ALGEBRAIC THEORY OF SPHERICAL HARMONICS

Transcription

1 ALGEBRAIC THEORY OF SPHERICAL HARMONICS Nicholas Wheeler, Ree College Physics Department Febrary 996 Introction To think of the partial ifferential eqations of physics is to think of eqations sch as the following: heat eqation : ϕ = D t ϕ schröinger eqation : ϕ = id t ϕ + Wϕ wae eqation : ϕ = D t ϕ poisson eqation : ϕ = W laplace eqation : ϕ = () The soltion of sch eqations (sbject to specifie sie conitions) is a task which as their names alreay sggest has engage the attention of leaing mathematicians for centries By separation of ariables (when it can be effecte) sch partial ifferential eqations gie rise to orinary ifferential eqations of n orer soil from which sprang classical material associate with the names of Gaß, Legenre, Legerre, Hermite, Bessel an many others, the stff of higher analysis, which fon synthesis in Strm-Lioille theory an the theory of orthogonal polynomials The ierse ramifications of the mathematical problems ths pose hae proen to be irtally inexhastible An central to the whole story has been the ifferential operator, the theory latent in () Nor is it ifficlt to gain an intitie sense of the reason is encontere so biqitosly Let ϕ(x, y) be efine on a neighborhoo containing the point (x, y) on the Ecliian plane At points on the bonary of a isk centere at (x, y) the ale of ϕ is gien by ϕ(x + r cos θ, y + r sin θ) =e r cos θ x +r cos θ x ϕ(x, y) = ϕ + r ( ϕ x cos θ + ϕ y cos θ ) + r( ϕ xx cos θ +ϕ xy cos θ sin θ + ϕ yy sin θ ) + () Notes for a Ree College Math Seminar presente 7 March 996

2 Algebraic theory of spherical harmonics The aerage of the ales assme by ϕ on the bonary of the isk is gien therefore by ϕ = π { right sie of () rθ πr = ϕ ++ 4 r{ ϕ xx + ϕ yy + So we hae ϕ ϕ = 4 r ϕ (3) in leaing approximation Laplace s eqation asserts simply that the ϕ-fnction is relaxe : ϕ = ϕ = ϕ eerywhere (4) If the ϕ-fnction escriptie of a physical fiel is, on the other han, not relaxe, it is natral to set restoring force = k { ϕ ϕ = mass element acceleration k 4 r ϕ =πr ρ t ϕ an ths to recoer the wae eqation, with D =8πρ/k The operator is actally qite a rbst constrct Within the exterior calcls one has = +( ) p(n p+) while on Riemannian manifols one has the Laplace-Beltrami operator { g ij ϕ ;i;j () ϕ = g gg ij x i x i ϕ (5) proie ϕ transforms as a scalar ensity of weight W = In the more general case (as, for example, in qantm mechanics, where in orer to presere ψ ψ n x = the wae fnction mst transform as a ensity of weight W = ) the explicit escription of (W ) is ery messy, bt practical work is mch simplifie by the following little-known ientity: (W ) g + W = g + W () (6) We hae achiee here by aeraging oer the srface of a small eneloping -sphere (which is to say: a small isk centere at the fiel-point in qestion) a reslt which is more commonly achiee by aeraging oer nearest-neighboring lattice points In n-imensions the lattice argment gies ϕ ϕ = n r ϕ which is precisely the reslt which, as I hae (with labor!) shown elsewhere, is obtaine when one aerages oer the srface of an eneloping n-sphere See p 7 of Electroynamical applications of the exterior calcls (996) for the origin an meaning of the sign factor

3 Introction 3 Seeral analogs of the shift rle (6) will be encontere in sbseqent pages So mch by way of general orientation To sty ϕ = is in Ecliean n-space to sty { ϕ(x,x,,x n )= x x x n The case n = is in many ways exceptional, for this familiar reason: if the complex-ale fnction f(z) = (x, y) + i(x, y) is analytic, then by irte of the Cachy-Riemann conitions x = y an y = x the real-ale fnctions (x, y) an (x, y) are conjgate harmonic = = It is therefore ery easy to exhibit soltions of the Laplace eqation in two imensions (thogh not so easy to satisfy impose bonary conitons) When n 3 one has access, nfortnately, to no sch magic carpet, an the sitation is, in some respects, more crystaline For example, L P Eisenhart establishe in 934 that there exist precisely eleen coorinate systems (of a certain type) in which the 3-imensional Laplace eqation { ( x separates; they are the ) + + ϕ(x, y, z) = y z Rectanglar Circlar-cylinrical Elliptic-cylinrical Parabolic-cylinrical Spherical Prolate spheroial Oblate spheroial Parabolic Conical Ellipsoial Paraboloial coorinate systems, the etaile escriptions of which are spelle ot in the hanbooks; see, for example, Fiel Theory Hanbook by P Moon & D Spencer (Springer, 96)

