Maxwell-Sylvester Multipoles and the Geometric Theory of Irreducible Tensor Operators of Quantum Spin Systems
aa r X i v : . [ qu a n t - ph ] M a r Maxwell-Sylvester Multipoles and the Geometric Theory ofIrreducible Tensor Operators of Quantum Spin Systems
Patrick Bruno ∗ European Synchrotron Radiation Facility,BP 220, 38043 Grenoble Cedex, France
Abstract
A geometric theory of the irreducible tensor operators of quantum spin systems. It is basedupon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. Inthe latter, an order- ℓ multipolar potential is represented by a collection of ℓ equal length vectors,i.e. by ℓ points on a sphere, instead of by its components on some fixed (but arbitrary) basis.The geometric representation offers a much more appropriate tool for getting physical insight onspecific characteristics of a multipole, such as its symmetries, or its departure from ideal symmetry.We derive explicit expressions enabling to perform any calculations we may need to perform onmultipoles. All relevant quantities are eventually expressed in terms of scalar products of pairs ofvectors (i.e., in terms of geometric quantities such as lengths and angles). The whole formalismis entirely independent of any particular choice of coordinate, and needs no use of the somehowabstract formalism traditionally used when dealing with angular momenta.The formalism is then applied to treat the problem of the irreducible tensor operators of quan-tum spin systems. It enables to completely dispense with the calculation and use of the Stevensoperators, which can be quite complicated even for moderate values of ℓ . Explicit expressions forthe calculation of expectations values of physical observables are derived. They essentially consistin combinations of scalar products of vector pairs. Together with the coherent state representa-tion of the quantum states of spin systems, this provides a complete geometric, coordinate-free,description of the states, dynamics and physical properties of these systems. ∗ [email protected] . INTRODUCTION For many problems in physics and outside physics, a geometric representation of infor-mation is frequently more insightful than a numeric formulation of the same information, nomatter how precise, accurate, and otherwise useful the latter may be. To use an analogy, alist of the GPS coordinates of the main cities of a country would hardly provide any insighton its geography and on the opportunities and challenges it implies, whereas a map of thecountry displaying the position of those cities would instantaneously convey to the observera global understanding of the geography and its implications.Since the seminal work of Heisenberg, Born and Jordan [1–3], the algebraic formulation ofquantum mechanics has been the major theoretical tool of quantum theory. This algebraicapproach has indeed a considerable number of advantages such as its formal elegance andits convenience of use. In particular, due to the availability of very efficient algorithm forperforming numerical linear algebra calculations (solving linear equation systems, eigenvalueproblems, determinants, etc.) it has been extremely successful for the quantitative numericalstudy of quantum mechanical systems. On the other hand, a major weakness of the algebraicapproach lies in the fact that the physical interpretation of information encoded as complexcomponents of a Hilbert space vector, or as complex matrix elements of an operator is(except in the simplest cases) not immediate at all. For example, if the quantum state of asystem exhibits some fundamental symmetry property (such as rotational symmetry, point-group spacial symmetry, time-reversal invariance, etc.) that symmetry may be completelyobscured and would require a precise numerical check to be revealed.By contrast with the algebraic approach, a geometric formulation of physical laws fre-quently offers a unique and powerful insight. The most striking example of the power of thegeometric approach is Einstein’s reformulation and relativistic generalization of the theoryof gravitation, in the framework of a pseudo-riemannian geometry of space-time. A furtherexample is the modern geometric formulation of classical mechanics in the language of sym-plectic geometry on phase space [4]. Actually, it is quite natural that physical (or moregenerally scientific) questions be addressed in some geometric setting, since most frequentlyone has to address such as ”how far is the outcome of experiment A from that of experimentB?” of ”how far is the outcome of experiments from the prediction of theory?”, which nat-urally calls for the introduction some appropriate metric to quantify to observed ”farness”.2uch questions become most acute when precision is an central issue, such as in metrology.A good survey of the importance of modern geometrical concepts in physics is given in thebooks of Nakahara and Frankel [5, 6].In the field of quantum mechanics, geometrical concepts have played an increasinglyimportant rˆole, since Berry’s discovery of the geometrical phase [7–11]. In particular, thespin coherent states [28–30] and the Majorana stellar representation [33–35] for quantumspin systems enable a coordinate-free geometric representation of the pure quantum states ofsuch systems. A more detailed presentation of these geometric description of spin quantumstates is given in Sec. IV.The description of operators representing physical observable has remained, so far, almostexclusively based upon their expansion on some basis set, such as the tabulated Stevensoperators [45]. The aim of the present paper is to present an alternative approach, inwhich both the quantum states and the operators acting on them are described in terms ofgeometrical concepts, and in which calculations of physical quantities can be performed interms of geometrical quantities