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Vvedensky D. Group theory in physics (lecture notes, 2001)(158s)

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Lecture notes on group theory for physicists, attributed to D. Vvedensky and dated 2001, kept in the archive's downloaded math books. The introduction covers symmetry in physical laws, Lorentz invariance of the wave equation, Noether's theorem, and symmetry in quantum mechanics and solids. It then works through parity and even/odd eigenfunctions for one-dimensional Schrödinger problems; later chapters are not shown in the excerpt.

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Chapter 1 Introduction As far as I can see, all a priori statements in physics have their origin in symmetry. |Hermann Weyl1 1.1 Symmetry in Physics Symmetry is a fundamental human concern, as evidenced by its pres-ence in the artifacts of virtually all cultures. Symmetric objects areaesthetically appealing to the human mind and, in fact, the Greekwork symmetros was meant originally to convey the notion of \well- proportioned" or \harmonious." This fascination with symmetry ﬂrstfound its rational expression around 400 B.C. in the Platonic solids andcontinues to this day unabated in many branches of science. 1.1.1 What is a Symmetry? An object is said to be symmetric, or to have a symmetry, if there isa transformation, such as a rotation or re°ection, whereby the objectlooks the same after the transformation as it did before the transforma-tion. In Fig. 1.3, we show an equilateral triangle, a square, and a circle.The triangle is indistinguishable after rotations of 1 3…and2 3…around its geometric center, or symmetry axis. The square is indistinguishable 1InSymmetry (Princeton University Press, 1952) 1 2 Introduction after rotations of1 2…,…, and3 2…, and the circle is indistinguishable after allrotations around their symmetry axes. These transformations are said to be symmetrytransformations of the corresponding object, which are said to be invariant under such transformations. The more symme- try transformations that an object admits, the more \symmetric" it issaid to be. One this basis, the circle is \more symmetric" than thesquare which, in turn, is more symmetric than the triangle. Anotherproperty of the symmetry transformations of the objects in Fig. 1.3that is central to this course is that those for the triangle and squarearediscrete , i.e., the rotation angles have only discrete values, while those for the circle are continuous . (a) (b) (c) Figure 1.1: An equilateral triangle (a), square (b) and circle (c). These ob- jects are invariant to particular rotations about axes that are perpendicularto their plane and pass through their geometric centers (indicated by dots). 1.1.2 Symmetry in Physical Laws In the physical sciences, symmetry is of fundamental because there are transformations which leave the laws of physics invariant. Such trans-formations involve changing the variables within a physical law suchthat the equations describing the law retain their form when expressedin terms of the new variables. The relationship between symmetryand physical laws began with Newton, whose equations of motion werefound to be the same in diﬁerent frames of reference related by Galileantransformations. Symmetry was also the guiding principle that en-abled Lorentz and Poincar¶ e to derive the transformations, now known as Lorentz transformations, which leave Maxwell’s equations invari-ant. The incompatibility between the Lorentz invariance of Maxwell’s Introduction 3 equations and the Galilean invariance of Newtonian mechanics was, of course, resolved by Einstein’s special theory of relativity. As an example of a symmetry in a physical law, consider the prop- agation of an impulse at the speed of light c. This is governed by the wave equation, which is obtained from Maxwell’s equations: 1 c2@2u @t2=@2u @x2+@2u @y2+@2u @z2: (1.1) The Lorentz transformation of space-time coordinates corresponding to a velocityv=(v;0;0) is x0=°(x¡vt);y0=y; z0=z; t0=°µ t¡v c2x¶ ; (1.2) where°=( 1¡v2=c2)¡1=2. When expressed in terms of the transformed coordinates ( x0;y0;z0;t0), the wave equation (1.1) is found to retain its form under this transformation: 1 c2@2u0 @t02=@2u0 @x02+@2u0 @y02+@2u0 @z02: (1.3) This implies that the wave propagates in the same way with the same velocity in two inertial frames that are in uniform motion with respectto one another. The Lorentz transformation is thus a symmetry trans-formation of the wave equation (1.1) and this equation is said to be covariant with respect to these transformations. In general, symmetry transformations of physical laws involve the space-time coordinates,which are sometimes called geometricalsymmetries , and/or internal co- ordinates, such as spin, which are called internal symmetries . 1.1.3 Noether’s Theorem Identifying appropriate symmetry transformations is one of the centralthemes of modern physics since their mathematical expression aﬁectsthe structure and predictions of physical theories. Work by both math-ematicians and physicists, culminating with Emmy Noether, led to thedemonstration that there was a deep relationship between symmetryand conservation laws. This is now known as Noether’s Theorem: 4 Introduction Noether’s Theorem. The covariance of the equations of motion with respect to a continuous transformation with nparameters implies the existence of nquantities, or constants of motion, i.e., conservationlaws. In classical mechanics, the conservation of linear momentum results from the translational covariance of Newton’s equations of motion, i.e.,covariance with respect to transformations of the form r 0=r+a, for any vectora. The conservation of angular momentum similarly results from rotational covariance, i.e., covariance with respect rotations inspace:r 0=Rr, whereRi sa3£3 rotation matrix. Finally, the con- servation of energy results from the covariance of Newton’s equationsto translations in time, i.e., transformations of the form t 0=t+¿. 