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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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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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