such as lengths, area, and angles.In Sec. II, a coordinate-free tensor formalism is presented. The one-to-one correspondencebetween symmetric tensors, homogenous polynomials, and psherical functions is stressed, aswell as the one-to-one correspondence between traceless symmetric tensors, harmonic poly-nomials, harmonic potentials, and spherical harmonics, is stressed. The canonical decompo-sition of symmetric tensors of arbitrary rank into their harmonic (or traceless, or irreducible)components is presented, and explicit formulas for this canonical decomposition are given.Various formulas are derived, enabling to perform calculations within the tensor formalism.Sec. III introduces the Maxwell-Sylvester geometric description of multipolar potentials.In particular, it physical meaning, in terms of the electrostatic interaction among multipoles,which had remained unnoticed so far, is highlighted.Sec. IV presents the geometric description of quantum spin states based upon the spincoherent states, and on the Majorana stellar representation.Sec. V is devoted to the geometric representation of quantum observable operators, interms of the Maxwell-Sylvester multipoles. Their use for actual calculations of physicalquantities is detailed.Finally, the Appendix details some notations and conventions used in the papers, as wellas formulae concerning Legendre polynomials, of which we make use throughout the paper.3
I. TENSORS, HOMOGENEOUS POLYNOMIALS, SPHERICAL FUNCTIONS,AND THEIR CANONICAL DECOMPOSITIONA. Symmetric Tensors and Homogeneous Polynomials
Let v i with i ∈ [1 , n ] be vectors in R . For short-hand notation, we introduce the n -tuple V ( n ) to describe the ordered collection of the v i , i.e., V ( n ) ≡ { v , v , . . . , v n } .A rank- n tensor T ( n ) is a (real or complex) function of V ( n ) which is linear with respectto each of the v i ’s, i.e., which satisfies T ( n ) ( λ v , . . . , λ n v n ) = λ . . . λ n T ( n ) ( v , . . . , v n ) . (1)A rank-1 tensor is of course just a vector. To make the link with the more familiar descriptionof tensors, which makes use of Cartesian coordinates, let us pick a direct orthonormal triadof (real) unit vectors (ˆ e , ˆ e , ˆ e ). The familiar Cartesian components of the tensor T ( n ) arethen given by T ( n ) µ ,...,µ n = T ( n ) ( e µ , . . . , e µ n ) (2)with µ i = 1 , , A ( n ) and B ( m ) , the tensor product A ( n ) ⊗ B ( m ) is the rank-( n + m ) tensordefined by (cid:0) A ( n ) ⊗ B ( m ) (cid:1) ( v , . . . , v n , v n +1 , . . . , v n + m ) ≡ A ( n ) ( v , . . . , v n ) B ( m ) ( v n +1 , . . . , v n + m ) . (3) (cid:0) A ( n ) ⊗ B ( m ) (cid:1) ( V ( n ) , W ( m ) ) ≡ A ( n ) ( V ( n ) ) B ( m ) ( W ( m ) ) . (4)Let S n be the symmetric group of index n ; for any permutation σ ∈ S n , we define σ V ( n ) ≡{ v σ (1) , v σ (2) , . . . , v σ ( n ) } . A tensor T ( n ) is said to be fully symmetric if for any permutation σ ∈ S n , it satisfies T ( n ) ( σ V ( n ) ) = T ( n ) ( V ( n ) ) . (5)To any tensor T ( n ) , we can associate a fully symmetric tensor Sym T ( n ) , defined bySym T ( n ) ( V ( n ) ) = 1 n ! X σ ∈ S n T ( n ) ( σ V ( n ) ) . (6)Let us introduce the fully symmetrized product A ( n ) J B ( m ) , defined by A ( n ) K B ( m ) ≡ Sym (cid:0) A ( n ) ⊗ B ( m ) (cid:1) , (7)4nd the tensor power (cid:0) A ( n ) (cid:1) ⊙ k ≡ A ( n ) ⊙ · · · ⊙ A ( n ) | {z } k factors , (8)which is a tensor of rank nk . A tensor of central importance is the rank-2 tensor δ (2) definedby δ (2) ( v , v ) = v · v . (9)Expressed in Cartesian coordinates, it is given by the Kronecker symbol. It is straightforwardto prove that it can also be expressed as δ (2) = 34 π Z S d ˆ n ˆ n ⊙ ˆ n , (10)where the integral runs over the 2-dimensional sphere of unit radius S . For any rank- n tensor A ( n ) , we define the rank-( n −
1) derived tensor A ( n ) [ v i ] obtained by fixing the i − thentry to v and leaving the n − n − p )derived tensor by fixing p entries of a rank- n tensor. The contraction over the pair of indices( i, j ) of two tensors A ( n ) and B ( m ) can thus be defined as A ( n ) : ( i,j ) B ( m ) ≡ π Z S d ˆ n A ( n ) [ˆ n i ] ⊗ B ( m ) [ˆ n j ] . (11)This is a rank-( n + m −
2) tensor. This process can be iterated to define the p -fold contractionover p pairs of indices. It is easy to get convinced that the contraction defined in this waycoincides with familiar definition in terms in terms of summation over Cartesian indices.Clearly, when dealing with fully symmetric tensors, the choice of the entries when definingderived tensors and contractions is unimportant, and we shall simplify the notations byomitting the indices i, j, . . . . So for the p -fold contraction of two symmetrical tensors willbe written A ( n ) : ( p ) B ( m ) ≡ (cid:18) π (cid:19) p p Y k =1 (cid:18)Z S d ˆ n k (cid:19) A ( n ) [ˆ n , . . . , ˆ n p ] ⊗ B ( m ) [ˆ n , . . . , ˆ n p ] . (12)It is a tensor of rank ( n + m − p ). The 1-fold contraction of two symmetric tensors is theinner product: A ( n ) · B ( m ) ≡ A ( n ) : (1) B ( m ) , (13)and one easily check that δ (2) is the unit for the inner product, i.e., A ( n ) · δ (2) = δ (2) · A ( n ) = A ( n ) . (14)5he notation of the full contraction of two symmetrical tensors (i.e., for p = min( n, m )),will be further simplified and written A ( n ) : B ( m ) , hich is a tensor of rank | n − m | . Fromthe definition of the contraction, one can easily observe that, given three symmetric tensors A ( n ) , B ( m ) , C ( n + m ) , one has the following property: (cid:0) A ( n ) ⊙ B ( m ) (cid:1) : C ( n + m ) = A ( n ) : D ( n ) (15)= B ( m ) : E ( m ) , (16)where D ( n ) ≡ B ( m ) : C ( n + m ) (17) E ( m ) ≡ A ( n ) : C ( n + m ) . (18)The ( i, j )-trace of a tensor A ( n ) is the rank-( n −
