Total Angular Momentum Waves for Scalar, Vector, and Tensor Fields

TL;DR

This work constructs complete total-angular-momentum (TAM) wave bases for scalar, vector, and tensor fields in three-dimensional space, providing fixed-$JM$ solutions to the Helmholtz equation and enabling cosmological calculations on the celestial sphere. It develops three equivalent TAM bases for vectors and five for traceless rank-2 tensors (OAM, L/E/B, and helicity), demonstrates how to obtain the TAM waves from scalar TAM waves via derivative operators, and establishes explicit projections onto vector and tensor spherical harmonics. The formalism yields practical tools for calculating observable quantities, such as lensing deflection-angle power spectra from density perturbations and gravitational waves, and holds promise for streamlined Boltzmann treatments of CMB fluctuations. By connecting 3D TAM waves to 2D sky observables, the paper lays the groundwork for applying TAM methods to higher-order correlations and broader fields in cosmology and field theory.

Abstract

Most calculations in cosmological perturbation theorydecompose those perturbations into plane waves (Fourier modes). However, for some calculations, particularly those involving observations performed on a spherical sky, a decomposition into waves of fixed total angular momentum (TAM) may be more appropriate. Here we introduce TAM waves, solutions of fixed total angular momentum to the Helmholtz equation, for three-dimensional scalar, vector, and tensor fields. The vector TAM waves of given total angular momentum can be decomposed further into a set of three basis functions of fixed orbital angular momentum (OAM), a set of fixed helicity, or a basis consisting of a longitudinal (L) and two transverse (E and B) TAM waves. The symmetric traceless rank-2 tensor TAM waves can be similarly decomposed into a basis of fixed OAM or fixed helicity, or a basis that consists of a longitudinal (L), two vector (VE and VB, of opposite parity), and two tensor (TE and TB, of opposite parity) waves. We show how all of the vector and tensor TAM waves can be obtained by applying derivative operators to scalar TAM waves. This operator approach then allows one to decompose a vector field into three covariant scalar fields for the L, E, and B components and symmetric-traceless-tensor fields into five covariant scalar fields for the L, VE, VB, TE, and TB components. We provide projections of the vector and tensor TAM waves onto vector and tensor spherical harmonics. We provide calculational detail to facilitate the assimilation of this formalism into cosmological calculations. As an example, we calculate the power spectra of the deflection angle for gravitational lensing by density perturbations and by gravitational waves. We comment on an alternative approach to CMB fluctuations based on TAM waves. Our work may have applications elsewhere in field theory and in general relativity.