Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

Perturbation Method for Particle-like Solutions of the Einstein–Dirac Equations

The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein–Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard’s equation generates solutions of the Einstein–Dirac equations.     

Extended Dirac factorization: ‘multi-mass’ special cases in clifford form

The hypercomplex approach to Dirac physics is here summarized. The new mass term is shown to give a ψ wave, in the extension of Dirac’s Klein-Gordon factorization (when the usual Dirac mass part equals zero), that is completely within the ‘complexified’ Hamilton-Pauli sub-algebraE. This may help with the physical interpretation of this new mass, with its five possible parts. New, spin 1/2 quanta also seem possible. These are all variations on Dirac’s factorization, where mass is generalized to several parts (multi-parts) or, instead, each new mass part, of the five, can be taken one at a time in nature, for individual wave equations. Lorentz symmetry is also naturally extended from 6 parameters to 8 parameters, and this has sweeping metaphysical implications. Dirac algebra is doubled also in a natural way using quaternions.     

Asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary

In this article we discuss the asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary when the boundary part is stretched. In [12] the author studied the same question under the assumption of no existence of L² - and extended L² -solutions of Dirac operators when the boundary part is stretched to infinite length. Therefore, the results in this article generalize those in [12]. Using the main results we obtain the formula describing the ratio of two zeta-determinants of Dirac Laplacians with the APS boundary conditions associated with two unitary involutions σ₁ and σ₂ on ker B satisfying Gσ_i = -σ_i G. We also prove the adiabatic decomposition formulas for the zeta-determinants of Dirac Laplacians on a closed manifold when the Dirichlet or the APS boundary condition is imposed on partitioned manifolds, which generalize the results in [10] and [11].     

Representation theoretic harmonic spinors for coherent families

Coherent continuation π ₂ of a representation π ₁ of a semisimple Lie algebra arises by tensoring π ₁ with a finite dimensional representation F and projecting it to the eigenspace of a particular infinitesimal character. Some relations exist between the spaces of harmonic spinors (involving Kostant’s cubic Dirac operator and the usual Dirac operator) with coefficients in the three modules. For the usual Dirac operator we illustrate with the example of cohomological representations by using their construction as generalized Enright-Varadarajan modules. In [9] we considered only discrete series, which arises as generalized Enright-Varadarajan modules in the particular case when the parabolic subalgebra is a Borel subalgebra.     

The Douglas-Kroll-Heß Method: Convergence and Block-Diagonalization of Dirac Operators

We show that the Douglas-Kroll block-diagonalization method for the Dirac operator with Coulomb potential is convergent in norm resolvent sense for coupling constant γ less than γ_c = 0.37758 which corresponds to atomic number 51. Moreover, we give an explicit expression for the corresponding block-diagonalized Dirac operator. Communicated by Vincent Rivasseau submitted 26/02/05, accepted 12/04/05     

Fundamental solutions of the equations of quantum mechanics as distributions and distinctive properties of the Dirac equation solutions

We show that the causal Green’s functions for interacting particles in external fields in both relativistic quantum mechanics (for the Dirac electron) and nonrelativistic quantum mechanics can be obtained as distributions if the free-particle Green’s functions are used and equations for the corresponding test functions are chosen. We study quantum properties of solutions of the Dirac equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 287–301, May, 2007.     

A Resolution for the Dirac Operator in Four Variables in Dimension 6

The Dirac operator in several operators is an analogue of the - operator in theory of several complex variables. The Hartog’s type phenomena are encoded in a complex of invariant differential operators starting with the Dirac operator, which is an analogue of the Dolbeault complex. In the paper, a construction of the complex is given for the Dirac operator in 4 variables in dimension 6 (i.e. in the non-stable range). A peculiar feature of the complex is that it contains a third order operator. The methods used in the construction are based on the Penrose transform developed by R. Baston and M. Eastwood. The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117.     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

Hua operator on vector bundle: Application to AdS/CFT correspondence of Dirac fields

Hua’s theory of harmonic functions on classical domains is generalized to the theory on holomorphic vector bundles over classical domains and further on vector bundles over the real classical domains and quaternion classical domains. In case of the simplest quaternion classical domain there is a relation between Hua operator and Dirac operator, by which an AdS/CFT correspondence of Dirac fields is established.     