2) tensor defined asTr ( i,j ) A ( n ) ≡ π Z S d n A ( n ) [ˆ n i , ˆ n j ] . (19)For fully symmetrical tensors, we can omit to specify the indices ( i, j ), and simply write A ( n, ≡ Tr A ( n ) ≡ A ( n ) : δ (2) . (20)The p -fold trace is then A ( n,p ) ≡ A ( n,p − : δ (2) = A ( n ) : ( δ (2) ) ⊙ p . (21)It is a tensor of rank ( n − p ).A tensor is said to be traceless if it satisfiesTr ( i,j ) A ( n ) = 0 (22)for any pair of indices ( i, j ).Let us pick some cartesian coordinates ( x, y, z ) in R . A function f ( r ) on R is a homo-geneous polynomial of degree n of the 3 variables x, y, z if it can be expressed as f ( r ) = X p,q,s ≥ f p,q,r x p y q z s , (23)with p + q + s = n . For any tensor A ( n ) , we can define a function on R via A ( n ) ( r ) ≡ A ( n ) ( r ⊙ n ) = A ( n ) : ( r ⊙ n ) . (24)6t is easy to see that it is a homogeneous polynomial of degree n , and there is actually a one-to-one correspondence between rank- n fully symmetrical tensors and order- n homogeneouspolynomials [12]. An the correspondence is given by A ( n ) ( V ( n ) ) = 1 n ! n Y k =1 ( v k · ∂ r ) A ( n ) ( r ) . (25)From now on, unless explicitly specified, we shall deal only with fully symmetrical tensors,and thus shall identify them with their corresponding homogeneous polynomial. So withoutany ambiguity, we shall use the same symbol to represent the tensor and the polynomial.The context should make clear which one is actually meant.Let us now express the p -fold trace A ( n,p ) of a symmetric tensor in terms of the action ofthe Laplace operator ∆ r on A ( n ) . From the definition of the trace and the above equation,one has A ( n, ( V ( n − ) = 1 n ! n − Y k =1 ( ∂ r · ∂ r ) ( v k · ∂ r ) A ( n ) ( r ) (26)= 1( n − n − Y k =1 ( v k · ∂ r ) (cid:18) ( n − n ! ∆ r A ( n ) ( r ) (cid:19) , (27)so that A ( n, ( r ) = ( n − n ! ∆ r A ( n ) ( r ) . (28)By iteration, we finally get A ( n,p ) ( r ) = ( n − p )! n ! (∆ r ) p A ( n ) ( r ) . (29) B. Traceless Symmetric Tensors and Harmonic Polynomials
A function f ( r ) over R is said to be harmonic if it satisfies Laplace’s equation∆ r f ( r ) ≡ ∂ r · ∂ r f ( r ) = 0 . (30)For an order- n polynomial P ( n ) ( r ) associated to the symmetric tensor P ( n ) , one can easilyshow that ∆ r P ( n ) ( r ) = n ( n − P ( n, ( r ) , (31)so that a polynomial is harmonic iff its associated tensor is traceless. By iteration of theabove formula, we get (∆ r ) p P ( n ) ( r ) = n !( n − p )! P ( n,p ) ( r ) , (32)7armonic polynomials are called regular solid harmonics. By Kelvin’s transformation the-orem [13], to each regular solid harmonic H ( ℓ ) ( r ), we can associate another solution ofLaplace’s equation, the irregular solid harmonic defined by V ( ℓ ) ( r ) ≡ r H ( ℓ ) (cid:16) r r (cid:17) = 1 r ℓ +1 H ( ℓ ) ( r ) . (33)As we shall discuss later, V ( ℓ ) ( r ) can be thought of as the electrostatic potential of an electric2 ℓ -pole located at r = 0. The spherical harmonics are defined as Y ( ℓ ) ( r ) ≡ H ( ℓ ) ( r ) r ℓ = r ℓ +1 V ( ℓ ) ( r ) . (34)The order- ℓ spherical harmonics satisfy∆ r Y ( ℓ ) ( r ) + ℓ ( ℓ + 1) r Y ( ℓ ) ( r ) = 0 . (35)Writing H ( ℓ ) ( r ) = r ℓ Y ( ℓ ) ( r ) and using the fact that r · ∂ r Y ( ℓ ) ( r ) = 0, one can show easilythat the solid harmonics satisfy∆ r (cid:0) ( r · r ) n H ( ℓ ) ( r ) (cid:1) = 2 n (2 n + 2 ℓ + 1)( r · r ) n − H ( ℓ ) ( r ) . (36)By iteration, we get(∆ r ) p (cid:0) ( r · r ) n H ( ℓ ) ( r ) (cid:1) = (2 n )!!(2( n − p ))!! (2( n + ℓ ) + 1)!!(2( n + ℓ − p ) + 1)!! ( r · r ) n − p H ( ℓ ) ( r ) for p ≤ n, (37)= 0 for p > n. (38)In view of the one-to-one correspondence between traceless symmetric tensors, regular andirregular solid harmonics, and spherical harmonics, we shall use them interchangeably, de-pending on which is more appropriate in the given context.We define later use, we introduce the inner product of two spherical functions f (ˆ n ) and g (ˆ n ) h f | g i ≡ π Z S d ˆ n f ⋆ (ˆ n ) g (ˆ n ) . (39) C. Canonical decomposition of symmetric tensors
Given a symmetric rank- n tensor A ( n ) , we seek a decomposition into its harmonic (trace-less) components as: A ( n ) = ⌊ n/ ⌋ X k =0 A ( n )( n − k ) ⊙ ( δ (2) ) ⊙ k . (40)8n the above expression, the rank-( n − k ) tensor A ( n )( n − k ) is the rank-( n − k ) harmonic(traceless) component of A ( n ) . The existence and unicity of such decomposition will beproven below.Rewriting (40) as A ( n ) ( r ) = ⌊ n/ ⌋ X k =0 ( r · r ) k A ( n )( n − k ) ( r ) . (41)Comparing with the harmonic decomposition of a spherical function, Eqs. (135,139), we get A ( n )( n − k ) ( r ) = r n − k n − k ) + 14 π Z S d ˆ n A ( n ) (ˆ n ) P n − k (cid:16) ˆ n · r r (cid:17) . (42)We now want to derive an explicit expression of A ( n )( n − k ) in terms of the p -fold traces of A ( n ) .Using Eqs. (29,37), and taking the p -fold trace of Eq. (41), we get A ( n,p ) ( r ) = ⌊ n/ ⌋ X k = p (2 k )!!(2( k − p ))!! (2( n − k ) + 1)!!(2( n − k − p ) + 1)!! ( r · r ) k − p A ( n )( n − k ) ( r ) , (43)i.e., ( δ (2) ) ⊙ ( p ) ⊙ A ( n,p ) = ⌊ n/ ⌋ X k = p (2 k )!!(2( k − p ))!! (2( n − k ) + 1)!!(2( n − k − p ) + 1)!! ( δ (2) ) ⊙ ( k ) ⊙ A ( n )( n − k ) . (44)We have thus obtained a set of linear equation relating the p -fold traces to the harmoniccomponents A ( n )( n − k ) of the canonical decomposition. The ( ⌊ n/ ⌋ + 1) × ( ⌊ n/ ⌋ + 1) matrix ofthe linear system is an upper triangular matrix, with strictly positive diagonal elements; it istherefore invertible, which proves the existence and unicity of the canonical decomposition.To obtain the explicit expression of the harmonic components, one would need to invertmatrix of the linear system. Instead, I shall use here a simpler approach. Let us specializeto the case of of a symmetric tensor given by B ( n ) ≡ b ⊙ n . (45)The p -fold traces are given by B ( n,p ) = ( a · b ) p b ⊙ ( n − p ) . (46)We have B ( n ) ( r ) = ( a · r ) n (47)= ( br ) n ⌊ n/ ⌋ X k =0 q n,k P n − k (cid:18) b · r br (cid:19) , (48)9here we have used the expansion of the monomial in Legendre polynomials Eqs.