1.1.4 Symmetry and Quantum Mechanics The advent of quantum mechanics and later quantum ﬂeld theory fos- tered entirely new avenues for investigating the consequences of sym-metry. London and Weyl introduced a type of transformation known asa gauge transformation into quantum theory, with total electric chargeas the conserved quantity. In the early 1960s, Gell{Mann and Ne’emanproposed the unitary symmetry SU(3) for the strong interactions. Thisled to the proposal by Gell{Mann and Zweig of a new, deeper, levelof quanta, \quarks," to account for this symmetry. Heisenberg, Gold-stone and Nambu suggested that the ground state (i.e., the vacuum)of relativistic quantum ﬂeld theory may not have the full global sym-metry of the Hamiltonian, and that massless excitations (Goldstonebosons) accompany this \spontaneous symmetry breaking." Higgs andothers found that for spontaneously broken gauge symmetries there areno Goldstone bosons, but instead massive vector mesons. This is nowknown as the Higgs phenomenon and its veriﬂcation veriﬂcation hasbeen the subject of extensive experimental eﬁort. Another aspect of symmetry, also due to the quantum mechanical nature of matter, arises from the arrangement of atoms in moleculesand solids. The symmetry of atomic arrangements, whether in a sim-ple diatomic molecule or a complex crystalline material such as a high-temperature superconductor, aﬁects many aspects of their electronic Introduction 5 and vibrational properties and especially their response to external thermal, mechanical, and electromagnetic perturbations. The trans-formation properties of wavefunctions in quantum mechanics are an example of what is known as Representation Theory , which was devel- oped by the mathematicians Frobenius and Schur near the turn of the20th century. This inspired a huge eﬁort by physicists and chemiststo determine the physical consequences of the symmetries of wavefunc- tions which continues to this day. Notable examples include Bloch’swork on wavefunctions in periodic potentials, which forms the basis ofthe quantum theory of solids, Pauling’s work on the chemical interpre-tation orbital symmetries, and Woodward and Hoﬁman’s work on howthe conservation of orbital symmetry determines the course of chemicalreactions. Recent scientiﬂc advances that highlight the prominent rolethat symmetry maintains in condensed-matter physics is the discov-ery of quasicrystals, which have rotational symmetries (e.g., ﬂvefold, asshown in Fig. 1.2) which are incompatible with the translational sym-metry of ordinary crystals and are thus sometimes called aperiodic, andthe C 60form of carbon, known as \Buckminsterfullerene," or \Buck- yballs", a name derived from its resemblance to structures (geodesicdomes) proposed by R. Buckminster Fuller as an alternative to conven-tional architecture. Figure 1.2: A section of a Penrose tile, which has a ﬂvefold rotational symmetry, but no translational symmetry. This two-dimensional structureshares a number of features with quasicrystals. 6 Introduction 1.2 Examples from Quantum Mechanics 1.2.1 One-Dimensional Systems To appreciate how symmetry enters into the description of quantum me- chanical systems, we consider the time-independent Schr˜ odinger equa- tion for the one-dimensional motion of a particle of mass mbound by a potential V(x): • ¡„h2 2md2 dx2+V(x)‚ ’(x)=E’(x); (1.4) where „h=h=2…,his Planck’s constant, ’is the wavefunction, and E is the energy eigenvalue. By writing this equation as H’=E’,w e identify the coordinate representation of the Hamiltonian operator as H=¡„h2 2md2 dx2+V(x): (1.5) In the following discussion, we will utilize the fact that the energy eigen- values of one-dimensional quantum mechanical problems such as thatin (1.4) are nondegenerate, i.e., each energy eigenvalue is associatedwith one and only one eigenfunction. 2 Suppose that the potential in (1.4) is an even function of x. The mathematical expression of this fact is the invariance of this potentialunder the inversion transformation x!¡x: V(¡x)=V(x): (1.6) Examples of such potentials are the symmetric square well and the harmonic oscillator (Fig. 1.3), but the particular form of the potentialis unimportant for this discussion. The kinetic energy term in (1.4) isalso invariant under the same inversion transformation as the potential,since d 2 d(¡x)2=d2 dx2(1.7) 2This follows directly from the fact that this equation, together with appropri- ate boundary conditions , constitute a Sturm{Liouville problem. Other well-known properties of solutions of Schr˜ odinger’s equation (real eigenvalues, discrete eigen- values for bound states, and orthogonality of eigenfunctions) also follow from theSturm{Liouville theory. Introduction 7 xE Figure 1.3: The ﬂrst four eigenfunctions of the Schr˜ odinger equation (1.4) for an inﬂnite square-well potential, V(x)=0f o