4 4 Algebraic theory of spherical harmonics Central to the qantm mechanics of a particle moing in a prescribe force fiel is the time-inepenent Schröinger eqation, which has the form ψ(x, y, z) = [ W (x, y, z)+λ ] ψ(x, y, z) (7) The presence of the W-factor seres to estroy separability except in faorable special cases For example, if the force fiel is rotationally inariant W (x, y, z) =U(r) with r = x + y + z (7) then (7) oes separate in spherical coorinates Moreoer U(r) =kr + U(r) =kr permits separation also in rectanglar coorinates permits separation also in parabolic coorinates The former is familiar as the isotropic spring potential, an the latter as the Kepler potential These are, as it happens (accoring to Bertran s theorem 3 ), the only central potentials in which all (bone) orbits close pon themseles; the connection between oble separability an orbital closre is a eep one, bt it is a story for another ay Stanar analytical constrction of spherical harmonics My main objectie toay is to escribe a noel approach 4 to the spherical separation of (7) a noel approach to the theory of spherical harmonics an it is to nerscore the noelty (an the merit!) of the metho that I pase now to otline the stanar approach to the spherical separation problem One writes x = r sin θ cos φ y = r sin θ sin φ (8) z = r cos θ giing (s) =(x) +(y) +(z) =(r) + r (θ) + r sin θ (φ) an from (5) obtains { = r sin θ r r sin θ r + θ sin θ θ + φ csc θ φ One writes ψ(x, y, z) =Ψ(r, θ, φ) an assmes Ψ(r, θ, φ) =R(r) Y (θ, φ) (9) 3 For an excellent iscssion see Appenix A in the n eition of Golstein s Classical Mechanics (98) 4 Can a metho first escribe nearly seenty years ago fairly be sai to be noel? Little-known an non-stanar is perhaps a better escription

5 Analytic theory of spherical harmonics 5 to obtain { r { sin θ r r r [ U(r)+λ ] α r R(r) = () Y (θ, φ) = αy (θ, φ) ( ) θ sin θ θ + sin θ φ where α is a separation constant It is notable that the particlars of the problem, as written onto U(r), enter into the strctre of the raial eqation (), bt are completely absent from (), which looks only to what we might call the sphericity of the problem To complete the separation, one writes Y (θ, φ) =Θ(θ) Φ(φ) () an obtains { sin θ θ sin θ θ + α β sin θ { φ + β Θ(θ) = () Φ(φ) = () where β is again a separation constant From () an the reqirement that soltions be reglar on the whole sphere one is le easily to the orthonormal fnctions Φ m (φ) e imφ where m =, ±, ±, (3) π an to the conclsion that β = m Retrning with the latter information to () one confronts a more intricate problem A change of ariables θ ω cos θ (4) proces D(m )P (ω) = (5) where P (cos θ) =Θ(θ) an the operator D(m ) is efine { D(m ( ) ω ) ω ω + α m ω { ( = ω ) ω ω ω + α m ω (6) Remarkably (compare the shift rle (6)), D(m ) ( ω ) m m = ( ω ) m m D() (7) ω ω with this implication: if P (ω) is a soltion of D()P (ω) = then P m (ω) ( ω ) m m P (ω) is a soltion of D(m )P m (ω) = (8) ω

6 6 Algebraic theory of spherical harmonics So one sties { ( ω ) ω ω + α P (ω) = (9) Highly non-triial analysis leas to the conclsion that soltions reglar on the sphere exist if an only if α = l(l + ) with l =,,, () in which case the soltions are in fact the famos Legenre polynomials, which can be escribe P l (ω) = l ( ω l ) l () l! ω Ths, when all the st has settle, is one le to the fnctions Y l m l +(l m )! (θ, φ) 4π (l + m )! P l m (cos θ)e imφ () where l =,,, an m =, ±, ±,, ±l These spherical harmonics are orthonormal on the sphere π π ) (Y l m (θ, φ) Y l m (θ, φ) sin θθφ= δ m m δ l l an pt one in position to o Forier analysis on the sphere, jst as the fnctions (3) permit one to o Forier analysis on the circle This is a wonerfl accomplishment, of high practical importance in a great ariety of applications Bt the argment which le s to the constrction of the fnctions Y l m (θ, φ) is notable for its opaqe intricacy, an has left s eeply inebte to Legenre, who was clearly no sloch! Harmonic polynomials: Kramers constrction It was by straightforwar application of precisely sch classical analysis (an its relatiely less well known parabolic conterpart) that Schröinger, in his ery first qantm mechanical pblication (96), constrcte the qantm theory of the hyrogen atom Almost immeiately thereafter the Dtch physicist H A Kramers, rawing inspiration from Schröinger s accomplishment, sketche an alternatie approach to the theory of spherical harmonics which has, in my iew, mch to recommen it, bt which remains relatiely little known It was in the (misplace) hope of rectifying the latter circmstance that H C Brinkman (formerly a stent of Kramers ) pblishe in 956 the slim monograph (Applications of Spinor Inariants in Atomic Physics, North-Hollan) which has been my principal sorce The germinal iea resies in two assmptions First we assme ψ(x, y, z) to hae compare (9) the factore strctre ψ(x, y, z) =F l (r) (a r) l {{ manifestly rotation-inariant (3)

