Relativistic Remnants of Non-Relativistic Electrons

Electrons obeying the Dirac equation are investigated under the non-relativistic $c \mapsto \infty$ limit. General solutions are given by derivatives of the relativistic invariant functions whose forms are different in the time- and the space-like region, yielding the delta function of $(ct)^2 - x^2$. This light-cone singularity does survive to show that the charge and the current density of electrons travel with the speed of light in spite of their massiveness.


Introduction
It is well known that the Dirac electron has a piece vibrating with the light velocity, called "Zitterbewegung" whose origin is supposed as a mixing of the positive and the negative components. Discussions have been made by the Heisenberg equation of motion for the Dirac Hamiltonian [1] or by the momentum representation for the Dirac field Ψ (x) [2]. Electron moving in the zigzag motion with the light speed also appears on the stage of the pilot-wave approach to quantum field theory [3] (Feynman had already discussed the zigzag motion with the light velocity in the context of path integral [4]).
We shall, in this paper, focus the electron field Ψ (x) itself, to show that electrons obeying the Dirac equation inevitably bear portions traveling with the light speed in the non-relativistic limit. As a preliminary, let us recall the Dirac Hamiltonian with {γ µ , γ ν } = 2η µν , diag(η µν ) = (1, −1, −1, −1) ; (µ, ν = 0, 1, 2, 3) , being the 4 × 4 gamma matrices represented as In order to investigate the non-relativistic limit, it is useful to perform a unitary (called the Foldy-Wouthuysen) transformation [5], with tan 2 |p| mc θ(p) = |p| mc , such that Here the lower component corresponds to the negative energy state which should be discarded in the non-relativistic world. Therefore we shall pick up the positive energy part and study its wave mechanical structure in the next section. In the following section 3, we shall treat covariant solutions of the free Dirac equation, which will be extended to interacting cases in sec. 4. The final section is devoted to the discussion. Some of the detailed calculations in sec.4 are relegated to the appendices.
2 Wave Mechanics of H = c p 2 + m 2 c 2 Consider a single component wave mechanics governed by the Hamiltonian (5), that is, a wave function Ψ (t, x) obeying the Schrödinger equation The solution reads where ψ(0, x ′ ) is an arbitrary function, is the kernel 1 withP designating the momentum operator and the integration range from −∞ to ∞ has been omitted here and hereafter unless otherwise specified.
In order to calculate the kernel (8), introduce k ≡ p/ and write µ ≡ mc , to obtain where use has been made of the polar coordinates to the final expression. Put k = µ sinh Θ to find whose exponent reads, with the aid of an addition theorem, as with Make sifts Θ → Θ + α, β and again utilize the addition theorem to obtain with (Here we have discarded the odd function part: and cosh β = r/ −x 2 µ in view of (13), the second term of (15) reads where we have used the relations, Finally the Bessel function formulas [9] ∞ −∞ dΘ cosh Θ e −iχ + cosh Θ = −πH lead us to Here note that I consists of different functions in the regions ct > r and ct < r, which causes the delta function singularity when a differentiation is made. (The µ-independent delta function emerges from the huge momentum domain p ≫ mc.) The kernel is, from (10), obtained, by differentiating (18) with respect to r, as where use has been made of and then (see Appendix A) as well as [10], d zdz By taking the non-relativistic limit c → ∞, that is, µ → ∞ (9), the kernel(19) reads, where we have employed the asymptotic expansion of the Bessel function [11] Z n (z) z →∞ Now put then substitute (23) into (7) with changing the variables as x ′ → ∆x to find which, by introducing the polar coordinates, yields When ct, |x| ≫ √ a, the second term fades away and around the peak, (26) becomes which apparently travels with the speed of light in spite of its massiveness. The origin lies in the delta function δ(ct − |x|) (19) emerging from the discontinuity of I (18) between the time-, ct > |x|, and the space-like, ct < |x|, regions. We shall call this as a light-cone singularity. (It should be emphasized that the light-cone singularity cannot become visible under the momentum representation [13]. ) As a necessary consequence, any wave written as Ψ (x) = d 4 yD(x, y)ψ(y) ; ∀ ψ(y) , must have the light-cone singularity, if D contains derivatives to some function with a discontinuity on the light-cone. The solutions of the Dirac equation meet with this, so, in the next sections, we shall study those.

