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Spherical-Harmonic Tensors Jm Spherical-harmonic tensors Francisco Gonzalez Ledesma1 and Matthew Mewes2 1Department of Physics, Florida State University, Tallahassee, Florida 32306, USA 2Physics Department, California Polytechnic State University, San Luis Obispo, California 93407, USA The connection between spherical harmonics and symmetric tensors is explored. For each spher- ical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin- weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity. I. INTRODUCTION Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields. While they are commonly written in terms of the spherical-coordinate polar angle θ and azimuthal angle φ, spherical harmonics can be expressed in terms of cartesian coordinates, which is convenient in certain applications. The cartesian versions involve rank-j symmetric trace-free tensors jm [1–5]. These form a basis for traceless tensors and provide a link between functions on the sphere and symmetricY traceless tensors in three dimensions. This work builds on the above understanding in several ways. We first develop a new method for calculating the scalar spherical harmonics Yjm in terms of components of the direction unit vector n = sin θ cos φ ex + sin θ sin φ ey + cos θ ez . (1) The result can be used to write the spherical harmonics in terms of cartesian coordinates, spherical-coordinate angles, or any other coordinates. We then extract the traceless jm tensors and study their properties. These are extended to rank-̺ tensors ̺ with nonzero trace, which canY be used to perform both a trace and angular-momentum Yjm decomposition of an arbitrary tensor. The formalism is then generalized to spin-weighted spherical harmonics sYjm [6–8] and tensor-valued function spaces. Spherical harmonics are eigenfunctions of angular momentum J = S + L, with eigenvalues J 2 = j(j + 1) and Jz = m, where m is limited by m j. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization| | of ≤ the rotational properties and direction dependence of a system. For a scalar function f(n), the spin S is zero, and J is purely orbital angular momentum L, which accounts for the functional dependence on n. The spherical decomposition f(n)= jm fjmYjm(n) involves quantum numbers j, m associated 2 2 { } with the compatible operators J ,Jz = L ,Lz . Each term in the expansion represents just one example of a 2 { } { } P structure with definite J and Jz. In contrast to scalar functions, constant tensors are pure-spin objects with zero orbital angular momentum. Con- sider, for example, a constant traceless symmetric tensor T of rank ̺. In this case, the total spin and total angular momentum are both j = ̺. The symmetry of T implies a total of 2j + 1 independent components, matching the number of m values for fixed j. It can be expanded in spin-eigenbasis tensors , T = T . Each term in Yjm m mYjm this expansion has the same total angular momentum as the corresponding Yjm term in the expansion of the scalar f(n), but in the form of spin rather than orbital angular momentum.. P arXiv:2010.09433v1 [physics.class-ph] 13 Oct 2020 It is not surprising that a connection exists between spherical harmonics Y (n) and the basis tensors. In jm Yjm fact, the contraction of the jm tensor with the n vector j times yields a scalar function proportional to Yjm(n). This provides a link betweenY scalar functions and constant tensors, a relation that can be generalized to tensor-valued functions. Contracting jm with a single n vector gives a traceless symmetric rank-(j 1) tensor function of n. This decreases the spin byY one and increases the orbital angular momentum by one, while− leaving the total angular momentum unchanged. Subsequent contractions with n continue to convert spin angular momentum to orbital angular momentum until we arrive at the scalar spherical harmonics Yjm(n). Consequently, each jm generates a set of j +1 2 Y tensor-valued eigenfunctions of J and Jz with different ranks. For example, the tensor 30 generates four different abc abc Y abc angular-momentum eigenfunctions: the spin-3 constant ( 30) , the spin-2 ( 30) nc, the spin-1 ( 30) nbnc, and abc Y Y Y the scalar ( 30) nanbnc. This procedure yields a natural set of tensor spherical harmonics of different ranks and spins. The componentsY of these tensors in a special helicity basis [9] are the spin-weighted spherical harmonics up to a normalization factor. 