Dirac open quantum system dynamics: formulations and simulations

Renan Cabrera,1, ∗ Andre G. Campos,1 Denys I. Bondar,1 and Herschel A. Rabitz1

1Department of Chemistry, Princeton University, Princeton, NJ 08544, USA(Dated: September 12, 2016)

We present an open system interaction formalism for the Dirac equation. Overcoming a complex-ity bottleneck of alternative formulations, our framework enables efficient numerical simulations(utilizing a typical desktop) of relativistic dynamics within the von Neumann density matrix andWigner phase space descriptions. Employing these instruments, we gain important insights into theeffect of quantum dephasing for relativistic systems in many branches of physics. In particular, theconditions for robustness of Majorana spinors against dephasing are established. Using the Kleinparadox and tunneling as examples, we show that quantum dephasing does not suppress negativeenergy particle generation. Hence, the Klein dynamics is also robust to dephasing.

PACS numbers: 03.65.Pm, 05.60.Gg, 05.20.Dd, 52.65.Ff, 03.50.Kk

INTRODUCTION.

The Dirac equation is a cornerstone of relativisticquantum mechanics [1]. It was originally developed todescribe spin 1/2 charged particles playing an essentialrole in the field of high energy physics [2–4]. Recently,there is resurging interest in the Dirac equation becauseit was found to be an effective dynamical model of unex-pectedly diverse phenomena occurring in high-intensitylasers [5], solid state [6–9], optics [10, 11], cold atoms[12, 13], trapped ions [14, 15], circuit QED [16], and thechemistry of heavy elements [17, 18]. However, there isa need to go beyond coherent dynamics offered by theDirac equation alone in order to model the effects of im-perfections, noise, and interaction with a thermal bath[19]. To construct such models, we will first review howthese effects are described without relativistic considera-tions [20].

In the non-relativistic regime, the Schrodinger equa-tion describes a quantum systems isolated from the restof the universe. This is a good approximation for certainconditions. For example, an atom in a dilute gas can beconsidered to be a closed system if the time scale of thedynamics is much faster than the mean collision time.If we would like to include collisions in the picture, weneed to keep track of the quantum phases of each atomin the gas. This is unfeasible. This type of dynamicsmotivated development of the theory of open quantumsystems [21], where a single particle picture is retainedalbeit with more general dynamical equations. There aretwo methods to introduce interactions with an environ-ment: (i) the Schrodinger equation with an additionalstochastic force, or (ii) the conceptually different densitymatrix formalism [20]. In the latter, a state of an openquantum system is represented by a self-adjoint densityoperator ρ with non-negative eigenvalues summing up toone. The master equation, governing evolution of ρ, reads

i~d

dtρ = [H, ρ] +D(ρ), (1)

where H is the quantum Hamiltonian and the dissipatorD(ρ) encodes the interaction with an environment. Thevon Neumann equation [20] describing unitary evolutionis recovered by ignoring the dissipator. When D(ρ) 6= 0,Eq. (1) generally does not preserve the von Neumannentropy S = −Tr (ρ log ρ), which measures the amount ofinformation stored in a quantum system. We note thateffective elimination of D(ρ) is a fundamental challengein order to develop many quantum technologies [22, 23].

The non-relativistic theory of open quantum systemsprovided profound insights into some fundamental ques-tions of physics such as the emergence of the classicalworld from the quantum one [24–30], measurement the-ory [24, 31–33], quantum chaos [27, 30, 34] and syn-chrotron radiation [35–37].

To study the quantum-to-classical transition, it is in-strumental to put both mechanics on the same mathe-matical footing [24, 25, 28, 31, 38–45]. This is achieved bythe Wigner quasi-probability distribution W (x, p) [46],which is a phase-space representation of the density op-erator ρ. Note that the Wigner function serves as a basisfor a self-consistent phase space representation of quan-tum mechanics [42, 47], which is equivalent to the densitymatrix formalism.

Previous attempts to construct the relativistic theoryof open quantum system relied on the relativistic ex-tension of the Wigner function without introducing thecorresponding density matrix formalism. In Sec. , wewill first present the manifestly covariant density ma-trix formalism for a Dirac particle and then constructthe Wigner representation. The development of the rela-tivistic Wigner function was motivated by applications inquantum plasma dynamics and relativistic statistical me-chanics [3]. The manifestly covariant relativistic Wignerformalism for the Dirac equation was put forth in Refs.[2, 48–50] (see Ref. [3] for a comprehensive review). Inaddition, exact solutions for physically relevant systemswere reported in Refs. [51, 52]. The following concep-tual difference between the non-relativistic and relativis-tic Wigner functions was elucidated in Ref. [53]: In non-

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relativistic dynamics, Hudson’s theorem states that theWigner function for a pure state is positive if and onlyif the underlying wave function is a Gaussian [54]. Inother cases, the Wigner function contains negative val-ues. However, this statement does not carry over to therelativistic regime. In particular, there are many physi-cally meaningful spinors whose Wigner function is pos-itive [53]. Note that the Wigner function’s negativityis an important resource in quantum information theory[55, 56].

The limit ~ → 0 of the non-relativistic Wigner func-tion is non-singular and recovers classical mechanics. Thesame limiting property is expected from the relativisticextension. However, the manifest covariance of the rela-tivistic Wigner function needed to be broken in order toperform the ~ → 0 limit [50, 57, 58]. From a differentperspective, the covariant classical limit was obtained inRefs. [41, 59]. In Appendix of the current work, weprovide a simpler manifestly-covariant derivation of theclassical limit. Contrary to the previous work, our deriva-tion recovers two decoupled classical equations of motion:one governing the dynamics of positive energy particlesand the other describing negative energy particles (i.e.,antiparticles). This classical limit of the Dirac equationis an example of classical Nambu dynamics [60].

An alternative quantum field theoretic formulation ofthe Wigner function for Dirac fermions has also been putforth [57, 61–66].

As mentioned before, the current interest in the Diracequation goes far beyond relativistic physics. These newopportunities come along with new challenges. It is theaim of the current Article to overcome some of thoseproblems by furnishing a new formulation of traditional(i.e., closed system) relativistic dynamics enabling effi-cient numerical simulations as well as physically consis-tent inclusion of open system interactions. We believethat the developed formalism and numerical methods willinfluence the following fields:

1. Understanding the role of the environment for theclassical world emergence. In particular, we eluci-date the influence of decoherence (i.e., loss of quan-tum phase coherence) on relativistic dynamics inSecs. and , where Klein tunneling [7] and the as-sociated paradox are analyzed along with the Ma-jorana fermion dynamics.

2. Development of the quantum relativistic theory ofenergy dissipation. Based on existing models ofnon-relativistic quantum friction [67, 68], we expecta relativistic model of energy damping to obey: (i)the mass-shell constraint, (ii) translational invari-ance (in particular, the dynamics should not de-pend on the choice of the origin), (iii) equilibra-tion (the model should reach a steady state at longtime propagation. In particular, the final energyat t→ +∞ should be bounded thereby preventing

runaway population of the negative energy contin-uum), (iv) thermalization (i.e., the achieved steadystate should represent thermal equilibrium), (v) rel-ativistic extension of Ehrenfest theorems (i.e., seethe dynamical constraints for expectation valuesencompassing energy drain in Ref. [68]). Somepreliminary steps towards the desired relativisticmodel are reported in Ref. [69].