rjxj•LandV(x)!1 forjxj>L (left), and a harmonic oscillator potential, V(x)=1 2kx2, where kis the spring constant of the oscillator (right). The abscissa is the spatial positionxand the ordinate is the energy E, with the vertical displacement of each eigenfunction given by its energy. The origins are indicated by brokenlines. Thus, the Hamiltonian operator in (1.5) is itself invariant under inver- sion, i.e., inversion is a symmetry transformation of this Hamiltonian.We now use this property of Hto change variables from xto¡xin (1.4) and thereby obtain the Schr˜ odinger equation for ’(¡x): • ¡„h2 2md2 dx2+V(x)‚ ’(¡x)=E’(¡x) (1.8) SinceEis nondegenerate, there can be only one eigenfunction associ- ated with this eigenvalue, so the ’(¡x) cannot be linearly independent of’(x). The only possibility is that ’(¡x)i sproportional to’(x): ’(¡x)=A’(x) (1.9) whereAis a constant. Changing xto¡xin this equation, ’(x)=A’(¡x) (1.10) and then using (1.9) to replace ’(¡x), yields ’(x)=A2’(x) (1.11) 8 Introduction This requires that A2= 1, i.e.,A=1o rA=¡1. Combining this result with (1.9) shows that the eigenfunctions ’of (1.4) must be either even ’(¡x)=’(x) (1.12) or odd ’(¡x)=¡’(x) (1.13) under inversion. As we know from the solutions of Schr˜ odinger’s equa- tion for square-well potentials and the harmonic oscillator (Fig. 1.3),both even and odd eigenfunctions are indeed obtained. Thus, not alleigenfunctions have the symmetry of the Hamiltonian, although theground state usually does. 3Nevertheless, the symmetry (1.6) does pro- vide a classiﬂcation of the eigenfunctions according to their parity under inversion. This is a completely general result which forms one of thecentral themes of this course. 1.2.2 Symmetries and Quantum Numbers The example discussed in the preceding section showed how symmetryenters explicitly into the solution of Schr˜ odinger’s equation. In fact, we can build on our discussion in Sec. 1.1.2, and especially Noether’s theo-rem, to establish a general relationship between continuous symmetriesand quantum numbers. Consider the time-dependent Schr˜ odinger equation for a free particle of massmin one dimension: i„h@’ @t=¡„h2 2m@2’ @x2: (1.14) The solutions to this equation are plane waves: ’(x;t)=ei(kx¡!t); (1.15) wherekand!are related to the momentum and energy by p=„hk andE=„h!. In other words, the quantum numbers kand!of the 3A notable exception to this is the phenomenon of spontaneous symmetry- breaking discussed in Sec. 1.1, where the symmetry of the equations of motion and the boundary conditions is not present in the observed solution for the groundstate. Introduction 9 solutions to Eq. (1.14) correspond to the momentum and energy which, because of the time- and space-translational covariance of this equation,correspond to conserved quantities. Thus, quantum numbers are asso-ciated with the symmetries of the system. Similarly, for systems withrotational symmetry, such the hydrogen atom or, indeed, anyatom, the appropriate quantum numbers are the energy and the angular mo-mentum, the latter producing two quantum numbers, as required byNoether’s theorem, because the transformations have two degrees offreedom. 1.2.3 Matrix Elements and Selection Rules One of the most important uses of symmetry is to identify the matrixelements of an operator which are required to vanish. Continuing withthe example in the preceding section, we consider the matrix elementsof an operatorH 0whose position representation H0(x) has a deﬂnite parity. The matrix elements of this operator are given by H0 ij=Z ’i(x)H0(x)’j(x)dx (1.16) where the range of integration is symmetric about the origin. If H0 has even parity, i.e., if H0(¡x)=H0(x), as in (1.6), then these matrix elements are nonvanishing only if ’i(x) and’j(x) are both even or both odd, since only in these cases is the integrand an even function of x. This is called a selectionrule , since the symmetry of H0(x) determines, or selects, which matrix elements are nonvanishing. Suppose now that H0(x) has odd parity, i.e., H0(¡x)=¡H0(x). The matrix elements in (1.16) now vanishes if ’i(x) and’j(x) are both even or both odd, since these choices render the integrand anodd function of x. In other words, the selection rule now states that only eigenfunctions of opposite parity are coupled by such an operator. Notice, however, that the use of symmetry only identiﬂes which matrixelements must vanish; it provides no information about the magnitude of the nonvanishing matrix elements. Suppose that H 0(x)=Ax (1.17) 10 Introduction whereAis a constant, i.e., H0(x) is proportional to the coordinate x. Such operators arise in the quantum theory of transitions induced byan electromagnetic ﬂeld. 4H0(x) clearly has odd parity, so the matrix elements (1.16) are nonvanishing only if ’i(x) and’j(x) have opposite parity. But, if H0(x)=¡„h2 2md2 dx2(1.18) which is the coordinate representation of the kinetic energy operator, then the matrix elements (1.16) are nonvanishing only if ’i(x) and ’j(x) have the same parity. Selection rules are especially useful if there are broken symmetries . For example, the Hamiltonian of an atom, which is the sum of thekinetic energies of the electrons and their Coulomb potentials, is in-variant under all rotations. But when an atom is placed in an electricor magnetic ﬂeld, the Hamiltonian acquires an additional term whichisnotinvariant under all rotations, since the ﬂeld now deﬂnes a pre- ferred direction. These are the Stark and Zeeman eﬁects, respectively.A similar situation is encountered in quantum ﬂeld theory when, be-ginning with a Lagrangian that is invariant under certain symmetryoperations, a term is added which does not have this invariance. Ifthe symmetry-breaking terms in these cases are small, then selectionrules enter into the perturbative calculation around the solutions of thesymmetric theory. 