7 Harmonic polynomials: Kramers constrction 7 Introction of (3) into (7) leas straightforwarly to the eqation { (a r) l (l +) + r r r [ W (r)+λ ] F l (r)+f l (r) (a r) l = The statement (a r) l = (4) is now not force by the sal separation argment, so will simply be assme; the implicit companion of that assmption is the moifie raial eqation { (l +) + r r r [ W (r)+λ ] F l (r) = (4) It is to (4) that we henceforth confine or attention Immeiately (a r) l = l(l )(a r) l (a a) (5) so to achiee (4) we mst hae a a = The impliction is that the nll 3-ector a mst be complex: a = b + ic with b = c an b c = (6) Since a + a + a 3 = entails a 3 = i (a + ia )(a ia ) it becomes fairly natral to introce complex ariables a + ia a ia Then ( a = + ) ( a = i ) a 3 = i The a(, ) ths efine has the property that (7) a(, ) =a(, ) (8) We concle that as (, ) ranges oer complex -space a(, ) ranges twice oer the set of nll 3-ectors With the ai of (7) we obtain (a r) l = [ ( + ) ( x i ) y + iz ] l = [ ( x iy ) + iz+ ( x + iy )] l ( r ) l [ = sin θe iφ +i cos θ + sin θe +iφ] l

8 8 Algebraic theory of spherical harmonics which, proie l is an integer, we can notate ( r l m=+l ) m= l l m l+m Q m l (θ, φ) (9) Since (9) is, by constrction, harmonic for all an we hae which by () entails an { r l Q m l (θ, φ) = { r r r r α r r l = giing back again α = l(l +) { sin θ θ sin θ θ + sin θ + l(l +) Q m φ l (θ, φ) = Or assignment now is to constrct explicit escriptions of the fnctions Q m l (θ, φ); or expectation, of corse, is that we will fin Proceeing in Kramer s cleer footsteps, we write Q m l (θ, φ) Y m l (θ, φ) (3) m=+l m= l l m l+m Q m l (θ, φ) = [ sin θe iφ +i cos θ + sin θe +iφ] l A change of ariable = ( e iφ) l [ sin θ +i (/) e iφ cos θ +( / ) e iφ sin θ ] l = ( e iφ) l [( Z ) sin θ +Z cos θ ] l Z i (/) e iφ ( e iφ ) l [( = Z ) sin θ +Z sin θ cos θ ] l sin θ ( e iφ ) l [ = ( cos θ Z sin θ ) ] l sin θ ( e iφ ) l l { k [ = sin θ k! Zk ( cos θ Z sin θ ) ] l (3) Z k= Z= Z Ω(Z) cos θ Z sin θ

9 Harmonic polynomials: Kramers constrction 9 gies Z =( sin θ) Ω whence k k ( Z = sin θ k Ω) so { k [ ( cos θ Z sin θ ) ] l Z Z= = ( sin θ ) { k k [ Ω ] l Ω = ( sin θ ) k k [ ω ] l ω Ω=Ω() where (consistently with prior sage) Ω() = cos θ ω Retrning with this information to (3) we obtain l m= l ( l m l+m Q m e iφ l (θ, φ) = sin θ ( e iφ ) l l = sin θ = m= l ) l l k= ) k k [ Z sin θ ω k!( ] l ω l+m l+m [ Z sin θ ω ] l (l + m)! {{ ω =( i) l+m l+m l m e i(l+m)φ (sin θ) l+m l l m l+m ( i) l+m ( (l + m)! eimφ sin θ ) l+m m [ ω ] l ω {{ ) l+m ( ω ) l m= l =( ) l( ω ) m ( ω =( ) l l l!p m l (ω) giing Q m l (θ, φ) =( ) l ( i) l+m l l! (l + m)! eimφ P l m (cos θ) (3) precisely as anticipate at (3) Remarkably, we hae achiee (3) withot haing ha to sole any n -orer ifferential eqations, withot imposing any reglarity conitions (these were latent in or initial assmption), withot acqiring inebteness to Legenre It is instrctie to compare the preceeing line of argment with its -imensional conterpart Since a +a = entails a = ia it becomes natral to write a = (33) a = i Then (a r) m =[x + iy] m =[r cos φ + ir sin φ] m = r m [ m e imφ ] (34)

10 Algebraic theory of spherical harmonics where I hae force myself to procee in peantic imitation of the argment that le to (9) The consists now of bt a single term Were I to contine in my peantry, I wol write Q m (φ) =e imφ (35) an obsere that { r m Q m (φ) = thogh this is harly a srprise; r Q (φ) =x + iy z so r m Q m (φ) =z m, which is an analytic fnction, an therefore is assrely harmonic Look now to the manifestly rotation-inariant expressions C mn π (A r) m (a r) n φ (36) = r m+n U m n In the -representation rotation entails π Q m(φ) Q n (φ) φ (36) e iϑ (37) ner which U m n is inariant if an only if m = n The implication is that π Q m(φ) Q n (φ) φ = nless m = n The fnctions Q m (φ) are, in other wors, orthogonal It follows moreoer from the φ-inepenence of (A r) (a r) =(U x iu y)(x + iy) =r U that C mm = [ r U ] m π φ = [ r U ] m π so in fact we hae π { Q π if m = n m(φ) Q n (φ) φ = (38) otherwise The integral relations jst establishe are, of corse, triial implications of the efinitions (35) of the fnctions Q m (φ) Note, howeer, that in arriing at (38) we i not hae actally to integrate anything; the moe of argment was entirely algebraic, roote in the transformational aspects of the formalism at han Moreoer, the line of argment sketche aboe has (as will emerge) the property that it amits of natral generalization 3 Rotational ramifications At (7) we set p an association between complex nll 3-ectors a = b + ic an the points of a complex -space: a a to a 3