3. Modeling environmental effects in Dirac materi-als such as topological insulators [8, 70, 71], Weylsemimetals [72, 73], and graphene [6]. In thesecases, open system dynamics models sample impu-rities and imperfections as well as external noise.Recently, the Dirac equation with an additionalstochastic force was utilized for this purpose [19].To the best of our knowledge, a more general mas-ter equation formalism is yet to be explored.

4. Understanding robustness of a Majorana particle,which is defined as being its own antiparticle. Ex-perimental implementation of solid-state analoguesof Majorana fermions [74–76] opens up possibilitiesto study the physics of these unusual states. Inparticular, Majorana bound states are well suitedcomponents of topological quantum computers [77].Due to its topological nature, Majorana states areexpected to be robust against perturbations andimperfections [78]. Dissipative dynamics modeledwithin a Lindblad master equation confirmed a sig-nificant degree of robustness in a specific opticallattice [79]. However, the robustness is not univer-sal [80] and there is a need for enhancement (e.g.,employing error correction techniques [81]). Notethat Majorana states studied in condensed matterphysics [74–76], do not strictly coincide with the au-thentic Majorana spinors [82], albeit sharing com-mon features. In the present paper, we consideroriginal Dirac Majorana spinors [82]. In Sec. , wedemonstrate that a single-particle Majorana spinorexhibits robustness even for strong couplings to thedephasing environment, which otherwise quicklywashes out interferences for particle-particle super-positions (aka, Schrodinger cat states). Moreover,this phenomenon has an intuitive explanation inthe phase-space representation, where quantum de-phasing turned out to be equivalent to Gaussianfiltering over the momentum axis (detailed expla-nation in Secs. and ). The applicability of thisinsight to condensed matter systems should be asubject of further studies.

5. Development of manifestly covariant quantum opensystem interaction. Coupling a Dirac particle to theenvironment generally introduces a preferred frameof reference, thereby breaking the Lorentz invari-ance. However, coupling to the vacuum, causing

3

spontaneous emission, Lamb shift etc. [83], and ra-diation reaction [84, 85], needs to be manifestly co-variant because the vacuum has no preferred frameof reference. Solid state physics holds a promise toimplement many exotic quantum effects experimen-tally not yet verified [86], e.g., the Unruh effect andHawking radiation. Solid state dynamics naturallyincludes the interaction with the environment, thusthe need to include open system interaction into thedynamics of interest. A relativistic quantum theoryof measurements also requires development of man-ifestly covariant master equations. Currently, ap-proaches based on axiomatics [87], stochastic Diracand Lindblad master equations [88] are explored.Nevertheless, the proposed equations are computa-tionally unfeasible at present. In the current work,we lay the ground for a computationally efficienttechnique by introducing a manifestly covariant vonNeumann equation (see Sec. ) based on Refs.[2, 3, 48–50].

This paper is organized in seven sections and two ap-pendices. Section provides the general mathematicalformalism including the manifestly relativistic covariantvon Neumann equation. Section is concerned with therelativistic Wigner function and related representations.Section introduces open system interactions by consid-ering a model of dephasing, environmental interactionleading to the loss of quantum phase. Numerical algo-rithms are developed in Sec. and illustrated for thedynamics of Majorana spinors and the Klein paradox inSecs. and , respectively. The final section provides theconclusions. Appendix treats the concept of relativisticcovariance, and Appendix elaborates the classical limit(~ → 0) of the Dirac equation in manifestly covariantfashion.

GENERAL FORMALISM

Note that throughout the paper, x and x denote dif-ferent variables; likewise, x and x denote different oper-ators. In addition, Greek characters (e.g., µ, ν), usedas indices for Minkowski vectors, are assumed to runfrom 0 to 3; while, Latin indices (e.g., j, k) run from1 to 3. The Minkowski metric is a diagonal matrixdiag(1,−1,−1,−1). This implies that x0 = x0 andxk = −xk.

The manifestly covariant Dirac equation reads

D(xµ, pµ)|ψ〉 = 0, (2)

where the Dirac generator D(xµ, pµ) and the commuta-tion relations are defined as

D(xµ, pµ) = γµ[cpµ − eAµ(x)]−mc2, (3)

[xµ, pν ] = −i~δµν . (4)

Note that the negative sign in the right hand side of Eq.(4) occurs due to the fact

[xk, pj ] = −i~δkj ←→ [xk, pj ] = i~δkj , (5)

in agreement with non-relativistic dynamics where themomentum is expressed in contravariant components pj .

From the well established work on relativistic statisti-cal quantum mechanics [2, 3, 48–50], the manifestly co-variant von Neumann equation can be written as

D(xµ, pµ)P = 0, PD(xµ, pµ) = 0, (6)

where P represents the density state operator actingon the Manifestly Covariant Spinorial Hilbert space(MCS). Equation (6) is the foundation for all the sub-sequent developments.

Following Ref. [89, 90], we introduce the ManifestlyCovariant Hilbert Phase space (MCP) where the algebraof observables consists of (x, pµ) [see Eq. (4)] along with

the mirror operators (x′µ, p′µ) obeying

[xµ, pν ] = −i~δµν , [x′µ, p′ν ] = i~δµν , (7)

and all the other commutators vanish. In MCP the roleof density operator P is taken over by the ket state |P 〉according to

O(xµ, pµ)P ←→ −→O (xµ, pµ)|P 〉, (8)

P O(xµ, pµ) ←→ |P 〉←−O ( x′µ, p′µ ), (9)

where the arrows indicate the direction of applicationof the operators O(xµ, pµ) and O( x′µ, p′µ ). Thus, therelativistic von Neumann equation (6) reads in MCP as

−→D(xµ, pµ)|P 〉 = 0, |P 〉←−D( x′µ, p′µ ) = 0. (10)

A summary of the two introduced formulations is givenin Table I.

The manifest covariance of Eq. (10) can be relaxed toimplicit covariance by separating the time according tothe 3 + 1 splitting xµ = (ct, xk) [91]. This means thatthe underlying relativistic covariance is maintained butit is no longer evident. In the spirit of the 3 + 1 schemewe define the Dirac Hamiltonian as

H = αk[cpk − eAk(t, xk)] +mc2γ0 + eA0(t, xk). (11)

The von-Neumann equation (10) in the ImplicitCovariant Hilbert Phase space (ICP) becomes

[c−→p0 −

−→H (t, xk, pk)

]|P 〉γ0 = 0, (12)

|P 〉γ0[c←−p′0 −

←−H (t

′, x′k, p′k)

]= 0. (13)

Inspired by the Bopp transformations in the non-relativistic quantum mechanical phase space [92, 93], a

4

Manifestly CovariantSpinorial Hilbert space

MCS

Manifestly CovariantHilbert Phase space

MCP

State P |P 〉

Operators O(xµ, pµ)−→O (xµ, pµ),

←−O (x′µ, p′µ)

Equation D(xµ, pµ)P = 0−→D(xµ, pµ)|P 〉 = 0

of motion

PD(xµ, pµ) = 0 |P 〉←−D( x′µ, p′µ ) = 0

TABLE I: Two manifestly covariant formulations of relativis-tic quantum mechanics.

