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Quantum Mechanics with Basic Field Theory by Bipin R. Desai

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31  Special theory of relativity: Klein–Gordon and Maxwell’s equations

We now enter the domain of relativity. First we modify and extend the nonrelativistic theories to be compatible with the postulates of the special theory of relativity. We express the previously defined physical quantities in a four-dimensional language and introduce the concept of covariance so that the equations look the same in any inertial frame. Then we introduce the Klein–Gordon equation, which is a relativistic extension of the Schrödinger equation. We discover that the solutions of this equation can have negative energies and the probability density is not necessarily positive definite. We also consider Maxwell’s equation and discuss the consequences of gauge invariance. ...

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