The Charge and the Current Density of Free Electrons
First let us summarize the relativistic invariant functions which participate in solving a relativistic equation. The D-dimensional scalar Klein-Gordon field is given as where µ is defined by (9) and J(x) is a source, complicated function of φ describing interactions. Here and hereafter the repeated indices always imply the summation. When J = 0, the solution is where ∆(x) is an invariant function defined by and ϕ(y) is an arbitrary function. The notations, with k and x being (D − 1)-dimensional vector, should be understood. When J = 0, the solution is where φ 0 (x) is (30) and ∆ F (x) is the Feynman propagator, These are shown as [14] ∆ By noting J −n (z) = (−) n J n (z), N −n (z) = (−) n N n (z) and K −n (z) = K n (z), they become in D = 4(ν = 1) [15], Note that they have a discontinuity on the light-cone. (Any relativistic invariant function does.) In view of (30) and (32), however, there are no derivative so that we cannot have light-cone singularities for scalar fields 2 . (We do not care the O(1) delta function in (36) and (37) under the non-relativistic limit µ → ∞.) The free electron field Ψ 0 (x) obeys the Dirac equation, where µ is (9) and Ψ 0 (x) is given as with ϕ 0 and χ 0 being two component spinor. The solution of (38) reads, where S(x) is the invariant function for the Dirac field, with ∆(x) (36) and ψ(y) being an arbitrary four component spinor. From (40) and (41), Ψ 0 (x) must own the light-cone singularity.
For the sake of simplicity, the initial electron configuration is assumed to be, where ξ 0 is a constant two component spinor. Since from (31) (with D = 4) (40) with (42) implies the initial condition, In the following we consider three cases; ; which is called as 3-, 2-, and 1-dimensional packets respectively 3 . In view of (41), the solution (40) becomes, In the following, we shall discuss the charge and the current densities defined by 4 Now take the non-relativistic limit c → ∞ (µ → ∞) to find that ∆(x) (36) reduces to whose derivatives are calculated as follows: first note by use of Thus whose second term reads has been considered in the final expression. Meanwhile the first term becomes, with the aid of J 1 (z)/z z=0 = 1/2 (128), yielding to Similarly by ∂ k = −2x k ∂/∂x 2 and (50) From these, we can convince that the leading terms in the non-relativistic approximation are nothing but the light-cone singularities. Therefore by noting that as well as µ∆(x) = O µ 3/2 , (46) reads as where we have noticed x 0 > 0 and made a shift y → y + x . Now proceed to individual cases: from (45) and (54) where which becomes with the aid of the polar coordinates Hence which further turns out to be around the peak, which apparently travels with the speed of light. Insert (59) into (47) to find where use has been made of the anti-commutation relations {σ j , σ k } = 2δ jk in J (3) . Those travel with the speed of light. It would be easier to prepare packets restricted in the 2-or 1-dimensional region. The former reads, by inserting (45) into (54), as where σ 2 ≡ (σ 1 , σ 2 ), x 2 ≡ (x 1 , x 2 ) and Here we have put x 3 = 0, since the observation should also be made in the x 3 = 0 plane. (63) becomes under the polar coordinates, to When |x 2 |, x 0 ≫ √ a, the saddle point θ = π gives a asymptotic value such that Thus (62) reads Here the final expressions has been obtained by putting x 0 /|x 2 | → 1 in the coefficient; since the peak is now given as whose velocity is again the light speed. The charge and the current densities (47) read Finally we discuss the 1-dimensional case written, after putting x 2 = 0, x 3 = 0, as where which turns out, by recalling x 0 > 0, to be Thus whose peak is around which the charge and the current densities (47) read In view of (61), (67) and (74), the free electrons travel with the speed of light in the non-relativistic world 5 .

