Pryce’s mass-center operators and the anomalous velocity of a spinning electron

In the present work, we develop a method to derive the anomalous velocity of a spinning electron. From Dirac equation, the relationships among the expectation values of the Pryce’s mass-center operator, the position operator, the spin operator and the canonical momentum operator are investigated. By requiring that the center of mass for a classical spinning electron is related to the expectation value of Pryce’s mass-center operator, one can obtain a classical expression for the position of the electron. With the classical equations of motion, the anomalous velocity of a spinning electron can be easily obtained. It is shown that two factors contribute to the anomalous velocity: one is dependent on the selection of Pryce’s mass-center operators and the other is a type-independent velocity expressed by the rotational velocity and the Lorentz force.     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

The hamiltonian path integrals and the uniform semiclassical approximations for the propagator

The generalized path expansion scheme is defined for path integration in phase-space. Within this framework we study the semiclassical limits to the propagator, both in the momentum and the coordinate representations. It is shown that the role played by the Morse operator in the Lagrangian formulation of the path integral method is taken by another differential operator of the Dirac type. The relevant properties of this operator are discussed. The semiclassical approximations are obtained by extending the results of catastrophe theory for the asymptotic evaluation of finite-dimensional integrals to the domain of path integration. Various forms of the uniform semiclassical approximations are obtained. Their validity and applicability are discussed. The method is illustrated by a solution of a simple example in which nongeneric catastrophe occurs.     

Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment

The Lagrangian and Hamiltonian formulations for the relativistic classical dynamics of a charged particle with dipole moment in the presence of an electromagnetic field are given. The differential conservation laws for the energy-momentum and angular momentum tensors of a field and particle are discussed. The Poisson brackets for basic dynamic variables, which form a closed algebra, are found. These Poisson brackets enable us to perform the canonical quantization of the Hamiltonian equations that leads to the Dirac wave equation in the case of spin 1/2. It is also shown that the classical limit of the squared Dirac equation results in equations of motion for a charged particle with dipole moment obtained from the Lagrangian formulation. The inclusion of gravitational field and non-Abelian gauge fields into the proposed formalism is discussed.Received: 4 June 2005, Published online: 27 July 2005     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

The Supersymmetry and Eigen Energy Spectrum of a Charged Dirac Particle in a Uniform Constant Magnetic Field

Based on the opinion that the γ-matrices in Dirac equation have structure and are decomposable, we decompose the γ-matrices into the direct product of the operators in the spin space and the particle-antiparticle space. By using this method, we attain a complete set of commutative operators, a set of quantum numbers and the correspondingly eigen solutions of the Hamiltonian for a charged Dirac particle moving in a uniform constant magnetic field. In addition, the dynamic supersymmetry of the Hamiltonian is unveiled. Spin symmetry breaking and particle-antiparticle symmetry breaking are discussed, and the supersymmetric group operator of the degenerate spin subspace resulting from the spin residual supersymmetry is found.     

WKB approximation in complex time

The WKB approximation to the one particle Schrödinger equation in time is used to obtain the wavefunction at a given point as a sum of semiclassical terms, each corresponding to a different classical trajectory (real or complex) but ending up at the same point. A method to find out reflection coefficient for processes involving one and two turning points is developed and it is shown that the semiclassical complex analysis reproduces exactly the reflection coefficient that is obtained through the exact solution of the problem. The connection between pair production and reflection amplitude is also shown. The pair production amplitude in a time dependent gravitational background is calculated and it is shown that the vacuum considered in complex trajectory WKB analysis refers to adiabatic vacuum.     

Semiclassical Expectation Values for Relativistic Particles with Spin 1/2

For relativistic particles with spin 1/2, which are described by the Dirac equation, a semiclassical trace formula is introduced that incorporates expectation values of observables in eigenstates of the Dirac-Hamiltonian. Furthermore, the semiclassical limit of an average of expectation values is expressed in terms of a classical average of the corresponding classical observable.     

Supersymmetrization of Quaternion Dirac Equation for Generalized Fields of Dyons

The quaternion Dirac equation in presence of generalized electromagnetic field has been discussed in terms of two gauge potentials of dyons. Accordingly, the supersymmetry has been established consistently and thereafter the one, two and component Dirac Spinors of generalized quaternion Dirac equation of dyons for various energy and spin values are obtained for different cases in order to understand the duality invariance between the electric and magnetic constituents of dyons.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

高维de Sitter时空中宇宙视界处的Fermi子隧穿

运用半经典近似理论,本文研究了来自静态高维deSitter时空和高维Schwarzschild-de Sitter时空宇宙视界处的Fermi子隧穿辐射.在文中,描述1/2自旋粒子行为的Dirac方程被简化为一个简单的形式,接着运用方程组有非平凡解的条件,可以得到了半经典的Hamilton-Jacobi方程,从而使得问题大大得以简化,最终得到了静态de Sitter时空中宇宙视界处的Fermi子隧穿率和Hawking温度.     

