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Vadim V. Varlamov

Publications and source records attributed to Vadim V. Varlamov.

13 recordsLinked to original sources
arXiv2004

Equations of Geodesic Deviation and the Inverse Scattering Transform

Solutions of equations of geodesic deviation in three- and four- dimensional spaces obtained by the inverse scattering transform are considered. It is shown that in the case of three-dimensional space solutions of geodesic deviation equations are reduced to solutions of the well-known Zakharov-Shabat problem. In four- dimensional space system of geodesic deviation equations is associated with $3\times 3$ matrix Schrödinger equation, and dependence on parameters defined by the nonlinear equations of three-wave interaction.

solv-int↗
arXiv2001

About Algebraic Foundations of Majorana-Oppenheimer Quantum Electrodynamics and de Broglie-Jordan Neutrino Theory of Light

An algebraic description of basic physical fields (neutrino field, electron-positron field and electromagnetic field) is studied. It is sown that the electromagnetic field can be described within a quotient representation of the proper orthochronous Lorentz group. The relation of such a description with Majorana-Oppenheimer quantum electrodynamics and de Broglie-Jordan neutrino theory of light is discussed.

math-ph↗
arXiv2001

Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field

A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.

math.DG↗
arXiv2001

Fundamental Automorphisms of Clifford Algebras and an Extension of Dabrowski Pin Groups

Double coverings of the orthogonal groups of the real and complex spaces are considered. The relation between discrete transformations of these spaces and fundamental automorphisms of Clifford algebras is established, where an isomorphism between a finite group of the discrete transformations and an automorphism group of the Clifford algebras plays a central role. The complete classification of Dabrowski groups depending upon signatures of the spaces is given. Two types of Dabrowski quotient groups are introduced in case of odd-dimensional spaces. Application potentialities of the introduced quotient groups in Physics are discussed.

math-ph↗
arXiv2001

Clifford Algebras and Lorentz Group

Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. The Clifford algebras are understood as an `algebraic covering' of a full system of the finite-dimensional representations of the Lorentz group. Space-time discrete symmetries P, T and PT, represented by fundamental automorphisms of the Clifford algebras, are defined on all the representation spaces. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. Physical fields of the different types are formulated within such representations. The Atiyah-Bott-Shapiro periodicity is defined on the Lorentz group. It is shown that modulo 2 and modulo 8 periodicities of the Clifford algebras allow to take a new look at the de Broglie-Jordan neutrino theory of light and the Gell-Mann-Ne'emann eightfold way in particle physics. On the representation spaces the charge conjugation C is represented by a pseudoautomorphism of the complex Clifford algebra. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of the massless physical fields. By way of example, neutrino field is described via the simplest quotient representation. Weyl-Hestenes equations for neutrino field are given.

math-ph↗
arXiv2001

Discrete Symmetries and Clifford Algebras

An algebraic description of basic discrete symmetries (space reversal P, time reversal T and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphisms groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dabrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah-Bott-Shapiro periodicity. Generalized Brauer-Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford-Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

math-ph↗
arXiv2000

Spinor Representations of Surfaces in 4-Dimensional Pseudo-Riemannian Manifolds

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.

math.DG↗
arXiv1999

Spinor Fields on the Surface of Revolution and their Integrable Deformations via the mKdV-Hierarchy

Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov- Novikov hierarchy in the case of the surface of revolution reduces to a well-known Zakharov-Shabat system. In the case of one-soliton solution an explicit form of the spinor fields is given by means of linear Bargmann potentials and is expressed via the Jost functions of the Zakharov-Shabat system. It is shown also that integrable deformations of the spinor fields on the surface of revolution are defined by a modified Korteweg-de Vries hierarchy.

math.DG↗
arXiv1998

Physical fields and Clifford algebras II. Neutrino field

The neutrino field is considered in the framework of a complex Clifford algebra $\C_3\cong\C_2\oplus\stackrel{\ast}{\C}_2$. The factor-algebras ${}^ε\C_2$ and ${}^ε\stackrel{\ast}{\C}_2$, which are obtained by means of homomorphic mappings $\C_3\to\C_2$ and $\C_3\to\stackrel{\ast}{\C}_2$, are identified with the neutrino and antineutrino fields, respectively. In this framework we have natural explanation for absence of right-handed neutrino and left-handed antineutrino.

hep-th↗
arXiv1997

Physical fields and Clifford algebras

The physical fields (electromagnetic and electron fields) considered in the framework of Clifford algebras $\C_2$ and $\C_4$. The electron field described by the algebra $\C_4$ which in spinor representation is realized by well-known Dirac $γ$-matrices, and by force of isomorphism $\C_{4}\cong\C_{2}\otimes \C_{2}$ is represented as a tensor product of two photon fields. By means of this introduced a system of electron field equations, which in particular cases is coincide with Dirac's and Maxwell's equations.

hep-th↗