irreducible representations of SO(2) and SO(3)

Chapter 8 Irreducible Representations of SO(2) and SO(3) The shortest path between two truths in the real domain passes through the complex domain. |Jacques Hadamard1 Some of the most useful aspects of group theory for applications tophysical problems stem from the orthogonality relations of charactersof irreducible representations. The widespread impact of these relationsstems from their role in constructing and resolving new representationsfrom direct products of irreducible representations. Direct products areespecially important in applications involving continuous groups, withthe construction of higher dimensional irreducible representations, thederivation of angular momentum coupling rules, and the characteriza-tion of families of elementary particles all relying on the formation anddecomposition of direct products. Although the notion of an irreducible representation can be carried over directly from our development of discrete groups through Schur’sflrst lemma, a transcription of Schur’s second lemma and the GreatOrthogonality Theorem to the language of continuous groups requiresa separate discussion. This is because proving the latter two theorems 1Quoted in The Mathematical Intelligencer 13(1), 1991. 125 126 Irreducible Representations of SO(2) and SO(3) necessitates performing summations over group elements and invoking the Rearrangement Theorem (Theorem 2.1). This theorem guaranteesthe following equality X gf(g)=X gf(g0g); (8.1) where the summation is over elements gin a groupG,g0is any other el- ement inG, andfis some function of the group elements. The crucial point is that the same quantities appear on both sides of the equa-tion; the only difierence is the order of their appearance. To proceedwith the proofs of these theorems for continuous groups requires anequality analogous to (8.1): Z f(R)dR=Z f(R0R)dR; (8.2) whereRandR0are the elements of a continuous group and fis some function of these elements. To appreciate the issues involved, we writethe integral on the left-hand side of (8.2) as an integral over the pa-rameters Z f(R)dR=Z f(R)g(R)da; (8.3) whereg(R) is the density of group elements in parameter space in the neighborhood of R. The equality in (8.2) will hold provided that the density of group elements is arranged so that the density of the pointsR 0Ris the same as that of the points R. Our task is to flnd the form of g(R) which ensures this. A related concept that will arise is the notion of the \order" of the continuous group as the volume of its elements inthe space deflned by the parameters of the group. This chapter is devoted to the characters and irreducible represen- tations of SO(2) and SO(3). For SO(2), we will show that the density ofgroup elements is uniform across parameter space, so the density func-tion reduces to a constant. But, for SO(3), we will need to carry outthe determination of the density function in (8.3) explicitly. This willillustrate the general procedure which is applicable to any group. Forboth SO(2) and SO(3), we will derive the basis functions for their irre-ducible representations which will be used to obtain the correspondingcharacters and to demonstrate their orthogonality Irreducible Representations of SO(2) and SO(3) 127 8.1 Orthogonality of Characters for SO(2) The structure of SO(2) is simple enough that many of the results ob- tained for discrete groups can be taken over directly with little or nomodiflcation. The basis of this claim is that the Rearrangement Theo-rem for this group is, apart from the replacement of the sum by an in-tegral, a direct transcription of that for discrete groups which, togetherwith this group being Abelian, renders the calculation of characters astraightforward exercise. 