11 Rotational ramifications By comptation a a = a a + a a + a 3a 3 = ( + ) = {( ) t (39) The right sie of (39) will be inariant ner = U if an only if U is nitary: U t U = I Withot loss of generality one can write U = e iϑ S with S nimolar: S t S = I an et S = It follows from (7) that = e iθ inces a a = e iϑ a (4) To sty the action a a similarly ince by S we fin it most efficient to procee infinitesimally; we write S = I + δϕ L (4) an obsere that the nimolarity of S entails the traceless anti-hermiticity of L : ( ) iλ L = 3 λ + iλ (4) λ + iλ iλ 3 where withot loss of generality we assme et L = λ +λ +λ 3 = Introcing (4) into (4) an (4) into = S = + δ we obtain δ = δϕ L = δϕ iλ3 + (λ + iλ ) ( λ + iλ ) iλ 3 which by (7) inces a a = a + δa

12 Algebraic theory of spherical harmonics with δa = a a δ + δ = δ + i i δ +i i = δϕ iλ + iλ 3 ( ) λ 3 ( + ) iλ iλ ( ) λ ( + ) after simplifications = δϕ λ a 3 λ 3 a λ 3 a λ a 3 by appeal once again to (7) λ a λ a Ths o we obtain δ a a = δϕ λ 3 λ λ 3 λ a a a 3 λ λ a 3 {{ We hae now in han an association of the form generates rotations abot the λ-axis S(ϕ, λ) R(ϕ, λ) between the elements S(ϕ, λ) = exp of SU() an the elements { ( ) ϕ iλ 3 λ + iλ λ + iλ iλ 3 R(ϕ, λ) = exp ϕ λ 3 λ λ 3 λ λ λ (43) (44) of O(3) In R(ϕ +π, λ) =R(ϕ, λ) bt S(ϕ +π, λ) = S(ϕ, λ) (I omit the easy proof) we see the sorce of the biniqeness of the association

13 Rotational ramifications 3 Look now again to (9), which we may notate ( r ) l m=+l (a r) l = ξ m (l)q m (l) (45) m= l with Q m (l) Q m l (θ, φ) ξ m (l) l m l+m Explicitly 3 ξ() =, ξ( )=,ξ() =, ξ( 3 )=, ξ() = 3 The object ξ ξ( ), with coorinates if µ = ξ µ ξ µ ( )= if µ = lies in a -imensional complex ector space S calle spin space complex nmbers ξ µµ µr ξ µ ξ µ ξ µr, are the components of a -imensional spinor of rank r, which lies in the space S S S The object ξ(l) proies a colmnar isplay of the components of the objects that lie in the symmetrize proct space S r S S S; r =l an im S r = r +=l + Spinor algebra an spinor analysis, generally conceie, can be nerstoo to be the straightforwar complex generalizations of tensor algebra an tensor analysis One sties the transformations in proct spaces which are ince by transformations in the base space, an pays special attention to objects which are transformationally inariant The base space can, in general, be n-imensional, bt in the literatre is freqently nerstoo to be (as for s it presently is) -imensional Sch, then, is the general context within which we ask this relatiely narrow qestion: What can we say concerning the transformations within S r which are ince by nimolar transformations within S? By way of preparation for an attack on the problem, we note that (4) entails L = I, so from (43) it follows that ( cos S(ϕ, λ) = cos ϕ I + sin ϕ L = ϕ + iλ 3 sin ϕ (λ + iλ ) sin ϕ ) (λ iλ ) sin ϕ cos ϕ iλ 3 sin ϕ The can be escribe α β S = β α with α α + β β = (46)

14 4 Algebraic theory of spherical harmonics where α = α(ϕ, λ) cos ϕ + iλ 3 sin ϕ β = β(ϕ, λ) (λ + iλ ) sin ϕ (46) are the so-calle Cayley-Klein parameters Consier now the transformation ξ µ ξ µ = S µ ν(α, β) ξ ν in S (47) Explicitly which in S l entails = α+ β = β + α (48) ξ m (l) ξ m (l) =(α + β) l m ( β + α ) l+m = polynomial of egree l in the ariables {,, expressible therefore as follows: = n=+l n= l S m n(α, β; l) ξ n (l) S m n(l) ξ n (l) (49) an gies back (47) at l = Explicit escription of the (l +) (l +) matrix S m n(l) is straightforwar in principle, if teios in practice In the case l = one obtains, for example, S() = S m n() = α αβ β αβ (α α β β) βα (5) β α β α Actally, the case l = acqires special interest from the following crios circmstance: (7) can be notate so a = C ξ() with C = i +i i ξ() ξ() = S() ξ() a a = CS() C {{ a (5) Noting that C amits of the ecomposition C = D U R matrix of (44)