(116,118).Thus, we have B ( n )( n − k ) ( r ) = q n,k ( b · b ) k ( br ) n − k P n − k (cid:18) b · r br (cid:19) . (49)It then simply remains to insert in the above equation the expression of the Legendre poly-nomials (111), and we finally get B ( n )( n − k ) = q n,k ⌊ n/ ⌋ X p = k p n − k,p − k ( δ (2) ) ⊙ ( p − k ) ⊙ B ( n,p ) , (50)which the announced explicit expression of the harmonic components of a symmetric tensorin terms of its traces. Although, we have used the specific case Eq. (45) to derive theabove result, it follows from the reasoning leading to Eq. (44) that it actually holds for anysymmetric tensor.Let us now use the above result, Eq. (50) to derive some results we shall need later. Letus consider the symmetric tensor A ( n ) given as a symmetrized tensor product of n vectors a i , i.e., A ( n ) ≡ n K i =1 a i . (51)We want to calculate the integral14 π Z S d ˆ n n Y i =1 ( a i · ˆ n ) = 14 π Z S d ˆ n A ( n ) (ˆ n ) . (52)Obviously, by symmetry, this is zero if n is odd, so we take n ≡ m . Using the canonicaldecomposition (50) and the fact that integral over the sphere of an order- ℓ spherical harmonicvanishes if ℓ = 0, we get 14 π Z S d ˆ n m Y i =1 ( a i · ˆ n ) = A (2 m )(0) (53)= q m,m p , A (2 m,m ) (54)So we finally get14 π Z S d ˆ n n Y i =1 ( a i · ˆ n ) = 1(2 m + 1) 1(2 m )! X σ ∈ S m m Y i =1 ( a σ ( i ) · a σ ( i + m ) ) . (55)There are actually (2 m − m vectors a i , so the above result can also be expressed as14 π Z S d ˆ n n Y i =1 ( a i · ˆ n ) = 1(2 m + 1) 1(2 m − X pairings Y pairs ( i,j ) ( a i · a j ) . (56)10s a byproduct, this yields the following useful formula14 π Z S d ˆ n ˆ n px ˆ n qy ˆ n rz = (2 p )! p ! (2 q )! q ! (2 r )! r ! ( p + q + r )!(2( p + q + r ) + 1)! . (57)Let us now consider two rank- n symmetric tensors A ( n ) and B ( n ) , and their respective order- n harmonic components A ( n )( n ) and B ( n )( n ) . We want to calculate D A ( n )( n ) B ( n )( n ) E S ≡ π Z S d ˆ n A ( n )( n ) (ˆ n ) B ( n )( n ) (ˆ n ) . (58)Using the Cartesian form of the tensors A ( n )( n ) and B ( n )( n ) , and using Einstein’s convention ofsummation over repeated indices, we have D A ( n )( n ) B ( n )( n ) E S ≡ A ( n )( n ) µ ,...,µ n B ( n )( n ) ν ,...,ν n π Z S d ˆ n (ˆ n µ . . . ˆ n µ n ) (ˆ n ν . . . ˆ n ν n ) . (59)We can calculate the integral using Eq. (56). We remark that, among the (2 n − µ indices are paired give a zerocontribution, because A ( n )( n ) and B ( n )( n ) are traceless. So we need to consider only the pairingsin which each µ index is paired with a ν index. There are n ! such pairings, and each overyields a contribution equal to A ( n )( n ) : B ( n )( n ) , so that we finally get D A ( n )( n ) B ( n )( n ) E S = n !(2 n + 1)!! A ( n )( n ) : B ( n )( n ) . (60)Using the canonical decomposition (41), we have D A ( n )( n ) B ( n )( n ) E S = (2 n + 1) (4 π ) Z S d ˆ n Z S d ˆ a Z S d ˆ b A ( n ) (ˆ a ) P n (ˆ a · ˆ n ) P n (ˆ n · ˆ b ) B ( n ) (ˆ b ) . (61)Using the reproducing kernel property of Legendre polynomials (134), this becomes D A ( n )( n ) B ( n )( n ) E S = 2 n + 1(4 π ) Z S d ˆ a Z S d ˆ b A ( n ) (ˆ a ) P n (ˆ a · ˆ b ) B ( n ) (ˆ b ) , (62)i.e., D A ( n )( n ) B ( n )( n ) E S = D A ( n )( n ) B ( n ) E S = D A ( n ) B ( n )( n ) E S . (63)On the other hand, using the canonical decomposition (40), we have A ( n ) : B ( n )( n ) = ⌊ n/ ⌋ X k =0 (cid:16) A ( n )( n − k ) ⊙ ( δ (2) ) ⊙ k (cid:17) : B ( n )( n ) . (64)Using the property (15) of the tensor contraction, we have (cid:16) A ( n )( n − k ) ⊙ ( δ (2) ) ⊙ k (cid:17) : B ( n )( n ) = A ( n )( n − k ) : (cid:16) ( δ (2) ) ⊙ k : B ( n )( n ) (cid:17) = 0 for k = 0 , (65)11ecause ( δ (2) ) ⊙ k : B ( n )( n ) is the k -fold trace of the harmonic (i.e., traceless) tensor B ( n )( n ) andtherefore vanishes. Thus we have A ( n )( n ) : B ( n )( n ) = A ( n ) : B ( n )( n ) = A ( n )( n ) : B ( n ) . (66)Inserting now the decomposition (50) of B ( n )( n ) into the above result, and using once more theproperty (15) of the tensor contraction, we get A ( n )( n ) : B ( n )( n ) = ⌊ n/ ⌋ X p =0 p n,p p n, A ( n,p ) : B ( n,p ) (67)= ⌊ n/ ⌋ X p =0 ( − p n ! (2 n − p − p p ! (2 n − n − p )! A ( n,p ) : B ( n,p ) . (68)We have thus been able to express the contraction of the order- n harmonic components oftwo rank- n tensors explicitly, in terms of the contractions of their traces. III. MAXWELL-SYLVESTER MULTIPOLESA. Heuristic Presentation of the Maxwell-Sylvester Multipoles
So far, we have discussed spherical harmonics using only their general properties withoutspecifying them. The traditional way of using them consists in expanding them on the basisof the familiar 2 ℓ + 1 complex Y mℓ or real Y ℓm spherical harmonics [21, 22]. In this approach,a order- ℓ spherical harmonic is then given by the 2 ℓ + 1 complex expansions coefficients.This cartesian-algebraic approach has the advantages and inconvenients discussed in theIntroduction.An alternative approach based upon coordinate-free geometric concepts has been intro-duced by Maxwell some 125 years ago [23]. Maxwell’s approach is most easily understood byconsidering the potential of electrostatic multipoles. As pointed out earlier (see Eq. (34)),each order- ℓ spherical harmonic is associated to the electrostatic potential of an order- ℓ mul-tipole (a 2 ℓ -pole). Maxwell’s construction can be physically understood in the following way:a dipolar electrostatic potential may be obtained by taking 2 opposite charges and shiftingthem by a small amount u . Similarly, a quadrupolar field may be obtained by taking 2opposite dipole moments and by shifting them by a small amount u . The process can beiterated further and a 2 ℓ -polar potential may be obtained by shifting 2 opposite 2 ℓ − -polesand shifting them by a small amount u ℓ . Without any restriction, the vectors u , u , ..., u ℓ q . So, Maxwell expresses an order- ℓ multipolar field as V ( ℓ ) U ( ℓ ) ( r ) = q ( − ℓ ℓ ! ℓ Y i =1 (ˆ u i · ∂ r ) 1 r , (69)where U ( ℓ ) ≡ { ˆ u , ˆ u , . . . , ˆ u ℓ } is an unordered collection (multiset) of (non-necessarily dis-tinct) unit vectors. That the above expression is actually a harmonic function follows im-mediately from the fact 1 /r is harmonic. The importance of Maxwell’s geometric multipolerepresentation stems from a theorem due to Sylvester [24] who proved that any 2 ℓ -polar po-tential (i.e., any real order- ℓ solid harmonic) can be expressed by Maxwell’s representation,which is actually unique (up to a change of sign of any pair of the unit vectors ˆ u i ). B. Maxwell-Sylvester Multipoles as a Tool for the Geometric Representation ofSpherical Harmonics