ICP operators Mirror ICP operators

Space-time

Momentum-energy

t = t− 12τ

xk = xk − ~2θk

p0 = Ω + 12cE

pk = pk + ~2λk

t′

= t+ 12τ

x′k = xk + ~2θk

p′0 = Ω− 12cE

p′k = pk − ~2λk

TABLE II: Operators in the Implicitly Covariant HilbertPhase space (ICP) where (t, τ , Ω, E, xk, pk, λk, θ

k) representthe ICP Bopp operators.

representation of the algebra (7) can be constructed in

terms of ICP Bopp operators (t, τ , Ω, E, xk, pk, λk, θk) in

Table II, obeying

[t, E] = −i~, [Ω, τ ] = −i~, (14)

[xj , λk] = −iδjk, [pj , θk] = −iδkj , (15)

where all the other commutators vanish, in particular[xk, pj ] = 0. A graphical illustration of the relation be-tween the time variables t−t′ and t−τ is shown in Fig. 1.Adding and substracting Eqs. (12) and (13), and uti-lizing the Bopp operators, we obtain the von-Neumann

t

τ

-4 -2 0 2 4

-4

-2

0

2

4

t

t'

FIG. 1: (Color online) Graphical illustration of the relationbetween the double time variables in the ICP space as de-fined in Table II. The color gradient is directed along the tcoordinate.

equation in the ICP space

E|P 〉γ0 =−→H

(t− τ

2, xk − ~

2θk, pk +

~2λk

)|P 〉γ0

(16)

− |P 〉γ0←−H(t+

τ

2, xk +

~2θk, pk −

~2λk

),

2cΩ|P 〉γ0 =−→H

(t− τ

2, xk − ~

2θk, pk +

~2λk

)|P 〉γ0

(17)

+ |P 〉γ0←−H(t+

τ

2, xk +

~2θk, pk −

~2λk

).

E and Ω can be realized in terms of differential operatorsas

t = t E = i~∂

∂t, (18)

τ = τ Ω = i~∂

∂τ, (19)

turning Eqs. (16) and (17) into a system of two differen-tial equations that can be solved by either propagatingalong t while keeping τ fixed, or moving along τ with tconstant. In particular, setting τ = 0 in Eq. (16), we ob-tain the relativistic von-Neumann equation in the SlicedCovariant Hilbert Phase space (SCP)

i~d

dt|P 〉γ0 =

−→H

(t, xk − ~

2θk, pk +

~2λk

)|P 〉γ0 (20)

− |P 〉γ0←−H(t, xk +

~2θk, pk −

~2λk

).

It is well known that a Lorentz transformation mixes thespace and time degrees of freedom, as recapitulated inAppendix . In particular, the time-evolution of the state

5

t

5

τ

t

FIG. 2: (Color online) Schematic illustration of a quantumstate propagating along time t within the slice = 0 accord-ing to Eq. (20). A di↵erent inertial reference frame wouldgenerate another slice.

III. RELATIVISTIC WIGNER FUNCTION

This section is devoted to study specific representa-tions of the von-Neumann equation in the SCP space(20) in order to derive the time-evolution of the relativis-tic Wigner function.

Following Table II, there are four representations ofinterest:

• The double configuration space is defined by setting

xk = xk, k = k, k = i@

@xk, pk = i

@

@k. (22)

Hence, the equation of motion (20) becomes

i~@B0

@t=!H

t, xk ~

2k, pk +

~2k

B0

B0 H

t, xk +~2k, pk

~2k

, (23)

where B is defined as the relativistic Blokhintsevfunction

B0 =1p~hxk, k|P i0 = hxk ~

2k|P0|xk +

~2ki.

(24)

For pure states, B is expressed in terms of the four-column Dirac spinor as

B(t, xk, k)0 = (t, xk ~2k) †(t, xk +

~2k). (25)

Therefore, B is a 44 complex matrix-valued func-tion of two degrees of freedom x . The non-relativistic version of the Blokhintsev function wasintroduced in Refs. [91–93].

*Fourier Transform

+

TABLE III: (Color online) Relation between the double con-figuration (xkk) and the double momentum (kpk) spacesas defined in Table II. The dashed axes along pk and k in-dicate that they are related via a direct Fourier transform.The solid axes along xk and k indicate a similar connection.These relations are also schematically presented in Eq. (39).

• The phase space is defined by

xk = xk, pk = pk, k = i@

@xk, k = i

@

@pk. (26)

The underlying equation of motion (20) reads

i~@W0

@t=!H

t, xk ~

2k, pk +

~2k

W0

W0 H

t, xk +~2k, pk

~2k

, (27)

where W is the sought relativistic Wigner function

W0 =1

2~hxk, pk|P i0, (28)

FIG. 2: (Color online) Schematic illustration of a quantumstate propagating along time t within the slice τ = 0 accord-ing to Eq. (20). A different inertial reference frame wouldgenerate another slice.

in a different reference frame corresponds to a differentslicing in the t − τ plane. Therefore, the state propa-gated by Eq. (20) with τ = 0 does not contain enoughinformation to deduce the observations from a differentinertial frame of reference. Nevertheless, Eq. (20) rep-resents a consistent relativistic equation of motion de-scribing dynamics from the particular frame of reference(corresponding to the τ = 0 slice) free of any nonphysicalartifacts, e.g., superluminal propagation. A schematic il-lustration of slicing dynamics at τ = 0 is shown in Fig.2. Note that equations of motion containing two timevariables also appear in non-relativistic dynamics [94].

Using Table II, we rewrite Eq. (20) in the HilbertSpinorial space

i~d

dtP γ0 =[H(t, xk, pk), P γ0]. (21)

Note that this equation resembles Eq. (1) with D = 0.In other words, we obtain a straightforward relativisticextension of the density matrix formalism for the Diracequation. Migdal [95] employed Eq. (21) to describe theeffect of multiple scattering on Bremsstrahlung and pairproduction.

RELATIVISTIC WIGNER FUNCTION

This section is devoted to study specific representa-tions of the von-Neumann equation in the SCP space(20) in order to derive the time-evolution of the relativis-tic Wigner function.

Following Table II, there are four representations ofinterest:

FIG. 3: (Color online) Relation between the double configu-ration (xk − θk) and the double momentum (λk − pk) spacesas defined in Table II. The dashed axes along pk and λk in-dicate that they are related via a direct Fourier transform.The solid axes along xk and θk indicate a similar connection.These relations are also schematically presented in Eq. (39).

• The double configuration space is defined by setting

xk = xk, θk = θk, λk = i∂

∂xk, pk = −i ∂

∂θk. (22)

Hence, the equation of motion (20) becomes

i~∂Bγ0

∂t=−→H

(t, xk − ~

2θk, pk +

~2λk

)Bγ0−

Bγ0←−H

(t, xk +

~2θk, pk −

~2λk

), (23)

where B is defined as the relativistic Blokhintsevfunction

Bγ0 =1√~〈xk, θk|P 〉γ0 = 〈xk − ~

2θk|P γ0|xk +

~2θk〉.

(24)

6

For pure states, B is expressed in terms of the four-column Dirac spinor ψ as

B(t, xk, θk)γ0 = ψ(t, xk − ~2θk)ψ†(t, xk +

~2θk). (25)

Therefore, B is a 4×4 complex matrix-valued func-tion of two degrees of freedom x − θ. The non-relativistic version of the Blokhintsev function wasintroduced in Refs. [96–98].