15 Rotational ramifications 5 where D U i +i i is iagonal an real, while is nitary we hae R = DUSU t D whence R T = R t = D US t U t D From R t R = I it therefore follows that S t U t DDU S = U t DDU; ie, that S S() is nitary with respect to an ince metric : S t GS= G where G U t DDU= C t C = (5) = ince metric matrix in S an that T CSC = DUSU t D is literally nitary: T t T = I The sitation jst encontere is (I assert withot proof) entirely general: S t (l) G(l) S(l) =G(l) (53) where the l + -imensional ince metric is real, iagonal, an symmetric abot the anti-iagonal: G(l) = G l G l G Gl G l (54) We obsere also that the matrix elements of S(l) are polynomials of egree l in {α, α,β,β,so S(l) + S(l) :l =,,, 3, {α, β { α, β inces (55) S(l) S(l) :l =, 3, 5, I am in position now to sketch the argment by which one might establish the orthonormality of the spherical harmonics Q m l (θφ) We look compare (36) to the expressions C l l π π (a r) l (b r) l sin θθφ (56)

16 6 Algebraic theory of spherical harmonics an note that these are on the one han manifestly rotation-inariant, bt (accoring to (45)) can on the other han be escribe ( r l C l l ) = +l m =+l m =+l ξ m m = l m = l π π (l ) η m (l ) Q m l (θ, φ)qm l (θ, φ) sin θθφ {{ (56) We arge that of necessity l = l (so we write l in place of both) an = constant G m m (l) Orthogonality then follows from the iagonality of G(l) To get a hanle on the ale of the mltiplicatie constant, we set a = b an obtain C ll = π π Withot loss of generality we set (a r) l (a r) l sin θθφ a = a i an obtain (a r) (a r) =a a (x iy)(x + iy) =a a (r z )=a a r cos θ, giing π C ll = r l (a a) l π cos l θ sin θθ {{ = l + Eiently l ξ t Gξ constant = (a a) l π l + (57) I am, howeer, in position to carry the argment to completion only in the case l =, where we hae ξ t Gξ = t = ( + ) = a a by (39) giing constant = 8π 3 in the case l =

17 Analytic theory of hyperspherical harmonics 7 Finally we note it to be an implication of reslts now in han that the spherical harmonics of gien orer l transform among themseles in sch a way as to establish a l + -imensional representation of the rotation grop O(3) In this sense: let the fnctions Q m (l) Q m l (θ, φ) : m = l, (l ),,,, +,,+(l ), +l relate in the familiar way to a cartesian frame in 3-space, an let Q m (l) Q m l ( θ, φ) : m = l, (l ),,,, +,,+(l ), +l relate in that same way to a rotate frame Looking to (45), we concle from the rotational inariance of the expression on the left that in association with the rotationally-ince contraariant transformation of the spinor components ξ m (l) is a coariant transformation of the fnctions Q m (l): ξ m (l) S(l) Q m (l) S (l) ξ m (l) Q m (l) (58) 4 Analytic theory of hyperspherical harmonics Interesting problems emerge when one looks as is from seeral points of iew qite natral to the N-imensional generalization of the preceeing material To write y = r sin φ x = r cos φ y = r sin θ sin φ x = r sin θ cos φ z = r cos θ y = r sin θ sin θ sin φ x = r sin θ sin θ cos φ z = r sin θ cos θ z = r cos θ is to see qite clearly the pattern of the neste constrction by means of which spherical coorinates {r, φ, θ,θ,,θ N are introce in the general case At eqatorial points (θ = π) on the 4-sphere we recoer the spherical coorinatization of 3-space, while at θ = θ = π we recoer the polar coorinatization of the plane Familiarly (s) -imensional =[(rcos φ)] +[(rsin φ)] =(r) + r (φ) (59)

18 8 Algebraic theory of spherical harmonics from which it follows that (s) 3-imensional =[(r cos θ)] + { [(r sin θ)] {{ +(r sin θ) (φ) =(r) + r (θ) =(r) +(r sin θ) (φ) + r (θ) (59) (s) 4-imensional =[(r cos θ )] + { [(r sin θ )] {{ +(r sin θ sin θ ) (φ) +(r sin θ ) (θ ) =(r) + r (θ ) =(r) +(r sin θ sin θ ) (φ) +(r sin θ ) (θ ) + r (θ ) (593) = r φ θ θ T (r sin θ sin θ ) (r sin θ ) r {{ = g ij (4) r φ θ θ (6) where g ij (4) is the Ecliian metric in hyperspherical coorinates Similarly (proceeing by what might be calle the metho of imensional ascent ) we hae (g ) g ij (5) = (g ) (g ) (g 3 ) where g r sin θ 3 sin θ sin θ g r sin θ 3 sin θ g r sin θ 3 g 3 r entails g(5) = g g g g 3 = r 4 sin 3 θ 3 sin θ sin θ Working from (5) we fin that the Laplacian in (for example) 5-space can be escribe 5 = r 4 r r4 r + r sin θ 3 sin θ sin θ φ + r sin θ 3 sin sin θ θ sin θ θ θ + r sin θ 3 sin sin θ θ θ θ + r sin 3 sin 3 θ 3 θ 3 θ 3 θ 3