Let us now discuss the geometric interpretation of the Maxwell-Sylvester representation.It associates to each order- ℓ solid harmonic (or harmonic polynomial, or traceless symmetrictensor) with a multiset U ( ℓ ) ≡ { u , . . . , u ℓ } . From the latter, one can construct the symmetrictensor U ( ℓ ) ≡ ℓ K i =1 u i , (70)which may vie as a ”skeleton” of the associated harmonic tensor U ( ℓ )( ℓ ) . The latter can beconstructed by the method described in Sec. II C above. While the solid harmonic U ( ℓ ) ℓ (ˆ n )may be a quite complicated function of ˆ n , its skeleton U ( ℓ ) (ˆ n ) = ℓ Y i =1 ( u i · ˆ n ) (71)is very simple as it vanishes on ℓ grand circles, which are the equators corresponding to thepoles u i . So, if we represent the function sgn( U ( ℓ ) (ˆ n )) on a sphere, we obtain a very simplegeometric representation which entirely determines the solid harmonic, up to a the scalefactor |U ( ℓ ) | ≡ ℓ Y i =1 u i , (72)which we may represent geometrically as the radius of the sphere. We note that, in the caseof multiply degenerate equators, we would have to specify their multiplicity as well.13t this point, it may be useful to make the link with the familiar (real) spherical har-monics Y ℓ,m , and specify how the latter are described in terms of the Maxwell-Sylvesterrepresentation. The zonal spherical harmonics Y ℓ, have all their ℓ great circles degenerateand located on the ”geographic” equator. The sectorial spherical harmonics Y ℓ, ± ℓ have theirgreat circles located on ”meridian lines” with an angle separation of 2 π/ℓ (the great circlesfor m = − ℓ are shifted by an angle πℓ from those for m = − ℓ ). Finally, for a general valueof m , there are | m | great circle in meridian planes, separated by an angle 2 π/m , and ℓ − | m | degenerate great circles in the equatorial plane.The complex sectorial spherical harmonics H ± ℓℓ are particularly simple. They are given(up to the prefactor) by H ± ℓℓ = (ˆ x ± iˆ y ) J ℓ (73)and are actually equal to their skeleton, because(ˆ x ± iˆ y ) · (ˆ x ± iˆ y ) = 0 , (74)so that all the traces in the expansion (50). Similarly, any tensor of the form(ˆ a ± iˆ b ) J ℓ (75)with a = b and ˆ a · ˆ b = 0 is a sectorial harmonic of axis ˆ a × ˆ b .Since the solid harmonic has the same symmetries as its skeleton, this representation isparticularly well suited for analyzing its symmetries, unlike the traditional description of areal spherical harmonic in terms of its expansion over the basis of the (real) H ml . Conversely,constructing a spherical harmonic that possesses some particular symmetry is immediate andreduces to a geometric construction on the sphere, whereas except in some simple cases, theconstruction of spherical harmonics can be a tedious exercise, and the symmetry of theobtained result is frequently not manifestly apparent. Furthermore, the Maxwell-Sylvestergeometric representation offers a very simple and natural way of constructing solid harmonicsthat deviates slightly from a given symmetry, by moving slightly the multipole vectors awayfrom their symmetric positions. C. Electrostatic Interpretation of the Maxwell-Sylvester Multipole Vectors
There is a very interesting electrostatic analogy associated with the Maxwell-Sylvesterrepresentation. Let us consider a sphere with a surface charge distribution σ ( ℓ ) A (ˆ n ) given14y the spherical harmonic A ( ℓ )( ℓ ) . From Poisson’s equation, it follows that the electrostaticenergy is (up to some, for us unimportant, factor) E A = 12 ℓ ( ℓ + 1) 14 π Z S d ˆ n | σ ( ℓ ) A (ˆ n ) | (76)= 12 ℓ ( ℓ + 1) D A ( ℓ )( ℓ ) |A ( ℓ )( ℓ ) E . (77)so the electrostatic energy of a multipole is expressed by its norm. So, if we build thesuperposition of two such multipoles A ( ℓ )( ℓ ) and B ( ℓ )( ℓ ) , the energy of the resultant multipole is E A + B = 12 ℓ ( ℓ + 1) 14 π Z S d ˆ n | σ ( ℓ ) A (ˆ n ) + | σ ( ℓ ) B (ˆ n ) | (78)= 12 ℓ ( ℓ + 1) (cid:16)D A ( ℓ )( ℓ ) |A ( ℓ )( ℓ ) E + D B ( ℓ )( ℓ ) |B ( ℓ )( ℓ ) E + 2 D A ( ℓ )( ℓ ) |B ( ℓ )( ℓ ) E(cid:17) . (79)So, the mechanical work ∆ E A , B needed to superpose the two multipoles by bringing oneonto another from infinite distance (i.e., their mutual interaction energy) is expressed by theinner product of the two multipoles.Let us consider a real sectorial harmonic C ( ℓ )( ℓ ) = Re (cid:20)(cid:16) (ˆ a + iˆ b )e i φ (cid:17) ⊙ ℓ (cid:21) , (80)with a = b and ˆ a · ˆ b = 0. As discussed earlier, it is a sectorial multipole of axis ˆ a × ˆ b , withan azimuthal angle determined by the phase φ .Let us consider the interaction energy between a generic multipole U ( ℓ )( ℓ ) given as thetraceless component of a symmetrized tensor product U ( ℓ )( ℓ ) ≡ ℓ K i =1 u i (81)and the above sectorial multipole. Using the formulae Eq. (60,67) and because all the tracesof the sectorial multipole vanish, we have∆ E U , C = 12 ℓ ( ℓ + 1) ℓ !