1.3 Summary The notion of symmetry implicit in all of the examples cited in thischapter is endowed with the algebraic structure of \groups." This is atopic in mathematics that had its beginnings as a formal subject onlyin the late 19th century. For some time, the only group that was knowand whose properties were studied were permutation groups. Cauchyplayed a major part in developing the theory of permutations, but itwas the English mathematician Cayley who ﬂrst formulated the notionof an abstract group and used this to identify matrices and quaternions 4E. Merzbacher, Quantum Mechanics 2nd edn. (Wiley, New York, 1970), Ch. 18. Introduction 11 as groups. In a later paper, Cayley showed that every ﬂnite group could be represented in terms of permutations, a result that we will prove inthis course. The fact that geometric transformations, as discussed inthis chapter, and permutations, share the same algebraic structure ispart of the richness of the subject and is rooted in its history as anadjunct to the study of algebraic solutions of equations. In the nextchapter, we discuss the basic properties of groups that form the basisof this course. 12 Introduction Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. |David Hilbert1 The importance of symmetry in physics, and for quantum mechanicsin particular, was discussed in the preceding chapter. In this chapter,we begin our development of the algebraic structure which enables usto formalize what we mean by \symmetry" by introducing the notionof a group and some related concepts. In the following chapters we willexplore the consequences of this algebraic structure for applications tophysics. 2.1 Groups: Deﬂnitions and Examples The motivation for introducing an algebraic structure to describe sym-metry in physical problems is based on transformations. But the def-inition of a group is based on a much more abstract notion of what a\transformation" entails. Accordingly, we ﬂrst set out the conditions 1As quoted in, N. Rose, Mathematical Maxims and Minims (Rome Press, Raleigh, North Carolina, 1988). 13 14 Elements of Abstract Group Theory that an abstract group must satisfy and then consider both abstract and concrete examples. Deﬁnition. AgroupGis a set of elements fa;b;c;:::gtogether with a binary composition law, called multiplication , which has the following properties: 1.Closure. The composition of any two elements aandbinG, called theproduct and written ab, is itself an element cofG:ab=c. 2.Associativity. The composition law is associative, i.e., for any elementsa,b, andcinG,(ab)c=a(bc). 3.Identity. There exists an element, called the unitoridentity and denoted by e, such that ae=ea=afor every element ainG. 4.Inverses. Every element ainGhas an inverse, denoted by a¡1, which is also in G, such that a¡1a=aa¡1=e. The closure property ensures that the binary composition law does not generate any elements outside of G. Associativity implies that the computation of an n-fold product does not depend on how the elements are grouped together.2For example, the product abcis un- ambiguous because the two interpretations allowed by the existence ofa binary composition rule, ( ab)canda(bc), are equal. As will be shown in Sec. 2.3, the left and right identities are equal and unique, as arethe left and right inverses of each element. Thus we can replace theexistence of an identity and inverses in the deﬂnition of a group withthe more \minimal" statements: 3 0.Identity. There exists a unique element, called the unitoridentity and denoted by e, such that ae=afor every element ainG. 40.Inverses. Every element ainGhas a unique inverse, denoted by a¡1, which is also in G, such that a¡1a=e. 2In abstract algebra (the theory of calculation), binary composition can be asso- ciative or non-associative. The most important non-associative algebras in physicsare Lie algebras, which will be discussed later in this course. Elements of Abstract Group Theory 15 The terms \multiplication," \product," and \unit" used in this def- inition are not meant to imply that the composition law corresponds toordinary multiplication. The multiplication of two elements is only anabstract rule for combining an ordered pair of two group elements toobtain a third group element. The diﬁerence from ordinary multiplica-tion becomes even more apparent from the fact that the compositionlaw need not be commutative, i.e., the product abneed not equal bafor distinct group elements aandb. If a group does have a commutative composition law, it is said to be commutative orAbelian . Despite the somewhat abstract tone of these comments, a moment’s re°ection leads to the realization that the structure of groups is ideallysuited to the description of symmetry in physical systems. The groupelements often correspond to coordinate transformations of either ge-ometrical objects or of equations of motion, with the composition lawcorresponding to matrix multiplication or the usual composition lawof functions, 3so the associativity property is guaranteed.4If two op- erations each correspond to symmetry operations, then their productclearly must as well. The identity corresponds to performing no trans-formation at all and the inverse of each transformation correspondsto performing the transformation in reverse, which must exist for thetransformation to be well-deﬂned (cf. Example 2.4). Example 2.1. Consider the set of integers, :::;¡3;¡2;¡1;0;1;2;3;::: with the composition rule being ordinary addition. The sum of any two integers is an integer, thus ensuring closure, addition is an associativeoperation, 0 is the identity, and the inverse of nis¡n, which is clearly an integer. Thus, the integers form a group under addition. This groupis denoted by Z(derived from the German word Zahlen for integers). 