• The phase space is defined by

xk = xk, pk = pk, λk = i∂

∂xk, θk = i

∂

∂pk. (26)

The underlying equation of motion (20) reads

i~∂Wγ0

∂t=−→H

(t, xk − ~

2θk, pk +

~2λk

)Wγ0−

Wγ0←−H

(t, xk +

~2θk, pk −

~2λk

), (27)

where W is the sought after relativistic Wignerfunction

Wγ0 =1

2π~〈xk, pk|P 〉γ0, (28)

which can be recovered from the Blokhintsev func-tion through a Fourier transform

W (t, xk, pk) =1

(2π)3

∫B(t, xk, θk) exp(ip · θ)d3θ. (29)

Note that only contravariant components are usedin Eqs. (28) and (29).

• The reciprocal phase space is defined as

xk = −i ∂

∂λk, pk = −i ∂

∂θk, λk = λk, θk = θk.

(30)

The corresponding equation of motion is

i~∂Aγ0∂t

=−→H

(t, xk − ~

2θk, pk +

~2λk

)Aγ0−

Aγ0←−H(t, xk +

~2θk, pk −

~2λk

), (31)

where A is the relativistic ambiguity function

Aγ0 =1√~〈λk, θk|P 〉γ0, (32)

which is recovered from the Blokhintsev functionaccording to

A(t, λk, θk) =

∫B(t, xk, θk) exp(−ix · λ)d3x. (33)

• The double momentum space is introduced as

xk = −i ∂

∂λk, pk = pk, λk = λ, θk = i

∂

∂pk. (34)

The corresponding equation of motion is

i~∂Zγ0

∂t=−→H

(t, xk − ~

2θk, pk +

~2λk

)Zγ0−

Zγ0←−H

(t, xk +

~2θk, pk −

~2λk

), (35)

where

Zγ0 =1√~〈λk, pk|P 〉γ0 = 〈pk +

~2λk|P γ0|pk − ~

2λk〉,

(36)

which is related with the Wigner function via

W (t, xk, pk) =1

(2π)3

∫Z(t, λk, pk) exp(ix · λ)d3λ. (37)

Similarly, we also have

A(t, λk, θk) =

∫Z(t, λk, pk) exp(−ip · θ)d3p. (38)

In summary, all these four functions are connected

through Fourier transforms as visualized in the follow-

ing diagram:

W (x, p)Fx→λ // Z(λ, p)

B(x, θ)

Fθ→p

OO

Fx→λ // A(λ, θ)Fθ→p

OO(39)

where vertical arrows denote the direct Fθ→p Fourier

transforms while horizontal arrows indicate the direct

Fx→λ Fourier transforms. A similar diagram can be

drawn in terms of the inverse Fourier transforms as

W (x, p)

Fp→θ

Z(λ, p)Fλ→xoo

Fp→θ

B(x, θ) A(λ, θ)Fλ→xoo

(40)

Since the relativistic Wigner function W is a 4 × 4complex matrix, its visualization is cumbersome. Never-theless, most of the information is contained in [53]

W 0(t, xk, pk) ≡ Tr [W (t, xk, pk)γ0]/4. (41)

In fact, this zero-th component is sufficient to obtain theprobability density j0 ≡ ψ†(t, xk)ψ(t, xk) as

∫W 0(t, xk, pk)d3p = ψ†(t, xk)ψ(t, xk) (42)

∫W 0(t, xk, pk)d3x = ψ†(t, pk)ψ(t, pk), (43)

7

where ψ is the Dirac spinor in the momentum represen-tation, i.e. the Fourier transform of ψ.

Equations (42) and (43) reveal that the zero-th com-ponent of the relativistic Wigner function (41) acts as aquasi-probability distribution – a real valued non-positivefunction, whose marginals coincide with the coordinateand momentum probability densities, respectively.

OPEN SYSTEM INTERACTIONS

Inspired by non-relativistic quantum mechanics [seeEq. (1)], we add a dissipator to the relativistic von Neu-mann equation (21) to account for open system dynamics

i~d

dtP γ0 = [H(t, xk, pk), P γ0] + i~D(P γ0). (44)

We note that Eq. (44) does not need to comply withrelativistic covariance. Nevertheless, this is not a defi-ciency when dealing with environments such as thermalbaths that are already furnished with a preferred frameof reference.

Motivated by the treatment of quantum dephasing innon-relativity [20], we propose to include the followingdissipator in Eq. (44)

D[P γ0] = −D~2

[xk, [xk, P γ0]], (45)

where D is the decoherence coefficient controlling thedephasing intensity and no summation on k is implied.In non-relativistic systems this interaction is utilized todescribe the loss of coherence due to the interactionwith an environment associated with a thermal bath[25, 29, 40, 99]. In addition, a system undergoing contin-uous measurements in position follows the same dynam-ics [28, 100]. A model similar to Eq. (45) describes theeffect of multiple scattering on Bremsstrahlung and pairproduction in high energy limit of the incident electron[95].

The dynamical effect of an interaction can be charac-terized by calculating the time derivative of the expecta-tion value of an observable O

d

dt〈O〉 = Tr

[d

dt(P γ0)O

]. (46)

Assuming that the equation of motion is of the form

d

dtP γ0 =M(P γ0), (47)

the time derivative of 〈O〉 is expressed as follows

d

dt〈O〉 = Tr

[M(P γ0)O

]= Tr

[P γ0M†(O)

], (48)

where M† is the adjoint operator of M with respect tothe Hilbert-Schmidt scalar product.

The particular dephasing dissipator (45) is self-adjoint,

D†[O] = D[O]; (49)

as a result,

D†[xk] = D†[pk] = 0. (50)

This means that the dephasing does not change theHeisenberg equations of motion for position and momen-tum observables. The open system interaction affects thedynamics of the second order momentum

D†[xkxj ] = 0 D†[pkpj ] = 2Dδkj , D†[xkpj ] = 0,(51)

which in turn leads to a momentum wavepacket broad-ening. Moreover, considering that the free Dirac Hamil-tonian (11) is linear in momentum, we obtain from Eqs.(50) and (48)

d

dt

⟨γ0γkpk +mcγ0

⟩= 0. (52)

In other words, the energy is conserved under the actionof the dephasing dissipator (45). This is in stark contrastto non-relativistic dephasing, which is characterized bymonotonically increasing energy.

The classical limit of dephasing (45) is diffusion. Rel-ativistic extensions of diffusion face fundamental chal-lenges [101]. For instance, large values of D may inducedynamics leading to superluminal propagation, whichbreaks down the causality of the Dirac equation (see,e.g., Theorem 1.2 of Ref. [102]). The length-scale of dif-fusion is

√〈x2〉 =

√2Dt; hence, the characteristic speed√

〈x2〉/t =√

2D/t must be smaller than the speed oflight. The shortest time interval for which the single par-ticle picture is valid t ∼ ~/(2mc2), i.e., the zitterbewegungtime scale. Considering all these arguments, we obtainthe constrain: D ~/(4m), or equivalently, 4D/c λ(where λ = ~/(mc) is the reduced Compton wavelength)in order to maintain causal dephasing dynamics.