(2 ℓ + 1)!! Re " e i ℓφ ℓ Y i =1 ( u i · (ˆ a + iˆ b )) . (82)So, if the axis ˆ a × ˆ b of the sectorial multipole is parallel to any of the of the multipole vectors u i , independently of the azimuthal angle φ , the interaction between the multipole U ( ℓ )( ℓ ) andthe sectorial multipole C ( ℓ )( ℓ ) vanishes. Thus, from Sylvester’s theorem we see that for anyreal 2 ℓ -pole, there are exactly ℓ (possibly) degenerate sectorial multipoles axes which havevanishing interaction energy with U ( ℓ )( ℓ ) . This interpretation shows that the Maxwell-Sylvester15ultipole vector have a clear physical reality and indicates (at least conceptually) how theycan be determined.Surprisingly, more than 150 years after its discovery by Maxwell and Sylvester, the geo-metric representation of multipoles has remained widely ignore by the physics community,and has found very little application in physics. The main use of the Maxwell-Sylvestermultipoles, so far, has been for the symmetry classification of elasticity tensors [12, 25].Recently, it has found some renewed interest and been used to analyze and interpret theanisotropy of the cosmic microwave background radiation temperature [26, 27].However, it has so far found no use in the quantum theory of atoms, nuclei and molecules,and their interaction with electromagnetic radiation, in spite of the prominent role playedby the concept of multipoles in this context. To provide the basis for a geometric theory ofquantum multipoles constitutes the main objective of the present work. IV. GEOMETRIC REPRESENTATION OF QUANTUM SPIN STATES
Let us now consider a system consisting of a single quantum spin of magnitude J (whichmay be integer or half-integer). We are interested only in the degrees of freedom associatedwith the angular momentum J , so that the Hilbert space is H ≡ { C (2 J +1) − } . Thetraditional way of describing a (pure) quantum state of the systems consists in specifyingits (2 J + 1) complex components on some basis of the Hilbert space (most frequently, this isthe familiar basis of the eigenstates | J, m i of the J z operator). Operators describing physicalobservables, on the other hand, are represented by (2 J + 1) × (2 J + 1) Hermitian matricesacting on the above vectors states. While it is convenient for calculations, such ”cartesian”description of the quantum states usually conveys very little physical insight, except in thesimplest cases.On a more fundamental level, a further weakness lies in the fact that since two Hilbertspace vectors differing only by a (non-zero) complex prefactor lead to the same expectationvalue for any physical observable, they should be considered as actually representing thesame physical state of the system; this means that the Hilbert space H is not the actualspace of physical states of the quantum system; instead the phase space of physical (pure)quantum states is the projective Hilbert space PH = H / ∼ , the quotient set of equivalenceclasses in H , where | ψ i ∼ | ψ i iff | ψ i = c | ψ i (with c ∈ C , c = 0). For example, for16 Hilbert space of finite dimension (2 J + 1), the projective Hilbert space is PH = C P J ,the complex projective space of complex dimension 2 J . Because of this redundancy of theHilbert space, using it to formulate quantum mechanics does not allow to reveal in full depththe mathematical structure of quantum mechanics.It is therefore desirable to have a geometric description of the quantum states of thesystem, that would be independent of a particular choice of coordinates and basis states,and that would provide a much more natural way of investigating its symmetries, while stillenabling to perform conveniently all physically relevant calculations. A. Spin Coherent States
One such geometric description is that of the spin coherent states (cid:12)(cid:12) ˆ n ( J ) (cid:11) [28–30], whichare obtained by rotating the fully polarized state (cid:12)(cid:12) ˆ z ( J ) (cid:11) ≡ | J, J i from the ˆ z axis to the ˆ n axis, i.e., (cid:12)(cid:12) ˆ n ( J ) (cid:11) is defined (up to a norm and phase convention) by( J · ˆ n ) (cid:12)(cid:12) ˆ n ( J ) (cid:11) = J (cid:12)(cid:12) ˆ n ( J ) (cid:11) . (83)Their scalar product is given by (cid:10) ˆ n ( J ) | ˆ n ′ ( J ) (cid:11) = (cid:18) n · ˆ n ′ (cid:19) e i J Σ(ˆ z , ˆ n , ˆ n ′ ) , (84)where Σ(ˆ z , ˆ n , ˆ n ′ ) is the oriented area of the spherical triangle (ˆ z , ˆ n , ˆ n ′ ). The above expressionfixes the norm and phase convention. The spin coherent states satisfy the following resolutionof unity J ≡ J + 14 π Z S d ˆn | ˆn ( J ) ih ˆn ( J ) | , (85)which enables to represent a pure quantum state (cid:12)(cid:12) Ψ ( J ) (cid:11) by its projection on the coherentstates Ψ ( J ) ( ˆn ) ≡ h ˆn ( J ) | Ψ ( J ) i , which is a wave function over the sphere S . Actually, becauseof the redundancy of the continuous coherent states (by contrast with the projection onthe (2J+1) states of an orthonormal basis), the phase information is not needed and allobservables can be obtained from from the Husimi function Q Ψ ( J ) ( ˆn ) ≡ | Ψ ( J ) ( ˆn ) | , (86)which is a probability distribution on the sphere. Clearly, the coherent states representationhas the desired geometric character. In particular all the symmetry properties of the state17 (cid:12) Ψ ( J ) (cid:11) will be manifestly displayed as symmetries of its representation on the sphere via Q Ψ ( J ) ( ˆn ).The coherent states representation has found considerable use and is a central tool in thetheory of quantum spin systems [31, 32]. In particular it is at the heart of the path integraltheory for quantum spin systems. B. Majorana’s Stellar Representation
On the other hand, except for spin-1 / ( J ) ( ˆn ) has exactly 2 J (possibly degenerate)zeros, and that these zeros entirely determine the quantum state represented by | Ψ ( J ) i . Sothe state | Ψ ( J ) i can be represented geometrically by a ”constellation” of 2 J ”stars” (theantipodal points of the zeros of Ψ ( J ) ( ˆn )) on the sphere. The Majorana representation maybe thought of as the generalization for a spin J > / / / (cid:12)(cid:12) ˆ n (1 / (cid:11) .Since a spin- J system may be constructed as a fully symmetrized (i.e., bosonic) state of acomposite system made of 2 J spins-1 /
