Wouthuysen Transformation Research Articles

Dirac equations with confining potentials

This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. Under certain conditions, despite the addition of radially symmetric, linear confining potentials, specific bound-state energies surprisingly preserve their exact Dirac–Coulomb values. The generality of the "preservation mechanism" is investigated. To this end, a Foldy–Wouthuysen transformation is used to calculate the corrections to the spin-orbit coupling induced by the linear confining potentials. We find that the matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the preservation mechanism. The result of the Foldy–Wouthuysen transformation is used to verify that the preservation is restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. The results derived from the nonrelativistic effective operators are compared to the fully relativistic radial Dirac equations. Furthermore, we show that the preservation mechanism does not affect antiparticle (negative-energy) states.

Nonrelativistic approximation of the Dirac equation for slow fermions in static metric spacetimes

We analyze the nonrelativistic approximation of the Dirac equation for slow fermions, having small kinetic energies compared to their rest energy $m$ and moving in spacetimes with a static metric, caused by the weak gravitational field of the Earth and a chameleon field, and derive the most general effective gravitational potential to order $1/m$, induced by a static metric of spacetime excluding possible rotations of the coordinate frame. The derivation of the nonrelativistic Hamilton operator of the Dirac equation is carried out by using a standard Foldy--Wouthuysen transformation. We discuss the chameleon field as source of a torsion field and torsion--matter interactions.

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’).

The Douglas–Kroll–Hess method based on vector-potential-including Foldy–Wouthuysen transformation: Application to NMR shielding tensor

We propose a novel Foldy–Wouthuysen transformation including a vector potential A, which can be used to introduce restricted magnetic balance in the Douglas–Kroll–Hess (DKH) method. We also demonstrate that the DKH method can be used in combination with the gauge-including atomic orbital method. For the numerical examination, we calculate the NMR shielding constants and anisotropies of noble gas atoms (Ne, Ar, Kr, Xe), halogens (F, Cl, Br, I), and chalcogens (O, S, Se, Te) in molecules without using extremely large basis sets.

EFFECTIVE ACTION FOR FERMIONS WITH ANOMALOUS MAGNETIC MOMENT FROM FOLDY–WOUTHUYSEN TRANSFORMATION

In this paper, we calculate the effective action for neutral particles with anomalous magnetic moment in an external magnetic and electric field. We show that we can take advantage from the Foldy–Wouthuysen transformation (FWT) for such systems, determined in our previous works: indeed, by this transformation we have explicitly evaluated the diagonalized Hamiltonian, allowing to present a closed form for the corresponding effective action and for the partition function at finite temperature from which the thermodynamical potentials can be calculated.

The cylindrical δ-potential and the Dirac equation

In this paper, we discuss the Dirac equation in the presence of an attractive cylindrical δ-shell potential V(ρ) = −aδ(ρ − ρ0), where ρ is the radial coordinate and a > 0. We present a detailed discussion on the boundary conditions the wavefunction has to satisfy when crossing the support of the potential, proceeding then to explore the dependence of the ground state on the parameter a, analyzing the occurrence of supercritical effects. We also apply the Foldy–Wouthuysen transformation, discussing the non-relativistic limit of this problem.

The effect of inertia on the Dirac electron, the spin Hall current and the momentum space Berry curvature

We have studied the spin dependent force and the associated momentum space Berry curvature in an accelerating system. The results are derived by taking into consideration the non-relativistic limit of a generally covariant Dirac equation with an electromagnetic field present, where the methodology of the Foldy–Wouthuysen transformation is applied to achieve the non-relativistic limit. Spin currents appear due to the combined action of the external electric field, the crystal field and the induced inertial electric field via the total effective spin–orbit interaction. In an accelerating frame, the crucial role of momentum space Berry curvature in the spin dynamics has also been addressed from the perspective of spin Hall conductivity. For time dependent acceleration, the expression for the spin polarization has been derived.

Foldy–Wouthuysen transformation applied to the interaction of an electron with ultrafast electromagnetic fields

By means of the Foldy–Wouthuysen transformation the nonrelativistic approximation of the external-electromagnetic-field Dirac equation to fifth order in powers of 1/m is obtained. Generalizing this result we postulate a general expression of the direct spin–field electronic Hamiltonian valid at any order in 1/m.

Effective field theory of a topological insulator and the Foldy–Wouthuysen transformation

Employing the Foldy–Wouthuysen transformation, it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern–Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2+1 dimensions is discussed and by means of a dimensional reduction approach the 1+1 dimensional descendant of the 2+1 dimensional Chern–Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derived through the Foldy–Wouthuysen transformation. Acquainted with it, the second Chern numbers are calculated for specific choices of the integration domain. A method is proposed to obtain 3+1 and 2+1 dimensional descendants of the effective field theory of the 4+1 dimensional time reversal invariant topological insulator theory. Inspired by the spin Hall effect in graphene, a hypothetical model of the time reversal invariant spin Hall insulator in 3+1 dimensions is proposed.

