Relativistic ponderomotive Hamiltonian of a Dirac particle in a vacuum laser field

# Relativistic ponderomotive Hamiltonian of a Dirac particle in a vacuum laser field

D. E. Ruiz, C. L. Ellison, and I. Y. Dodin Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA
August 25, 2019
###### Abstract

We report a point-particle ponderomotive model of a Dirac electron oscillating in a high-frequency field. Starting from the Dirac Lagrangian density, we derive a reduced phase-space Lagrangian that describes the relativistic time-averaged dynamics of such a particle in a geometrical-optics laser pulse propagating in vacuum. The pulse is allowed to have an arbitrarily large amplitude provided that radiation damping and pair production are negligible. The model captures the Bargmann-Michel-Telegdi (BMT) spin dynamics, the Stern-Gerlach spin-orbital coupling, the conventional ponderomotive forces, and the interaction with large-scale background fields (if any). Agreement with the BMT spin precession equation is shown numerically. The commonly known theory in which ponderomotive effects are incorporated in the particle effective mass is reproduced as a special case when the spin-orbital coupling is negligible. This model could be useful for studying laser-plasma interactions in relativistic spin- plasmas.

## I Introduction

In recent years, many works have been focused on incorporating quantum effects into classical plasma dynamics [1; 2]. In particular, various models have been proposed to marry spin equations with classical equations of plasma physics. This includes the early works by Takabayasi [3; 4] as well as the most recent works presented in Refs. [5; 6; 7; 8; 9; 10; 11; 12; 13]. Of particular interest in this regard is the regime wherein particles interact with high-frequency electromagnetic (EM) radiation. In this regime, it is possible to introduce a simpler time-averaged description, in which particles experience effective time-averaged, or “ponderomotive,” forces [14; 15; 16]. It was shown recently that the inclusion of spin effects yields intriguing corrections to this time-averaged dynamics [17; 18]. However, current “spin-ponderomotive” theories remain limited to regimes where (i) the particle de Broglie wavelength is much less than the radiation wavelength and (ii) the radiation amplitude is small enough so that it can be treated as a perturbation. These conditions are far more restrictive than those of spinless particle theories, where non-perturbative, relativistic ponderomotive effects can be accommodated within the effectively modified particle mass [19; 20; 21; 22; 23]. One may wonder then: is it possible to derive a fully relativistic, and yet transparent, theory accounting also for the spin dynamics and the Stern-Gerlach-type spin-orbital coupling?

Excitingly, the answer is yes, and the purpose of this paper is to propose such a description. More specifically, what we report here is a point-particle ponderomotive model of a Dirac electron [24]. Starting from the Dirac Lagrangian density, we derive a phase-space Lagrangian (75) in canonical coordinates with a Hamiltonian (76) that describes the relativistic time-averaged dynamics of such particle in a geometrical-optics (GO) laser pulse propagating in vacuum [25]. The pulse is allowed to have an arbitrarily large amplitude (as long as radiation damping and pair production are negligible) and, in case of nonrelativistic interactions, a wavelength comparable to the electron de Broglie wavelength. The model captures the spin dynamics, the spin-orbital coupling, the conventional ponderomotive forces, and the interaction with large-scale background fields (if any). Agreement with the Bargmann-Michel-Telegdi (BMT) spin precession equation [26] is shown numerically. The aforementioned “effective-mass” theory for spinless particles is reproduced as a special case when the spin-orbital coupling is negligible. Also notably, the point-particle Lagrangian that we derive has a canonical structure, which could be helpful in simulating the corresponding dynamics using symplectic methods [27; 28; 29].

This work is organized as follows. In Sec. II the basic notation is defined. In Sec. III the main assumptions used throughout the work are presented. To arrive at the point-particle ponderomotive model, Secs. IV-VII apply successive approximations and reparameterizations to the Dirac Lagrangian density. Specifically, in Sec. IV we derive a ponderomotive Lagrangian density that captures the average dynamics of a Dirac particle. In Sec. V we obtain a reduced Lagrangian model that explicitly shows orbital-spin coupling effects. In Sec. VI we deduce a “fluid” Lagrangian model that describes the particle wave packet dynamics. In Sec. VII we calculate the point-particle limit of such “fluid” model. In Sec. VIII the ponderomotive model is numerically compared to a generalized non-averaged BMT model. In Sec. IX the main results are summarized.