This dephasing interaction (45) can be expressed in theSCP space, leading to a very simple expression [90]

∂

∂t〈xjθj |P 〉 = −Dθkθkδkj〈xjθj |P 〉, (53)

which is convenient for numerical propagation, as shownin Sec. .

NUMERICAL ALGORITHM

Stimulated by the resurgent interest in the Dirac equa-tion, a plethora of propagation methods were recently de-veloped [103–108]. However, to the best our knowledgeRef. [109] is the only work devoted to propagation of therelativistic von Neumann equation (21), albeit without

8

open system interactions. The purpose of this section isto develop an effective numerical algorithm to propagatethe master equation (44) describing quantum dephasing(45). The computational effort with the proposed algo-rithm scales as the square of the Dirac equation propa-gation complexity. This algorithmic development enablesthe relativistic Wigner function simulations, which werepreviously hindered by the complexity of the underlyingintegro-differential equations [48, 50].

The evolution governed by Eq. (21)

i~d

dtQ =[H(t,xk, pk), Q], (54)

with Q = P γ0 is equivalent to

Qt+dt = e−idtH(t,x,p)/~QteidtH(t,x,p)/~, (55)

where dt is an infinitesimal time step.Considering that the Hamiltonian can be decomposed

as

H = K(p) + V (x), (56)

K(p) = cαkpk +mc2γ0/2, (57)

V (x) = eA0(t, xk)− eαkAk(t, xk) +mc2γ0/2, (58)

where the mass term contributes to both K(p) and V (x).The first order splitting with error O(dt2) is then

Qt+dt = e−idtV (x)/~e−idtK(p)/~QteidtK(p)/~eidtV (x)/~,

(59)

which implies a two step propagation

Q1/2 = e−idtK(p)/~QteidtK(p)/~ (60)

Qt+dt = e−idtV (x)/~Q1/2eidtV (x)/~ (61)

Using Eqs. (8) and (9) we move to SCP

|Q1/2〉 = e−idt−→K(p)/~|Qt〉eidt

←−K(p′)/~, (62)

|Qt+dt〉 = e−idt−→V (x)/~|Q1/2〉eidt

←−V (x′)/~. (63)

Note that |Qt〉 is a complex 4 × 4 matrix reflecting thespinor degrees of freedom. The arrows can be eliminatedby choosing suitable bases

〈pp′|Q1/2〉 = e−idtK(p)/~〈pp′|Qt〉eidtK(p′)/~, (64)

〈xx′|Q1/2〉 = Fpp′→xx′〈pp′|Q1/2〉, (65)

〈xx′|Qt+dt〉 = e−idtV (x)/~〈xx′|Q1/2〉eidtV (x′)/~, (66)

〈pp′|Qt+dt〉 = Fxx′→pp′〈xx′|Qt+dt〉, (67)

where Fpp′→xx′ and Fxx′→pp′stand for Fourier trans-

forms from the momentum representation to the posi-tion representation and vice versa. Considering that thestate is a 4 × 4 matrix, the Fourier transform is inde-pendently applied to each matrix component. From the

FIG. 4: (Color online) Schematic representation of the iter-ative steps to propagate the quantum state according to Eqs.(71)-(74).

computational perspective, the fast Fourier transform isemployed. Further details about the phase space prop-agation via the fast Fourier transform can be found inSec. III of Ref. [90].

Having described the propagation algorithm in SCP(xk, xk′, pk, pk′), one can apply a similar strategy to the

Bopp operators (xk, pk, θk, λk) (see Table II). There aremultiple advantages of the latter representation. Impor-tantly, some open system interactions (e.g., the dephas-ing model explained in detail in Sec. ) take simpler forms

in terms of (xk, pk, θk, λk). The momentum and coordi-nate grids in (xk, xk′, pk, pk′) are interdependent suchthat if the discretization step size dx and the grid ampli-tude of x are specified, then the momentum incrementdp and the amplitude of p are fixed and vice versa. How-ever, the momentum and position grids in (xk, pk, θk, λk)are independent, thus allowing the flexibility to choosedx, dp, and amplitudes of x and p, in order to resolve thequantum dynamics of interest.

The following equation of motion is obtained from Eq.(20):

i~d

dt|Q〉 =

−→K

(pk +

~2λk

)|Q〉 − |Q〉←−K

(pk −

~2λk

),

+−→V

(xk +

~2θk

)|Q〉 − |Q〉←−V

(xk −

~2θk

).

(68)

The first order splitting leads to the two step propagation

|Q1/2〉 = e−idt~−→K(p+ ~

2 λ)|Qt〉eidt~←−K(p− ~

2 λ), (69)

|Qt+dt〉 = e−idt~−→V (x− ~

2 θ)|Q1/2〉e idt~←−V (x+ ~

2 θ). (70)

The employment of the appropriate basis at each step

9

removes the need for arrows

〈λp|Q1/2〉 = e−idt~ K(p+ ~

2 λ)〈λp|Qt〉eidt~ K(p− ~

2 λ), (71)

〈xθ|Q1/2〉 = Fλp→xθ〈λp|Q1/2〉, (72)

〈xθ|Qt+dt〉 = e−idt~ V (x− ~

2 θ)〈xθ|Q1/2〉e idt~ V (x+ ~2 θ), (73)

〈λp|Q1/2〉 = Fxθ→λp〈λp|Q1/2〉, (74)

where the Fourier transform conform with Eq.(39) andEq. (40) according to

Fxθ→λp ≡ Fx→λFθ→p = Fθ→pFx→λ, (75)

Fλp→xθ ≡ Fλ→xFp→θ = Fp→θFλ→x. (76)

A schematic view of the sequence of steps (71)-(74) isshown in Fig. 4. Note that to maintain consistency,the propagator must be solely expressed in terms of con-travariant components, e.g.,

K(p± ~2λ) = cαk

(pk ± ~

2λk)

+mc2γ0/2. (77)

The matrix exponentials in Eq. (71) can be evaluatedanalytically. For instance, assuming a two dimensionalquantum system (ignoring x3 and p3) we obtain

e−idt~ [cαkpk+mc2γ0] =

K11 0 0 K14

0 K11 K23 00 K32 K∗11 0K41 0 0 K∗11

, (78)

with

K11 = cos(cdtF/~)− imc sin(cdtF/~)

F, (79)

K14 =sin(cdtF/~)

F

(−ip1 − p2

), (80)

K23 = −U∗14, (81)

K32 = U14, (82)

K41 = −U∗14, (83)

F =√

(mc)2 + (p1)2 + (p2)2. (84)

Likewise, the exponential in Eq. (73) yields

e−idt~ [αµeAµ+mc2γ0] = e−

ieA0dt~

A11 0 A13 A14

0 A11 A23 A24

A31 A32 A∗11 0A41 A42 0 A∗11

,

(85)

with

A11 = cos(dtG/~)− imc2 sin(dtG/~)

G, (86)

A31 = A13 = iA3 sin(dtG/~)

G, (87)

A41 = A23 = (−A2 + iA1)sin(dtG/~)

G, (88)

A32 = A14 = −A∗41, (89)

A42 = A24 = A∗31, (90)

G =√

(mc2)2 + (A1)2 + (A2)2 + (A3)2. (91)

Having described the propagation for closed systemDirac evolution, we now proceed to introduce quantumdephasing (45), a particular open system interaction. Ac-cording to Eq. (53), the dephasing dynamics enters intothe exponential of the potential energy, thereby modify-ing the propagation step (73) as

〈xθ|Qt+dt〉 = e−idt~ V (x− ~

2 θ)〈xθ|Q1/2〉e idt~ V (x+ ~2 θ), (92)

with

− idt~V

(x± ~

2θ

)= − idt

~V

(x± ~

2θ

)− Ddt

2θ2. (93)

The replacement of Eq. (73) by Eq. (92) is mathemat-ically equivalent to Gaussian filtering along the momen-tum axis (i.e., convolution with a Gaussian in momen-tum) of the coherently propagated W (t, x1, p1). Thissimple interpretation of the dephasing dynamics playsa crucial role in Sec. .

