Four vectors like electric or charge particle currents transform under Lorentz boost obeying certain transformation laws in classical physics  . It is expected that the quantum mechanical charge currents for elementary particles (and/or antiparticles) as derived from the probability currents  , by multiplying the particles’ charge with it, follow these transformation laws if the correspondence principle  relating quantum and classical physics is to remain valid. As is stated in reference  the Dirac equation describing elementary particles like the electron give rise to terms in the probability current four vector which have rapid oscillation or “zitterbewegung” in cases where states are formed by combining particle and antiparticle wave functions. In this present letter we show that these zitterbewegung terms do not follow the Lorentz transformation law. It is also demonstrated that the wave function for a scalar particle (along with its antiparticle) which follows the Klein Gordon equation  gives rise to no such zitterbewegung terms at least in the zeroth component of charge currents which violate its Lorentz transformation properties.
2. Lorentz Transformation Properties of Classical and Quantum Mechanical Currents
Any classical four vector transforms from a reference frame to another which is moving with respect to with a constant velocity along the direction according to (we are writing the zeroth that is the component only)
as given for example by Jackson  (see p. 526 of this reference). If in this frame the value of (the component of current) is zero then the ratio of charge density in to its value in is given by
On the other hand the Dirac particle-antiparticle pair, like the electron and positron can be represented in frame in which it is assumed to be at rest by the wave function as given for example by Bjorken and Drell  (see p. 10 of this book) to be
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  . Also we need not normalize the wave function to study the Lorentz transformation properties of the charge current four vector (that is the electronic charge e multiplied by the probability current four vector). This quantity can be expressed as
following the prescriptions of pages 23 and 9 of reference  . Here is the complex conjugate of the transpose of . The zeroth component of this vector is the charge density and can be evaluated in both and frames to see if they follow Equation (1) above provided in the frame is given by Equation (2) implying that particle and antiparticle are both at rest in this frame. Thus in the frame
into which substitution of the value of from Equation (2) yields
Before applying Equation (4) to the frame we must transform the spinor to using the matrix operator which appears in Equation (3.5) of reference  that is
Thus from Equations (3) and (6) we get
where is the complex conjugate of . Thus the second term in Equation (7) is the zitterbewegung term and if the equation is divided by Equation (5) we get
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  ). Furthermore charge density is an observable in quantum mechanics and so this will lead to some consequences as far as experimental results are concerned.
3. Scalar Particle Charge Density and Its Lorentz Transformation
The quantum mechanical wave function of a scalar particle follows the Klein-Gordon equation and can be written following the guidelines of reference  as
where we have retained the velocity c of light explicitly. The charge density can be expressed following Equation (3.17) of this above reference as
We evaluate this in both the and frames to obtain from and the expressions
The expressions for above are obtained from by using the
Lorentz transformation equation for which is .
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 and as in Equation (1) if we divide Equation (12) by Equation (11).
4. Conclusion and Response to the Constructive Criticism of the Referee
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 and 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  it is made clear that the concept of charged current for the particle-antiparticle pair as given in the case of the zeroth component by Equation (10) above was introduced for the first time by Pauli and Weisskopf in 1934 which consists of terms of both signatures. The Dirac equation fails to show such consistency if we are to accept his definition of the particle (and antiparticle) current and this as we know is a direct result of the desire to keep probability density “positive”. However we would like to add that the Dirac equation has explained the fine structure of hydrogen atom along with a host of experiments in particle physics so it would be premature to put a question mark on this equation.