## Ii Notation

The following notation is used throughout the paper. The symbol “” denotes definitions, “h. c.” denotes “Hermitian conjugate,” and “c. c.” denotes “complex conjugate.” Unless indicated otherwise, we use natural units so that the speed of light and the Plank constant equal unity (). The identity matrix is denoted by . The Minkowski metric is adopted with signature . Greek indices span from to and refer to spacetime coordinates with corresponding to the time variable . Also, and . Latin indices span from to and denote the spatial variables, i.e., and . Summation over repeated indexes is assumed. In particular, for arbitrary four-vectors and , we have and . The Feynman slash notation is used: , where are the Dirac matrices (see below). The average of an arbitary complex-valued function with respect to a phase is denoted by . In Euler-Lagrange equations (ELEs), the notation “” means that the corresponding equation is obtained by extremizing the action integral with respect to .

## Iii Basic Formalism

As for any quantum particle or non-dissipative wave [30], the dynamics of a Dirac electron [24] is governed by the least action principle , where is the action

 A=∫Ld4x, (1)

and is the Lagrangian density given by [31]

 L=i2[¯ψγμ(∂μψ)−(∂μ¯ψ)γμψ]−¯ψq⧸Aψ−¯ψmψ. (2)

Here and are the particle charge and mass, is a complex four-component wave function, and is its Dirac conjugate. The Dirac matrices satisfy

 γμγν+γνγμ=2gμνI4, (3)

where is the Minkowski metric tensor. Hence,

 (4) ⧸a⧸a=a2I4, (5)

for any pair of four-vectors and . In this work, the standard representation of the Dirac matrices is used:

 γ0=(I200−I2),γ=(0σ−σ0), (6)

where are the Pauli matrices. Note that these matrices satisfy

 (γμ)†=γ0γμγ0. (7)

We consider the interaction of an electron with an EM field such that the four-vector potential has the form

 A(ϵx,Θ)=Abg(ϵx)+Aosc(ϵx,Θ). (8)

Here describes a background field (if any) that is slow, as determined by a small dimensionless parameter , which is yet to be specified. The other part of the vector potential,

 Aosc(ϵx,Θ)=Re[Aosc,c(ϵx)eiΘ], (9)

describes a rapidly oscillating EM wave field, e.g., a laser pulse. Here is a complex four-vector describing the laser envelope with a slow spacetime dependence, and is a rapid phase. The EM wave frequency is defined by , and the wave vector is . Accordingly, . We describe within the GO approximation [32] and assume that the interaction takes place in vacuum. Then, the four-wavevector satisfies the vacuum dispersion relation

 k2=ω2−k2=0, (10)

which can also be expressed as

 ⧸k⧸k=k2I4=0. (11)

Furthermore, a Lorentz gauge condition is chosen for the oscillatory field such that

 ∂μAμosc=0. (12)

In this work, we neglect radiation damping and assume

 ω′/ωc≪1, (13)

where is the Compton frequency and is the frequency in the electron rest frame. Then, pair production (and annihilation) can be neglected. We also assume

 ϵ≐max(1ωτ,1|k|ℓ)≪1, (14)

where and are the characteristic temporal and spatial scales of , , and . Using this ordering and the Lagrangian density (2), we aim to derive a reduced Lagrangian density that describes the ponderomotive (-averaged) dynamics of an electron accurately enough to capture the spin-orbital coupling effects to the leading order in . As shown in Refs. [33; 34], this requires that terms be retained when approximating the Lagrangian density (2). Such reduced Lagrangian density is derived as follows.

## Iv Ponderomotive Model

In this section, we derive a ponderomotive Lagrangian density for the four-component Dirac wave function.

