Foldy Wouthuysen Research Articles

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.

Calculation of Higher-Order Foldy-Wouthuysen Transformation Hamiltonian

The Foldy—Wouthuysen Hamiltonian of a light atomic system that has an mα8 contribution to energy levels is calculated. The case of a Dirac—Coulomb field is discussed. The results can be used for relativistic and radiative corrections to energy levels in the low-energy part. A divergent operator δ2(r) emerges. This is probably due to the nature of the point-like charge source. The effective method of radiation calculation may be re-checked.

What is the relativistic spin operator?

Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin operator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious zitterbewegung (quivering motion) or violation of the angular momentum algebra. Only the Foldy–Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.

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’).

Dirac particle spin in strong gravitational fields

Dynamics of the Dirac particle spin in general strong gravitational fields is discussed. The Hermitian Dirac Hamiltonian is derived and transformed to the Foldy-Wouthuysen (FW) representation for an arbitrary metric. The quantum mechanical equations of spin motion are found. These equations agree with corresponding classical ones. The new restriction on the anomalous gravitomagnetic moment (AGM) by the reinterpretation of Lorentz invariance tests is obtained.

NONCOMMUTATIVE PATH INTEGRAL FOR SPINLESS RELATIVISTIC EQUATION IN THE TWO-COMPONENT THEORY

In this paper, we have constructed the Green function of the Feshbach–Villars (FV) spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables. Subsequently, for the perform calculations, we diagonalize the Hamiltonian governing the dynamics of the system via the Foldy–Wouthuysen (FW) canonical transformation. The exact calculations have been done in the cases of free particle and magnetic field interaction. In both cases, the energy eigenvalues and their corresponding eigenfunctions are deduced.

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.

Inertial spin Hall effect in non-commutative space

In the present paper the study of inertial spin current(that appears in an accelerated frame of reference) is extended to Non-Commutative (NC) space. The $\theta$-dependence, ($\theta$ being the NC parameter), of the inertial spin current is derived explicitly. We have provided yet another way of experimentally measuring $\theta$. Our bound on $\theta$ matches with previous results. In Hamiltonian framework, the Dirac Hamiltonian in an accelerating frame is computed in the low energy regime by exploiting the Foldy-Wouthuysen scheme. The NC $\theta$-effect appears from the replacement of normal products and commutators by Moyal *-products and *-commutators. In particular, the commutator between the external magnetic vector potential and the potential induced by acceleration becomes non-trivial. Expressions for $\theta$-corrected inertial spin current and conductivity are derived. The $\theta$ bound is obtained from the out of plane spin polarization, which is experimentally observable.

Validation of the Eriksen method for the exact Foldy-Wouthuysen representation

The Eriksen method is proven to yield a correct and exact result when a sufficient condition of exact transformation to the Foldy-Wouthuysen (FW) representation is satisfied. Therefore, the Eriksen method is confirmed as valid. This makes it possible to establish the limits within which the approximate “step-by-step” methods are applicable. The latter is done by comparing the relativistic formulas for a Hamiltonian operator in FW representation (obtained using those methods) and the known expression for the first terms of a series, which defines the expansion of this operator in powers of v/c as found by applying the Eriksen method.

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.

Classical limit of relativistic quantum mechanical equations in the Foldy-Wouthuysen representation

It is shown that, under the Wentzel-Kramers-Brillouin approximation conditions, using the Foldy-Wouthuysen (FW) representation allows the problem of finding a classical limit of relativistic quantum mechanical equations to be reduced to the replacement of operators in the Hamiltonian and quantum mechanical equations of motion by the respective classical quantities.

Momentum and spin dynamics of Dirac particles at effective dimensional reduction

We consider the dynamics of Dirac particles moving in the curved spaces of variable dimension interpolating smoothly between 3- and 2-dimensional spaces and considered as a toy model for 2-dimensional structures in solid state physics. Performing the Foldy-Wouthuysen (FW) transformation of Dirac equation and passing to the classical limit, we derive the equations of motion of momentum and spin. The spin precesses with the variable angular velocity and may "flick" appearing in the remnant 2-dimensional space twice during the period.

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.

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy–Wouthuysen and the Pauli–Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s = ( 1 , 0 ) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford–Dirac algebra is put into consideration.

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.