Semiclassical approximation of the Dirac equation with supersymmetry

The general scheme of the successive construction of semiclassical approximation for the classical Dirac equation in a background Yang-Mills field, where the usual Dirac operator is replaced by that with supersymmetry, is suggested. The first two terms of the semiclassical expansion in Planck’s constant are derived in an explicit form. It is shown that supersymmetry of the initial Dirac operator leads to appearance of new additional terms in the classical equation of motion for spin of a particle and ipso facto requires appropriate modification for the Lagrangian of the spinning particle. The result obtained is used for the construction of one-to-one mapping between two Lagrangians of a classical color-charged spinning particle, one of which possesses local supersymmetry, and another doesn’t. It is demonstrated that for recovery of the one-to-oneness the additional terms obtained above in the semiclassical approximation of the Dirac operator with supersymmetry should be added to the Lagrangian without supersymmetry.     

Some classical solutions of the sourceless SO(4,1) gauge field equations

Some new classical solutions of the sourceless SO(4,1) gauge field equations are found by identifying the internal symmetry indices with the space-time indices as in the cases of the instanton or the meron solutions. This identification of the internal and the space-time indices takes the simplest form when the gauge field equation is expressed in (4,1) de Sitter space, which is conformal to the Minkowski space having the de Sitter group SO(4,1) as a group of motions. The form of the solutions is close to the de Sitter ‘plane wave’ solutions found recently, i.e. the solutions of the Klein-Gordon, Dirac and Maxwell-Proca equations in de Sitter space. The group theoretical structure of the new solutions is discussed and their relations to the Iwasawa decomposition of the non-compact semisimple group SO(4,1) are pointed out.     

Path algebras,wave-particle duality,and quantization of phase space

Semigroup algebras admit certain ‘coherent’ deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie’s wave-particle duality. This theory leads to a model of “physical” phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be described as a random walk. The wavelength associated to a particle moving in this space need not be constant and its phase can change discontinuously.     

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     

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F_α^K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F_α^K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.     

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.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

Nonlinear Spin and Pseudo-Spin Symmetric Dirac Equations

Nonlinear Dirac equations are obtained by variation of the spinor action whose Lagrangian components have the same conformal degree and the coupling parameter of the self-interaction term is dimensionless. In 1+1 space-time, we show that these requirements result in the “conventional” quartic form of the nonlinear interaction and present the general equation for various coupling modes. These include, but not limited to, the Thirring and Gross-Neveu models. We consider the spin and pseudo-spin symmetric models and obtain a numerical solution. We also propose a two-component “minimal” pseudo-scalar coupling model.     

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.     

Symmetry and invariance

The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.     

整数自旋粒子的方程和波函数

从高自旋态的Bargmann-Wigner方程出发,建立了整数自旋粒子的运动方程,通过求解方程得到了一套整数自旋粒子波函数,并建立了等效Largrange形式.     

The motion of a classical spinning particle in a Riemann-Cartan space-time

A consistent set of equations of motion for classical charged particles with spin and magnetic dipole moment in a Riemann-Cartan space-time is generated from a constrained Lagrangian formalism. The equations avoid the spurious free helicoidal solutions and at the same time conserve the canonical condition of normalization of the 4-velocity. The 4-velocity and the mechanical moment are parallel in this theory, where the condition of orthogonality between spin and 4-velocity is treated as a nonholonomic constraint. A generalized BMT precession equation is obtained as one of the results of the formalism.     

On Spinning Particles

There exist classical systems whose canonical quantization yields relativistic wave equations. As a constructive proof, the classical mechanics of a translating-rotating five-frame is considered. Its quantization yields the Dirac, Weyl, Klein-Gordon, Maxwell-Proca, and higher spin equations, together with a rotational mass spectrum for the states predicted.     

Wave functions in geometric quantization

A geometrical way is described to associate quantum states in the sense of geometric quantization to wave functions in the quantum mechanical sense for each relativistic elementary particle. Explicit computations are made in a number of cases: Klein-Gordon and Dirac equations, neutrino and antineutrino Weyl equations, and very general cases of massive and massless particles of arbitrary spin. In this later case one is led in a canonical way to Penrose wave equations.     

Wave function of a free electron in a laser plasma via Riemannian geometry

The wave function of a free electron in a laser plasma described via Riemannian geometry is derived by solving the Dirac equation in the associated curved space-time. If the laser field vanishes, the wave function naturally reduces to the case in flat space-time.     

有限空间中经典场的正则量子化

从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.