### iv.1 Wave function parameterization

Consider the following representation for the four-component wave function:

 ψ(x)=ξeiθ. (15)

Here is a fast real phase, and is a complex four-component vector slow compared to . In these variables, the Lagrangian density (2) is expressed as

 L=i2[¯ξγμ(∂μξ)−(∂μ¯ξ)γμξ]+¯ξ(⧸π−q⧸Aosc−mI4)ξ, (16)

where

 πμ(ϵx)≐pμ−qAμbg, (17) pμ(ϵx)≐−∂μθ. (18)

It is convenient to parameterize in terms of the “semiclassical” Volkov solution (Appendix A) since the latter becomes the exact solution of the Dirac equation in the limit of vanishing . Specifically, we write

 ξ(ϵx,Θ)=Ξei~θφ. (19)

Here is a near-constant function with an asymptotic representation of the form

 φ(ϵx,Θ)=∞∑n=−∞ϵ|n|φn(ϵx)einΘ. (20)

(so that at ), the real phase is given by

 ~θ(ϵx,Θ)≐qπ⋅k∫Θπ⋅AoscdΘ′−q22(π⋅k)∫Θ(A2osc−⟨A2osc⟩)dΘ′ (21)

and has the property , and is a matrix defined as follows:

 Ξ(ϵx,Θ)≐I4+q2(π⋅k)⧸k⧸Aosc. (22)

Notice also that the Dirac conjugate of is given by

 ¯ξ=(Ξei~θφ)†γ0=e−i~θφ†Ξ†γ0=e−i~θ¯φγ0Ξ†γ0, (23)

where

 γ0Ξ†γ0= γ0[I4+q2(π⋅k)⧸A†oscγ0γ0⧸k†]γ0 = I4+q2(π⋅k)⧸Aosc⧸k. (24)

Here we used Eqs. (3), (7), and (22).

### iv.2 Lagrangian density in the new variables

Inserting Eqs. (19) and (23) into Eq. (16) leads to

 L= ¯φγ0Ξ†γ0⧸πΞφ=L1−¯φmγ0Ξ†γ0Ξφ=L2−¯φγ0Ξ†γ0q⧸AoscΞφ=L3 +i2[¯φγ0Ξ†γ0γμΞ(∂μφ)−c.\,c.]=L4−¯φγ0Ξ†γ0(⧸∂~θ)Ξφ=L5 +i2¯φ[γ0Ξ†γ0γμ(∂μΞ)−h.\,c.]φ=L6. (25)

Let us explicitly calculate each term in Eq. (25). Substituting Eqs. (22) and (24) into leads to

 L1= ¯φγ0Ξ†γ0⧸πΞφ = ¯φ[I4+q2(π⋅k)⧸Aosc⧸k]⧸π[I4+q2(π⋅k)⧸k⧸Aosc]φ = ¯φ[⧸π+q2(π⋅k)(⧸Aosc⧸k⧸π+⧸π⧸k⧸Aosc) +q24(π⋅k)2⧸Aosc⧸k⧸π⧸k⧸Aosc]φ = ¯φ[⧸π+q⧸Aosc+qAosc⋅kπ⋅k⧸π−qAosc⋅ππ⋅k⧸k +q2Aosc⋅kπ⋅k⧸Aosc−q2A2osc2(π⋅k)⧸k]φ, (26)

where we used Eqs. (4), (5), and (11). Similarly,

 L2= −¯φmγ0Ξ†γ0Ξφ = −¯φm[I4+q2(π⋅k)⧸Aosc⧸k][I4+q2(π⋅k)⧸k⧸Aosc]φ = −¯φm[I4+q2(π⋅k)(⧸Aosc⧸k+⧸k⧸Aosc)]φ = −¯φm(1+qAosc⋅kπ⋅k)φ, (27)

where we used Eq. (11) to obtain the third line and Eq. (4) to get the last line. For , one obtains

 L3= −¯φγ0Ξ†γ0q⧸AoscΞφ = −¯φ[I4+q⧸Aosc⧸k2(π⋅k)]q⧸Aosc[I4+q⧸k⧸Aosc2(π⋅k)]φ = −¯φ[q⧸Aosc+q2π⋅k⧸Aosc⧸k⧸Aosc +q34(π⋅k)2⧸Aosc⧸k⧸Aosc⧸k⧸Aosc]φ = −¯φ[q⧸Aosc+2q2Aosc⋅kπ⋅k⧸Aosc−q2A2oscπ⋅k⧸k +q3Aosc⋅k2(π⋅k)2⧸Aosc⧸k⧸Aosc]φ. (28)