8.1.1 The Rearrangement Theorem We flrst show that the rearrangement theorem for this group is Z2… 0R(’0)R(’)d’=Z2… 0R(’)d’: This implies that the weight function appearing in (8.3) is unity, i.e., the density of group elements is uniform in the space of the parameter’. Using the fact that R(’ 0)R(’)=R(’0+’), we have Z2… 0R(’0)R(’)d’=Z2… 0R(’0+’)d’: We now introduce a new integration variable µ=’0+’. Since’0is flxed, we have that d ’=dµ. Then, making the appropriate changes in the upper and lower limits of integration, and using the fact thatR(’+2…)=R(’), yields Z2… 0R(’0+’)d’=Z’0+2… ’0R(µ)dµ =Z2… ’0R(µ)dµ+Z’0+2… 2…R(µ)dµ =Z2… ’0R(µ)dµ+Z’0 0R(µ)dµ =Z2… 0R(µ)dµ; which verifles our assertion. 128 Irreducible Representations of SO(2) and SO(3) 8.1.2 Characters of Irreducible Representations We can now use Schur’s flrst lemma for SO(2). Since SO(2) is an Abelian group, this flrst lemma requires all of the irreducible represen-tations to be one-dimensional (cf. Problem 4, Problem Set 5). Thus,every element is in a class by itself and the characters must satisfy thesame multiplication rules as the elements of the group: ´(’)´(’ 0)=´(’+’0): (8.4) The character corresponding to the unit element, ´(0), which must map onto the identity for ordinary multiplication, is clearly unity forall irreducible representations: ´( 0 )=1: (8.5) Finally, we require the irreducible representations to be single-valued, i.e., an increase in the rotation angle by 2 …does not change the efiect of the rotation. Thus, ´(’+2…)=´(’): (8.6) The three conditions in (8.4), (8.5), and (8.6) are su–cient to determine the characters of all of the irreducible representations of SO(2). We will proceed by writing Eq. (8.4) as a difierential equation and using (8.5) as an \initial condition" and (8.6) as a \boundary condi-tion." In (8.4), we set ’ 0=d’, ´(’)´(d’)=´(’+d’); and expand both sides of this equation to flrst order in d ’: ´(’)" ´(0) +d´ d’flflflfl ’=0d’# =´(’)+d´ d’d’: Then, using (8.5) and cancelling common terms, this equation reduces to a flrst-order ordinary difierential equation for ´(’): d´ d’=´0 0´(’); Irreducible Representations of SO(2) and SO(3) 129 where´0 0=´0(0) is to be determined. The general solution to this equation is ´(’)=Ae´0 0’; whereAis a constant of integration which is also to be determined. In fact, by setting ’= 0 and invoking (8.5), we see that A= 1. The requirement (8.6) of single-valuedness, when applied to this solution,yields the condition that e ´0 0(’+2…)=e´0 0’; or, e2…´0 0=1: The most general solution of this equation is ´0 0=im, wherei2=¡1 andmis any integer. This produces an inflnite sequence of characters of the irreducible representations of SO(2): ´(m)(’)=eim’;m =:::;¡2;¡1;0;1;2;::: (8.7) The identical representation corresponds to m= 0. In contrast to the case of flnite groups, we see that SO(2) has an inflnite set of irreduciblerepresentations, albeit one that is countably inflnite. 8.1.3 Orthogonality Relations Having determined the characters for SO(2), we can now examine thevalidity of the orthogonality theorems for characters which were dis-cussed for discrete groups in Theorem 5.1. We proceed heuristicallyand begin by observing that the exponential functions in (8.7) are or-thogonal over the interval 0 •’<2…: Z2… 0ei(m0¡m)’d’=2…–m;m0: By writing this relation as Z2… 0´(m)⁄(’)´(m0)(’)d’=2…–m;m0; (8.8) 130 Irreducible Representations of SO(2) and SO(3) we obtain an orthogonality relation of the form in Eq. (5.4), once we identify the \order" of SO(2) as the quantity Z2… 0d’=2…: This is the \volume" of the group in the space of the parameter ’, which lies in the range 0 •’<2…, given that the density function is unity, according to the discussion in thew