디랙 방정식

Let there be light.

Start with the relativistic wave equation (Dirac equation).

\(i \gamma^\mu \partial_\mu \psi - m \psi =0\)

Choose a Lagrangian.

\(\mathcal{L} = i \bar{\psi} \gamma^\mu \partial_\mu \psi -m \bar{\psi} \psi\)

We need a \(U(1)\) - local gauge invariance.

\(\psi(x) \to e^{i\alpha(x)}\psi(x)\)

To obtain the local gague invariance, introduce a new gauge field and get

\(\mathcal{L} = i \bar{\psi} \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma_\mu A^\mu \psi -m \bar{\psi} \psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\)

and there was light. God saw that the light was good.

1927, Dirac adopted new equation for relativistic Schrodinger equation to remove negative energy states.
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