Foldy Wouthuysen Research Articles

Effective theories and non-minimal couplings in low-dimensional systems

Dimensionality aspects of non-minimal electromagnetic couplings are investigated. By means of the Foldy–Wouthuysen transformation, we attain (non-)relativistic interactions related to the non-minimal coupling in three-dimensional spacetime, for both the bosonic and fermionic fields. Next, we establish some comparisons and analyze particular situations in which the external electromagnetic fields are described either by Maxwell or Maxwell–Chern–Simons Electrodynamics. In addition, we consider the situation of a non-minimal coupling for the fermionic field in four dimensions, carry out its dimensional reduction to three dimensions and show that the three-dimensional scenario previously worked out can be recovered as a particular case. Finally, we discuss a number of structural aspects of both procedures.

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.

Supersymmetry and Stable Dirac Sea in Carbon Nanotubes

We consider the construction of exactly solvable models of twisted carbon nanotubes by applying Darboux transformation to the square of the Dirac-like Hamiltonian, describing the low-energy dynamics of charge carriers. It is shown that obtained two pseudo-Schrodinger expressions are intertwined, demonstrating supersymmetry structure. It is proved, that for the considered class of pseudo-scalar external perturbations, the unitary disentanglement of negative/ positive states as within Foldy-Wouthuysen (FW) approach, so by Eriksen or SU(2) methods, can be carried out exactly, thus demonstrating the stability of the Dirac sea. The necessity of introduction of the probability density operator into the theory, while considering the problem in FW representation, is underlined.

Statistical properties of the two-dimensional Dirac oscillator with spin–orbit coupling

We investigate the statistical properties of a two-dimensional Dirac oscillator (2D DO) in the presence of a spin–orbit coupling, breaking its supersymmetry. The Hamiltonian of the problem is shown to admit an exact Foldy–Wouthuysen transformation, which allowed us to discard negative-energy states in carrying out our study. The main thermodynamical functions: the Helmholtz free energy, the mean energy, the entropy and the specific heat, are obtained from the canonical partition function, and their behaviours are analysed. The high temperature limit of different thermodynamical functions is probed analytically by utilizing the Euler–Maclaurin formula.

Supersymmetry of Relativistic Hamiltonians for Arbitrary Spin

Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary but fixed spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. Here, the supercharges transform between energy eigenstates of positive and negative energy. For such supersymmetric Hamiltonians, an exact Foldy–Wouthuysen transformation exists which brings it into a block-diagonal form separating the positive and negative energy subspaces. The relativistic dynamics of a charged particle in a magnetic field are considered for the case of a scalar (spin-zero) boson obeying the Klein–Gordon equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation. In the latter case, supersymmetry implies for the Landé g-factor g=2.

Dynamics of the relativistic electron spin in an electromagnetic field

A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy–Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy–Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy–Wouthuysen spin dynamics additionally accounts for the relativistic particle-antiparticle coupling. We conclude that the Pryce spin operator provides a suitable description of the relativistic spin dynamics in a weak-to-intermediate external field, whereas the Foldy–Wouthuysen spin operator is more suitable in the strong field regime.

Paraxial wave function and Gouy phase for a relativistic electron in a uniform magnetic field

A connection between relativistic quantum mechanics in the Foldy–Wouthuysen representation and the paraxial equations is established for a Dirac particle in external fields. The paraxial form of the Landau eigenfunction for a relativistic electron in a uniform magnetic field is determined. The obtained wave function contains the Gouy phase and significantly approaches the paraxial wave function for a free electron.

Nonrelativistic expansion of the Dirac equation with spherical scalar and vector potentials by a reconstituted Foldy-Wouthuysen transformation

Inspired by the reconstituted similarity renormalization group method, the reconstituted Foldy-Wouthuysen (FW) transformation is proposed. Applied to the Dirac equation in the covariant density functional theory, the reconstituted FW transformation shows a fast convergence of the spectrum of the single-particle energy. The single-particle densities and the single-particle scalar densities obtained by this new method are also investigated. In particular, the relativistic corrections to the densities from the picture-change error between the Schr\"odinger and Dirac pictures are discussed in detail. Taking these relativistic corrections into account, both the single-particle densities and the single-particle scalar densities are almost identical to their exact values.

Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics

First, we examine how spin is treated in special relativity and the necessity of introducing spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a spinning particle. Next, we discuss quantum electrodynamics and the Foldy–Wouthuysen transformation which we note is a position operator identical to the Pryce–Newton–Wigner position operator. The classical version of the operators are shown to be essential for the treatment of classical relativistic particles in general relativity, of special interest being the case of binary systems (black holes/neutron stars) which emit gravitational radiation.

Symmetries of Relativistic Hydrogen Atom

The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

The nth-order moment of the nuclear charge density and contribution from the neutrons

AbstractThe relativistic expression for the $n$th-order moment of the nuclear charge density is presented. For the mean square radius (msr) of the nuclear charge density, the non-relativistic expression, which is equivalent to the relativistic one, is also derived consistently up to $1/M^2$ with use of the Foldy–Wouthuysen transformation. The difference between the relativistic and non-relativistic expressions for the msr of the point proton density is also discussed. The $n(\ge 4)$th-order moment of the nuclear charge density depends on the point neutron density. The fourth-order moment yields useful information on the msr of the point neutron density, and is expected to play an important role in electron scattering off neutron-rich nuclei.

