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arXiv:1108.4143 · Physical Review A

Nonlocality of Foldy-Wouthuysen and related transformations for the Dirac equation

Tomasz M. Rusin, Wlodek Zawadzki

Published Dec 29, 2011• 6 citations
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Abstract

Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength ${\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc$.

DOI
10.1103/physreva.84.062124

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Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength \ensuremathλc=\ensuremathℏ/mc{\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc\ensuremathλc​=\ensuremathℏ/mc.
We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.
The spectrum of the Dirac Hamiltonian H D has a positive and a negative part, the first corresponds to the energy levels of the electron, the second to minus the energy levels of the positron. The Foldy-Wouthuysen transformation is a tool to obtain a decomposition of H D in a direct sum H D =H e- (+)-H e+ , where H e- is the Hamiltonian of the electron and H e+ is that of the positron. Unfortunately, this decomposition has been obtained in a closed form only when the external field is purely magnetic, while, in the presence of an electric field, only a perturbative expansion is available. The authors give a path integral representation of the semigroup phi (x, t)=exp(-(t/)H e- ) phi 0 (x) in an external electromagnetic field. Their formula is the relativistic version, for Dirac particles, of the well known Feynman-Kac-Ito formula.
A nonrelativistic reduction of the second-order Dirac equation is investigated. An equation K2+[Π0; K−Π̂]=m2+Π̂2 is obtained for a ‘‘kinetic Hamiltonian’’ K such that Φ obeys a nonrelativistic Schrödinger-type equation Π0Φ=KΦ in addition to obeying the second-order Dirac equation. Non-self-adjoint terms in K are removed by a change of representation from Φ to ΦT ≡PΦ, and a closed expression P≡[ 1/2 +(K† −Π̂)(K−Π̂)/2(m)2]1/2 is obtained for P. Similar results are obtained for antiparticle states, and a type of ‘‘factorization theorem’’ for the second-order Dirac equation is obtained in which the second-order Dirac equation is replaced by a pair of uncoupled first-order equations. Since nonperturbative techniques are used throughout, the factorization theorem holds independently of the question of convergence of the Foldy–Wouthuysen series and is valid whenever the equation K2+[Π0; K−Π̂]=m2+Π̂2 admits a solution for K. The connection with the usual linear Dirac equation is investigated, and closed expressions are described for a transformation to a representation of that equation which is ‘‘even’’ in the language of Foldy and Wouthuysen. The transformation described is equivalent to two consecutive ‘‘odd’’ unitary transformations.
The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.