Generalized Foldy-Wouthuysen transformation and pseudodifferential operators

Abstract

We show that the Foldy-Wouthuysen transformation and its generalizations are simplified if the methods of pseudodifferential operators are used, which also allow estimating the exactness of the transition from the Dirac equation to the reduced equations for electrons and positrons. The methods and techniques used can be useful not only in studying asymptotic solutions of the Dirac equation but also in other problems. Diagonalization of the Dirac equation for a free particleGiven the Dirac equation, the Foldy-Wouthuysen transformation [1] allows obtaining nonrelativistic equations of the type of the Pauli equation for the quantum dynamics of electrons and positrons whose kinetic energy is much smaller than mc 2 (v c). This transformation is applicable under the assumption that the electric and magnetic fields are small and amounts to a procedure of consecutively applying unitary transformations. There are generalizations of the Foldy-Wouthuysen transformation applicable in a wide range of energies up to relativistic ones (see, e.g., [2]).To make the subsequent calculations and transformations more transparent, we recall some well-known facts about the Dirac equation [3]. The Dirac equation describes the quantum relativistic motion of electrons and positrons. We assume that the electric and magnetic fields E and H are described by a four-dimensional vector potential (A, Φ), A = (A 1 , A 2 , A 3 ). In accordance with the Maxwell equations, the electric and magnetic fields are found fromThe Dirac equation in the coordinate-time space R 3 x × R t has the form i ∂Ψ ∂t = HΨ, H = cαπ + βmc 2 + eΦ, (1.1) where α = {α 1 , α 2 , α 3 },

Electronic optics in graphene in the semiclassical approximation

“…We therefore obtain It is clear that both the first and the second term in this equation vanish, since X * 2,τ = J * = 0. In section 4.4, we derived that J * τ = (C 4 ) * J * , see equation (108). This equation remains valid in our rotated coordinate system.…”

“…[58,59], see also Ref. [105], which finds its origin in the ideas of the Foldy-Wouthuysen transformation [106,107,108]. For simplicity, we confine ourselves to the case where all the modes are scalar, although this is not a fundamental limitation of the method.…”

A Flow Equation Approach Striving Towards an Energy-Separating Hamiltonian Unitary Equivalent to the Dirac Hamiltonian with Coupling to Electromagnetic Fields

“…Actually, the result H (∞) being an even operator was obtained in [11] merely for a special case of perturbation theory, akin (but not identical) with the outcome of the perturbative procedure of consecutive step by step canonical transformations aiming at eliminating the odd terms in the Dirac Hamiltonian due to Foldy and Wouthuysen (FW) [2]. Comparing the results obtained with the BP-method with the ones obtained by the FW-method a discrepancy arises in the 6 th -order of perturbation theory expanding in the small parameter κ = v c [12].…”

“…µ (r, t) being a probability amplitude for particles just like in nonrelativistic Schrödinger quantum mechanics, which is plainly wrong, as has been first revealed by the analysis of the meaning of locality in quantum mechanics by Newton and Wigner [1]. For a brilliantly witty discussion of this point, already elucidated in pioneering work by Foldy and Wouthuysen [2], we refer to Costella and McKellar [3]. Indeed interpreting Ψ (D) µ (r, t) as a probability amplitude gives cause to several well known absurdities, for instance the components of the "velocity" operator in the Heisenberg picture do not commute and have eigenvalues equal to ±c , see [16], [17], [20].…”

“…Based on insight into the fundamental meaning of locality obtained by Newton and Wigner (NW) in 1949 for (freely moving) relativistic fermions [1], Foldy and Wouthuysen (FW) established in 1950 with their unitary transformation of the free Dirac Hamiltonian an interpretation of concepts like position or velocity or spin or orbital angular momentum [2], void of the well known paradoxical properties of operators in the Dirac representation * nils.schopohl@uni-tuebingen.de (for instance the velocity operator having eigenvalues ±c) and thus agreeable to the physical intuition ascribed to a moving single particle [3].…”