3For two functions f(x) andg(x), the application of f, followed by the application ofgisg[f(x)], and the application of gfollowed by the application of fisf[g(x)]. 4The associativity of linear operations in general, and matrices in particular, is discussed by Wigner in Group Theory (Academic, New York, 1959), along with other group properties. 16 Elements of Abstract Group Theory Since the order in which two integers are added is immaterial, Zis an Abelian group. Example2.2. The importance of the composition law for determining whether a set of elements forms a group can be seen by again consideringthe integers, but now with ordinary multiplication as the compositionrule. The product of any two integers is again an integer, multiplicationis associative, the unit is 1, but the inverse of nis 1=n, which is notan integer ifn6= 1. Hence, the integers with ordinary multiplication do not form a group. Example2.3. Consider the elements f1;¡1gunder ordinary multipli- cation. This set is clearly closed under multiplication and associativityis manifestly satisﬂed. The unit element is 1 and each element is itsown inverse. Hence, the set f1;¡1gis a two-element group under mul- tiplication. Example 2.4. Consider the set of 2 £2 matrices with real entries ˆab cd! ; (2.1) such that the determinant, ad¡bc, is non-zero. The composition law is the usual rule for matrix multiplication: ˆa1b1 c1d1!ˆa2b2 c2d2! =ˆa1a2+b1c2a1b2+b1d2 c1a2+d1c2c1b2+d1d2! : To determine if this set of matrices forms a group, we must ﬂrst show that the product of two matrices with non-zero determinant is also amatrix with non-zero determinant. This follows from that fact thatfor any pair of 2£2 matrices AandB, their determinants, denoted by det(A) and det(B), satisfy det( AB) = (detA)(detB). Associativity can be veriﬂed with a straightforward, but laborious, calculation. Theidentity is ˆ10 01! Elements of Abstract Group Theory 17 and the inverse of (2.1) is 1 ad¡bcˆd¡b ¡ca! ; which explains the requirement that ad¡bc6= 0. This group is denoted by GL(2, R), for general linear group of 2£2 matrices with real entries. Note that the elements of this group form a continuous set, so GL(2, R) is a continuous group. 2.2 Permutation Groups A permutation of nobjects is a rearrangement of those objects. When combined with the usual rule for function composition for successivepermutations (see below), these permutations are endowed with thestructure of a group, which is denoted by S n. At one time, permuta- tion groups were the only groups studied by mathematicians and theymaintain a special status in the subject through Cayley’stheorem , which establishes a relationship between Snandevery group with nelements. In this section, we will examine the structure of S3, both as an abstract group and as the symmetry group of an equilateral triangle. The groupS3is the set of all permutations of three distinguishable objects, where each element corresponds to a particular permutationof the three objects from a given reference order. Since the ﬂrst objectcan be put into any one of three positions, the second object into eitherof two positions, and the last object only into the remaining position,there are 3£2£1 = 6 elements in the set. These are listed below: e= ˆ123 123! a=ˆ123 213! b=ˆ123 132! c=ˆ123 321! d=ˆ123 312! f=ˆ123 231! In this notation, the top line represents the initial, or reference, order of the objects and the bottom line represents the eﬁect of the per-mutation. The composition law corresponds to performing successive 18 Elements of Abstract Group Theory permutations and is carried out by rearranging the objects according to the ﬂrst permutation and then using this as the reference order to rear-range the objects according to the second permutation. As an example,consider the product ad, where we will use the convention that opera- tions are performed from right to left, i.e., permutation dis performed ﬂrst, followed by permutation a. Element dpermutes the reference order (1;2;3) into (3;1;2). Element athen permutes this by putting the ﬂrst object in the second position, the second object into the ﬂrstposition, and leaves the third object in position three, i.e., a= ˆ123 213! =ˆ312132! : Notice that it is only the permutation of the distinct objects, not their labelling, which is important for specifying the permutation. Hence, ad=ˆ123 132! =b; An analogous procedure shows that da=ˆ213 321!