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This is something I've been unsure of for a while but still don't quite get.

How does one tell whether an expression (e.g. the Dirac equation) is covariant or not? I get it for a single tensor, but how is it defined when there is no overall up/down index to base it on? Any advice?

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up vote 3 down vote accepted

The covariance of Dirac equation in the ordinary Hamiltonian form

$$i\hbar \frac{\partial \Psi}{\partial t} = H \Psi$$

is far from evident. The 'trick' consists on rewriting it in terms of invariant/covariant quantities such as the Dirac matrices $\gamma_\mu$ and kinetic four-momenta $\pi^\mu$

$$\gamma_\mu \pi^\mu \Psi = m \Psi$$

This rewritten form can be found in any standard textbook (check e.g. Feynman's Quantum electrodynamics) and its covariance is rather evident.

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