A super-twisted Dirac operator and Novikov inequalities

A super-twisted Dirac operator is constructed and deformed suitably. Following Shubin’s approach to Novikov inequalities associated to the deformed de Rham-Hodge operator, we give a for mula for the index of the super-twisted Dirac operator, and Novikov type inequalities for the deformed operator. In particular, we obtain a purely analytic proof of the Hopf index theorem for general vector bundles.     

The HVZ Theorem for a Pseudo-Relativistic Operator

The localization of the essential spectrum of a relativistic two-electron ion is provided. The analysis is performed with the help of the pseudo-relativistic Brown–Ravenhall operator which is the restriction of the Coulomb–Dirac operator to the electrons’ positive spectral subspace. Submitted: September, 2005. Revised: March 27, 2006. Accepted: August 3, 2006.     

Dynamical lower bounds for 1D Dirac operators

Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli–Dirac one and, in contrast to the discrete case, critical energies are also found for the continuous Dirac case with positive mass. R. A. Prado was supported by FAPESP (Brazil). C. R. de Oliveira was partially supported by CNPq (Brazil).     

A Discrete Bochner�CMartinelli Formula

In the even dimensional case the discrete Dirac equation may be reduced to the so-called discrete isotonic Dirac system in which suitable Dirac operators appear from both sides in half the dimension. This is an appropriated framework for the development of a discrete Martinelli–Bochner formula for discrete holomorphic functions on the simplest of all graphs, the rectangular \mathbbZ^m{\mathbb{Z}^m} one. Two lower-dimensional cases are considered explicitly to illustrate the closed analogy with the theory of continuous variables and the developed discrete scheme.     

Asymptotic expression for the correlation function of twisted fields in the two-dimensional Dirac model on a lattice

A determinant representation is obtained for the correlation function of twisted fields in the two-dimensional Dirac model on a lattice. These fields are determined by twisted boundary conditions for the Dirac fermions. The asymptotic expression is calculated for the correlation function at large distances (the vacuum expectation of the twisted field) at the critical point and in the scaling region. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2 pp. 329–346, Nember, 1999.     

Atiyah—Patodi—Singer type index theorems for manifolds with splitting of $ \eta $-invariants

We construct self-adjoint extensions of Dirac operators on manifolds with corners of codimension 2, which generalize the Atiyah—Patodi—Singer boundary conditions. The boundary conditions are related to geometric constructions, which convert problems on manifolds with corners into problems on manifolds with boundary and wedge singularities. In the case, where the Dirac bundle is a super-bundle, we prove two general index theorems, which differ by the splitting formula for -invariants. Further we work out the de Rham, signature and twisted spin complex in closer detail. Finally we give a new proof of the splitting formula for the -invariant. Submitted: October 1999, Revised version: March 2001.     

On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval

We study representations of solutions of the Dirac equation, properties of spectral data, and inverse problems for the Dirac operator on a finite interval with discontinuity conditions inside the interval. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 601–613, May, 2005.     

The Generalized Dolbeault Complex in Two Clifford Variables

The Penrose transform is used to construct a complex starting with the Dirac operator in two Clifford variables. The corresponding relative BGG complex and its direct image is computed for cohomology with values in line bundles induced by representations in singular infinitesimal character. The limit of the induced spectral sequence is computed in cases connected with the Dirac operator in two Clifford variables. The work presented here is a part of the research project MSM 0021620839 and was supported also by the grants GAUK 447/2004 and GA ČR 201/05/2117.