ˆ123213! =ˆ123321! =c; which shows that the composition law is not commutative, so S3is a non-Abelian group. A geometric realization of S3can be established by considering the symmetry transformations of an equilateral triangle (Fig. 2.1). Theelementsa,b, andccorrespond to re°ections through lines which in- tersect the vertices at 3, 1, and 2, respectively, and dandfcorrespond to clockwise rotations of this triangle by 2 3…and4 3…radians, respec- tively. The eﬁects of each of these transformations on the positionsof the vertices of the triangle is identical with the corresponding ele-ment ofS 3. Thus, there is a one-to-one correspondence between these transformations and the elements of S3. Moreover, this correspondence is preserved by the composition laws in the two groups. Consider forexample, the products adanddacalculated above for S 3. For the equi- lateral triangle, the product adcorresponds to a rotation followed by Elements of Abstract Group Theory 19 32 1c31 2d23 1f12 3e21 3a13 2b Figure 2.1: The symmetry transformations of an equilateral triangle labelled by the corresponding elements of S3. The lines in the diagram corresponding to the identity are lines through which re°ections of transformations a,band care taken. The transformations dandfare rotations. a re°ection. Thus, beginning with the standard order shown for the identity the successive application of these transformations is shownbelow: 12 31 2 33 2 1da By comparing with Fig. 2.1, we see that the result of these transfor- mations is equivalent to the transformation b. Similarly, one can show thatda=cand, in fact, that all the products in S3are identical to those of the symmetry transformations of the equilateral triangle. Twosuch groups that have the same algebraic structure are said to be iso- morphic to one another and are, to all intents and purposes, identical. This highlights the fact that it is the algebraic structure of the group 20 Elements of Abstract Group Theory which is important, not any particular realization of the group. Further discussion of this point will be taken up in the next chapter. 2.3 Elementary Properties of Groups The examples in the preceding section showed that all groups are en-dowed with several general properties. In this section, we deduce someadditional properties which, although evident in particular examples,can be shown generally to follow from the properties of abstract groups. Theorem 2.1. (Uniqueness of the identity) The identity element in a groupGis unique. Proof. Suppose there are two identity elements eande 0inG. Then, according to the deﬂnition of a group, we must have that ae=a and e0a=a for allainG. Settinga=e0in the ﬂrst of these equations and a=e in the second shows that e0=e0e=e; soe=e0. This theorem enables us to speak of theidentityeof a group. The notationeis derived from the German word Einheit for unity. Another property common to all groups is the cancellation of com- mon factors within equations. This property owes its existence to theassociativity of the group composition rule. Elements of Abstract Group Theory 21 Theorem2.2. (Cancellation) In a groupG, the left and right cancel- lation laws hold, i.e., ab=acimpliesb=candba=caimpliesb=c. Proof. Suppose that ab=ac. Leta¡1be an inverse of a. Then, by left-multiplying by this inverse, a¡1(ab)=a¡1(ac) and invoking associativity, (a¡1a)b=(a¡1a)c; we obtain eb=ec; sob=c. Similarly, beginning with ba=caand right-multiplying by a¡1shows that b=cin this case also. Notice that the proof of this theorem does not require the inverse of a group element to be unique; only the existence of aninverse was required. In fact, the cancellation theorem can be used to prove thatinverses are, indeed, unique. Theorem 2.3. (Uniqueness of inverses) For each element ain a groupG, there is a unique element binGsuch thatab=ba=e. Proof. Suppose that there are two inverses bandcofa. Thenab=e andac=e. Thus,ab=ac, so by the Cancellation Theorem, b=c. As in the case of the identity of a group, we may now speak of the inverse of every element in a group, which we denote by a¡1.A s w a s discussed in Sec. 2.1, this notation is borrowed from ordinary multipli-cation, as are most other notations for the group composition rule. Forexample, the n-fold product of a group element gwith itself is denoted 22 Elements of Abstract Group Theory bygn. Similarly gngm=gn+m, which conforms to the usual rule of ex- ponents for real numbers. However, there are some notable exceptions.For two group elements aandb, the equality of ( ab) nandanbndoes not generally hold. As the examples in Sec. 2.1 demonstrated, as longas this notation is interpreted in the context of the appropriate groupcomposition rule, no confusion should arise. 2.4 Discrete and Continuous Groups Groups are divided into two general categories: discrete and continuous.The basis deﬂnitions apply to both types of group, but the discussionof a number of properties depends sensitively on the discrete or con-tinuous nature of the group. In this course, we will focus our attentionon discrete groups ﬂrst, to establish a conceptual base, and considercontinuous groups later in the course. 