The terms , , and in Eq. (25) involve spacetime derivatives of , , , which have slow spacetime and rapid dependences. For convenience, let us write the derivative operator as follows:

 ∂μf(ϵx,Θ)=ϵdμf(ϵx,Θ)−kμ∂Θf(ϵx,Θ), (29)

where is an arbitrary function and indicates a derivation with respect to the first argument of . In this notation, can be written as follows:

 L4 =i2[¯φγ0Ξ†γ0γμΞ(∂μφ)−c.\,c.] =−i2[¯φγ0Ξ†γ0⧸kΞ(∂Θφ)−c.\,c.] +iϵ2[¯φγ0Ξ†γ0γμΞ(dμφ)−c.\,c.] =−i2[¯φ⧸k(∂Θφ)−(∂Θ¯φ)⧸kφ] +iϵ2[¯φγ0Ξ†γ0γμΞ(dμφ)−c.\,c.], (30)

where in the third line, we used Eqs. (11), (22), and (24). Similarly, substituting Eq. (21) into leads to

 L5 =−¯φγ0Ξ†γ0(⧸∂~θ)Ξφ =¯φ[I4+q⧸Aosc⧸k2(π⋅k)]⧸k(∂Θ~θ)[I4+q⧸k⧸Aosc2(π⋅k)]φ −ϵ¯φ[I4+q⧸Aosc⧸k2(π⋅k)](⧸d~θ)[I4+q⧸k⧸Aosc2(π⋅k)]φ =¯φ⧸kφ⎡⎢ ⎢⎣qAosc⋅ππ⋅k−q2A2osc2(π⋅k)+q2⟨A2osc⟩2(π⋅k)⎤⎥ ⎥⎦ −ϵ¯φ[I4+q⧸Aosc⧸k2(π⋅k)](⧸d~θ)[I4+q⧸k⧸Aosc2(π⋅k)]φ. (31)

Finally, the last term gives

 L6 =i2¯φ[γ0Ξ†γ0γμ(∂μΞ)−h.\,c.]φ =iϵ2¯φ{[I4+q⧸Aosc⧸k2(π⋅k)]⧸d[q⧸k⧸Aosc2(π⋅k)] −dμ[q⧸Aosc⧸k2(π⋅k)]γμ[I4+q⧸k⧸Aosc2(π⋅k)]}φ. (32)

Substituting Eqs. (26)-(32) into Eq. (25) leads to

 L=−i2[¯φ⧸k(∂Θφ)−(∂Θ¯φ)⧸kφ]+¯φ⎡⎢ ⎢⎣⧸π+⧸kq2⟨A2osc⟩2(π⋅k)−mI4⎤⎥ ⎥⎦φ+F+G, (33)

where

 F≐iϵ2[¯φγ0Ξ†γ0γμΞ(dμφ)−c.\,c.] (34)

and

 G≐¯φ[qχπ⋅k(⧸π−mI4)−q2χπ⋅k⧸Aosc−q3χ2(π⋅k)2⧸Aosc⧸k⧸Aosc]φ−ϵ¯φ[I4+q⧸Aosc⧸k2(π⋅k)](⧸d~θ)[I4+q⧸k⧸Aosc2(π⋅k)]φ+iϵ2¯φ{[I4+q⧸Aosc⧸k2(π⋅k)]⧸d[q⧸k⧸Aosc2(π⋅k)]−dμ[q⧸Aosc⧸k2(π⋅k)]γμ[I4+q⧸k⧸Aosc2(π⋅k)]}φ. (35)

Here we introduced . From Eqs. (12) and (29), one has , so

 χ=ϵ∫ΘdμAμosc(ϵx,Θ′)dΘ′. (36)

Hence, it is seen that , so .