The presented algorithm can be implemented with theresources of a typical desktop computer and are wellsuited for GPU computing [110]. In particular, the illus-tration in the next section were executed with a Nvidiagraphics card Tesla C2070.

MAJORANA SPINORS

Hereafter, assuming a one dimensional dynamics, theWigner function takes the functional form W (t, x1, p1).Furthermore, natural units (c = ~ = 1) are used through-out. In this section we employ a 512 × 512 grid for x1

and p1 as well as a time step dt = 0.01. Animations ofsimulations can be found in Ref. [111].

Majorana spinors, characterized for being their ownantiparticles, are the subject of interest in a broad rangeof fields including high energy physics, quantum informa-tion theory and solid state physics [112]. In particular,the solid state counterpart of the relativistic Majoranaspinors is known to be robust against perturbations andimperfections due to peculiar topological features [78].

In this section we study the dynamics of the originalMajorana spinor [82] in the presence of dephasing noise(45). Let

ψ =

ψ1

ψ2

ψ3

ψ4

(94)

be an arbitrary spinor, then there are two underlyingMajorana states (see, e.g., Chapter 12, page 165 of Ref.[113])

ψM± =

ψ1

ψ2

ψ3

ψ4

±

−ψ∗4ψ∗3ψ∗2−ψ∗1

. (95)

10

an#par#cle

par#cle

FIG. 5: (Color online) The relativistic Wigner functionW 0(t, x1, p1) for a Majorana spinor ψM+ associated with thespinor in Eq. (96) at (a) t = 0 and (b) at t = 12. Note thatthe particle undergoes dephasing with coefficient D = 0.01,without an external electromagnetic field. An animated illus-tration can be found in [111].

In particular, we propagate the Majorana spinor ψM+[shown in Fig. 5(a)] obtained from

ψ0 = e−(x1)2

2 +ix1p1(p0 +mc, 0, 0, p1)T , (96)

with p0 =√

(p1)2 + (mc)2 and the numerical values p1 =5, m = 1 and the dephasing coefficient D = 0.01 innatural units. The resulting time propagation of ψM+ isshown in Fig. 5 (b).

Figure 5 reveals that the particle-antiparticle super-position of the Majorana state generates a strong inter-ference in the phase space, which survives an even veryintense dephasing interaction. The reason of such robust-ness is that both the particle component (with a posi-tive momentum) and the antiparticle component (with anegative momentum) move in parallel along the positivespatial direction. This is in agreement with the interpre-tation of antiparticles as particles moving backwards intime. The interference fringes, consisting of negative andpositive stripes, also remain parallel to the momentumaxis. Considering the remark after Eq. (92), the actionof dephasing is equivalent to the Gaussian filtering alongthe p1 axis only. This mixes negative values with neg-ative, positive values with positive, but never positivewith negative values of the Wigner function. Hence, thisleaves the interference stripes invariant. In other words,free Majorana spinors evolve in a decoherence-free sub-space [114].

FIG. 6: (Color online) The relativistic Wigner functionW 0(t, x1, p1) for a particle-particle superposition correspond-ing to the spinor in Eq. (97) at (a) t = 0 and (b) at t = 12.Note that the particle undergoes dephasing with coefficientD = 0.01, without an external electromagnetic field. An ani-mated illustration can be found in [111].

The described Majorana state dynamics is fundamen-tally different from the evolution of a cat-state, i.e., aparticle-particle superposition. For example, up to a nor-malization factor, consider the following initial cat-state,composed of mostly particles:

ψ0 = e−(x1)2

2

[eix

1p1 + e−ix1p1]

(p0 +mc, 0, 0, p1)T .

(97)

Figure 6 depicts the evolution of this state under the in-fluence of the same dephasing interaction as in Fig. 5.Contrary to the Majorana case, the negative momentumcomponents of the cat state are made of particles; there-fore, we observe in Fig. 6 that they move along the nega-tive spatial direction. The interference stripes connectingthe positive (moving to the right) and negative (movingto the left) momentum components no longer remain par-allel with respect to the p1 axis. Thus, dephasing occursas the Gaussian filtering averages over positive and neg-ative stripes, thereby washing interferences out.

We note that the distortion from the original Gaussiancharacter of particle and atiparticle states at initial timein Figs. 5 and 6 is due to the momentum dispersion.

The total integrated negativity of the Wigner function

N(t) =

∫

W 0(t,x1,p1)<0

W 0(t, x1, p1)dx1dp1 (98)

11

Majorana State Cat State

0 1 2 3 4 5 6-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

t (natural units)

NegativeArea

FIG. 7: (Color online) Negativity (98) of the Majorana state(a particle-antiparticle superposition) in solid line correspond-ing to the free evolution presented in Fig. 5 in comparisonwith the negativity of the cat state (a particle-particle super-position) in dashed lines corresponding to Fig. 6.

is widely regarded as a measurement of the quantum co-herence because interferences are associated with Wignerfunction’s negative values. Figure 7 shows that the neg-ativity of the cat state reduces, while the negativity ofthe Majorana state is constant. Moreover, the negativ-ity of the free Majorana spinor remains constant even forextreme values of the decoherences. Therefore, this ro-bustness is not a perturbative effect with respect to thedephasing coefficient D. Note that Majorana spinor’s ini-tial negativity is more pronounced than that of the catstate (Fig. 7). Hence, Majorana states are more coherentthan cat-states.

Having studied free evolution, we now proceed to a Ma-jorana state evolving under the influence of the spatiallymodulated mass m → m + 0.05(x1)2. This type of sys-tem also maintains a high coherence despite significantdephasing D = 0.01. The initial Majorana state is shownin Fig. 8 (a) while the propagated state at time t = 14.is shown in Fig. 8 (b). The latter figure shows that inter-ference is preserved. A comparison of the negativities forMajorana and cat-states as functions of time are shownin Fig. 9, where the Majorana state negativity oscil-lates albeit with some decay, which is much slower thanthe cat-state decay. Figure 10, showing the full Wignerdynamics, sheds light on the revival of the Majorana’snegativity: When the particle and antiparticle compo-nents merge and separate, the negativity disappears andappears, respectively. Furthermore, Majorana’s dynam-ics seems to be approximately constrained to a surfacein the phase space and time, in contrast to the cat-statedynamics shown in Fig. 11.

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(a)

(b)

an#par#cle

par#cle

FIG. 8: (Color online) (a) Initial Majorana state extractedfrom (96), along with its marginal distribution in positionwhere the gray area represents the underlying mass modu-lated potential m→ m+0.05(x1)2. (b) Propagated Majoranastate at time t = 14. An animated illustration can be foundin [111].

