The concept of vector spherical harmonics is generalized for symmetric and traceless Cartesian tensor fields of arbitrary rank. Differential relations of these functions are derived as generalizations of the gradient formula for scalar, and the divergence and curl formulas for vector spherical harmonics.
The irreducible representations of the rotation group by vector fields, known as vector spherical harmonics [1,2], form an appropriate tool for angular-momentum classification of space-dependent vector quantities, such as electromagnetic fields [3] or velocity fields in fluid mechanics [4]. Their extension to more general tensor fields is straightforward, as long as one contents oneself with the usual coupling of orbital angular momentum to an internal angular momentum represented by a standard set of spin eigenfunctions.
An elaborate instrument for analyzing fields of higher spins, which does not imply a coupled angular momentum, has been developed as spin-weighted spherical harmonics [5,6]. On the other hand, to our knowledge, no general explicit calculus exists for integer-number representations of a definite total angular momentum, where the internal rotational behaviour is characterized by Cartesian tensors. Special tensor spherical harmonic functions of rank 2 have been constructed in the framework of gravitation theory [7]. A more general formalism might be of use, for instance, in kinetic theories, where the Cartesian representation of higher moments of the local momentum distribution functions, generalizations of the velocity-field, seems to be more convenient that the spherical one (e.g., see Reference [8]).
In this letter we propose an extension of the concept of vector spherical harmonics to Cartesian tensor fields of arbitrary rank, which, of necessity, must be symmetrical and traceless. In particular, we derive elementary relations, which express the coupling of the coordinate and gradient vector to tensor spherical harmonics by these same functions, thus allowing apphcation of the calculus to expansions in differential equations, such as kinetic equations for spherically-symmetric situations.
Starting from Cartesian unit-vectors ea, with a = 1,2, 3 for x, y, and z direction, and in the s-fold tensor-product space generated by tiffs set, we define hnear combinations of the tensor basis that form a 2s + 1-dimensional irreducible representation of the rotation group* csms = csms e( O ~),