Maxwell-Like Equations for Free Dirac Electrons

1 Introduction

There have been many attempts to construct classical models for a spin-1/2 particle [1], [2], [3], [4], [5], [6], [7], [8], [9]. This interest arises from the need to more fully understand the somehow intriguing internal structure of the electron. In the Barut-Zanghi theory [9], the electron is characterised by a Lagrangian whose equations of motion exhibit a classical analogue to the phenomenon of Zitterbewegung (ZB). Based on these ideas, Salesi and Recami [10], [11], [12], [13] present a field theory of the spinning electron that constructs a classically intelligible description of the electron. Different approaches have also been proposed [14], [15], [16], [17], [18]. In particular, Campolattaro [19] shows that the Dirac equation is equivalent to Maxwell’s equations for the electromagnetic field generated by two currents: one electric in nature and one magnetic-monopolar.

This article gives a different representation to this problem. It is well known [20], [21], [22], [23], [24] that Maxwell’s equations can be written in spinor notation which resembles that used for the Dirac electron. Here, we want to proceed in the opposite direction, by writing down Maxwell-like equations for the free Dirac electron bispinor components. If this is really possible, we need to find (electric-like) vector and (magnetic-like) pseudo-vector fields describing the electron bispinor wavefunction in a way similar to electrodynamic formalism. At first sight, this appears quite reasonable in view of the fact that the eigenvalue of the electron velocity operator is ±c, which resembles a photon-like behaviour.

2 Maxwellian Description of a Free Dirac Electron

To begin with, we consider the well-known [25] basic free-field Lorentz scalar Dirac Lagrangian density from which the field equations are derived (hereafter ħ=c=1):

where x=(t, x), with metric signature g(+−−−). Here, we choose the standard Dirac representation

in which σ_i, i=1, 2, 3, are Pauli spin matrices. Varying ℒ with respect to Ψ¯=Ψ†γ0, we find the (Euler-Lagrange) Dirac equation for a free electron of mass m_e:

The Dirac bispinor Ψ can be written in terms of two-component spinors ϕ and χ:

where, in general, we can write

with ϕ_R, _I and χ_R, _I spinors with real components. Let ψ denote either ϕ or χ. Thus, we define

with complex spinors ψ_N and ψ_M symmetric and antisymmetric under space inversion x→−x, respectively.

If the spinor (4) describes either a positive-energy free electron or a negative-energy free electron, ϕ and χ must have opposite parity. Motivated by [26], [27], for a state of positive parity (an electron), we set

where M_i, N_j, U_S, and U_P, i=1, 2, 3, are eight independent real fields, which under space inversion transform as

The parity 𝒫 and the charge conjugation 𝒞 operators for the Dirac field Ψ are defined as usual:

with ϕC(x)=σ2χM∗(x) and χC(x)=−σ2ϕN∗(x).

Inserting (7) and (4) into (1), we can rewrite the Lagrangian density in the form

where G^αβ(x) and G˜αβ(x) are second-rank tensors defined by

Here, K_αβ=g_αγK^γδg_δβ is the analogue of the antisymmetric EM field-strength tensor whose components are

with K˜αβ=(1/2)ϵαβγδKγδ, where ϵ_αβγδ is the Levi-Civita pseudo-tensor (ϵ⁰¹²³=1). The symmetric fields (in the indices α and β) g^αβU_S and g^αβU_P transform like tensor and pseudo-tensor, respectively. In (10), q^α is a constant spacelike four-vector (it does not depend on the space-time coordinates) under restricted Lorentz transformations, which will be specified for a particular inertial reference frame below. Thus, the electromagnetic-like tensors (11) are no longer traceless:

We derive the (coupled) differential equations of motion for M_i, N_j, U_S, and U_P from (3) by directly using the bispinor representation (7). After a straightforward algebra, we find a set of eight (formally) simple Maxwell-like equations:

In the above equations, we have chosen a reference frame where (q^α)=(0, 0, 0, 1)=(0, q), which is consistent with the Dirac representation (2). From (3), we find that F_a={M_i, N_j, U_S, U_P} also satisfies the Klein-Gordon equations:

Note that on similar grounds, the massless case is discussed in [26], [27].

