Showing papers on "Ricci decomposition published in 1971"

A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold

Abstract: A new scalar‐tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well‐known Brans‐Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.

The Dimensions of Irreducible Tensor Representations of the Orthogonal and Symplectic Groups

Abstract: It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ 1, σ 2, …, σp ) with σ 1 + σ 2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.

Classification of the Ricci tensor

Abstract: This paper contains a classification of the Ricci tensorRαβ. The method of derivation is analogous to the spinor version of the Petrov classification of the Weyl tensor. It is shown how the various classes are related to the number and type of eigenvectors and eigenvalues ofRαβ. The classification is useful in the geometrization of various fields. The case of a real scalar field is treated in detail.

Simple Procedure for Determining the Number of Components of an Irreducible Tensor

Abstract: We show that the number of linearly independent components of a tensor in n dimensions with specified symmetry properties is given by a polynomial in n This polynomial can be determined in a simple way from the Young diagram associated with the tensor

Poincaré‐Irreducible Tensor Operators for Positive‐Mass One‐Particle States. II

Abstract: The main objective of this article is the relativistic generalization of the ordinary SO(3)‐irreducible spin tensor operators for particles with positive mass. Two classes of relativistic one‐particle tensor operators are constructed. The tensor operators of the first class transform according to those representations of the Poincare group that are induced by the one‐valued unitary irreducible representations of the pseudo‐unitary group SU(1, 1) which belong to the continuous principal and the discrete principal series. These tensors are operator‐valued functions of a spacelike 4‐momentum transfer. The tensor operators of the second class correspond to vanishing 4‐momentum transfer. They transform according to those representations of the Poincare group that are induced by the unitary irreducible representations of the pseudo‐orthogonal group SO(3, 1) or its universal covering group SL(2C) which belong to the principal series. Both classes of Poincare‐irreducible tensor operators are constructed in a spin helicity basis for timelike 4‐momentum by means of projection operators which are continuous linear superpositions of unitary operator realizations for the groups SU(1, 1) and SL(2C). The Clebsch‐Gordan coefficients associated with the reduction into the two classes of Poincare‐irreducible tensor operators of a dyadic product of spin‐helicity basis vectors are calculated.

Determination of operator equivalents from irreducible tensors

Abstract: A new method of derivation of operator equivalents is proposed, in which individual irreducible tensor components are expressed in terms of both real and complex unit tensors derived from real and complex (or bi) vectors. The method is extended to irreducible tensor operators which do not have full permutation symmetry and it is found that these are equivalent to totally symmetric tensor operators of rank equal to the angular momentum quantum number.