preceding section. Note thatthe integration over ’is efiectively a sum over classes. Example8.1. Consider the representation of SO(2) derived in Section 7.2: R(’)=ˆcos’¡sin’ sin’ cos’! : (8.9) Since SO(2) is an Abelian group, this representation must be reducible. We can decompose this representation into its irreducible componentsby using either the analogue of the Decomposition Theorem (Section5.3) for continuous groups or, more directly, by using identities betweencomplex exponential and trigonometric functions: ´(’)·tr[R(’)] = 2 cos’ =e i’+e¡i’: A comparison with (8.7) yields ´(’)=´(1)(’)+´(¡1)(’); so the representation in (8.9) is a direct sum of the irreducible repre- sentations corresponding to m= 1 andm=¡1.2 2This example illustrates the importance of the fleld used in the entries of the matrices for SO(2). If we are restricted to realentries, then the representation in (8.9) is irreducible . But, if the entries are complex , then this example shows that this representation is reducible . Irreducible Representations of SO(2) and SO(3) 131 8.2 Basis Functions for Irreducible Rep- resentations We were able to determine the characters for all of the irreducible rep- resentations of SO(2) without any knowledge of the representationsthemselves. But this is not the typical case for continuous groups. Wewill see, for example, when determining the characters for SO(3) thatwe will be required to construct explicit representations of rotationscorresponding to difierent classes. The action of these rotations on thebasis functions will determine the representation of that class and thecharacter will be calculated directly from this representation. As anintroduction to that discussion, in this section we will determine thebasis functions of the irreducible representations of SO(2). We begin by calculating the eigenvalues of the matrix in (8.9) from det(R¡‚I)=0 : flflflflflcos’¡‚¡sin’ sin’ cos’¡‚flflflflfl = (cos’¡‚)2+ sin2’ =‚2¡2‚cos’+1=0: Solving for ‚yields ‚= cos’§isin’=e§i’: (8.10) The corresponding eigenvectors are proportional to x§iy. Thus, op- erating on these eigenvectors with R(’) (see below) generates the irre- ducible representations corresponding to m= 1 andm=¡1 in (8.7), i.e., the characters ´(1)(’) and´(¡1)(’). Obtaining the basis functions for the other irreducible representa- tions of SO(2) is now a matter of taking appropriate direct products,since ´ (m)(’)´(m0)(’)=´(m+m0)(’): In particular, the m-fold products ( x§iy)mgenerate irreducible repre- sentations for the m-fold direct product, as discussed in Sec. 6.5. This 132 Irreducible Representations of SO(2) and SO(3) can be verifled directly from the transformation (8.9) applied to xand y: x0=xcos’¡ysin’; y0=xsin’+ycos’: Then, (x0§iy0)m=h xcos’¡ysin’§i(xsin’+ycos’)im =h x(cos’§isin’)§iy(cos’§isin’)im =h (x§iy)e§i’im =(x§iy)me§im’: Therefore, we can now complete the character table for SO(2), including the basis functions which generate the irreducible representations: SO(2) ER (’) ¡§m:(x§iy)m1e§im’ We note for future reference that the basis functions ( x§iy)mcould have been derived in a completely difierent manner. Consider Laplace’sequation in two dimensions: @ 2u @x2+@2u @y2=0: This equation is invariant under all the elements of SO(2), as can be easily verifled. The general solution to this equation is u(x;y)=F(x+iy)+G(x¡iy); whereFandGare arbitrary functions. Thus, if we are interested in solutions which are homogeneous polynomials of degree m, we can Irreducible Representations of SO(2) and SO(3) 133 choose in turn solutions with F(s)=smandG(s) = 0 and then with F(s)=0a n dG(s)=sm. We thereby obtain the expressions u(x;y)=(x§iy)m(8.11) as solutions of Laplace’s equations which are also the basis functions of the irreducible representations of SO(2). These functions are the ana-logues in two dimensions of spherical harmonics, which are the solutionsof Laplace’s equations in three dimensions. These will be discussed laterin this chapter. 