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My impression is that there are two ways of thinking about the Dirac equation:

  1. Quantum Mechanically: Here we think of the spinor $\phi$ as a generalization of the Schrodinger wave function which includes spin. I believe this is the way that Dirac first thought of it.
  2. Quantum Field Theoretically: Here we think of $\phi$ as a ${classical~ field}$, like the electromagnetic $4$-potential $A$ which we then quantize to produce the quantum field that "electrons are excitations of."

What is the relationship between these two uses of $\phi$?

My impression is that the second way is the "more right" way of thinking about things-- but it is still true that we can consider the electron as a quantum mechanical particle which obeys Schrodinger's equation. So how do we recover the QM picture from the QFT picture?

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