19 Analytic theory of hyperspherical harmonics 9 = r 4 r r4 r + { r sin 3 sin 3 θ 3 θ 3 θ 3 θ 3 + [ sin θ 3 sin sin θ (6) θ θ θ + sin θ [ sin θ θ sin θ θ + sin θ [ φ ]]] A stanar line of argment leas from (6) to the conclsion that the flly-separate fnction F = R(r) Φ(φ) Z (θ ) Z (θ ) Z 3 (θ 3 ) will be harmonic ( 5 F = ) if an only if { r r r4 r α 4 R(r) = sin 3 θ 3 + α 4 sin θ 3 α 3 Z 3 (θ 3 )= θ 3 θ 3 { sin θ + α 3 sin θ α Z (θ )= θ θ { sin θ 3 { sin θ θ sin θ + α sin θ α θ { φ + α Z (θ ) = (6) Φ(φ) = where α, α, α 3 an α 4 are separation constants Eqialently we hae, in reerse orer (ie, in the orer in which the eqations are serially to be sole), { ω { ω { ω 3 ω ω ω 3 { φ + α Φ(φ) = (63) ω + α α ω ω P (ω ) = (63) 3ω + α 3 α ω ω P (ω ) = (63) 4ω 3 + α 4 α 3 ω 3 ω3 P 3 (ω 3 ) = (633) { r r r4 r α 4 R(r) = (633) where ω k cos θ k Soltions of (63) are of the form Φ(φ) e imφ an a reglarity conition entails (compare (3)) α = m with m =, ±, ±,

20 Algebraic theory of spherical harmonics Eqation (63) eqialently (6)/ sin θ is precisely the eqation which (in the iscssion sbseqent to (5)) was seen alreay to entail α = l(l + ) with l = m, m +,m+, an to gie rise to the Legenre fnctions Pl m (ω) ( ω ) l+m m ( ω ) l ω Eqation (63) assmes therefore the strctre { ( ω ) ω 3ω l(l +) + α ω ω P (ω) = (64) In preparation now for a chain of argment (63) (63) (633) we obsere that the following statement (which proies yet another instance of a shift rle ) ( ω ) { ( m D m ω ) D ( +k ) ωd + α { ( = ω ) D ( +k ) ωd + α m(m + k) ω ( ω ) m D m (65) (here D ω ) hols as an operator ientity (an gies back (7) in the case k = ), an has this implication: if G(ω) is a soltion of { ( ω ) D ( +k ) ωd + α G(ω) = (66) then F (ω) ( ω ) m D m G(ω) is a soltion of { ( ω ) D ( +k ) ωd + α It becomes eient on this basis that m(m + k) ω F (ω) = (66) F l (ω) ( ω ) l D l G(ω) (67) will be a soltion of (64) if G(ω) is a soltion of (66) with k = Bt eqations of the type (66) were stie in the 89 s by Gegenbaer, who fon that reglar soltions exist if an only if α = l(l ++k) with l =,,, (67) an are in sch cases gien by the Gegenbaer polynomials G l (ω; k), which are generate as follows [ ] +k = G l (ω; k)x l (673 xω + x n= G l (ω; k) ( ω ) k ( ω ) l ( ω ) l+ k {{ polynomial of egree l

21 Analytic theory of hyperspherical harmonics an rece to the Legenre polynomials at k = What I call the associate Gegenbaer fnctions are efine G m l (ω; k) ( ω ) m D m G l (ω; k) : m =,,,,l which gie back the associate Legenre fnctions at k = We are in position now to procee irectly to the following istillation of the implications of (63): { φ + m e ±imφ = { ω m ω G m l (ω ;)= { ( ω ω ) ω { ω 3 ω 3 + l (l +) ω ω 3ω + l (l +) l (l +) ω ω 4ω 3 + l 3 (l 3 +3) l (l +) ω 3 ω3 { r G l G l r r4 r l 3(l 3 +3) l (ω ;)= l 3 (ω 3 ;)= R(r) = (68) The general soltion of the last eqation is seen easily to hae the form R l3 (r) =Ar l3 + B r l3+3 (69) The orthogonality (an normalization) of the fnctions Yl m p l (θ l p,,θ,θ,φ) = normalization factor G lp l p (θ p ; p ) G l l (θ ;) G m l (θ ;)e imφ l p l p l l m p=n can be extracte from these two facts: ifferential srface area on the nit sphere in N-space is gien by Ω = sin p θ p sin θ sin θ θ θ θ p an the Gegenbaer polynomials hae this orthogonality property: { π if m n G m (cos θ; k)g n (cos θ; k) sin k+ θθ= complicate factor if m = n I mst, howeer, refer reaers to the stanar hanbooks for the etails The hyperspherical harmonics Yl m p l (θ l p,,θ,θ,φ) permit one to o Forier analysis on the srface of an N = p + -imensional hypersphere In a crios sense we hae labore harer an harer to o less an less, for the srface area of an N-sphere of nit rais is gien by S N =π N /Γ ( N ), which is maximal at N = 7 anishes in the limit N

