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Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation | | Tsung-Wei Chen
; Dah-Wei Chiou
; | | Date: |
31 Oct 2013 | | Abstract: | The classical dynamics for a charged spin particle is governed by the Lorentz
force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi
(T-BMT) equation for spin precession. In static and homogeneous electromagnetic
fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the
Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a
spin-1/2 particle, is consistent with the classical Hamiltonian (with both the
orbital and spin parts) up to the order of $1/m^{14}$ ($m$ is the particle’s
mass) in the low-energy/weak-field limit. In this paper, we extend this
correspondence to the case of inhomogeneous fields. Regardless of the field
gradient (e.g. Stern-Gerlach) force, the T-BMT equation is unaltered and thus
the classical Hamiltonian remains the same, but subtleties arise and need to be
clarified. For the relativistic quantum theory, we apply Erikson’s method to
obtain the exact FW transformations for the two special cases, which in
conjunction strongly suggest that, in the low-energy limit, the FW transformed
Dirac-Pauli Hamiltonian is in agreement with the classical Hamiltonian plus the
Darwin term, provided classical variables are promoted to quantum operators by
a specific prescription of Weyl ordering. The Darwin term is shown to have no
classical correspondence. | | Source: | arXiv, 1310.8513 | | Services: | Forum | Review | PDF | Favorites | |
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