Equivalence principle for antiparticles and its limitations

We investigate the particle–antiparticle symmetry of the gravitationally coupled Dirac equation, both on the basis of the gravitational central-field problem and in general curved space–time backgrounds. First, we investigate the central-field problem with the help of a Foldy–Wouthuysen transformation. This disentangles the particle from the antiparticle solutions, and leads to a “matching relation” of the inertial and the gravitational mass, which is valid for both particles as well as antiparticles. Second, we supplement this derivation by a general investigation of the behavior of the gravitationally coupled Dirac equation under the discrete symmetry of charge conjugation, which is tantamount to a particle[Formula: see text]antiparticle transformation. Limitations of the Einstein equivalence principle due to quantum fluctuations are discussed. In quantum mechanics, the question of where and when in the Universe an experiment is being performed can only be answered up to the limitations implied by Heisenberg’s Uncertainty Principle, questioning an assumption made in the original formulation of the Einstein equivalence principle. Furthermore, at some level of accuracy, it becomes impossible to separate nongravitational from gravitational experiments, leading to further limitations.

Non-relativistic expansion of single-nucleon Dirac equation: Comparison between Foldy-Wouthuysen transformation andsimilarity renormalization group * * Partially supported by the JSPS Grant-in-Aid for Early-Career Scientists (18K13549), the JSPS-NSFC Bilateral Program for Joint Research Project on Nuclear mass and life for unravelling mysteries of the r-process and RIKEN Pioneering Project: Evolution of Matter in the Universe

By following the Foldy-Wouthuysen (FW) transformation of the Dirac equation, we derive the exact analytic expression up to the 1/M4 order for general cases in the covariant density functional theory. The results are compared with the corresponding ones derived from another novel non-relativistic expansion method, the similarity renormalization group (SRG). Based on this comparison, the origin of the difference between the results obtained with the FW transformation and the SRG method is explored.

Fermionic Schwinger model for spinning Feshbach–Villars particle in magnetic field

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using fermionic coherent state path integral. The fermionization of the charge-spin symmetry is presented following the Schwinger recipe. The explicit calculations are done in the free case and magnetic interaction using the Foldy–Wouthuysen tranformation. The spectrum and wave functions are deduced.

The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.

Proca Particle in Riemannian Spacetimes

Relativistic quantum-mechanical description of electromagnetic, inertial, and gravitational interactions of a Proca (spin-1) particle is presented. Covariant equations defining electromagnetic interactions of a Proca particle with the anomalous magnetic and electric dipole moments in Riemannian spacetimes are formulated. The relativistic Foldy–Wouthuysen transformation with allowance for only terms proportional to the zero power of the Planck constant is performed as an example. The Hamiltonian obtained agrees with the corresponding Hamiltonians derived for scalar and Dirac particles and with their classical counterpart.

Self-Adjoint Second-Order Equations for Fermions and the Foldy–Wouthuysen Representation

Self-adjoint second-order equations with spinor wave functions in an external electromagnetic field are obtained for fermions. The relationship is found between these equations and the Dirac equations in the Foldy–Wouthuysen representation. Closed chiral symmetric equations are written for left and right spinors by using the Dirac matrices in chiral representation. A closed chiral symmetric Foldy–Wouthuysen equation is obtained for an arbitrary electromagnetic field. There are no summands mixing left and right spinors in these equations irrespective of the presence or absence of fermion mass.

Influences of the coordinate dependent noncommutative space on charged and spin currents

We study the charged and spin currents on a coordinate dependent noncommutative space. Starting from the noncommutative extended relativistic equation of motion, the nonrelativistic approximation is obtained by using the Foldy–Wouthuysen transformation, and then the charged and spin currents are derived by using the extended Drude model. We find that the charged current is twisted by modifying the off-diagonal elements of the Hall conductivity, however, the spin current is not affected up to leading order of the noncommutative parameter.

Generalisation of Gilbert damping and magnetic inertia parameter as a series of higher-order relativistic terms

The phenomenological Landau–Lifshitz–Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to first-order relativistic effects. To include magnetic inertial effects the LLG equation has previously been extended with a supplemental inertia term; the arising inertial dynamics has been related to second-order relativistic effects. Here we start from the relativistic Dirac equation and, performing a Foldy–Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that contains relativistic correction terms to any higher order. Using the Heisenberg equation of spin motion we derive general relativistic expressions for the tensorial Gilbert damping and magnetic inertia parameters, and show that these tensors can be expressed as series of higher-order relativistic correction terms. We further show that, in the case of a harmonic external driving field, these series can be summed and we provide closed analytical expressions for the Gilbert and inertial parameters that are functions of the frequency of the driving field.

Raman scattering of photons by hydrogenlike ions I: A semirelativistic and nondipole approach

Going beyond the usual nonrelativistic treatments of Raman scattering of light by a ground-state electron in hydrogenic ions, we propose in this work a semirelativistic and nondipole approach in the case of inelastic process under the ionization threshold. This scheme is based, on the one hand, on the discrete semirelativistic Coulomb Sturmian basis sets, and on the other hand, on the direct Foldy–Wouthuysen interaction Hamiltonian derived recently from the Dirac equation with electromagnetic potentials (Hinschberger and Hervieux 2012) and confirmed by us to seventh order in powers of 1∕m. It includes the paramagnetic, the diamagnetic, the magnetic field–spin and the electric field–spin interactions. Moreover, the photon fields are expanded in electric and magnetic multipoles and nondipole effects are fully taken into account. On utilizing these ingredients, we build in detail the multipole expansion of the semirelativistic Raman scattering cross section when the photon energy is under the ionization threshold. For each type of the allowed multipole components, namely EE, MM and EM in the standard notations, suitable closed forms of total scattering cross sections are calculated for numerical computations. Such analytical expressions enable the evaluation of the contribution of each of these interaction operators and field multipoles in the scattering process.