A Reexamination of Lorentz-Covariant Normalization and Propagator Structure in the Square-Root Klein–Gordon Equation

Mohammad Javad Kazemi, Jafar Khodagholizadeh, Mohammad Hosein Barati

In this paper, we investigate the Lorentz invariance of the square root Klein-Gordon equation. This equation with interpreting $|\psi|^{2}$ as position probability density is not Lorentz invariance but we can make this equation Lorentz invariant with the help of new proper interpretation of the wave function. In this regard we consider the relation between the probability density and the wave function as$\rho=\psi \psi^{\ast}+ (\hat{D}\psi)^{\ast}\hat{D}\psi$ and show appropriate selection of the linear operator$\hat{D}$the square root Klein-Gordon equation will be Lorentz invariant. In fact the solutions of this equation is equivalent to the positive energy solutions of the Dirac equation. \end{abstract}

Michael B. Heaney

Kazemi, Mohammad Javad, Mohammad Hosein Barati, Jafar Khodagholizadeh +1 more

We show that the Born's rule is incompatible with Lorentz symmetry of the Square Root Klein-Gordon equation (SRKG equation). It has been demonstrated that the Born rule must be modified in relativistic regime if one wishes to keep the SRKG equation as the correct equation for describing quantum behavior.

K. B. Wharton

Kazemi, Mohammad Javad, Mohammad Hosein Barati, Rokni, Yahya +1 more

We have shown that the Born rule $(ρ=|ψ|^{2})$ is inconsistent with Lorentz symmetry of the Salpeter equation (square root Klein-Gordon equation). So we find relativistic modification of the Born rule as $ρ=\frac{1}{2mc^{2}}(|\sqrt{\widehat{H}-mc^{2}}ψ|^{2}+|\sqrt{\widehat{H}+mc^{2}}ψ|^{2})$ , which is consistent with Lorentz symmetry of this equation

C. Best, P. Górnicki, Walter Greiner