8.3 Axis{Angle Representation of Proper Rotations in Three Dimensions The three most common parametrizations of proper rotations were dis- cussed in Section 7.4. For the purposes of obtaining the orthogonal-ity relations for the characters of SO(3), the representation in termsof a flxed axis about which a rotation is carried out|the axis{anglerepresentation|is the most convenient. We begin this section by show-ing how this representation emerges naturally from the basic propertiesof orthogonal matrices. 8.3.1 Eigenvalues of Orthogonal Matrices LetAbe any proper rotation matrix in three dimensions. Denoting the eigenvalues of Aby‚1,‚2, and‚3, and the corresponding eigenvalues byu1,u2, andu3, we then have Aui=‚iui fori=1;2;3. We can also form the adjoint of each equation: uy iAt=‚⁄uy i: These eigenvalue equations imply uy iui=uy iAtAui=j‚2 ijuy iui; 134 Irreducible Representations of SO(2) and SO(3) which shows that j‚2 ij= 1, i.e., that the modulus of every eigenvalue of an orthogonal matrix is unity [cf. (8.10)]. The most general formof such a quantity is a complex number of the form e i’for some angle ’. But these eigenvalues are also the roots of the characteristic equa- tion det(A¡‚I) = 0 so, according to the Fundamental Theorem of Algebra,3if they are complex, they must occur in complex conjugate pairs (because the coe–cients of this polynomial, which are obtainedfrom the entries of A, are real). Hence, the most general form of the eigenvalues of an orthogonal matrix in three dimensions is ‚ 1=1;‚ 2=ei’;‚ 3=e¡i’: (8.12) The eigenvector corresponding to ‚1= 1, which is unafiected by the action ofA, thereby deflnes the axis about which the rotation is taken. The quantity ’appearing in ‚2and‚3deflnes the angle of rotation about this axis. 8.3.2 The Axis and Angle of an Orthogonal Matrix In this section, we show how the axis and angle of an orthogonal matrixcan be determined from its matrix elements. We take the axis of therotation to be a unit vector n, which is the eigenvector corresponding to the eigenvalue of unity: An=n: (8.13) This equation and the orthogonality of A(AA t=AtA= 1) enables us to write Atn=AtAn=n: (8.14) Subtracting (8.14) from (8.13) yields (A¡At)n=0: 3K. Hofiman and R. Kunze, Linear Algebra 2nd edn (Prentice{Hall, Englewood Clifis, NJ, 1971), p. 138. Irreducible Representations of SO(2) and SO(3) 135 In terms of the matrix elements aijofAand the components niofn, we then have (a12¡a21)n2+(a13¡a31)n3=0; (a21¡a12)n1+(a23¡a32)n3=0; (a31¡a13)n1+(a32¡a23)n2=0: Notice that these equations involve only the ofi-diagonal elements of A. The solution of these equations yield the relations n2 n1=a31¡a13 a23¡a32;n3 n1=a12¡a21 a23¡a32; (8.15) which, when combined with the normalization condition n¢n=n2 1+n2 2+n2 3=1 determinesnuniquely. The angle of the rotation can be determined from the invariance of the trace of Aunder similarity transformations. Noting that the trace is the sum of the eigenvalues, and using (8.12), we have a11+a22+a33=1+ei’+e¡i’=1+2c o s ’; (8.16) so’is determined only by the diagonal elements of A. 8.3.3 Normal Form of an Orthogonal Matrix We conclude this section by deriving the form of a rotation matrix in an orthogonal coordinate system which naturally manifests the axis andangle. The diagonal form of a rotation matrix is clearly given by ⁄=0 BB@10 0 0ei’0 00ei’1 CCA: The eigenvector ncorresponding to ‚1= 1 is the axis of the rotation and can always be chosen to be real. However, the eigenvectors of ‚2= 136 Irreducible Representations of SO(2) and