2, the Majorana representation of the spin- J systemsis the constellation of the 2 J Bloch vectors of its 2 J spin-1 / J integer, the time-reversal-invariant quantum states are given by real spherical harmonics,so that Majorana’s problem reduces to the Maxwell-Sylvester problem. In this case, theconstellation of the 2 J Majorana stars constitutes in a multiset of J pairs of antipodal stars,whose great circles (to which they are perpendicular) coincide with the Maxwell-Sylvesterrepresentation [36]. The Majorana representation has been used in particular to shed light onthe geometric phase and dynamics of quantum spins [37] and to discuss the ”anti-coherence”of symmetric multi-qubits systems [38, 39]. We shall not discuss it further here, and use18ssentially the coherent state representation. V. GEOMETRIC REPRESENTATION OF THE IRREDUCIBLE TENSOR OP-ERATORS
Besides the description of the quantum states themselves, we also need a geometric de-scription of the operators acting in the Hilbert space and representing physical observablequantities. The traditional approach to this relies on Stevens’ construction of irreducibletensor operators [41–45]. In this approach, physical observables (such as the Hamiltonian)are constructed by quantizing a classical function of the angular momentum: A cl ( j ). Thelatter can be expressed as a polynomial of degree n of the components j x , j y , j z of the classicalangular momentum, which can be expanded in harmonic homogeneous polynomials: A cl ( j ) = n X ℓ =0 A cl( ℓ ) ( j ) . (87)where A cl( ℓ ) is an order- ℓ solid harmonic, which is usually expanded on the basis of thefamiliar (real) solid harmonics H ℓ,m . The corresponding quantum operator is obtained bysubstituting the classical angular momentum by the quantum mechanical operator ˆ J , i.e.,ˆ A ( J ) = n X ℓ =0 ˆ A ( ℓ ) (ˆ J ) . (88)In the above expression, a symmetrization of all products of the angular momentum com-ponents is meant. As the various components of ˆ J do not commute, the symmetrizationensures that (i) the resulting operator is Hermitian as required for a physical observable,and (ii) that the degree of the polynomial cannot be lowered by use of the commutationrelations. Thus, one has to carry out the task of performing explicitly this symmetrizationfor the solid harmonics H ℓ,m , in order to generate the Stevens operators. This is fairly easyfor ℓ ≤ ℓ (see e.g. [45]).The geometric approach relies on a description in terms of the coherent states, instead ofa fixed basis as in the ”cartesian” approach. To the operator ˆ A ( J ) we associate the sphericalfunction a (ˆ n ) ≡ h J, ˆ n | ˆ A ( J ) | J, ˆ n i , (89)which is called the Q-representation of the operator ˆ A ( J ) [46]. There is also a (non-unique)19-representation A (ˆ n ) defined byˆ A ( J ) ≡ (2 J + 1)4 π Z S d ˆ n | J, ˆ n i A (ˆ n ) h J, ˆ n | . (90)The non-unicity of the P-representation can be fixed by requiring Z S d ˆ n A (ˆ n ) P ℓ (ˆ n · ˆ u ) = 0 , ∀ ˆ u ∈ S , if ℓ > J. (91)Unlike the representation of the observable in terms of its matrix elements in a fixed basis,in which all matrix elements (on-diagonal and off-diagonal) are needed to fully specify theobservable, in the coherent state representation, due to the redundancy of the coherentstates, the observable is fully specified by its diagonal elements only.let us expand a (ˆ n ), and A (ˆ n ) into their harmonic components: a (ˆ n ) = X ℓ a ℓ (ˆ n ) (92) A (ˆ n ) = X ℓ A ℓ (ˆ n ) , (93)where a ℓ (ˆ n ) = 2 ℓ + 14 π Z S d ˆ u a (ˆ u ) P ℓ (ˆ u · ˆ n ) (94) A ℓ (ˆ n ) = 2 ℓ + 14 π Z S d ˆ u A (ˆ u ) P ℓ (ˆ u · ˆ n ) . (95)The problem is now to related the a ℓ (ˆ n ) and A ℓ (ˆ n ) to the harmonic components of theclassical potential A cl ℓ (ˆ n ) of the classical potential. From considerations of their rotationalsymmetry behavior, it follows that must must be proportional to each other, i.e., a ℓ (ˆ n ) = α J,ℓ A cl ℓ (ˆ n ) (96) A ℓ (ˆ n ) = β J,ℓ A cl ℓ (ˆ n ) , (97)where the coefficients αJ, ℓ and βJ, ℓ are given by [46]. α J,ℓ = (2 J )!2 ℓ (2 J − ℓ )! for ℓ ≤ J, (98) β J,ℓ = (2 J + ℓ + 1)!2 ℓ (2 J + 1)! for ℓ ≤ J, (99) α J,ℓ = β J,ℓ = 0 for ℓ > J. (100)If we now use the Maxwell-Sylvester geometric representation described in Sec. III B A ℓ (ˆ n )(or equivalently, for a ℓ (ˆ n ), or A cl ℓ (ˆ n )), if we represent them in terms of the skeleton on thesphere, we now describe any operator by a collection of geometrical patterns consisting ofsets of great circles on spheres, that contain all the information we may need concerning ouroperator. At the same time, this representation displays directly all the symmetries of theoperator. 20sing this, we can now express the expectation value of the operator for a quantum state (cid:12)(cid:12) Ψ ( J ) (cid:11) (which we take normalized to 1) (cid:10) Ψ ( J ) (cid:12)(cid:12) ˆ A ( J ) (cid:12)(cid:12) Ψ ( J ) (cid:11) = (2 J + 1)4 π Z S d ˆ n (cid:10) Ψ ( J ) | J, ˆ n i A (ˆ n ) h J, ˆ n (cid:12)(cid:12) Ψ ( J ) (cid:11) (101)= (2 J + 1)4 π Z S d ˆ n Q ( J )Ψ ( ˆn ) A (ˆ n ) (102)= J X ℓ =0 (2 J + 1) 14 π Z S d ˆ n Q Ψ ( J ) ( ℓ ) ( ˆn ) A ( ℓ ) (ˆ n ) . (103)We now use the Maxwell-Sylvester representation for both Q Ψ ( J ) ( ℓ ) and A ( ℓ ) and can applythe formulae (60,67). The skeletons of A ( ℓ ) and Q ( J )Ψ( ℓ ) are, respectively A ( ℓ ) ≡ ℓ K i =1 a ( ℓ ) i (104) Q ( ℓ )Ψ ( J ) ≡ ℓ K i =1 q ( ℓ ) i . (105)The final expression for the expectation value of the operator is (cid:10) Ψ ( J ) (cid:12)(cid:12) ˆ A ( J ) (cid:12)(cid:12) Ψ ( J ) (cid:11) = (2 J + 1) J X ℓ =0 ℓ !(2 ℓ + 1)!! ⌊ ℓ/ ⌋ X p =0 p ℓ,p p ℓ, Q ( ℓ,p )Ψ ( J ) : A ( ℓ,p ) , (106)where the traces are given by A ( ℓ,p ) = 1 ℓ ! X σ ∈ S ℓ p Y i =1 ( a ( ℓ ) σ ( i ) · a ( ℓ ) σ ( i + p ) ) ! ℓ K j =2 p +1 a ( ℓ ) σ ( j ) ! , (107)with analogous expression for Q ( ℓ,p )Ψ ( J ) . The contraction of the traces is Q ( ℓ,p )Ψ ( J ) : A ( ℓ,p ) = 1( ℓ !) X σ,σ ′ ∈ S ℓ p Y i =1 ( q ( ℓ ) σ ( i ) · q ( ℓ ) σ ( i + p ) ) ! p Y j =1 ( a ( ℓ ) σ ′ ( j ) · a ( ℓ ) σ ′ ( j + p ) ) ! × (108) × ℓ Y k =2 p +1 ( q ( ℓ ) σ ( k ) · a ( ℓ ) σ ′ ( k ) ) ! , (109)which completes the announced result. APPENDIXA. Notations and conventions
