_{1}

The Lorentz transformation properties of charge current four vector for Dirac spinor particles are examined once more especially for the zitterbewegung terms which are integral parts of this theory.

Four vectors like electric or charge particle currents transform under Lorentz boost obeying certain transformation laws in classical physics [

Any classical four vector J μ transforms from a reference frame S to another S ′ which is moving with respect to S with a constant velocity v along the x direction according to (we are writing the zeroth that is the μ = 0 component only)

J 0 ′ = 1 1 − v 2 c 2 ( J 0 − v c J 1 ) = J 0 cosh ω − J 1 sinh ω

as given for example by Jackson [

ρ ′ ρ = cosh ω (1)

On the other hand the Dirac particle-antiparticle pair, like the electron and positron can be represented in frame S in which it is assumed to be at rest by the wave function as given for example by Bjorken and Drell [

ψ ( x , t ) = A e − i m c 2 ℏ t [ 1 0 0 0 ] + B e i m c 2 ℏ t [ 0 0 0 1 ] (2)

Here the complex coefficients A and B represent the proportion in which the particle and antiparticle components are present in the combined state. There should not be any objection to forming such combined states with the help of Dirac spinors where both particle and antiparticle wave functions are involved as such states are included in forming wave packets as given for example by Equation (3.30) of reference [

J μ ( x , t ) = c e ψ + ( x , t ) γ 0 γ μ ψ ( x , t ) (3)

following the prescriptions of pages 23 and 9 of reference [

c ρ ( x , t ) = J 0 ( x , t ) = c e ψ + ( x , t ) ψ ( x , t ) (4)

into which substitution of the value of ψ ( x , t ) from Equation (2) yields

c ρ ( x , t ) = c e ( | A | 2 + | B | 2 ) (5)

Before applying Equation (4) to the frame S ′ we must transform the spinor ψ ( x , t ) to ψ ′ ( x ′ , t ′ ) using the matrix operator which appears in Equation (3.5) of reference [

ψ ′ ( x ′ , t ′ ) = [ cosh ω / 2 0 0 − sinh ω / 2 0 cosh ω / 2 − sinh ω / 2 0 0 − sinh ω / 2 cosh ω / 2 0 − sinh ω / 2 0 0 cosh ω / 2 ] [ A e − i m c 2 ℏ t 0 0 B e i m c 2 ℏ t ] (6)

Thus from Equations (3) and (6) we get

c ρ ′ ( x ′ , t ′ ) = e c ψ + ′ ( x ′ , t ′ ) ψ ′ ( x ′ , t ′ ) = c e ( | A | 2 + | B | 2 ) cosh ω − c e ( A * B e 2 i m c 2 ℏ t + A B * e − 2 i m c 2 ℏ t ) sinh ω (7)

where A * is the complex conjugate of A . Thus the second term in Equation (7) is the zitterbewegung term and if the equation is divided by Equation (5) we get

ρ ′ ρ = cosh ω − A * B e 2 i m c 2 ℏ t + A B * e − 2 i m c 2 ℏ t | A | 2 + | B | 2 sinh ω (8)

Comparison of Equations (1) and (8) shows that classical Lorentz covariance is violated by the zitterbewegung terms which are essential features of the Dirac description of a particle even though these may exist in a small proportion (see p 39 of reference [

The quantum mechanical wave function ϕ ( x , t ) of a scalar particle follows the Klein-Gordon equation and can be written following the guidelines of reference [

1 c 2 ∂ 2 ϕ ( x , t ) ∂ t 2 − ∇ 2 ϕ ( x , t ) + c 2 ℏ 2 m 2 ϕ ( x , t ) = 0 (9)

where we have retained the velocity c of light explicitly. The charge density c ρ ( x , t ) can be expressed following Equation (3.17) of this above reference as

c ρ ( x , t ) = c e i [ ϕ ( x , t ) * ∂ ϕ ( x , t ) ∂ t − ϕ ( x , t ) ∂ ϕ ( x , t ) * ∂ t ] (10)

We evaluate this in both the S and S ′ frames to obtain from ϕ ( x , t ) = A e − i m c 2 ℏ t + B e i m c 2 ℏ t and ϕ ′ ( x ′ , t ′ ) = A e − i m c 2 ℏ ( t ′ cosh ω + x ′ c sinh ω ) + B e i m c 2 ℏ ( t ′ cosh ω + x ′ c sinh ω ) the expressions

c ρ ( x , t ) = − 2 m c 3 e ℏ ( − | A | 2 + | B | 2 ) (11)

c ρ ′ ( x ′ , t ′ ) = − 2 m c 3 e ℏ ( − | A | 2 + | B | 2 ) cosh ω (12)

The expressions for ϕ ′ ( x ′ , t ′ ) above are obtained from ϕ ( x , t ) by using the

Lorentz transformation equation for t which is t = t ′ cosh ω + x ′ c sinh ω .

Here the complex numbers A and B have the same meaning as in Section 2. Now we are in a position to check from Equations (11) and (12) how the scalar field charge density transform. Indeed one obtains the same relation between ρ and ρ ′ as in Equation (1) if we divide Equation (12) by Equation (11).

We would like to thank the referee for his constructive comments especially on the need to make a contrastive analysis of the results obtained in sections 2 and 3. The immediate point that is to be noted is the fact that Equations (5) and (7) in contrast to Equations (11) and (12) have the terms involving | A | 2 and | B | 2 with the same signature. Antiparticles are supposed to be oppositely charged as compared to particles and this fact is made explicit in Equations (11) and (12). In fact on page 76 of reference [

Roy, R. (2017) Lorentz Transformation Properties of Currents for the Particle-Antiparticle Pair Wave Functions. Open Access Library Journal, 4: e3730. https://doi.org/10.4236/oalib.1103730