Complex Rotation Group Research Articles

Majorana–Oppenheimer Approach to Maxwell Electrodynamics. Part II. Curved Riemannian Space

The Riemann–Silberstein–Majorana–Oppenheimer approach to the Maxwell electrodynamics in the presence of electrical sources, arbitrary media and curved space-time is investigated within the matrix formalism and tetrad method. Symmetries of the matrix Maxwell equation under transformations of the local gauge complex rotation group SO(3,C) is demonstrated explicitly. Equivalence of the approach to general covariant Proca technique and spinor formalism is shown.

Quantum spin coverings and statistics

SLq(2) at primitive odd roots of unity qℓ = 1 is studied as a quantum cover of the complex rotation group SO(3, ), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case ℓ = 3 is discussed in detail.

Simplifying Kaufman's solution of the two-dimensional Ising model.

We considerably simplify Kaufman's solution of the two-dimensional Ising model by introducing two commuting representations of the complex rotation group SO(2n,C). All eigenvalues of the transfer matrix and therefore the partition function are found in a straightforward way.

On space–time carrying a total hyperbolic skew symmetric Killing vector field

The complex vectorial formalism on a general space–time ( M, g) was constructed by Cahen, Debever and Defrise. This formalism is based on the local isomorphism I: L(4)→SO 3( C) , where L(4) is the four-dimensional Lorentz group acting on the tangent spaces T p M and SO 3( C) is the three-dimensional complex rotation group. In this framework, the congruence of Debever plays a distinguished role. Its properties determine the general space–time M, in terms of Petrov’s classification. In the present paper, we assume that any hyperbolic vector field X on M is a skew symmetric Killing vector field having a spatial vector field Y as generative. The existence of such a vector field X is determined by an exterior differential system in involution. It is shown that M is the local Riemannian product M= M h× M s, where M h (resp. M s) is a totally geodesic and totally pseudo-isotropic hyperbolic (resp. spatial) surface (the Gauss map is ametric). Any such M is a space–time of type D in Petrov’s classification. It is proved that the congruence of Debever is of electric type; in particular, it is geodesic and shear 1-free. Other geometric properties on such a general space–time are obtained.

Electromagnetism in anisotropic chiral media.

We present here for homogeneous anisotropic chiral media an electromagnetic formalism manifestly covariant under the complex rotation group O(3,C). The constitutive relations, the electromagnetic field equations, and the boundary conditions are discussed. To display the main features of this formalism, we discuss the propagation of plane waves in a medium that is anisotropic and chiral but simple enough to make calculations tractable

Manifestly covariant formalism for electromagnetism in chiral media.

A formalism of electromagnetism covariant under the complex rotation group is first described and then discussed thoroughly, in the case of two-dimensional electromagnetic fields, in homogeneous, isotropic chiral media. As a result, we obtain the generalized Descartes-Snell laws, the Fresnel relations for reflection and refraction, as well as the Brewster condition for plane waves incident on a surface of discontinuity between chiral and achiral media.

Euler angle parametrization of the complex rotation group and its subgroups. I

A parametrization of the n-dimensional complex rotation group O+(n, C) by a set of (1/2)n(n−1) complex Euler angles is obtained in exactly the same way as that for the n-dimensional real rotation group O+(n, R). Certain subgroups L̂(n, r) of O+(n, C) are then considered and found isomorphic with the ‘‘generalized Lorentz groups’’ L(n, r), of which the special case L(4, 1) is the usual Lorentz group L of special relativity. Distinguishing features, in the form of reality nature, of Euler angles of L̂(n, r) are obtained for r=1, 2, and it is proved that these lead naturally to the definition of ‘‘real’’ Euler angles of the corresponding L(n, r), which, of course, include the Lorentz group L.

Euler angle parametrization of the complex rotation group and its subgroups. II

The work of Paper I is extended here by obtaining the reality nature of Euler angles of the subgroups L̂(n,r) (defined there) of O+(n,C), for arbitrary r, and then showing that these again enable one to define ‘‘real’’ Euler angles of the generalized Lorentz groups L(n,r), r arbitrary.

Irreducible representations of the complex rotation group in terms of the axis and angles of rotations

The expression for matrix elements corresponding to finite complex rotations have been derived in terms of the complex Euler angles of the complex rotations. Using this expression the values of matrix elements for special cases of eigenvalues of complex angular momentum operator have been computed.

Unitarity and On-Mass-Shell Analyticity as a Basis for S-Matrix Theories. III

The singularity structure of many-particle scattering amplitudes in momentum transfer variables is investigated in terms of the many-particle unitarity conditions. Cauchy-type kernels for analytic functions defined on complex rotation group in three dimensions are constructed and related to a theory of local representations of the rotation group, corresponding to complex angular momenta.

Rotations and Lorentz transformations

Any complex three-dimensional rotation is determined by a complex vector and by a complex angle of rotation. New, short proofs are given of the homomorphisms between the three-dimensional complex rotation group, the group of unimodular quaternions (or unimodular 2 × 2 matrices) and the restricted Lorentz group. A correspondence is established between certain complex three-dimensional rotation vectors and two-dimensional subspaces of Lorentz vectors. The two-dimensional subspaces which are invariant under a given restricted Lorentz transformation are shown to be determined by those eigenvectors of the corresponding three-dimensional rotation matrix which belong to real eigenvalues. For non-null restricted Lorentz transformations this leads to a proof of Synge's theorem.

New isotopic spin space and classification of elementary particles

We start from the new idea that isotopic spin space is invariant under the group of three dimensional complex conjugate rotations and justify this assumption on the basis of a classical model of extended particles described by relativistic rotators. We then express in terms of Euler parameters (a relativistic generalization of real three dimensional Euler angles) the infinitesimal operators of the Lie algebra associated with this group and obtain, in the general space of the corresponding eigenfunctions, finite dimensional sub-spaces associated with irreducible representations of the complex rotation group. In each subspace one then readily defines a set of « isotopic » or « internal state vectors » corresponding to different types of elementary particles so that each particle family is connected with an irreducible representation of this group. In this scheme isotopic strangeness and fermionic numbers correspond to certain infinitesimal rotation operators and one justifies (with new qualitative aspects) the empirical Nishijima-Gell-Mann classification of elementary particles.