Irreducible Cartesian Tensors. III. Clebsch-Gordan Reduction

Abstract

The reduction of products of irreducible Cartesian tensors is formulated generally by means of 3-j tensors. These are special cases of the invariant mappings discussed in Part II [J. A. R. Coope and R. F. Snider, J. Math. Phys. 11, 993 (1970)]. The 3-j formalism is first developed for a general group. Then, the 3-j tensors and spinors for the rotation group are discussed in detail, general formulas in terms of elementary invariant tensors being given. The 6-j and higher n-j symbols coincide with the familiar ones. Interrelations between Cartesian and spherical tensor methods are emphasized throughout.

Three-dimensional rotational averages in radiation-molecule interactions: an irreducible cartesian tensor formulation

“…'~~~ will only be non-vanishing when j * = O , and this can only occur when j=j". Similar reasoning shows that j ' = j"', and hence the substitution of equations (2.3)-(2.6) into (1.2), must give the result Equation (2.7) can be Le-expressed in a natural form [12,13] In order to develop (2.12) further, we note that by projecting the two invariant mappings given expression in (2.8) and (2.9) to their dual bases, through use of the metric tensor g,",", i.e. The invariant tensor mapping operator E:' k,,m, represents a mapping from the rank-j space to the natural rank-j, and hence weight-j, subspace.…”

“…Since the integrand in (1.5) must be rotationally invariant, it forms a basis for a totally symmetric irreducible representation of the rotation-inversion group SO(3). We can ascertain the exact form of this basis by expressing each tensor as a sum of its embedded irreducible tensor components [11][12][13]?.…”

Kinetic cross sections in the infinite order sudden approximation

“…Besides the n -j symbols, Q(kl1') is a normalization factor for 3 -j tensors. 29 In the ES approximation, the j dependence of the transition matrix is determined solely by a 3 -j symbol.…”

A Detailed Proof of the Fundamental Theorem of STF Multipole Expansion in Linearized Gravity

“…The demonstration, that the metric in (3) which depends on 10 multipolesF αβ L can be reduced to the form in (6) where the metric depends only on 2 multipoles (M L ,Ŝ L ), is a rather ambitious assignment of a task and makes extensive use of irreducible Cartesian tensor techniques originally introduced in [16][17][18]. Damour & Iyer (1991) [36] have also demonstrated that to order O (c −4 ) their post-Minkowskian multipoles coincide with the post-Newtonian multipoles of Blanchet & Damour (1989) [21] (Eqs.…”

“…However, the use of so-called Cartesian symmetric and tracefree (STF) multipole moments [12][13][14][15][16][17][18] instead of spherical harmonics simplifies considerably the calculations in gravitational physics [19][20][21][22]: the mathematical relations and expressions in gravitational theory become simpler, the numerical algorithms can be performed more efficiently, and the whole approach of gravitational theory becomes more elegant. By now, the STF multipole expansion, in post-Newtonian approximation ("weak-field slow-motion approximation", i.e.…”