Dirac equation in external electromagnetic fields

The Dirac equation, with minimal coupling to an external electromagnetic field a x) can be written 

A gauge transformation applied to both Dirac spinors and electromagnetic potentials, 

The Dirac equation (42) and its adjoint can also be derived by requiring that 

This confirms that = —ecy/y yr is a conserved current, as in (39). Other conserved quantities, such as the energy-momentum tensor can be derived in similar fashion. 

The Hamiltonian for an electron interacting with an external electromagnetic field is found to be 

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In this section, we derive the two-component Pauli equation from the Dirac equation in external electromagnetic fields. It is also desirable to recover the Schrodinger equation in order to see the connection between relativistic theory and nonrelativistic quantum mechanics. For this purpose, we rewrite Eq. 

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