2.4.1 Finite Groups One of the most fundamental properties of a group Gis number of elements contained in the group. This is termed the order ofGand is denoted byjGj. The group Zof integers under addition, has inﬂnite order and the order of S3, the group of permutations of three objects, is 6. We will be concerned initially with ﬂnite groups which, apart from their applicability to a range of physical problems, have a numberinteresting arithmetic properties. Finite groups also have properties which are not shared by either inﬂnite or continuous groups. For example, if an element gof a ﬂnite groupGis multiplied by itself enough times, the unit eis eventually re- covered. Clearly, multiplying any element gby itself a number of times greater thanjGjmust eventually lead to a recurrence of the product, since the number of distinct products is bounded from above by jGj. To show this explicitly, we denote a recurring product by aand write a=g p=gq; wherep=q+n. Then, by using the associativity of the composition Elements of Abstract Group Theory 23 law,gq+n=gqgn=gngq,s o gp=gqgn=gngq=gq; and, from the deﬂnition of the identity and its uniqueness, we conclude that gn=e: Thus, the set of elements g;g2;g3;:::represents a recurring sequence. Theorder of an element g, denoted byjgj, is the smallest value ofk such thatgk=e. Theperiod of such an element gis the collection of elementsfe;g;g2;:::;gk¡1g. Example 2.5. UsingS3as an example,jaj=jbj=jcj= 2 and jdj=jfj= 3. The corresponding periods are fe;ag,fe;bg,fe;cg, andfe;d;f =d2g. Theorem 2.4. (Rearrangement Theorem) Iffe;g1;g2;:::;gngare the elements of a group G, and ifgkis an arbitrary group element, then the set of elements Ggk=fegk;g1gk;g2gk;:::;gngkg contains each group element once and only once. Proof. The setGgkcontainsjGjelements. Suppose two elements of Ggkare equal:gigk=gjgk. By the Cancellation Theorem, we must have thatgi=gj. Hence, each group element appears once and only once in Ggk, so the sets GandGgkare identical apart from a rearrangement of the order of the elements if gkis not the identity. 24 Elements of Abstract Group Theory 2.4.2 Multiplication Tables One application of this theorem is in the representation of the compo- sition law for a ﬂnite group as a multiplication table . Such a table is a square array with the rows and columns labelled by the elements of thegroup and the entries corresponding to the products, i.e., the elementg ijin theith row and jth column is the product of the element gila- belling that row and the element gjlabelling that column: gij=gigj. To see how the construction of multiplication tables proceeds by utiliz-ing only the abstract group properties, consider the simplest nontrivialgroup, that with two distinct elements fe;ag. We clearly must have the productse 2=eandea=ae=a. The Rearrangement Theorem then requires that a2=e. The multiplication table for this group is shown below: ea eea aae Note that the entries of this table are symmetric about the main diag-onal, which implies that this group is Abelian. Now consider the group with three distinct elements: fe;a;bg. The only products which we must determine explicitly are ab,ba,a 2, andb2 since all other products involve the unit e. The product abcannot equal aorb, since that would imply that either b=eora=e, respectively. Thus,ab=e. The Cancellation Theorem then requires thata2=e, b2=a, andba=e. The multiplication table for this group is shown below: eab eeab aabe bbea Because the entries of this table are symmetric about the main diagonal,this group is also Abelian. Our procedure shows that every group with two or three elements must have the multiplication tables just Elements of Abstract Group Theory 25 calculated, i.e., the algebraic structures of group with two and three elements are unique ! Thus, we can speak of thegroup with two elements and thegroup with three elements. A similar procedure for groups with four elements fe;a;b;cgyields twodistinct multiplication tables (Problem Set 2). As a ﬂnal example, the multiplication table for S3is shown below: eabcdf eeabcdf aaedfbc bbfedca ccdfeab ddcabfe ffbcaed As is immediately evident from this table, S3notAbelian. 2.5 Subgroups and Cosets If, from a group G, we select a subset Hof elements which themselves form a group under the same composition law, His said to be a sub- group ofG. According to this deﬂnition, the unit element fegforms a subgroup of G, andGis a subgroup of itself. These are termed improper subgroups. The determination of proper subgroups is one of the cen- tral concerns of group theory. In physical applications, subgroups arisein the description of symmetry-breaking, where a term is added to aHamiltonian or a Lagrangian which lowers the symmetry to a subgroupof the original symmetry operations. Example2.6. The groupS 3has a number of proper subgroups: fe;ag, fe;bg,fe;cg, andfe;d;fg. The identiﬂcation of these subgroups is most easily carried out by referring to the symmetry operations of anequilateral triangle (Fig. 2.1). 26 Elements of Abstract Group Theory IfH=fe;h1;h2;:::;hrgis a subgroup of a group G, andgis an element ofG, then the set Hg=feg;h 1g;h 2g;:::;hrgg is aright coset ofH. Similarly, the set gH=fge;gh 1;gh 2;:::;ghrg is aleftcoset ofH. A coset need not be a subgroup; it will be a subgroup only ifgis an element of H. Theorem 2.5. Two cosets of a subgroup either contain exactly the same elements or else have no common elements. Proof. These cosets either have no common elements or have at least one common element. We will show that if there is a single incommon, then all elements are common to both subgroups. Let Hg 1 andHg2be two right cosets. If one common element of these cosets is hig1=hjg2, then g2g¡1 1=h¡1 jhi sog2g¡1 1is inH. But also contained in Hare the elements Hg2g¡1 1=feg2g¡1 1;h1g2g¡1 1;h2g2g¡1 1;:::;hrg2g¡1 1g since, according to the Rearrangement Theorem, each element of H appears once and only once in this sequence. Therefore, the elementsofHg 1are identical to those of (Hg2g¡1 