I'm puzzling with the following contradiction in the quantum description of electron in electro-magnetic fields.
Performing Foldy-Wouthuysen transformation of the Dirac equation (with arbitrary vector potential $A^\mu $) one can obtain the Pauli Hamiltonian. The latter contains spin-orbital term $H_{so}~(\vec{\sigma},\vec{E},\vec{P})$, where $\vec{P}=\vec{p}-\vec{A}$. In particular, this derivation of spin-orbital interaction is valid in the case of slowly varying fields (which is stated explicitly in the original paper of Foldy and Wouthuysen). However, in this case $H_{so}$ (as well the total Hamiltonian) becomes non-Hermitian, in a contrast to explicit Hermicity of the Dirac Hamiltonian, $H_d=(\vec{\alpha},\vec{P})+\beta m+ A_0$. Higher order corrections to the Pauli Hamiltonian can not restore its Hermicity. Does this mean that the expectation value of energy can be a complex number? If $H_d$ is considered as the operator of energy, its expectation values are real.
I believe this depends on the coordinate representation you use. In cartesian coordinates the Pauli Hamiltonian is hermitian term by term. In polar coordinates just by trivially making the coordinate change you arrive at a non-hermitian Hamiltonian that must be "repaired" to be hermitian. I have one article dedicated to this problem for the spin-orbit interaction, and how to show non-hermiticity and perform the corresponding "hermitization".
Dear Ernesto, thanks for the answer. Could you provide also reference for this article?
Do you consider the case of stationary fields? In the case of non-stationary fields
the anti-Hermitian part of the spin-orbital interaction is proportional to the rotor of electric field.
The paper I mention is
Mesoscopic rings with Spin-Orbit interactions
it is in full text at researchgate. I don't know how to make a link to it here.
It only treats time independent fields.
Hope it helps for your problem.
Ernesto
See Bjorken Drell Vol. 1 Eq. (4.5). There are two terms in (4.5) proportional to $\sigma \rot E$ and $\sigma E \times p$. Their sum is Hermitian. Besides, the FW transform is unitary and so there should not be any problems with Hermicity by construction.
Peter, thanks a lot. There is a difference between expressions in
Bjorken & Drell and in the Foldy-Wouthuysen paper. In the F-W paper $\sigma \rot E$ were missed
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