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.

Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry

We study supersymmetry of a self-isospectral one-gap Poschl-Teller system in the light of a mirror symmetry that is based on spatial and shift reflections. The revealed exotic, partially broken nonlinear supersymmetry admits seven alternatives for a grading operator. One of its local, first order supercharges may be identified as a Hamiltonian of an associated one-gap, non-periodic Bogoliubov-de Gennes system. The latter possesses a nonlinear supersymmetric structure, in which any of the three non-local generators of a Clifford algebra may be chosen as the grading operator. We find that the supersymmetry generators for the both systems are the Darboux-dressed integrals of a free spin-1/2 particle in the Schrodinger picture, or of a free massive Dirac particle. Nonlocal Foldy- Wouthuysen transformations are shown to be involved in the supersymmetric structure.

A generalization of the Baker–Hausdorff lemma

Two operator identities involving a q-commutator, [A,B]≡AB+qBA, where A and B are two arbitrary (generally noncommuting) linear operators acting on the same linear space and q is a variable that commutes with these two operators, are formulated, proved and discussed. The first identity is a direct generalization of the Baker–Hausdorff lemma, whereas the second involves the time derivative of the exponential function of a time-dependant operator. It is indicated how these two identities can be used to good advantage to treat the Foldy–Wouthuysen transformation of the Dirac Hamiltonian for a particle in an external electromagnetic field.

The origin of the hidden supersymmetry

The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, the Dirac delta potential problem and the Aharonov–Bohm effect (planar, bound state, and tubule models) are explained by a special nonlocal unitary transformation, which for the usual N=2 supercharges has a nature of Foldy–Wouthuysen transformation. We show that in general case, the bosonized supersymmetry of nonlocal, parity even systems emerges in the same construction, and explain the origin of the unusual N=2 supersymmetry of electron in three-dimensional parity even magnetic field. The observation extends to include the hidden superconformal symmetry.

Paraxial spin transport using the Dirac-like paraxial wave equation

In weakly inhomogeneous media, Maxwell equations assume a Dirac-like form that is particularly apt for the study of paraxial propagation. Using this form, and via the Foldy–Wouthuysen transformation technique of the Dirac equation, we study the spin transport of paraxial light beams in weakly inhomogeneous media. We derive the Berry effect terms and establish the spin Hall effect and the Rytov rotation law for polarized paraxial beam transport.

Foldy–Wouthuysen transformation for a spinning particle with anomalous magnetic moment

In this paper we study the Foldy–Wouthuysen transformation for a pseudoclassical particle with anomalous magnetic moment in an external classical stationary electromagnetic field. We show that the transformation can be expressed in a closed form for neutral particles in purely electrostatic fields and for neutral and charged particles in external magnetostatic fields. The explicit expressions of the diagonalized Hamiltonians are calculated.

Spacetime path formalism for massive particles of any spin

Earlier work presented spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three-space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the non-scalar propagator in terms of a path integral over the Poincaré group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well-known generalized Foldy–Wouthuysen transformation.

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.

Spin–rotation couplings: spinning test particles and Dirac field

The hypothesis of coupling between spin and rotation introduced long ago by Mashhoon is examined in the context of “1 + 3” and “3 + 1” space-time splitting techniques, either in special or in general relativity. Its content is discussed in terms of classical (Mathisson–Papapetrou–Dixon–Souriou model) as well as quantum physics (Foldy–Wouthuysen transformation for the Dirac field in an external field), reviewing and discussing all the relevant theoretical literature concerning the existence of such effect. Some original contributions are also included.

ABSENCE OF BCS CONDENSATION OF ATOMS IN A REPULSIVE FERMI GAS OF ATOMS

Bose–Einstein condensation of molecules and BCS condensation of Cooper pairs of atoms in an ultra-cold Fermi gas of atoms with a repulsive two-body interaction are studied by using the path integral representation of the grand partition function. From the self-consistent equations obtained in the present work for the order parameters, we have found that BCS condensation of atoms cannot occur in such a Fermi gas of atoms and that the condensate observed experimentally is composed of condensed molecules or possibly of preformed Cooper pairs. To substantiate our conclusions from the self-consistent equations for the order parameters, the effective atom–atom interaction mediated through molecules has also been computed with the Foldy–Wouthuysen transformation and has been found to be repulsive, which implies a net repulsive two-body interaction between atoms and hence the absence of BCS condensation of atoms.

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.