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional \({{\mathcal N} = 2}\) supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kähler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generate special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston’s complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kähler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.     

Equations of motion for a classical color particle in external non-Abelian fermionic and bosonic fields

On the basis of principles of gauge and reparametrization invariance of a real-valued Lagrangian, construction of the action describing the dynamics of a classical color-charged particle interacting with background non-Abelian gauge and fermion fields is considered. The cases of the linear and quadratic dependence of a Lagrangian on the background fermion field are discussed. It is shown that, in both cases, there exist an infinite number of interaction terms. From an iteration scheme, examples of construction of the first few currents and sources induced by a moving particle with non-Abelian charge are given. It is shown that these quantities by a suitable choice of parameters exactly reproduce additional currents and sources obtained previously in [1] on the basis of heuristic considerations.     

高维de Sitter时空中宇宙视界处的Fermi子隧穿

运用半经典近似理论,本文研究了来自静态高维de Sitter时空和高维Schwarzschild-de Sitter时空宇宙视界处的Fermi子隧穿辐射.在文中,描述1/2自旋粒子行为的Dirac方程被简化为一个简单的形式,接着运用方程组有非平凡解的条件,可以得到了半经典的Hamilton-Jacobi方程,从而使得问题大大得以简化,最终得到了静态de Sitter时空中宇宙视界处的Fermi子隧穿率和Hawking温度.     

Supersymmetry in the problem of a fermion in an external nonabelian gauge field

The quantum-mechanical problem was considered of the motion of a Dirac particle in a nonabelian chromomagnetic field of the local isospin group SU(2). It was shown that the problem possesses supersymmetry. Supercharges were constructed, satisfying the standard superalgebra. The fermionic number operator coincides with the invariant spin operator of the Dirac equation. We clarified the features, in the given case, of supersymmetry, appearing in the anomaly of the commutation relations for the bosonic operators that was caused by the mutual tie of the bosonic (spatial and isospin) and the fermionic (spin) degrees of freedom for motion in the gauge field considered. It was demonstrated that the degeneracy of the vacuum (ground state) is removed by means of the spontaneous symmetry-breaking SU(2)U(1). The choice of a solution in the group U(1) is determined by the sign of the charge (color) of the particle in the ground state.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 40–44, February, 1989.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Spin Hall effect and Berry phase of spinning particles

We consider the adiabatic evolution of the Dirac equation in order to compute its Berry curvature in momentum space. It is found that the position operator acquires an anomalous contribution due to the non-Abelian Berry gauge connection making the quantum mechanical algebra noncommutative. A generalization to any known spinning particles is possible by using the Bargmann–Wigner equation of motions. The noncommutativity of the coordinates is responsible for the topological spin transport of spinning particles similarly to the spin Hall effect in spintronic physics or the Magnus effect in optics. As an application we predict new dynamics for nonrelativistic particles in an electric field and for photons in a gravitational field.     

Supersymmetric Dirac particles in Riemann-Cartan space-time

A natural extension of the supersymmetric model of Di Vecchia and Ravndal yields a nontrivial coupling of classical spinning particles to torsion in a Riemann-Cartan geometry. The equations of motion implied by this model coincide with a consistent classical limit of the Heisenberg equations derived from the minimally coupled Dirac equation. Conversely, the latter equation is shown to arise from canonical quantization of the classical system. The Heisenberg equations are obtained exact in all powers of and thus complete the partial results of previous WKB calculations. We touch also on such matters of principle as the mathematical realization of anticommuting variables, the physical interpretation of supersymmetry transformations, and the effective variability of rest mass.     

Fermion Tunneling from an Axis-Symmetric Black Hole beyond the Semiclassical Approximation

Based on the Hamilton-Jacobi method beyond the semiclassical approximation proposed by R. Baberjee and B.R. Majhi, Hawking radiation of Dirac particle as tunneling through the event horizon is calculated. It is shown that all quantum corrections in the fermion particle action are proportional to the usual semiclassical contribution. Under the conception of irreducible mass and the first law of thermodynamics, the modifications to Hawking temperature and Bekenstein-Hawking entropy are given for a Kerr-Newmann black hole.     

Superspace mechanics as the classical limit of superfield theory

The semi-classical superspace pseudomechanics of Casalbuoni and others is shown to be the classical limit of a scalar superfield theory by a WKB type of approximation. By avoiding second class constraints, the relationship between the mechanics and the field theory is clarified. The spin1/2sector is then examined to find the (semiclassical) WKB limit of the Dirac equation.