Electrons in a Laboratory
In a realistic situation, electrons interact with the electromagnetic field A µ (x) such that whose solution is where Ψ 0 (x) is the free field discussed in the previous section and S F (x) is the Feynman propagator for the Dirac field, with ∆ F (x) given in (37). The interaction is assumed to take place for a finite interval, giving where 3-, 2-, and 1-dimensional packets are the general solution (76) reads as with Ψ (k) 0 given by (46). In the non-relativistic limit µ → ∞, ∆ F (x) (37) reduces to where we have utilized (125) to the final expression whose form reminds us (18) in sec.2. The derivative reads where use has been made of (50) and the asymptotic behavior (24). Noting (128) and (131) in Appendix A we have the light-cone singularity, Contrary to the previous situation, we now need both terms in (53), since y 0 cannot always be positive, so that Then (80) turns out to be where a shift y → y + x has been made. Let us exam individual cases: with the 3-dimensional packet (86) read, where which, with the aid of the polar coordinates and the error function [16], yields to Applying the differentiations, in (87) and assuming around the peak (60). (Details are relegated to Appendix B.) Therefore in view of Ψ where we have introduced a two component spinor The charge and the current densities are Since each term has a peak at the light-cone, the electron signal traveling with the light speed would be observed. Next we consider the 2-dimensional case: again we restrict x to be x 3 = 0 so that (86) with (79) are found as where which yields Therefore (Details are relegated to Appendix B.) This further reduces to around the peak (66). From (65) and (99), the total Ψ (2) (81) is given by The charge and the current densities are thus found as Finally we consider the 1-dimensional case: by restricting x to be x 2 = x 3 = 0 (79) brings (86) to After a little calculation (see Appendix B) it reads, when From (72) and (105) the total Ψ (1) (81) is obtained as The charge and the current densities are 6

Discussion
In this paper, first we discuss the wave mechanics of H = c p 2 + m 2 c 2 , which tells us that solutions of the Schrödinger equation inevitably possess the light-speed portion in the non-relativistic limit c → ∞. The reason is that the kernel contains a derivative acting to a function which owns the discontinuity on the light-cone. The solutions of the Dirac equation are also expressed by differentiations to the invariant functions ∆(x) and ∆ F (x) which consist of different functions in the time-and the space-like region, thus yield the light-cone singularity, which was the contents of sec.3 and 4. In relativistic field theories, c appears as µ = mc/ so that the non-relativistic limit implies µ → ∞ which, however, also interprets the semiclassical → 0 or an infinite mass limit m → ∞. According to the last case we can convince ourselves of survival of the light-cone singularity for massive particles. We should emphasize that our conclusion has been derived exclusively in the x-representation of Ψ (x) not in the momentum representation. The situation is unchanged if the source (78) would have a velocity v: consider, for example, Then from (86), (The spinor part is irrelevant.) Since the zeros in the delta function are given by which again shows that the maximum signal travels with the speed of light.
In order to widen the possibility we finally consider the case that the initial configuration is given with a definite momenta p, that is, instead of the packets (42) take in the solution Since at x 0 = 0, as was in (44), (112) implies an initial configuration, whose momentum reads as Following a similar procedure from (46) to (56), we find with which is further rewritten as with the help of (57). By noting and that exp − ia p · n ∂ ∂|x| is a shift operator, it reads Therefore When x 0 , |x| ≫ √ a ; |x| − iap · n/ ≈ |x|, it yields to a plane wave, around the peak x 0 − |x| ≈ 0, which implies that the energy-momentum relation is given by Therefore an alternative way to observe the relativistic remnant is a measurement for the energy and the momentum of electrons in the vacuum. then π 2 where we have used ln(r − ct) = iπ + ln(ct − r) to the final expression. Therefore (124) reads h and Ψ h .
(90): With the aid of the polar coordinates, (88) becomes where the addition theorem for the hyperbolic function has been used. In view of the error function formula (89) we have ∞ 0 dye −Ay 2 sinh By = 1 2 so that (133) becomes (90).
(98) Ψ (2) h : Apply differentiations in (95) and note the relation (135) by putting x → x 2 to obtain where we have omitted terms of .