Wigner function for free relativistic particles

Zavialov, O. I.; Malokostov, A. M.

doi:10.1007/bf02557343

Cited by 11 publications

(16 citation statements)

References 4 publications

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“…We now summarize the basic facts about the Wigner function introduced in ref. 2. The point of departure in [2] was the following form of the nonrelativistic Wigner function W (x, p, t) = 1 (2π) 3 1 6 d 3 p 1 d 3 p 2φ * (p 1 , t)φ(p 2 , t)δ(p − 1 2 (p 1 + p 2 ))e i(p 2 −p 1 )·x , (2.1) whereφ(p, t) is the Fourier transform of the wave function φ(x, t), that is φ(p, t) = 1 (2π) 3 2 d 3 x e −i p·x φ(x, t).…”

Section: Definition Of the Wigner Functionmentioning

confidence: 99%

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The Wigner function in the relativistic quantum mechanics

Kowalski

Rembieliński

2016

Annals of Physics

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Section: Definition Of the Wigner Functionmentioning

confidence: 99%

“…Now we have the parametric form of the Wigner function introduced in ref. [2] that can be easily obtained from (2.20) by switching to coordinates p 1,2 = mc sinh γ 1,2 on the mass-shell hyperboloid:…”

Section: )mentioning

confidence: 99%

The Wigner function in the relativistic quantum mechanics

Kowalski

Rembieliński

2016

Annals of Physics

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“…is not manifestly Lorentz invariant. Here: ψ(p, t) = ψ(p) e −itε(p) , ε(p) = p 2 + m 2 , d 3 p (2π) 3 |ψ(p, t)| 2 = d 3 p (2π) 3 1 2ε(p) |Ψ(p, t)| 2 = d 3 x d 3 p (2π) 3 n(r, p, t) = 1 = inv. (2) Along with a "non-relativistic" wave function ψ(p), we have also introduced a "relativistic" one, Ψ(p) = 2ε(p) ψ(p).…”

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confidence: 99%

On Wigner function of a vortex electron

Karlovets

2019

J. Phys. A: Math. Theor.

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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,

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“…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%

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

Przanowski

Tosiek

Turrubiates

2020

Fortschritte der Physik

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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.

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Wigner function for free relativistic particles

Abstract: A generalization of the Wigner function for the case of a free particle with the "relativistic" Hamiltonian p 2 + m 2 is given.

Cited by 11 publications

References 4 publications

The Wigner function in the relativistic quantum mechanics

The Wigner function in the relativistic quantum mechanics

On Wigner function of a vortex electron

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

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