Foldy-Wouthuysen Transformations Research Articles

Pseudo- symmetric Dirac equation: effect of a new mean spin angular momentum operator on Gilbert damping

The pseudo- symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy–Wouthuysen transformations. A new spin operator symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization—which is the quantity usually measured experimentally—is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phenomenological model of the Landau–Lifshitz–Gilbert equation (LLG).

Spin-dependent electron–radiation interaction

Spin-dependent electron–radiation interaction is derived from the Foldy–Wouthuysen transformations of the Dirac equation. A spin magnetic moment term is identified both with spin-preserving and spin-flipping transitions and inspected for atoms and condensed matter specifically of GaAs on the basis of first-principles calculation. The connections to spin relaxation and spin-resolved photoemission are also presented.

The Zeeman, spin-orbit, and quantum spin Hall interactions in anisotropic and low-dimensional conductors

To construct a microscopic theory of electrons and holes in anisotropic conductors that self-consistently treats their band effective mass anisotropy with their interactions with applied electric and magnetic fields, the Dirac equation is extended for an electron or hole in an orthorhombically-anisotropic conduction band. Its covariance is established both by a modified version of the Klemm–Clem transformations to a space in which it is isotropic, and also in its fully anisotropic form by making the most general proper and improper Lorentz transformations, proving its validity in both the relativistic and non-relativistic limits. The appropriate Foldy–Wouthuysen transformations are extended to expand about the non-relativistic Hamiltonian limit to fourth order in the inverse of the particle’s Einstein rest energy. The results have important consequences for magnetic measurements of many classes of clean anisotropic semiconductors, metals, and superconductors. In all of these cases, the Zeeman interaction is found to depend strongly upon the effective mass anisotropy. When an electron or hole is traveling in an atomically thin one-dimensional conduction band, its Zeeman, spin–orbit, and quantum spin Hall interactions are vanishingly small. Accurate expressions for the Zeeman, spin–orbit and quantum spin Hall interactions for two-dimensional conductors are provided.

Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in the weak-field limit by the method of direct perturbation theory

We apply the method of direct perturbation theory for the Foldy-Wouthuysen (FW) transformation upon the Dirac-Pauli Hamiltonian subject to external electromagnetic fields. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, by mathematical induction on the orders of $1/c$ in the power series, we rigorously prove the long-held speculation: the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.

Foldy–Wouthuysen transformation for non-Hermitian Hamiltonians

Two non-Hermitian fermion models are proposed and analyzed by using Foldy–Wouthuysen transformations. One model has Lorentz symmetry breaking and the other has a non-Hermitian mass term. It is shown that each model has real energies in a given region of parameter space, where they have a locally conserved current.

High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-Pauli Hamiltonians in the weak-field limit

The low-energy and weak-field limit of the Dirac equation can be obtained by an order-by-order block diagonalization approach to any desired order in the parameter $\ensuremath{\pi}/mc$ $(\ensuremath{\pi}$ is the kinetic momentum and $m$ is the mass of the particle). In previous work, it has been shown that, up to the order of ${(\ensuremath{\pi}/mc)}^{8}$, the Dirac-Pauli Hamiltonian in the Foldy-Wouthuysen (FW) representation may be expressed as a closed form and consistent with the classical Hamiltonian, which is the sum of the classical relativistic Hamiltonian for orbital motion and the Thomas-Bargmann-Michel-Telegdi Hamiltonian for spin precession. In order to investigate the exact validity of the correspondence between classical and Dirac-Pauli spinors, it is necessary to proceed to higher orders. In this paper, we investigate the FW representation of the Dirac and Dirac-Pauli Hamiltonians using Kutzelnigg's diagonalization method. We show that the Kutzelnigg diagonalization method can be further simplified if nonlinear effects of static and homogeneous electromagnetic fields are neglected (in the weak-field limit). Up to the order of ${(\ensuremath{\pi}/mc)}^{14}$, we find that the FW transformation for both Dirac and Dirac-Pauli Hamiltonians is in agreement with the classical Hamiltonian with the gyromagnetic ratio given by $g=2$ and $g\ensuremath{\ne}2$, respectively. Furthermore, with higher-order terms at hand, it is demonstrated that the unitary FW transformation admits a closed form in the low-energy and weak-field limit.

Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation

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 Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.

Foldy–Wouthuysen transformation, scalar potentials and gravity

We show that care is required in formulating the nonrelativistic limit of generalized Dirac Hamiltonians which describe particles and antiparticles interacting with static electric and/or gravitational fields. The Dirac–Coulomb and the Dirac–Schwarzschild Hamiltonians, and the corrections to the Dirac equation in a non-inertial frame, according to general relativity, are used as example cases in order to investigate the unitarity of the standard and ‘chiral’ approaches to the Foldy–Wouthuysen transformation, and spurious parity-breaking terms. Indeed, we find that parity-violating terms can be generated by unitary pseudo-scalar transformations (‘chiral’ Foldy–Wouthuysen transformations). Despite their interesting algebraic properties, we find that ‘chiral’ Foldy–Wouthuysen transformations change fundamental symmetry properties of the Hamiltonian and do not conserve the physical interpretation of the operators. Supplementing the discussion, we calculate the leading terms in the Foldy–Wouthuysen transformation of the Dirac Hamiltonian with a scalar potential (of the 1/r-form and of the confining radially symmetric linear form), and obtain compact expressions for the leading higher-order corrections to the Dirac Hamiltonian in a non-inertial rotating reference frame (‘Mashhoon term’).

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

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 ${\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc$.

Gauge potential formulations of the spin Hall effect in graphene

Two different gauge potential methods are engaged to calculate explicitly the spin Hall conductivity in graphene. The graphene Hamiltonian with spin–orbit interaction is expressed in terms of kinematic momenta by introducing a gauge potential. A formulation of the spin Hall conductivity is established by requiring that the time evolution of this kinematic momentum vector vanishes. We then calculated the conductivity employing the Berry gauge fields. We show that both of the gauge fields can be deduced from the pure gauge field arising from the Foldy–Wouthuysen transformations.

Bogoliubov transformations and fermion condensates in lattice field theories

We apply generalized Bogoliubov transformations to the transfer matrix of relativistic field theories regularized on a lattice. We derive the conditions these transformations must satisfy to factorize the transfer matrix into two terms which propagate fermions and antifermions separately, and we solve the relative equations under some conditions. We relate these equations to the saddle point approximation of a recent bosonization method and to the Foldy–Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian.

The contribution of spin torque to the spin Hall coefficient and spin motive force in aspin–orbit coupling system

We derive rigorously the relativistic angular momentum conservation equationby means of quantum electrodynamics. The novel nonrelativistic spincurrent and torque in the spin–orbit coupling system, up to the order of1/c4, are exactly investigated using Foldy–Wouthuysen transformations. We find a perfect spinHall coefficient including the contribution of the spin torque dipole. A novel spin motiveforce, analogous to the Lorentz force, is also obtained to help us understand the spin Halleffect.

Bound states of Dirac particles in gravitational fields

We investigate the quantum motion of a neutral Dirac particle bouncing on a mirror in curved spacetime. We consider different geometries: Rindler, Kasner-Taub, and Schwarzschild, and show how to solve the Dirac equation by using geometrical methods. We discuss, in a first-quantized framework, the implementation of appropriate boundary conditions. This leads us to consider a Robin boundary condition that gives the quantization of the energy, the existence of bound states and of critical heights at which the Dirac particle bounces, extending the well-known results established from the Schr\"odinger equation. We also allow for a nonminimal coupling to a weak magnetic field. The problem is solved in an analytical way on the Rindler spacetime. In the other cases, we compute the energy spectrum up to the first relativistic corrections, exhibiting the contributions brought by both the geometry and the spin. These calculations are done in two different ways. On the one hand, using a relativistic expansion and, on the other hand, with Foldy-Wouthuysen transformations. Contrary to what is sometimes claimed in the literature, both methods are in agreement, as expected. Finally, we make contact with the GRANIT experiment. Relativistic effects and effects that go beyond the equivalence principle escape the sensitivity of such an experiment. However, we show that the influence of a weak magnetic field could lead to observable phenomena.