### iv.3 Approximate Lagrangian density

The reduced Lagrangian density that governs the time-averaged, or ponderomotive, dynamics can be derived as the time average of , as usual [35; 36]. In our case, the time average coincides with the -average, so

 L≐⟨L⟩. (37)

Remember that we are interested in calculating with accuracy up to . Using Eqs. (11) and (20) and also the fact that is shifted in phase from by [cf. Eq. (36)], it can be shown that . Therefore, the contribution of to can be neglected. Similarly, we can also neglect the first term in Eq. (33) since

 −i2⟨¯φ⧸k(∂Θφ)−(∂Θ¯φ)⧸kφ⟩=∞∑n=−∞nϵ|2n|¯ϕn⧸kϕn=O(ϵ2), (38)

where we substituted the asymptotic expansion (20). The second term in Eq. (33) gives

 (39)

By following similar considerations, we also calculate , namely as follows. Averaging the first term in , we obtain

 ⟨¯φ0γ0Ξ†γ0γμΞdμφ0⟩ =¯φ0Γμdμφ0, (40)

where we used Eqs. (3) and (11). We also introduced the modified Dirac matrices

 Γμ(ϵx)≐ γμ+kμq22(π⋅k)2⟨⧸Aosc⧸k⧸Aosc⟩ = γμ+kμq22(π⋅k)2⟨⧸Aosc(2χ−⧸Aosc⧸k)⟩ = γμ−kμq22(π⋅k)2⟨⧸Aosc⧸Aosc⟩⧸k = γμ−kμq2⟨A2osc⟩2(π⋅k)2⧸k. (41)

Gathering the previous results, we obtain the following reduced Lagrangian density

 L=i2[¯ϕΓμ(∂μϕ)−(∂μ¯ϕ)Γμϕ]+¯ϕ⎡⎢ ⎢⎣⧸π+⧸kq2⟨A2osc⟩2(π⋅k)−mI4⎤⎥ ⎥⎦ϕ+O(ϵ2), (42)

where . Since only slow spacetime dependences appear in Eq. (42), we dropped the “” notation for slow spacetime derivatives and returned to the “” notation.

## V Reduced Model

In this section, the Lagrangian density (42) is further simplified by considering only positive-kinetic-energy particle states. The resulting model describes two-component wave functions instead of four-component wave functions, which leads to explicit identification of the spin-coupling term.

### v.1 Particle and antiparticle states

First let us briefly review the case when is vanishingly small so that can be neglected. In this case, Eq. (42) can be approximated as

 L0[θ,ϕ,¯ϕ]=¯ϕ⎡⎢ ⎢⎣⧸π+⧸kq2⟨A2osc⟩2(π⋅k)−mI4⎤⎥ ⎥⎦ϕ. (43)

where , , and can be treated as independent variables. [The Lagrangian density depends on in the sense that it depends on , which is defined through (Sec. IV.1).] When varying the action with respect to , the corresponding ELE is

 δ¯ϕ:(⧸λ−mI4)ϕ=0, (44)

where

 λμ(ϵx)≐πμ+αkμ (45)

is a quasi four-momentum [37] and

 α(ϵx)≐q2⟨A2osc⟩2(π⋅k). (46)

The local eigenvalues are obtained by solving

 det(⧸λ−mI4)=0. (47)

Since the local dispersion relation (47) has the same form as that of the free-streaming Dirac particle [38], one has

 λ⋅λ=π⋅π+q2⟨A2osc⟩=m2, (48)

where we used Eq. (10). Solving for leads to

 π0=−∂tθ−qVbg=±√(∇θ−qAbg)2+m2eff. (49)

Here is the “effective mass” [19; 20; 21] given by

 m2eff(ϵx)≐m2−q2⟨A2osc⟩(ϵx). (50)

(The minus sign is due to the chosen metric signature.) Equation (49) is the well known Hamilton-Jacobi equation that governs the ponderomotive dynamics of a relativistic spinless particle interacting with an oscillating EM vacuum field and a slowly varying background EM field [39; 40; 41; 42]. The two roots in Eq. (49) represent solutions for the particle and the antiparticle states.

