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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?

Qmechanic
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Phil Tosteson
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  • More on reduction from QFT to QM: http://physics.stackexchange.com/q/26960/2451 , http://physics.stackexchange.com/q/4156/2451 , https://physics.stackexchange.com/q/208615/2451 and links therein. – Qmechanic May 20 '18 at 17:35
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    This is done in detail in Weinberg's QFT book, if you're interested. He shows that the wavefunctions of 1-particle states satisfy the Dirac equation, IIRC. – Javier May 20 '18 at 17:47
  • @Javier I'm interested, do you know approx where it is? – Phil Tosteson May 20 '18 at 18:39
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    It's in Volume 1, chapter 14: Bound states in external fields. – Javier May 20 '18 at 20:03

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