Question

Relativistic Von Neumann equation?

Answer 1

Dirac equation. In particle physics, the Dirac equation is a relativistic wave equationderived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-12 massive particles such as electrons and quarks for which parity is a symmetry

Answer 2

In non-relativistic quantum theory, Schrödinger's equation can be re-expressed using the density matrix ρ=|ψ⟩⟨ψ|ρ=|ψ⟩⟨ψ| as the Von Neumann equation:

i∂tρ=1ℏ[H,ρ]i∂tρ=1ℏ[H,ρ]

Now for spin-1212 fermions relativistic theory introduces the Dirac equation, to which no exposition on the relativistic counterpart to the equation for the density matrix is given (or, at least, I have never seen one). Out of curiosity I derived what I think is the analogue:

iγμ∂μρ=1ℏc[H0,ρ]iγμ∂μρ=1ℏc[H0,ρ]

Where H0=mc2+VH0=mc2+V is the "rest hamiltonian." The issue is that the mc2mc2 term is not an operator, so it should be canceled in the commutator, but that would lead to the nonsensical conclusion that mass has no effect on the matrix. My first thought is to replace that term with an energy operator as H0→iℏ∂tH0→iℏ∂t or H0→H^H0→H^, the "standard" hamiltonian used in the schrodinger equation, but I do not know how to make that notion rigorous, nor if that substitution can reproduce the Dirac equation.

Is it valid to make such a substitution or otherwise express the Dirac equation in a similar way?