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Consider the Dirac field before quantisation. The evolution equation of this field does not describe any real world classical system because we know that there's no classical limit of the fermionic field (due to Pauli's exclusion principle).

But we do know of one macroscopic field that describes charged particles: The four-current field we use in the classical Maxwell's equations. These charges obey the four-vector Lorentz Force law in their time evolution.

My question is, since the Dirac equation describes all charges particles at the fundamental level, doesn't this mean that the Lorentz force law is, in some sense, a sort-of classical limit of the non-quantum Dirac field evolution equation? If so, how exactly are they related?

Qmechanic
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Ryder Rude
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I'm not sure what you are referring to as the Lorentz' force law, but the version $f = \rho E+j\times B$ can be derived only using classical electromagnetism with sources. It simply arises from Poynting's theorem for momentum. Let $$\Pi = \epsilon_0 E\times B$$ be the momentum density of the EM field and $$\sigma = \epsilon_0E\otimes E+\frac{1}{\mu_0}B\otimes B-\frac{1}{2}( \epsilon_0E^2+\frac{1}{\mu_0}B^2)\mathbb 1$$ be Maxwell's stress tensor, you have from Maxwell's equations: $$ \frac{\partial \Pi}{\partial t}-\nabla\cdot \sigma+ f = 0 $$ which is how you interpret $f$ as a volume force.

Hope this helps.

LPZ
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  • I was asking how this force law arises from the Dirac equation, since this law describes the motion of charges, and Dirac equation describes charged particles at the fundamental level – Ryder Rude Jul 17 '22 at 10:39
  • Well, then it’s not really the Lorentz force you’re interested in but rather the expression of the charge and current density $\rho,j$, is it not? – LPZ Jul 17 '22 at 11:33