Lorentz invariance of (14) can easily be checked as one does in the case of Maxwell’s equations for dyons [27]. From (14), we define electric and magnetic-like (self) four-currents

where, as represented above, q transforms as the spatial component of a four-vector. Hence, we can write (14) in the covariant form

These equations of motion can be derived by varying the Lagrangian (10) with respect to the four-vectors χ_η=q^αG_ηα and χ˜η=qαG˜ηα, so that (17) are deduced from the Euler-Lagrange equations

Here, we should emphasise that, in general, the following terms do not vanish

Note that the presence of U_S and U_P (identified as scalar and pseudo-scalar fields, respectively) induces longitudinal components in M and N, since from (14) we can obtain the (Klein-Gordon) second-order differential equations

On the left-hand side of (20), the terms in U_S and U_P vanish because [∇, ∂_t]=0.

It is worth mentioning that jMβ(x) and jNβ(x) are, generally, not conserved four-currents as it is easily determined from (17). In fact

However, from (15), we can define conserved four-currents by making

Starting with (17), it follows that the scalar product of jMβ(x) and jNβ(x) is the pseudo-scalar (invariant) quantity

3 Plane Wave Solutions and ZB

Next, we want to study plane wave solutions for (14). First, we choose a positive-energy solution of (3):

where

for spin-up (down) wavefunctions, respectively, where φ_p(x)=p·x−E_pt is the invariant phase, with Ep=+me2+p2. Identifying Ψ→ΨEp±, we get, for spin-up solutions,

with Cp=(me+Ep)/Ep. Correspondingly, for spin-down solutions,

where p→−p under space inversion x→−x. Thus, in both cases, j_M·j_N=0. For the latter case, M·N=0 in any inertial reference frame because U_S(x)=0.

In what follows, we regard a particularly interesting expression for the flux density associated with the Dirac field, as represented by (4) and (7). Consider a basic non-definite parity Dirac bispinor Φ as a linear superposition of positive- and negative-energy contributions:

The corresponding flux density is given by

Here, α_i are the Dirac matrices and ϵ^i,i=1, 2, 3, is the usual Cartesian canonical basis. The flux S(x) contains two pieces: (i) a longitudinal component in curl bracket {…}, and (ii) a ZB component in squared bracket […]. Note that S(x) represents the analogue of the Poynting vector for the electron [27]. Let us consider the motion of a spin-down electron along the z axis (p₁=0=p₂). From either (24) or (26) together with (29), we find that

Equation (30) shows that the electron motion follows, as one would expect, a helical path [1], [9], [10], with the radius of the circle decreasing as the energy increases (see Fig. 1). Thus, for Ep3→∞, ZB fades out, although its frequency increases linearly with Ep3. Notice that, by inserting ħ and c into the phase 2φp3(z, t) in the rest frame system (p₃=0), we get 2|∂tφp3(z, t)|=2mec2​​/​ℏ, which is the ZB frequency.

Figure 1:

Schematic graph (not to scale) of the flux density (30) when the electron is in the superposition state Φ(x) for spin-down solutions ΨEp3−(x).

4 Concluding Remarks

To summarise, we have derived Maxwell-like equations for free Dirac electrons starting with the associated Lagrangian density. For completeness, it was necessary to incorporate appropriate scalar and pseudo-scalar fields U_S and U_P into the Dirac bispinor (4), according to both (7) and (8). We then studied plane wave solutions associated to this system. To this end, we defined a Dirac bispinor as a linear superposition of positive- and negative-energy contributions obtaining the corresponding flux density, which exhibits a helical trajectory due to the electron ZB, as depicted in Figure 1.

A natural ensuing task is to attempt (second) quantisation of the free fields (11) which satisfy the equations of motion (14). Here, we have to follow, as much as possible, the quantisation procedure we know for the electron case, considering that the occupation number of a particular electron state is at most one. This problem will be treated in a future article.

Finally, note that (to some extent conversely) for a realistic, classical version of ZB, see [28], where this “trembling” motion at constant speed c is used to derive the Schrödinger equation. For the physical origin of ZB as a due to self-reaction, see [29], [30], [31]. It is noteworthy to mention that ZB of a free particle has never been directly observed [32], [33], [34], [35]. For one author [36], ZB is not even observable. Nonetheless, it has been already simulated. First, it was simulated with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has a similar mathematical form as the Dirac equation (although the physical condition is different) [32], [33], [34]. Second, it was simulated in a configuration with Bose-Einstein condensates [37]. Further research could lead to applications of (14) in these and some other physical systems whose Hamiltonians are listed in [33].