### v.2 Eigenmode decomposition

Corresponding to the eigenvalues given by Eq. (49), there exists four orthonormal eigenvectors which are obtained from Eq. (44) and represent the particle and the antiparticle states. Since form a complete basis, one can write , where are scalar functions. Recall also that pair production is neglected in our model due to the assumption (13). Let us hence focus on particle states, merely for clarity, which correspond to positive kinetic energies

 εeff=√π2+m2eff (51)

in the limit of vanishing . We will assume that only such states are actually excited (we call these eigenmodes “active”), whereas the antiparticle states acquire nonzero amplitudes only through the medium inhomogeneities (we call these eigenmodes “passive”). When designating the active mode eigenvectors by and the passive mode eigenvectors by , we have

 ηq={O(ϵ0),q=1,2O(ϵ1),q=3,4. (52)

As shown in LABEL:Ruiz:2015hq, due to the mutual orthogonality of all , the contribution of passive modes to is , so it can be neglected entirely. In other words, for the purpose of calculating , it is sufficient to adopt . It is convenient to write this active eigenmode decomposition in a matrix form

 ϕ(ϵx)=Ψη, (53)

where

 Ψ(ϵx)=√m+λ02εeff(I2σ⋅λm+λ0) (54)

is a matrix having and as its columns and

 η(ϵx)≐(η1η2). (55)

Note that and describe wave envelopes corresponding to the spin-up and spin-down states.

When inserting the eigenmode representation (53) into Eq. (42), one obtains [33]

 L=K−η†(E−U)η+o(ϵ), (56)

where

 K≐i2[η†Ψ†γ0ΓμΨ(∂μη)−c.\,c.], (57) E≐∂tθ+εeff+qVbg, (58) U≐i2[Ψ†γ0Γμ(∂μΨ)−h.\,c.]. (59)

The terms and , which are of order , represent corrections to the lowest-order (in ) Lagrangian density. Specifically, for one obtains (Appendix B.1)

 K=i2[η†(dtη)−(dtη†)η], (60)

where is a convective derivative associated to the zeroth-order velocity field

 v0(ϵx)≐∂εeff∂p=πεeff. (61)

Regarding , one obtains the ponderomotive spin-orbit coupling Hamiltonian (Appendix B.2)

 U=12σ⋅Ωeff, (62)

where

 Ωeff(ϵx)=qεeff(Bbg−λ×Ebgm+λ0)+q22εeff(π⋅k)⎡⎢ ⎢⎣k×∇⟨A2osc⟩−(λ×k)∂t⟨A2osc⟩m+λ0−k0λ×∇⟨A2osc⟩m+λ0⎤⎥ ⎥⎦+q2⟨A2osc⟩2εeff(π⋅k)2{(k0λm+λ0−k)×[k0qEbg+k×qBbg−(πμ∂μ)k]−λ×km+λ0[k⋅qEbg−(πμ∂μ)k0]} (63)

and .

When substituting Eqs. (51), (60), and (62) into Eq. (56), one obtains the following effective Lagrangian density

 L=−η†(∂tθ+√π2+m2eff+qVbg)η+i2[η†(dtη)−(dtη†)η]+12η†σ⋅Ωeffη. (64)

The first line of Eq. (64) represents the zeroth-order Lagrangian density that would describe a spinless relativistic electron. The second line, which is of order , introduces spin-orbit coupling effects. Also note that the Lagrangian density (64) is analogous to that describing circularly-polarized EM waves in isotropic dielectric media when polarization effects are included [43].

## Vi Continuous wave model

Here we construct a “fluid” description of the Dirac electron described by Eq. (64). Let us adopt the representation , where is a real function (called the action density) and is a unit vector such that . [From now on, we drop in the function arguments to simplify the notation, but we will continue to assume that the corresponding functions are slow.] Since the common phase of the two components of can be attributed to , we parameterize in terms of just two real functions and :

 z(ϑ,ζ)=(e−iϑ/2cos(ζ/2)eiϑ/2sin(ζ/2)). (65)

As in the case of the Pauli particle [34], determines the relative fraction of “spin-up” and “spin-down” quanta. Note that, under this reparameterization, the spin vector is given by

 S≐12z†σz=⎛⎜⎝sinζcosϑsinζsinϑcosζ⎞⎟⎠, (66)

and .

Expressing Eq. (64) in terms of the four independent variables leads to

 (67)

where one can immediately recognize the first line of Eq. (67) as Hayes’ representation of the Lagrangian density of a GO wave [44]. Four ELEs are yielded. The first one is the action conservation theorem

 δθ:∂tI+∇⋅(IV)=0. (68)

The flow velocity is given by , where