22 Algebraic theory of spherical harmonics 5 Concling remarks & open qestions Sch then, in bal otline, is the inexhastibly rich (an intricate!) analytic theory of hyperspherical harmonics, which I hae reiewe here in orer to be in position to pose this qestion: Can Kramers techniqe be generalize in sch a way as to yiel an algebraic theory of hyperspherical harmonics? In (for example) 5-space (a r) l3 is clearly rotationally inariant, an harmonic if a is nll Can one, in imitation of (7), complex-parameterize the set of nll 5-ectors an ths, from an analog of (9), recoer the fnctional ata isplaye in (68)? On the eience only of my many failres, I hae come to the ery tentatie conclsion that Kramers techniqe oes not generalize; constrctions imitatie of (9) are impossible for N>3 What I wol like to see is either (i) a constrctie soltion of the problem, or (ii) a clear inication of why sch a constrction is impossible Retrning now to the 3-imensional context of or initial iscssion, we fon that the transformational theory, as it emerge from the algebraic line of argment, was nexpectely richer than the classical theory of spherical harmonics Looking specifically to (55), we note that while (a r) l is a polynomial for l =,,, (a r) l is not a polynomial if l =, 3, 5, In qantm mechanics the half-integral representations (the spin representations ) of O(3) o play an important role, an annonce their presence by the characteristic appearance of mlti-component wae fnctions, with nmber of components = l +=, 4, 6, There is, howeer, an alternatie which (thogh precle by the qantm mechanical reqirement that the wae fnction be single-ale) is aailable in principle to some applications For the fnctions Y l m (θ, φ) e imφ( ω ) l+m m ( ω ) l ω remain meaningfl een when l an m are (both) half-integral Sch fnctions the constrction of which (since l + m is integral-ale) oes not een entail the concept of fractional ifferentiation 5 wol appear to permit one to perform Forier analysis on the oble-sphere In qantm mechanics one has sometimes to istintingish 7 rotations (which are eqialent to the ientity) from 36 rotations (which aren t), an it was to illstrate this fact that Dirac inente his famos spinor spanner There hae been two principal actors in my story, as I hae tol it: algebra an analysis A thir actor grop representation theory wol hae heay 5 We toch here, interestingly, on yet another sbject of which Laplace was a foning father; see Chapter I of K S Miller & B Ross, An Introction to the Fractional Calcls an Fractional Differential Eqations (993)

23 Concling remarks & open qestions 3 contribtions to make in any more complete accont, an the potential for cross-talk seems inexhastible Concerning the grop-theoretic aspect of my topic I mst on the present occasion be content to recor only a few inciental remarks The sense in which the hyperspherical harmonics Yl m p l (θ l p,,θ,θ,φ) can be expecte to fol among themseles in sch a manner as to proie representations of O(N) is most transparently eient in the case N =, where the spherical harmonics are (we isregar all normalization factors) Y (φ) = an Y ±m (φ) =e ±imφ : m =,, which organize natrally into an array of this esign: Action of the elements of O() can be escribe Y (φ) Y ( φ) =Y (φ + α) which in real terms entails cos mφ cos m φ cos mα sin mα cos mφ sin mφ sin m φ = sin mα cos mα sin mφ {{ R(α; m) Eiently (certain real linear combinations of) the -imensional spherical harmonics of leaing non-triial orer m = transform in irect imitation of the elements of O() ( x y ) x = R(α) ỹ x y with R(α) R(α; ) while those of higher orer m>proie a poplation of alternatie matrix representations R(α; m) =R m (α) ofo() Trning from O() to O(3), the spherical harmonics organize into an array of the esign ascening l

24 4 Algebraic theory of spherical harmonics Here again, (certain real linear combinations of) the 3-imensional spherical harmonics of leaing non-triial orer l = transform in irect imitation = R of the geometrical action of the elements of O(3) x y x ỹ = R x y z z z while the spherical harmonics of higher orer l>proie O(3) representations of ascening o imension Interleae among those are the een-imensional spinor representations of ascening imension = S One expects the pattern of these remarks to be repeate in the N-imensional case, bt I look here only to the nmerological aspects of the sitation Enlarging pon prior sage, let s agree to call l l p the orer of the hyperspherical fnction Yl m p l (θ l p,,θ,θ,φ), an let #(l; N) nmber of Y -fnctions of orer l in the N-imensioinal case Familiarly #(l;)= { for l = for l =,, 3, an #(l;3)=l + for l =,,, 3, An from l l p l p l we obtain k=l #(l; N) = #(k; N ) k= which gies rise to the following self-explanatory table:

25 Notes & references 5 Dimension N marches own the secon colmn; it appears therefore plasible that a statement of the form (certain real linear combinations of) the N-imensional spherical harmonics of leaing non-triial orer l = transform in irection imitation of the geometrical action of the elements of O(N) will hol generally, not jst in the cases N = an N = 3 I hae marke the interstices which in the case N = 3 are occpie by the spin representations of O(3) The absence of sch simply-patterne interstices for N>3 lens seeming weight to my conclsion that Kramers constrction oes not generalize We notice, howeer, that the preceing isplay is strongly reminiscent of what might be calle the table of hypertranglar nmbers N =: N = : N = 3 : N = 4 : N = 5 : N = 6 : N = 7 : an that if one (i) obles eery entry an (ii) makes the replacement N N+ one oes obtain a plasible table of spinorial interstitials While I attach no great weight to reslts achiee by sch mere nmerology, I o note with sharpene interest that sch constrctions o in fact occr in acconts of the irrecible representations of O(N) 6 Two final obserations: Kramers metho brings to min some aspects of a line of argment e to Maxwell 7 An it is iily eocatie of an operator-algebraic qantm theory of anglar momentm which was eise (bt neer properly pblishe) by J Schwinger 8 It wol be amsing to work ot the etaile interconnections, an to iscoer more particlarly what Maxwell/Schwinger might hae to say if the worl were N-imensional Notes & references Laplace s eqation V (x, y, z) = appears for the first time in a paper of 789 concerne with the stability of the rings of Satrn a problem which later was to engage the attention also of Maxwell Henrik Anthony Kramers (894-95) was a stent of Pal Ehrenfest, an in 934 scceee Ehrenfest at the Uniersity of Leien Kramers is remembere by physicists toay mainly as the K in the WKB metho, 6 See, for example, Chapter V, 7 of H Weyl, The Classical Grops: Their Inariants an Representations (Secon eition, 946) 7 See, for example, Volme I, Chapter VII, 5 of R Corant & D Hilbert, Methos of Mathematical Physics (953) ormyownelectroynamics (97), pp On anglar momentm, pblication NYO 37(6 Janary 95) of the U S Atomic Energy Commission

26 6 Algebraic theory of spherical harmonics bt ring his short life he mae eep contribtions all notable for their mathematical sophistication to a wie assortment of topical areas The work reporte here was apparently base on Weyl s treatment of the rotation grop an on work then crrent on the theory of inariants The theory of spinors was roghly contemporaneos with the work of Kramers; an er Waeren s Spinoranalyse appeare in 99, an in 935 R Braer & H Weyl pblishe an accont of É Cartan s Spinors in n Dimensions (American Jornal of Mathematics, 57, 45) A goo moern sorce is Cartan s The Theory of Spinors (Doer, 98, translate from the French eition of 937) The Gegenbaer polynomials (also calle ltraspherical polynomials ) are relatiely late aitions to the poplation of special fnctions of mathematical physics They generalize the theory of Legenre polynomials, an hae ery close associations with the Lagerre, Hermite an Tschebyscheff polynomials All are special cases of Gaß hypergeometric fnction F (a, b; c; z) =+ ab c z! a(a +)b(b +) z + c(c +)! + which, interestingly, preates most of the classic theory of special fnctions (Gaß pblishe in 83) an erie their name by allsion not to geometry in hyperspace bt (ia Eler to a sage introce by Wallis in 655) to generalize geometric series My Gegenbaer notation is eccentric (intene to simplify expression of the reslts of most immeiate interest to me) I hae fon A Erélyi s Higher Transcenental Fnctions (Bateman Manscript Project, McGraw-Hill, 953), 35; the Appenix to the Forth Chapter of W Magns & F Oberhettinger s Formlas & Theorems for the Fnctions of Mathematical Physics (Chelsea, 954); an B C Carlson s Special Fnctions of Applie Mathematics (Acaemic Press, 977) to be particlarly helpfl An elaborate accont of the theory of Spherical & Hyperspherical Harmonic Polynomials can be fon in Erélyi s Chapter XI (which was reportely base on npblishe corse notes by G Herglotz); for an alternatie accont (withot attribtion) of Kramers metho, see 5 A wonerflly engaging accont in erse yet! of the theory of Dirac s spinor spanner can be fon on pp of L H Kaffman s On Knots (Annals of Mathematics Sties Nmber 5 (Princeton 987)) Kafman, writing ner the title Qaternions an the Belt Trick, makes explicit contact with the Pali matrices which come to light when (4) is written L = i { λ σ + λ σ + λ 3 σ 3 Also of interest in this connection is A Jrsi sić, The Mercees knot problem, (Amer Math Monthly 3, 756 (996)) Kramer s algebraic metho is selom encontere in textbooks A fairly etaile accont of the metho, an sefl references, can, howeer, be fon in 7 of J L Powell & B Crassmann, Qantm Mechanics (96) When I ha occasion (998) with Crassmann, he respone Oh, that was Powell s contribtion He ha learne of the metho when a stent of E P Wigner, an was always fon of it