SO(3) ei’and‚3=e¡i’are inherently complex quantities. An orthonormal set can be chosen as n2=1 2p 2(0;1;i);n2=1 2p 2(0;1;¡i); respectively. Since we are interested in transformations of real coor- dinates, we must perform a unitary transformation from this complexbasis to a real orthogonal basis, in which case our rotation matrix ⁄ willno longer be diagonal. The required unitary matrix which accomplishesthis is U= 0 BBB@10 0 01 2p 21 2ip 2 01 2p 2¡1 2ip 21 CCCA: Thus, R=U¡1⁄U=0 BB@10 0 0 cos’¡sin’ 0 sin’ cos’1 CCA: (8.17) When expressed in this basis, the rotation matrix clearly displays the axis of rotation through the entry R11= 1, and the angle of rotation through a 2£2 rotational submatrix in a plane perpendicular to this axis. 8.3.4 Parameter Space for SO(3) The axis-angle representation of three-dimensional rotations providesa convenient parametrization of all elements of SO(3). We have seenthat every element of SO(3) can be represented by a unit vector corre-sponding to the rotation axis and a scalar corresponding to the rotationangle. Thus, consider the space deflned by the three quantities (n 1’;n 2’;n 3’); (8.18) wheren2 1+n2 2+n2 3= 1. Every direction is represented by a point on the unit sphere. Thus, deflning an azimuthal angle `and a polar angle Irreducible Representations of SO(2) and SO(3) 137 AB CD EO Figure 8.1: Two-dimensional representation of the parameter space of SO(3) as the interior of a sphere of radius …. The point Arepresents a rotation whose axis is along the direction OAand whose angle is the length of OA. The points at A,BandCcorrespond to rotations with the same angle but about axis along difierent directions. This deflnes the classes of SO(3). Thediametrically opposite points at DandEcorrespond to the same operation. µaccording to the usual conventions in spherical polar coordinates, the parameter space of SO(3) can be represented as (’cos`sinµ;’sin`sinµ;’cosµ); (8.19) where 0•’•…; 0•`•2…; 0•µ•…: We can now see directly that this parameter space corresponds to the interior of a sphere of radius …(Fig. 8.1). For every point within the sphere, there is a unique assignment to an element of SO(3): the direc- tion from the radius to the point corresponds to the direction of the rotation axis and the distance from the point to the origin represents the rotation angle. Two diametrically opposed points on the surface ofthe sphere ( ’=…) correspond to the same rotation, since a rotation by…aboutnis the same as a rotation by ¡…about this axis which, in turn, is the same as a rotation by …about¡n(whatever the sense of rotation). 138 Irreducible Representations of SO(2) and SO(3) Another useful feature of the axis-angle parametrization is the rep- resentation of classes of SO(3). Consider two elements of SO(3) whichhave the same angle of rotation ’but about difierent axes nandn 0.W e denote these operations by R(n;’) andR(n0;’). LetU(n;n0) denote the rotation of ninton0. The inverse of this operation then rotates n0 inton. The relationship between R(n;’),R(n0;’), andU(n;n0) is, therefore, R(n;’)=h U(n;n0)i¡1R(n0;’)U(n;n0); i.e.,R(n;’) andR(n0;’) are related by a similarity transformation and, therefore, belong to the same equivalence class . Referring to Fig. 8.1, equivalence classes of SO(3) correspond to operations whichlie on the same radius. Thus, a summation over the classes of SO(3) is equivalent to a sum over spherical shells . 