Vectors are noted using boldface l.c. letters ( a , b , etc.). Unit vectors, i.e., vectors be-longing to the 2-sphere of unit radius S , will be indicated by a ”hat” ˆ n . Capital boldfaceletters are used for ordered collections of vectors, e.g. V ( n ) ≡ { v , v , . . . , v n } . Caligraphic21apitals A ( n ) are used for tensors, the rank of which is indicated by the upper index ( n ). Weuse the symbol ⊗ to represent the tensor product of two tensors, e.g. A ( n ) ⊗ B ( m ) , and thesymbol ⊙ to represent the fully symmetrized tensor product, as defined explicitly in Eq. (7). B. Legendre Polynomials
As Legendre polynomials play an important role in this paper, some of their properties arereminded here [14]. The Legendre polynomial of order ℓ can be defined by Rodrigues’formula[15] P ℓ ( x ) ≡ ℓ d ℓ d x ℓ (cid:0) x − (cid:1) ℓ . (110)Their explicit expression is [16] P ℓ ( x ) ≡ ⌊ ℓ/ ⌋ X k =0 p ℓ,k x ℓ − k , (111)where the floor function is defined by ⌊ x ⌋ ≡ n, for n ≤ x < n + 1 , ( n ∈ Z ) (112)and where p ℓ,k ≡ ( − k (2 ℓ − k − k k !( ℓ − k )! . (113)Legendre polynomial or even (odd) order contain only monomials of even (odd) order. Theyare orthogonal to each other [17]:12 Z − d x P ℓ ( x ) P ℓ ′ ( x ) = δ ℓℓ ′ ℓ + 1 . (114)From the latter relation, we also obtain the completeness relation δ ( x − y ) = ∞ X ℓ =0 (cid:18) ℓ + 12 (cid:19) P ℓ ( x ) P ℓ ( y ) . (115)We shall also need the reciprocal of Eq. (111), the expression of monomials x n in termsof Legendre polynomials. Using Eqs.(114,115) and taking into account the parity of theLegendre polynomials, we obtain x n = ⌊ n/ ⌋ X k =0 q n,k P n − k ( x ) , (116)22ith q n,k ≡ n − k ) + 12 Z − d x x n P n − k ( x ) . (117)The calculating explicitly the integral [19], we finally obtain q n,k = (2 n − k + 1) n !2 k k ! (2 n − k + 1)!! . (118)Note that p n, = 1 q n, = (2 n − n ! . (119)We shall also need the expansion of (1 + x ) n in Legendre polynomials. Proceeding likeabove, and using the formula [20] Z − dx (1 + x ) n P ℓ ( x ) = 2 n +1 ( n !) ( n + ℓ + 1)! ( n − ℓ )! , (120)we obtain (1 + x ) n = 2 n ( n !) n X ℓ =0 l + 1( n + ℓ + 1)! ( n − ℓ )! P ℓ ( x ) . (121)Using the generating function [18] 1 √ − tx + t = ∞ X ℓ =0 t ℓ P ℓ ( x ) , (122)one can show easily that P ℓ (cid:18) r · r ′ rr ′ (cid:19) = ( − ℓ ℓ ! r ′ ℓ +1 r ℓ (cid:18) r · ∂∂ r ′ (cid:19) ℓ (cid:18) r ′ (cid:19) (123)= ( − ℓ ℓ ! r ℓ +1 r ′ ℓ (cid:18) r ′ · ∂∂ r (cid:19) ℓ (cid:18) r (cid:19) , (124)where r and r ′ are vectors in R . Since the electrostatic potential of a point charge satisfiesLaplace’s equation (except at the charge’s location), i.e,∆ r (cid:18) r (cid:19) = 0 , (125)it follows immediately that the function Y ℓ ( r , r ′ ) ≡ P ℓ (cid:18) r · r ′ rr ′ (cid:19) (126)is a spherical harmonic of order ℓ of r and r ′ , and H ℓ ( r , r ′ ) ≡ r ℓ r ′ ℓ Y ℓ ( r , r ′ ) (127)23s a solid harmonic of order ℓ of r and r ′ , i.e.,∆ r Y ℓ ( r , r ′ ) + ℓ ( ℓ + 1) r Y ℓ ( r , r ′ ) = ∆ r ′ Y ℓ ( r , r ′ ) + ℓ ( ℓ + 1) r ′ Y ℓ ( r , r ′ ) = 0 (128)∆ r H ℓ ( r , r ′ ) = ∆ r ′ H ℓ ( r , r ′ ) = 0 . (129)Let us consider the function f ℓ,ℓ ′ (ˆ u , ˆ v ) of (ˆ u , ˆ v ) ∈ S defined by f ℓ,ℓ ′ (ˆ u , ˆ v ) ≡ Z S d ˆ n P ℓ (ˆ u · ˆ n ) P ℓ ′ (ˆ n · ˆ v ) . (130)It is obviously invariant by rotation, i.e., for any rotation R ∈ SO (3), we must have f ℓ,ℓ ′ ( R (ˆ u ) , R (ˆ v )) = f ℓ,ℓ ′ ( u , v ) . (131)In particular, if we chose the rotation which permutes ˆ u and ˆ v , we have f ℓ,ℓ ′ (ˆ v , ˆ u ) = f ℓ,ℓ ′ (ˆ u , ˆ v ) = f ℓ ′ ,ℓ (ˆ u , ˆ v ) . (132)By construction f ℓ,ℓ ′ (ˆ u , ˆ v ) is a spherical harmonic of order ℓ with respect to its first argumentand a spherical harmonic of order ℓ ′ with respect to its second argument. Thus, we have ℓ ( ℓ + 1) f ℓ,ℓ ′ (ˆ u , ˆ v ) = ℓ ′ ( ℓ ′ + 1) f ℓ,ℓ ′ (ˆ u , ˆ v ) , (133)which implies that f ℓ,ℓ ′ (ˆ u , ˆ v ) = 0 if ℓ = ℓ ′ . We have thus established the orthogonality ofspherical harmonics of different orders, and in particular14 π Z S d ˆ n P ℓ (ˆ u · ˆ n ) P ℓ ′ (ˆ n · ˆ v ) = δ ℓℓ ′ ℓ + 1 P ℓ (ˆ u · ˆ v ) (134)For any function f (ˆ n ) defined on S , where S is the unit-radius sphere in R , the function f ℓ (ˆ n ) ≡ ℓ + 14 π Z S d ˆ u f (ˆ u ) P ℓ (ˆ u · ˆ n ) (135)is obviously a spherical harmonic of order ℓ . Thus, the above convolution of a sphericalfunction with the order- ℓ Legendre polynomial enables to extract its order- ℓ harmonic com-ponent. Thus, a function Y ℓ (ˆ n ) is a spherical harmonic iff it satisfies Y ℓ (ˆ n ) ≡ ℓ + 14 π Z S d ˆ u Y l (ˆ u ) P ℓ (ˆ u · ˆ n ) . (136)In particular, we have Y ℓ (ˆ n ) ≡ ℓ + 14 π Z S d ˆ u Y l (ˆ u ) P ℓ (ˆ u · ˆ n ) . (137)24o, the Legendre polynomial P ℓ (ˆ u · ˆ n ) is the reproducing kernel of the space of order- ℓ spherical harmonics. From the completeness relation (115), we obtain δ (ˆ u − ˆ v ) = 14 π ∞ X ℓ =0 (2 ℓ + 1) P ℓ (ˆ u · ˆ v ) . (138)So any function on S may be decomposed in a unique way into its harmonic components f (ˆ n ) = ∞ X ℓ =0 f ℓ (ˆ n ) . (139) [1] W. Heisenberg, Z. Phys. , 879–893 (1925).[2] M. Born and P. Jordan, Z. Phys., , 858–888 (1925).[3] M. Born, W. Heisenberg, and P. Jordan, Z. Phys., , 557-615 (1925)[4] V.I. Arnold, Mathematical Methods of Classical Mechanics (2nd ed.), Springer-Verlag (1989).[5] M. Nakahara,
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