The CPT Group in the de Sitter Space

$P$-, $T$-, $C$-transformations of the Dirac field in the de Sitter space are studied in the framework of an automorphism set of Clifford algebras. Finite group structure of the discrete transformations is elucidated. It is shown that $CPT$ groups of the Dirac field in Minkowski and de Sitter spaces are isomorphic.Comment: 14 pages, LaTeX2

Universal Coverings of the Orthogonal Groups

Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion $P$, time reversal $T$, charge conjugation $C$ and their combinations $PT$, $CP$, $CT$, $CPT$) is given. Discrete subgroups $\{1, P, T, PT\}$ of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a complex conjugation pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group ($CPT$-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of $CPT$-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of $CPT$-structures which include well-known Shirokov-D\c{a}browski $PT$-structures as a particular case. Quotient Clifford-Lipschitz groups and quotient representations are introduced. It is shown that a complete classification of the quotient groups depends on the structure of various subgroups of the extended automorphism group.Comment: 87 pages, LaTeX2

Group Theoretical Description of Space Inversion, Time Reversal and Charge Conjugation

A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. 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 automorphism groups is established for the Clifford algebras over the fields of real and complex numbers. Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. The Atiayh-Bott-Shapiro periodicity is defined on the Lorentz group. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of massless physical fields. An algebraic construction of basic physical fields is presented.Comment: 100 pages, LaTeX2

Group Theoretical Interpretation of the CPT-theorem

An algebraic description of basic discrete symmetries (space inversion P, time reversal T, charge conjugation C and their combinations PT, CP, CT, CPT) is studied. Discrete subgroups {1,P,T,PT} of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group (CPT-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of CPT-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CPT-structures which include well-known Shirokov-Dabrowski PT-structures as a particular case.Comment: 45 pages, LaTeX2

Lorentz group and mass spectrum of elementary particles

Mass spectrum of localized states (elementary particles) of single quantum system is studied in the framework of Heisenberg's scheme. Localized states are understood as cyclic representations of a group of fundamental symmetry (Lorentz group) within a Gelfand-Neumark-Segal construction. It is shown that state masses of lepton (except the neutrino) and hadron sectors of matter spectrum are proportional to the rest mass of electron with an accuracy of $0,41\%$.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1608.0459

Towards the Quantum Electrodynamics on the Poincare Group

A general scheme of construction and analysis of physical fields on the various homogeneous spaces of the Poincar\'{e} group is presented. Different parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski spacetime and two-dimensional complex sphere is the most suitable homogeneous space for the subsequent physical applications. The Lagrangian formalism and field equations on the Poincar\'{e} group are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this problem are expressed via an expansion in hyperspherical harmonics on the complex two-sphere. A physical sense of the boundary conditions is discussed. The boundary value problems of the same type are studied for the Dirac and Maxwell fields. In turn, general solutions of these problems are expressed via convergent Fourier type series. Field operators, quantizations, causal commutators and vacuum expectation values of time ordered products of the field operators are defined for the Dirac and Maxwell fields, respectively. Interacting fields and inclusion of discrete symmetries into the framework of quantum electrodynamics on the Poincar\'{e} group are discussed.Comment: 69 pages, LaTeX2e, to appear in "Progress in Mathematical Physics" (Nova Science Publishers, New York

Hyperspherical Functions and Harmonic Analysis on the Lorentz Group

Matrix elements of spinor and principal series representations of the Lorentz group are studied in the basis of complex angular momentum (helicity basis). It is shown that matrix elements are expressed via hyperspherical functions (relativistic spherical functions). In essence, the hyperspherical functions present itself a four-dimensional (with respect to a pseudo-euclidean metrics of Minkowski spacetime) generalization of the usual three-dimensional spherical functions. An explicit form of the hyperspherical functions is given. The hypespherical functions of the spinor representations are represented by a product of generalized spherical functions and Jacobi functions. It is shown that zonal hyperspherical functions are expressed via the Appell functions. The associated hyperspherical functions are defined as the functions on a two-dimensional complex sphere. Integral representations, addition theorems, symmetry and recurrence relations for hyperspherical functions are given. In case of the principal and supplementary series representations of the Lorentz group, the matrix elements are expressed via the functions represented by a product of spherical and conical functions. The hyperspherical functions of the principal series representations allow one to apply methods of harmonic analysis on the Lorentz group. Different forms of expansions of square integrable functions on the Lorentz group are studied. By way of example, an expansion of the wave function, representing the Dirac field $(1/2,0)\oplus(0,1/2)$, is given.Comment: 51 pages, LaTeX2

Relativistic wavefunctions on the Poincare group

The Biedenharn type relativistic wavefunctions are considered on the group manifold of the Poincar\'{e} group. It is shown that the wavefunctions can be factorized on the group manifold into translation group and Lorentz group parts. A Lagrangian formalism and field equations for such factorizations are given. Parametrizations of the functions obtained are studied in terms of a ten-parameter set of the Poincar\'{e} group. An explicit construction of the wavefunction for the spin 1/2 is given. A relation of the proposed description with the quantum field theory and harmonic analysis on the Poincar\'{e} group is discussed.Comment: 11 page

General Solutions of Relativistic Wave Equations

General solutions of relativistic wave equations are studied in terms of the functions on the Lorentz group. A close relationship between hyperspherical functions and matrix elements of irreducible representations of the Lorentz group is established. A generalization of the Gel'fand-Yaglom formalism for higher-spin equations is given. It is shown that a two-dimensional complex sphere is associated with the each point of Minkowski spacetime. The separation of variables in a general relativistically invariant system is obtained via the hyperspherical functions defined on the surface of the two-dimensional complex sphere. In virtue of this, the wave functions are represented in the form of series on the hyperspherical functions. Such a description allows to consider all the physical fields on an equal footing. General solutions of the Dirac and Weyl equations, and also the Maxwell equations in the Majorana-Oppenheimer form, are given in terms of the functions on the Lorentz group.Comment: 47 pages, LaTeX2

Relativistic Spherical Functions on the Lorentz Group

Matrix elements of irreducible representations of the Lorentz group are calculated on the basis of complex angular momentum. It is shown that Laplace-Beltrami operators, defined in this basis, give rise to Fuchsian differential equations. An explicit form of the matrix elements of the Lorentz group has been found via the addition theorem for generalized spherical functions. Different expressions of the matrix elements are given in terms of hypergeometric functions both for finite-dimensional and unitary representations of the principal and supplementary series of the Lorentz group.Comment: 21 page