Majorana State Cat State

0 5 10 15 20 25 30-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

t (natural units)

NegativeArea

FIG. 9: (Color online) Negativity of the Majorana state ofFig. 8 in solid line, compared to the negativity of the corre-sponding cat state.

12

FIG. 10: (Color online) Time stacked relativistic Wigner func-tion (0 ≤ t ≤ 20) for the Majorana dynamics shown in Fig.8. The interferences, located in the middle, remain robust allalong the evolution despite of the presence of significant quan-tum decoherence. The inteferences contain regions of negativevalue in blue. The integrated negativity (98) as a function intime is shown in Fig. 9.

KLEIN TUNNELING

As the second numerical example, we examine theKlein paradox [115], an unexpected consequence of theDirac equation, predicting that a positive energy parti-cle colliding with a sharp potential barrier of the heightV > mc2 is transmitted as a negative energy state. Forexample, the initial state (96) with p1 = 5, m = 1 isshown in Fig. 12 (a) along with the potential A0 =10(1 + tanh[4(x − 5)])/2. We observe in Fig. 12 (b)that most of the wavepacket has been transmitted as an-tiparticles.

An important extension of the Klein paradox is theKlein tunneling, where the step potential is replaced bya finite width barrier. In this case, the theoretical pre-diction specifies a high transmission even for a wide bar-rier. Condensed matter analogies of this phenomenonare a subject of active research [7, 116]. Three snapshotsof the Klein tunneling dynamics are shown in Fig. 13,where (a) corresponds to the positive energy initial state,(b) the state penetrating the potential barrier as antipar-ticle, and (c) the final state emerging from the barrier asparticle.

FIG. 11: (Color online) Time stacked relativistic Wigner func-tion (0 ≤ t ≤ 20) for a cat state evolving in the same potentialas the Majorana spinor in Fig. 10. The interferences, fadeshortly after the initiation of the propagation due to the ac-tion of quantum decoherence. The integrated negativity (98)as a function in time is shown in Fig. 9.

The Dirac particle has a spinorial as well as a configu-rational degree of freedom. The Klein tunneling can beviewed as an interband transition between positive andnegative energy states [117]. Analogous effects exist innon-relativistic dynamics. In particular, compared to thestructureless case, non-relativitic systems with many de-grees of freedom manifest many unique peculiarities suchas, e.g., transmission rate enhancement [118, 119] anddirectional symmetry breaking [120]. Thus, the energyexchange between different degrees of freedom underliesthe counterintuitive dynamics of both the Klein and thenon-relativistic tunneling of particles with internal struc-ture.

Furthermore, the Klein tunneling can be interpreted asthe Landau-Zener transition between positive and neg-ative energy states. This conclusion is obtained, e.g.,by comparing Eqs. (128) and (129) (setting Aµ = 0)with Eqs. (19)-(21) in Ref. [121]. This observation un-derscores an analogy between solid state and relativisticphysics.

Simulations with different values of the dephasing coef-ficient D have been performed in order to investigate theeffect of decoherence on the final transmission. Figure14 depicts the integrated negativity (98) as a function of

13

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(b)

(a)

an#par#cle

par#cle

FIG. 12: (Color online) Illustration of the Klein paradox interms of the relativistic Wigner function. The step potentialA0 = 10(1 + tanh[4(x − 5)])/2. is depicted as a gray area.The height of the step potential is V0 = 10 while the energyof the initial wavepacket is E = 5.01. (a) The initial stateW 0(t = 0, x1, p1) from Eq. (96) with p1 = 5. aimed towardsthe barrier. (b) Final state of the relativistic Wigner functionat t = 12 made of mostly of a negative energy wavepacket(antiparticle) being transmitted through the barrier.

time for three different values of D. The evolution with-out decoherence generates high negativity that indicatesinterference between the larger transmitted and smallerreflected wavepackets. In the same figure we observe thatthe decoherence eliminates negativity at later stages ofthe propagation. Nevertheless, the effect of decoherenceon the final transmission rate is small in Fig. 15, wherethe transmission as a function of time nearly coincides fordifferent values of D. We also note a weak dependence ofthe antiparticle generation on the dephasing coefficientas shown in Fig. 16. Contrary to non-relativistic quan-tum dynamics [24, 25, 28–31, 40, 90], decoherence in therelativistic regime does not recover a single particle clas-sical description. Furthermore, we show in Appendixthat the limit ~ → 0 of the Dirac equation leads to twoclassical Hamiltonians: One describing particles with aforward advancing clock (i.e., particles), while the other– a particle with backward flowing proper time (i.e., an-

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

(a)

5

0

-5

x1 (natural units)

p1(naturalunits)

par$cle

(a)

(a)

(b)

(c)

an$par$cle

par$cle

FIG. 13: (Color online) Illustration of the Klein tunnelingin terms of the relativistic Wigner function with the poten-tial barrier A0 = 5(tanh[4(x + 4)] + tanh[4(−x + 4)]) de-picted as a gray area. (a) The relativistic Wigner functionW 0(t = 0, x1, p1) for the initial state in Eq. (96) with p1 = 5.and positioned around x1 = −10. (b) The relativistic Wignerfunction at t = 6 in the process of entering the potential andtransforming into a negative energy wavepacket (antiparti-cle). (c) The final relativistic Wigner function at t = 24,where most of the initial wavepacket has been transmitted asa positive energy wavepacket (particle).

tiparticles). (This limit of the Dirac equation representsan example of classical Nambu dynamics [60].) This ex-plains the persistence of positive energy states even forstrong dephasing. We believe that the latter observationshould also hold in condensed matter physics.

14

D=0.0 D=0.01 D=0.05

0 5 10 15 20 25

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

t (natural units)

Negativity

FIG. 14: The integrated negative area in Eq. (98) as a func-tion of time for the Klein tunneling process. Three differ-ent values of the decoherence coefficient are considered forthe same initial state depicted in Fig. 13 (a). The first dipcorresponds to the first contact of the wave packet with thebarrier as shown in Fig. 13 (b). The second dip correspondsto the main wavepacket emerging from the barrier. Othersmaller dips appear as a contribution of the smaller reflectedwavepackets.

D=0.0 D=0.01 D=0.05

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t (natural units)

Transmission

FIG. 15: (Color online) The Klein transmission across the po-tential barrier as a function of time for the initial wavepacketshown in Fig. 13 (a), indicating a weak dependence on thedephasing intensity.

CONCLUSIONS

We introduced the density matrix formalism for rel-ativistic quantum mechanics as a generalization of thespinorial description of the Dirac equation. This formal-ism is employed to describe interactions with an envi-ronment. Moreover, we presented concise and effectivenumerical algorithms for the density matrix as well as

D=0.0 D=0.01 D=0.05

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t (natural units)

Antiparticleproportion

FIG. 16: (Color online) The antiparticle proportion as a func-tion of time for three different values of the decoherence coef-ficient in the Klein tunneling process. The initial state com-posed of mostly particles is shown in Fig. 13 (a). The firsthigh plateau corresponds to the period of time when mostof the wavepacket travels within the potential barrier as anantiparticle.

the relativistic Wigner function propagation.As a particularly important case, a Lindbland model

of quantum dephasing was studied. While decoherenceeliminated interferences, the particular structure of a freeMajorana spinor remained robust. Partial robustnesswas also observed for a coordinate dependent mass termin the Dirac equation. This robustness represents yet an-other remarkable attribute of Majorana spinors [122] notpresently acknowledged, which may be important exper-imentally. Moreover, the dynamics of the Klein paradoxas well as Klein tunneling turned out to be weakly af-fected by quantum dephasing.