Explicit Expression of Foldy-Wouthuysen Transformation

The general, explicit and formally closed expression of arbitrary n-times Foldy-Wouthuysen transformations is clearly and strictly derived out. It is proved that if transformed Hamiltonian needs to be approximated to the order 1/mK or mv2K when to involve the orders of the operators, then to make N = [(K + 1)/2]-times Foldy-Wouthuysen transformations is just enough ("[...]" means to take the part of integer). An example in non-relativistic quantum chromodynamics is given.

Parasupersymmetries and Non-Lie Constants of Motion for Two-Particle Equations

We search for hidden symmetries of two-particle equations with oscillator-equivalent potential, proposed by Moshinsky with collaborators. We proved that these equations admit hidden symmetries and parasupersymmetries which enable one to easily find the Hamiltonian spectra using algebraic methods and to construct exact Foldy–Wouthuysen transformations. Moreover, we demonstrate that these equations are reducible and generate Hamiltonians for pararelativistic or Kemmer oscillators. We also establish equivalence relations between different approaches to Kemmer oscillator and propose new one- and two-particle equations with oscillator-equivalent potentials.

Dirac Field, Gravity, Inertial Effects, and Computer Algebra

The article presents some new results obtained for the non-relativistic approximation of the Dirac equation in a non-inertial reference frame — rotated and accelerated — and in Schwarzschild gravitational field. These results are obtained with new routines of algebraic programming in REDUCE + EXCALC language for the Dirac equation in a non-inertial reference frame and after three successive Foldy–Wouthuysen transformations.

Quark effective theory at large orders in 1/ m

The existing derivations of a heavy quark effective theory (HQET) are presented from a unified point of view. It is shown that a correct normalization of the heavy quark field in the effective theory is crucial for making explicit the mass dependence of the matrix elements. A possible variant of HQET which satisfies this requirement and which is constructed from a sequence of Foldy-Wouthuysen transformations is presented. The renormalization of the resulting effective Lagrangian to order 1/ m 2 is performed including also effects arising from vacuum polarization.

Dirac particle, gravity, and inertial effects

AbstractDirac's equation with gravity for a noninertial observer is derived using local coordinate methods. Calculations for the equation are carried out to second order in the local coordinates. For easy application to interference experiments, the Schrödinger form of the Dirac equation with a well defined Hamiltonian in the local coordinates is presented. The presence of gravitational weighting factors in the scalar product lead to hermitian and antihermitian sectors for the Hamiltonian. The antihermitian part depends directly on the curvature and vanishes for zero curvature. The hermitian part which is important for the determination of phases is studied in detail and the nonrelativistic case is obtained by the application of three successive Foldy‐Wouthuysen transformations. The results also give local currents and interactions which have pure inertial, pure gravity and mixed sectors. The pure inertial terms are the ones obtained by Hehl and Ni. The pure gravity and mixed sectors have contributions which are electric, magnetic and double magnetic in character. The focus is on the curvature contributions. Some are well within reach of the anticipated accuracy of atomic interferometers currently under consideration and other terms may follow if improvements can be made.

Supersymmetry, Foldy-Wouthuysen transformations, and relativistic oscillators.

Dirac Hamiltonians expressed in terms of supercharges are analyzed in connection with supersymmetric quantum mechanics. Unitary Foldy-Wouthuysen transformations lead to equivalent diagonalized Hamiltonians, functions of supersymmetric (nonrelativistic) ones. Such considerations permit the definition of relativistic oscillators described by Dirac equations.

Covariance, CPT and the Foldy-Wouthuysen transformation for the Dirac oscillator

The covariance properties and the Foldy-Wouthuysen and Cini-Touschek transformations for the recently proposed Dirac oscillator are found. The charge conjugation, parity and time reversal properties are established. A physical picture as an interaction of an anomalous (chromo) magnetic dipole with a specific (chromo) electric field emerges for this system. This interaction suggests an alternative confinement potential for heavy quarks in QCD.