8.4 Orthogonality Relations for SO(3) The axis-angle representation of rotations provides, in addition to a conceptual simplicity of elements of SO(3) in parameter space, a naturalframework within which to discuss the integration over the elementsof SO(3) and thereby to obtain the Rearrangement Theorem for thisgroup. In this section, we derive the density function gin (8.3) for this group and then use this to identify the appropriate form of theorthogonality relations for characters 8.4.1 The Density Function As discussed in the introduction, one of the basic quantities of interestfor continuous groups is the density of group elements as a functionof position in parameter space. To determine this function for SO(3),we flrst consider the elements in the neighborhood of the identity andthen examine the behavior of these points under an arbitrary elementof SO(3). Referring to the discussion in Section 7.4.2, these elementscorrespond to rotations by inflnitesimal angles ’ 1,’2, and’3about each of the three coordinate axes. The rotation matrix associated with Irreducible Representations of SO(2) and SO(3) 139 this transformation is –R=0 BB@1¡’3’2 ’3 1¡’1 ¡’2’1 11 CCA: The identity of SO(3) corresponds to the origin in the three-dimensional parameter space, ’1=’2=’3= 0, and is indicated by the point O in Fig. 8.1. For inflnitesimal rotation angles, the parameter spacespanned by –Ris associated with an inflnitesimal volume element in the neighborhood of the origin. We now follow the inflnitesimal transformation –Rby a flnite trans- formationR(n;’), i.e., we form the product R–R . This generates a volume element in the neighborhood of Rand the product R–R can be viewed as transformation of the volume near the origin to that near R. The Jacobian of this transformation is the relative change of volumenear the origin to that near Ror, equivalently, is the relative change of the density of operations near the origin to that near R. According to the discussion in the introduction, this is the information requiredfrom the density function for SO(3). We have already seen that equivalence classes of SO(3) are com- prised of all rotations with the same rotation angle, regardless of thedirection of the rotation axis. Thus, the density function is expectedto depend only on ’. Referring to Fig. 8.1, this means that the density of elements depends only on the \radial" distance from the origin, noton the direction, so we can choose Rin accordance with this at our convenience. Therefore, in constructing the matrix R–R , we will use forRa matrix of the form in (8.17). Thus, R–R = 0 BB@10 0 0 cos’¡sin’ 0 sin’ cos’1 CCA0 BB@1¡’3’2 ’3 1¡’1 ¡’2’1 11 CCA =0 BB@1¡’3 ’2 ’3cos’+’2sin’cos’¡’1sin’¡’1cos’¡sin’ ’3sin’¡’2cos’sin’¡’1cos’¡’1sin’+ cos’1 CCA: 140 Irreducible Representations of SO(2) and SO(3) We can now use (8.15) and (8.16) to determine the axis n0and angle ’0of this product. The angle is determined from 1+2c o s’0=1+2c o s ’¡2’1sin’; which, upon cancelling common factors, becomes cos’0= cos’¡’1sin’: Using the standard trigonometric formula for the cosine of a sum, we flnd, to flrst order in ’1, that ’0=’+’1: The unnormalized components of n0are determined from (8.15) to be n0 1=¡2’1cos’¡2 sin’; n0 2=’3sin’¡’2(1 + cos’); n0 3=¡’2sin’¡’3(1 + cos’): To normalize the axis, we flrst determine the length based on these components. To flrst order in the ’i,w efl n d jn0j=2’1cos’+ 2 sin’: Thus, the components of the normalized rotation axis of R–R are n0 1=1; n0 2=¡1 2’3+1 2’21 + cos’ sin’; n0 3=1 2’2+1 2’31 + cos’ sin’: Expressed in terms of the parametrization in (8.18), R–R is given by (n0 1’0;n02’0;n03’0)= ( ’+’1;1 2’ˆ ¡’3+’21 + cos’ sin’! ;1 2’ˆ ’2+’31 + cos’ sin’!) : Irreducible Representations of SO(2) and SO(3) 141 This deflnes the transformation from the neighborhood of the origin to the neighborhood near R–R . The Jacobian Jof this transformation, obtained from J= detflflflflfl@(n0 i’0) @’jflflflflfl ; (8.20) determines how the density of elements of SO(3) near the origin is transformed to the density of points near R. By taking the derivatives in (8.20) to obtain the entries ( i;j) in the Jacobian matrix, we obtain J=flflflflflflflflflflflflfl10 0 0’1 + cos’ 2 sin’¡1 2’ 01 2’’1 + cos’ 2 sin’flflflflflflflflflflflflfl =’2 2(1¡cos’): Notice that lim ’!0J=1; so that the normalization of the volume in parameter space is such that the volume near the unit element is unity. Hence, the density of elements in parameter space is the reciprocal ofJ, so the density functiongin (8.3) is g(’)=2 ’2(1¡cos’): (8.21) 8.4.2 Integrals in Parameter Space The density function in (8.21) now permits us to carry out integral over the group. Thus, for a function F(’;›), where › denotes the angular variables in the parametrization in (8.19), we have ZZ g(’)F(’;›)’2d’d›; where we have used the usual volume element for spherical polar coor- dinates. Using the density function in (8.21), this integral becomes ZZ 2(1¡cos’)F(’;›)d’d›: 142 Irreducible Representations of SO(2) and SO(3) We can now establish the orthogonality relation for characters. If we denote the characters for two irreducible representations of SO(3)by´ „(’) and´”(’), then we have ZZ 2(1¡cos’)´„(’)´”(’)d’d› =–„;”ZZ 2(1¡cos’)d’d›: The integral on the right-hand side of this equation, which has the value 8…2, corresponds to the volume of SO(3) in parameter space. The integral over the angular variables on the left-hand side yields 2 £4…, so cancelling common factors, we obtain Z… 0(1¡cos’)´„(’)´”(’)d’=…–„;”: (8.22) This is the orthogonality relation for characters of SO(3). 8.5 Irreducible Representations and Char- acters for SO(3) For SO(2), we were able to determine the characters of the irreducible representations directly, i.e., without having to determine the basisfunctions of these representations. The structure of SO(3), however,does not allow for such a simple procedure, so we must determine thebasis functions from the outset. 8.5.1 Spherical Harmonics We proceed as in Section 8.2 by determining the homogeneous polyno-mial solutions of Laplace’s equation, now in three dimensions: @ 2u @x2+@2u @y2+@2u @z2=0: We seek solutions of the form u(x;y;z )=X a;bcab(x+iy)a(x¡iy)bz‘¡a¡b; Irreducible Representations of SO(2) and SO(3) 143 which are homogeneous polynomials of degree ‘. In spherical polar coordinates, x=rcos`sinµ; y =rsin`sinµ; z =rcosµ; where 0•`<2…and 0•µ•…, these polynomial solutions transform to u(r;µ;` )=X a;bcabr‘sina+bµcos‘¡a¡bµei(a¡b)`: (8.23) Alternatively, Laplace’s equation in spherical polar coordinates is 1 r2@ @rµ r2@u @r¶ +1 r2sinµ@ @µµ sinµ@u @µ¶ +1 r2sin2µ@2u @`2=0: When the method of separation of variables is used to flnd solutions of this equation of the form u(r;µ;` )=R(r)£(µ)'(`), the stipulation that the solution be single-valued with respect changes in `by 2…, u(r;µ;` +2…)=u(r;µ;` ); requires that '(`)/eim`; wheremis an integer. Comparing this expression with the correspond- ing factor in (8.23), we see that a¡b=m. Since the ranges of both aandbare between 0 and ‘, we see that there are 2 ‘+ 1 values of mconsistent with homogeneous polynomial solutions of degree ‘. The corresponding values of mare¡‘•m•‘. The 2‘+ 1 independent homogeneous polynomials of degree ‘are called the sphericalharmonics and denoted by Y‘m(µ;`). Their functional form is Y‘m(µ;`)/P‘ m(µ)eim`; (8.24) whereP‘ m(µ)i saLegendrefunction . In the following discussion, we will utilize only the exponential factor in the spherical harmonics. 