2 This paper is organized as follows. The basic theory is given in Sec. II. Section II A establishes some notation and conventions. A new expression for scalar spherical harmonics Yjm in terms of the components of n is derived in Sec. II B. This expression is used in Sec. II C to construct the traceless rank-j spherical-harmonic tensors jm. Section IID extends the to rank-̺ tensors ̺ with nonzero trace. The connection between the ̺ and spin-weightedY Yjm Yjm Yjm spherical harmonics sYjm is derived in Sec. II E. Some simple illustrative examples are given in Sec. III. An application involving Lorentz-invariance violation in Newtonian gravity is discussed in Sec. IV. Spin weight and spin-weighted spherical harmonics are reviewed in Appendix A. Appendix B provides a brief overview of Young symmetrizers. II. CONSTRUCTION A. Notation and conventions This section establishes some basic notation used throughout this paper. First, Latin indices a, b, c, . on tensor components indicate spatial dimensions in one of the coordinate systems described below. Greek letters α, β, γ, . are used in Sec. IV to indicate spacetime indices. x y z Several different special sets of basis vectors are useful. In addition to the cartesian basis ex, ey, ez = e , e , e , z { } { } we define Jz-basis vectors e , e , ez = e↓, e↑, e , where { ↑ ↓ } { } 1 1 e = e↓ = ex + iey , e = e↑ = ex iey . (2) ↑ √2 ↓ √2 − The standard spherical-coordinate basis vectors are denoted as e , e , e = er, eθ, eφ , where { r θ φ} { } e = n , e = cos θ cos φ e + cos θ sin φ e sin θ e , e = sin φ e + cos φ e . (3) r θ x y − z φ − x y r + Finally, we define a helicity basis er, e+, e = e , e−, e , where { −} { } 1 e = e∓ = eθ ieφ . (4) ± √2 ± Note that raising and lowering indices in the Jz basis exchanges “up” and “down” labels, while raising and lowering b indices in the helicity basis exchanges “plus” and “minus” labels. All bases are defined to be orthonormal: ea e = b · δa. We denote the direction cosines between two vectors, not necessarily from the same basis, as gaa′ = ea ea′ , aa′ a a′ a′ a′ aa′ · g = e e , and g = e e . Note that these are the components of the euclidean metric g = g e e ′ relative · a a · a ⊗ a to row basis ea and column basis ea′ . In addition to defining the inner product, the metric components can be used to transform tensor components between bases, including the raising and lowering of indices. We denote the symmetrized tensor product using . The symmetrized product of two vectors v and u is defined 1 ⊙ k to be v u = 2 (v u + u v). The k-fold symmetrized product of a vector v will be written as v⊙ , which in index ⊙ ⊗k a1...ak⊗ 1 (a1 ak ) a1 ak notation reads (v⊙ ) = k! v ...v = v ...v . This is the simple k-fold tensor product. The product of k l a ...a 1 (a a a a ) products is written (v⊙ u⊙ ) 1 k+l = v 1 ...v k u k+1 ...u k+l . More generally, the product of a rank-k ⊙ (k+l)! a1...ak+l 1 (a1...ak ak+1...ak+l) tensor T and a rank-l tensor S is (T S) = (k+l)! T S . We write the k-fold symmetric product k ⊙ of a tensor T as T ⊙ . The inner product of two equal-rank tensors T and S is defined as the invariant contraction a1...ak q T S = T Sa1...ak . Finally, a tensor index written with an exponent, such as a , indicates q copies of the index · 2 3 a. For example, T x y z = T xxyyyz is a cartesian-basis component of a rank-6 tensor T . B. Scalar spherical harmonics In this section, we develop a new method for calculating the scalar spherical harmonics Yjm in terms of the components of n. The derivation favors the Jz basis, in which the components of n are given by 1 1 +iφ 1 1 iφ n = (nx + iny)= sin θe , n = (nx iny)= sin θe− , nz = cos θ . (5) ↑ √2 √2 ↓ √2 − √2 The result of the calculation that follows is q↑ q↓ qz m (2j+1)(j+m)!(j m)! n n nz n − ↑ ↓ Yjm( ) = ( sgn m) 4π 2|m| qb , (6) − ( 2) ↑↓ q !q !qz! q ↑ ↓ q X|jm − 3 where q jm is the restriction to all nonnegative powers q = q , q , qz that sum to j and obey q q = m, and | { ↑ ↓ } ↑ − ↓ q = min(q , q ). In practice, this can be accomplished by summing over qz = j m , j m 2, j m 4 ..
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