j.aop.2016.09.016

Cited by 10 publications

(9 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have shown that such a program can be easily implemented within quantum mechanics for photon developed by I. Bia lynicki-Birula [2][3][4] and J. E. Sipe [5], and with the use of the continuous-discrete Wey -Wigner -Moyal formalism [1,9] (see also the wide bibliography therein). The result reinforces our believe that one can successfully apply this formalism to other relativistic particles thus obtaining a promising approach to searching for the relativistic Wigner functions [8,[26][27][28][29][30][31][32], although the problems with interpretation of the vector x are to be expected since for relativistic particles the operator x does not represent the position observable [23,25,[33][34][35][36][37][38][39]. So the natural question is if one can construct the Weyl -Wigner -Moyal formalism for photon employing the position operator introduced by Margaret Hawton [38].…”

Section: Discussionsupporting

confidence: 83%

“…The results that were found, reinforce our belief that one can successfully apply this formalism to other relativistic particles in searching for their relativistic Wigner functions. [ 10,27–33 ] On the other hand the problems with interpretation of the vector

\overset{x}{true}

are to be expected since for relativistic particles the operator

\overset{\vec{x}}{true}

does not represent the position observable. [ 16,24,26,34–39 ] So the natural question arises if one can construct the Weyl – Wigner – Moyal formalism for photon employing the position operator introduced by Margaret Hawton.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

The Weyl – Wigner – Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

Przanowski

Tosiek

Turrubiates

2020

Fortschritte der Physik

View full text Add to dashboard Cite

Classical model of light in helicity formalism is presented. Then quantum point of view at photons -construction and interpretation of photon wave function is proposed. Quantum mechanics of photon is investigated. The Bia lynicki -Birula scalar product Ψ 1 |Ψ 2 BB and the generalized Hermitian conjugation γ + + of linear operator γ are discussed. Quantum description of light on a phase space is developed. A photon Wigner function is built.

show abstract

Section: Discussionsupporting

confidence: 83%

\overset{x}{true}

are to be expected since for relativistic particles the operator

\overset{\vec{x}}{true}

Section: Discussionmentioning

confidence: 99%

The Weyl – Wigner – Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

Przanowski

Tosiek

Turrubiates

2020

Fortschritte der Physik

View full text Add to dashboard Cite

show abstract

“…Such a Wigner function can be made finite in the massless limit, analogously to Ref. [5] where a simplest example of this kind has been found.…”

mentioning

confidence: 78%

On Wigner function of a vortex electron

Karlovets

2019

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We derive a relatively simple and Lorentz-invariant expression for a Wigner function of a paraxial vortex electron described as a Laguerre-Gaussian wave packet.While relativistic generalizations of a Wigner function [1] of a quantum system have been shown to exist [2-5], these functions may still suffer from a lack of Lorentz invariance [4]. This is all the more so for wave packets with the non-Gaussian spatial profiles, such as a vortex state with an orbital angular momentum (OAM) ℓ [6], an Airy beam [7,8], and their different generalizations. Although there are several works where the Wigner functions of such packets are derived (see, for instance, [9-12]) and even studied experimentally for twisted photons [11,13], the problem of invariance of these functions under the Lorentz transformations for massive wave packets (first of all for vortex electrons, as they have recently been obtained experimentally [6]) remains unsolved.One of the reasons why it has not been done for vortex electrons yet is that the very wave functions describing these states (the so-called Bessel beams and the Laguerre-Gaussian packets [6]) are usually written in terms of the non-invariant quantities and the transformation properties of these functions remain unclear. As noted by , this lack of explicit invariance per se does not violate correct transformation properties of the observables. Nevertheless, aiming at relativistic applications in particle and hadronic physics [14], it is desirable to know how the Wigner functions of the wave packets behave under the Lorentz boosts. To achieve this goal, the packets themselves need to be described in terms of the variables with definite transformation laws, which has been done for Gaussian packets of the massive neutrinos in [15,16] and recently for vortex electrons in [17].Here we derive a Wigner function of a massive particle with a phase vortex (say, the twisted electron) described via the paraxial Laguerre-Gaussian packet. The latter is Lorentz invariant for longitudinal boosts (along a propagation axis) and represents a massive generalization of the corresponding state of a twisted photon. Correct relativistic transformation of all variables was made possible by using an invariant approach to describe wave packets adopted from [15,16]. As we argue in [17], it is convenient to choose the condition of paraxiality for massive particles to be invariant too, in contrast to the twisted photons. In this paper, we imply that the packet is wide in x-space compared to a Compton wavelength, σ ⊥ ≫ λ c = /mc ≡ 1/m, or that it is narrow in p-space, σ ∼ 1/σ ⊥ ≪ m. It is this Lorentz invariant condition of paraxiality that significantly simplifies the calculations and the resultant expression for the Wigner function. Indeed,

show abstract

“…From this point of view, Spohn's equation emerges as the classical Koopman-von Neumann theory corresponding to the Dirac equation. The develop methodology can be readily apply to the analysis of other relativistic dynamical systems (e.g., governed by the Klein-Gordon equation [61,62]).…”

Section: Discussionmentioning

confidence: 99%

Operational dynamical modeling of spin 1/2 relativistic particles

Cabrera

Campos

Rabitz

et al. 2019

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

The formalism of Operational Dynamical Modeling [Phys. Rev. Lett. 109, 190403 (2012)] is employed to analyze dynamics of spin half relativistic particles. We arrive at the Dirac equation from specially constructed relativistic Ehrenfest theorems by assuming that the coordinates and momenta do not commute. Forbidding creation of antiparticles and requiring the commutativity of the coordinates and momenta lead to classical Spohn's equation [Ann. Phys. 282, 420 (2000)]. Moreover, Spohn's equation turns out to be the classical Koopman-von Neumann theory underlying the Dirac equation.

show abstract

The Wigner function in the relativistic quantum mechanics

Abstract: A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.

Cited by 10 publications

References 16 publications

The Weyl – Wigner – Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

The Weyl – Wigner – Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

On Wigner function of a vortex electron

Operational dynamical modeling of spin 1/2 relativistic particles

Contact Info

Product

Resources

About