The presented numerical approach opens new hori-zons in a number of fields such as relativistic quantumchaos [123], the quantum-to-classical transition, and ex-perimentally inspired relativistic atomic and molecularphysics [124–126]. Additionally, our method can be usedto simulate effective systems modeled by relativistic me-chanics, e.g., graphene [127, 128], trapped ions [14], op-tical lattices [129], and semiconductors [130, 131]. Fi-nally, the developed techniques can be generalized totreat Abelian [50, 132, 133] as well as non-Abelian [2, 134](e.g., quark gluon) plasmas.

Acknowledgments. The authors thank Wojciech Zurekfor insightful comments. R.C. is supported by DOEDE-FG02-02ER15344, D.I.B., and H.A.R. are partiallysupported by ARO-MURI W911NF-11-1-0268. A.C. ac-knowledges the support of the Fulbright Program. D.I.B.was also supported by 2016 AFOSR Young InvestigatorResearch Program.

15

Lorentz covariance of the Dirac equation

A vector in Feynman’s slash notation reads

u/ = uµγµ, (99)

where the gamma matrices obey the following Cliffordalgebra

γµγν + γνγµ = 2gµν1, (100)

with gµν = diag(1,−1,−1,−1). The restricted Lorentztransform does not carry out reflections and preservesthe direction of time and belongs to the group referredas SO+(1, 3). In the present case the transformation forthe vector u/ is carried out in terms of Lorentz spinorsL belonging to the double cover group of SO+(1, 3), ac-cording to

u/→ u′/ = Lu/L−1. (101)

The concept of a spinor as an operator can be found forexample in chapter 10 of Ref. [113]. The double coverof SO+(1, 3) is known as the Spin+(1, 3) group and isprecisely defined as

Spin+(1, 3) = L ∈Matrices(4,C)|Lγ0L†γ0 = 1(102)

For this type of Lorentz transform the inverse can beobtained as [113]

L−1 = γ0L†γ0. (103)

The restricted Lorentz transform can also be carriedout by the action of the complex special linear groupSL(2,C) ' Spin+(1, 3) [113, 135, 136], which is madeof 2 × 2 complex matrices with determinant one. Theproper orthochronous Lorentz transformations can beparametrized by 6 variables denoting rotations and boots

L = exp

(1

2ηkγ

0γk)

exp

(1

4εjklθ

jγkγl), (104)

where θj represent three rotation angles, ηk three boosts(rapidity variables) and γµ = γ−1µ . The proper velocitycan be obtained as the active boost of the proper velocityof a particle initially at rest with proper velocity u/rest =γ0. This means that in general it is possible to find aLorentz spinor L such that

u/ = Lu/restL−1 = LL†γ0. (105)

This expression indicates that the information storedin the 4-vector u/ can be carried out by the associatedLorentz rotor L and the fixed reference 4-vector u/rest.

The Lorentz transformation in Eq. (101) implies that

uµγµ = Luµγ

µL−1. (106)

Considering that uµ transforms as the components of acovariant tensor, we obtain

uν∂xν

∂x′µγµ = uνLγ

νL−1, (107)

which implies that

LγνL−1 =∂xν

∂x′µγµ. (108)

The Lorentz transformation of a vector field that de-pends on the spacetime position x is carried out in asimilar manner as (101)

A(x)→ A(x) = LA(x)L−1. (109)

Moreover, assuming that the origins of the referenceframes coincide,

A(x) = LA(L−1xL)L−1. (110)

The Lorentz transformation of a spinorial field is con-sistent accordingly

ψ(x)→ ψ(x) = Lψ(x) (111)

The manifestly covariant Dirac equation is

ic~γµ∂

∂xµψ(x)− γµeAµ(x)ψ(x)−mc2ψ(x) = 0, (112)

such that applying the Lorentz rotor L on the left weobtain

ic~Lγµ∂

∂xµL−1Lψ(x)− LγµeAµ(x)L−1Lψ(x)−mc2Lψ(x) = 0,

(113)

Employing Eq. (108), the first term of this equation canbe written as

i~Lγµ∂

∂xµL−1Lψ(x) =i~

∂xµ

∂xνγν

∂

∂xµψ(x) (114)

=i~γν∂

∂xνψ(x). (115)

Therefore, maintaining the form for the Dirac equationand demonstrating its relativistic covariance

i~γµ∂

∂xµψ(x)− γµeAµ(x)ψ(x) = mcψ(x). (116)

Furthermore, it follows that the relativistic density ma-trix P (x, x′) = ψ(x)ψ†(x′)γ0 transforms as

P (x, x′)→ P (x, x′) = ψ(x)ψ†(x′)γ0 (117)

= Lψ(x)ψ†(x′)L†γ0 (118)

= Lψ(x)ψ†(x′)γ0γ0L†γ0 (119)

= LP (x, x′)L−1. (120)

16

The classical limit of the Dirac equation

The Dirac equation reads

Dψ =[γ0γµ(cpµ − eAµ(x))− γ0mc2

]ψ = 0. (121)

In the classical limit, we understand the situation whenthe operators of the momenta pµ and coordinates xµ com-mute [44, 137, 138]. Following the Hilbert phase spaceformalism [44, 89], we separate the commutative and non-commutative parts of the Dirac generator D by introduc-ing the algebra of classical observables

[xµ, pν ] = 0, [pµ, θν ] = −iδνµ, (122)

[xµ, λν ] = −iδµν , [λµ, θν ] = 0, (123)

which is connected with the quantum observables as

xµ = xµ − ~θµ/2, pµ = pµ + ~λµ/2. (124)

Substituting Eq. (124) into Eq. (121) and keeping theterms up to the zero-th order in ~, we get a function ofxµ and pµ. Considering that xµ and pµ commute, wedrop the hat hereafter such that

D = γ0γµ(cpµ − eAµ)− γ0mc2 +O(~). (125)

Utilizing the following unitary operator U

U =

√Ep +mc2

2Ep

(1− γk(cpk − eAk)

Ep +mc2

), (126)

Ep =√

(mc2)2 + (cp− eA)k · (cp− eA)k, (127)

we finally obtain

lim~→0

UDU† =

H+ 0H+

H−0 H−

, (128)

with

H± = cp0 − eA0 ± Ep. (129)

According to Eq. (129), the Dirac generator D in theclassical limit corresponds to a decoupled pair of classi-cal time-extended Hamiltonians. The Hamiltonian H+

describes the dynamics of a classical relativistic parti-cle; while, H− governs the dynamics of a particle travel-ing backwards in time, which resembles an antiparticle.These conclusions confirm the results of numerical sim-ulations in the main text, where a Dirac particle wascoupled to a bath causing decoherence that physicallyrealizes the ~→ 0 limit.

∗ Electronic address: rcabrera@princeton.edu

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