144 Irreducible Representations of SO(2) and SO(3) 8.5.2 Characters of Irreducible Representations TheY‘m(µ;`)f o r ma( 2 ‘+ 1)-dimensional representation of SO(3). Thus, for a general rotation R,w eh a v e RY‘m(µ;`)=‘X m0=¡‘Y‘m0(µ;`):¡‘ m0m(R) To determine the character of this representation, it is convenient to again invoke the fact that the classes of SO(3) are determined only bythe rotation angle, not by the direction of the rotation axis. Thus, wecan choose a rotation axis at our convenience and we therefore focus onrotations through an angle ’about thez-axis. In this case, the form of (8.24) allows us to write R z(’)Y‘m(µ;`)=Y‘m(µ;`¡’)=e¡im’Y‘m(µ;`): Thus, the corresponding transformation matrix is given by ¡‘[Rz(’)] =0 BBBBBB@e¡i‘’0¢¢¢ 0 0e¡i(‘¡1)’¢¢¢ 0 ............ 00¢¢¢ e i‘’1 CCCCCCA: (8.25) The character ´(‘)(’) of this class is obtained by taking the trace of this matrix: ´(‘)(’)=e¡i‘’+e¡i(‘¡1)’+¢¢¢+ei‘’ =e¡i‘’µ 1+ei’+e2i’+¢¢¢+e2‘i’¶ =e¡i‘’1¡e¡(2‘+1)i’ 1¡ei’ =e(‘+1=2)i’¡e¡(‘+1=2)i’ ei’=2¡e¡i’=2 =sin [(‘+1 2)’] sin (1 2’): Irreducible Representations of SO(2) and SO(3) 145 The orthogonality integral for these characters takes the form Z… 0(1¡cos’)sin [(‘+1 2)’] sin [(‘0+1 2)’] sin2(1 2’)d’: Using the trigonometric identity 2 sin2(1 2’)=1¡cos’ enables us to write the orthogonality integral as Z… 0sin [(‘+1 2)’] sin [(‘0+1 2)’]d’=1 2…–‘;‘0; where the right-hand side of this equation follows either from (8.22) or from the orthogonality of the sine functions over (0 ;…). It is possible to show directly, using Schur’s flrst lemma, that the spherical harmonics form a basis for (2 ‘+ 1)-dimensional irreducible representations of SO(3). However, this requires invoking properties ofthe Legendre functions in (8.24). If we conflne ourselves to the matricesin (8.25) then we can show that a matrix that commutes with all suchrotation matrices must reduce to a diagonal matrix. If we then considerrotations about any other direction, which requires some knowledge ofthe Legendre functions, we can then show that this constant matrixmust, in fact, be a constant multiple of the unit matrix. Hence, accord-ing to Schur’s flrst lemma, these representations are irreducible. We cannow construct the character table for SO(3) with the basis functionswhich generate the irreducible representations: SO(3) ER (’) ¡‘:Y‘m(µ;`)1sin [(‘+1 2)’] sin (1 2’) 8.6 Summary In this chapter, we have shown how the orthogonality relations devel- oped for flnite groups must be adapted for continuous groups, using 146 Irreducible Representations of SO(2) and SO(3) SO(2) and SO(3) as examples. For SO(2), which is a one-parameter Abelian group, this proved to be a straightforward matter. However,the corresponding calculations for SO(3) required us to determine ex-plicitly the density function to produce the appropriate form of theorthogonality relations. We found that the there are an inflnite se-quence of irreducible representations of dimensionality 2 ‘+ 1, where ‘‚0. Because of the connection between SO(3) and angular mo- mentum, the structure of these irreducible representations has severalphysical consequences: †For systems that possess spherical symmetry, the energy eigen- states have degeneracies of 2 ‘+1. The fact that there is a greater degeneracy for the hydrogen atom is due to a \hidden" SO(4)symmetry. 4 †The formation and decomposition of direct products of the irre- ducible representations of SO(3) forms the basis of angular mo-mentum coupling rules (Problem 6, Problem Sets 10) and theclassiflcation of atomic spectra. 5 †When atoms are placed within crystals, the original spherical symmetry is lowered to the symmetry of the crystal. This causeslevels which were degenerate in the spherically-symmetric envi-ronment to split. Such \crystal-fleld" efiects are important formany aspects for electrons in crystalline solids. 6 4H.F. Jones, Groups, Representations and Physics (Institute of Physics, Bristol, 1998), pp. 124{127. 5E.P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), pp. 177{194. 6M. Tinkham, Group Theory and Quantum Mechanics (McGraw{Hill, New York, 1964), pp. 65{80.