1)g1=Hg2(g¡1 1g1)=Hg2 so these two cosets have only common elements. Example2.7. Consider again the group S3and its subgroup H=fe;ag (Example 2.6). The right cosets of this subgroup are fe;age=fe;ag;fe;aga=fa;eg;fe;agb=fb;dg fe;agc=fc;fg;fe;agd=fd;bg;fe;agf=ff;cg Elements of Abstract Group Theory 27 We see that there are three distinct right cosets of fe;ag, fe;ag;fb;dg;fc;fg of which only the ﬂrst is a subgroup (why?). Similarly, there are three leftcosets offe;ag: fe;ag;fc;dg;fb;fg Notice that the left and right cosets are notthe same. Theorem 2.6 (Lagrange’s theorem). The order of a subgroup Hof a ﬂnite group Gis a divisor of the order of G, i.e.,jHjdividesjGj. Proof. Cosets either have all elements in common or they are dis- tinct (Theorem 2.5). This fact, combined with the Rearrangement The-orem, means that every element of the group must appear in exactlyone distinct coset. Thus, since each coset clearly has the same numberof elements, the number of distinct cosets, which is called the index of the subgroup, multiplied by the number of elements in the coset, isequal to the order of the group. Hence, since the order of the coset andthe subgroup are equal, the order of the group divided by the order ofthe subgroup is equal to the number of distinct cosets, i.e., an integer. Example2.8. The subgroupfe;agofS3is of order 2 and the subgroup fe;d;fgis of order 3. Both 2 and 3 are divisors of jS3j=6 . Lagrange’s theorem identiﬂes the allowable orders of the subgroups of a given group. But the converse of Lagrange’s theorem is notgener- ally valid, i.e., the orders of the subgroups of a group Gneed not span the divisors of G. 28 Elements of Abstract Group Theory 2.6 The Quotient Group 2.6.1 Conjugacy Classes Two elements aandbof a groupGare said to be conjugate if there is an element gin the group, called the conjugating element, such that a=gbg¡1. Conjugation is an example of what is called an equivalence relation , which is denoted by \ ·," and is deﬂned by three conditions: 1.a·a(re°exive). 2. Ifa·b, thenb·a(symmetric). 3. Ifa·bandb·c, thena·c(transitive). To show that conjugacy corresponds to an equivalence relation we consider each of these conditions in turn. By choosing g=eas the conjugating element, we have that a=eae¡1=a,s oa·a.I fa·b, thena=gbg¡1, which we can rewrite as g¡1ag=g¡1a(g¡1)¡1=b sob·a, withg¡1as the conjugating element. Finally, to show tran- sitivity, the relations a·bandb·cimply that there are elements g1 andg2such thatb=g1ag¡1 1andc=g2bg¡1 2. Hence, c=g2bg¡1 2=g2g1ag¡1 1g¡1 2=(g2g1)a(g2g1)¡1 socis conjugate to awith the conjugating element g1g2. Thus, conju- gation fulﬂlls the three conditions of an equivalency class. One important consequence of equivalence is that it permits the assembly of classes , i.e., sets of equivalent quantities. In particular, aconjugacy class is the totality of elements which can be obtained from a given group element by conjugation. Group elements in thesame conjugacy class have several common properties. For example,all elements of the same class have the same order. To see this, webegin with the deﬂnition of the order nof an element aas the smallest integer such that a n=e. An arbitrary conjugate bofaisb=gag¡1. Hence, bn=(gag¡1)(gag¡1)¢¢¢(gag¡1)| {z } nfactors=gang¡1=geg¡1=e Elements of Abstract Group Theory 29 sobhas the same order as a. Example2.9. The groupS3has three classes: feg,fa;b;cg, andfd;fg. As we discussed in Example 2.5, the order of a,b, andcis two, and the order ofdandfis 3. The order of the unit element is 1 and is always in a class by itself. Notice that each class corresponds to a distinct kindof symmetry operation on an equilateral triangle. The operations a,b, andccorrespond to re°ections, while dandfcorrespond to rotations. In terms of operations in S 3, the elements dandfcorrespond to cyclic permutations of the reference order, e.g., 1 !2, 2!3, and 3!1, whilea,b, andccorrespond to permutations which are not cyclic. 2.6.2 Self-Conjugate Subgroups A subgroup HofGisself-conjugate if the elements gHg¡1are identical with those of Hfor all elements gofG. The terms invariant subgroup andnormal subgroup are also used. A group with no self-conjugate proper subgroups is called simple .I fgHg¡1=Hfor allginG, then given an element h1inH, for anya, we can ﬂnd an element h2inHsuch thatah1a¡1=h2, which implies that ah1=h2a, or thataH=Ha. This last equality yields another deﬂnition of a self-conjugate subgroupas one whose left and right cosets are equal. From the deﬂnition of self-conjugacy and of classes, we can furthermore conclude that a subgroupHofGis self-conjugate if and only if it contains elements of Gin complete classes, i.e., Hcontains either all or none of the elements of classes ofG. The cosets of a self-conjugate subgroup are themselves endowed with a group structure, with multiplication corresponding to an element-by-element composition of two cosets and discounting duplicate products.We show ﬂrst that the multiplication of the elements of two right cosetsof a conjugate subgroup yields another right coset. Let Hbe a self- conjugate subgroup of Gand consider the two right cosets HaandHb. Then, the multiplication of HaandHbproduces products of the form h iahjb=hi(ahj)b 30 Elements of Abstract Group Theory The product ahjcan be written as hkafor somehkinH, sinceHis assumed to be self-conjugate. Thus, we have hi(ahj)b=hi(hka)b=(hihk)(ab) which is clearly an element of a right coset of H. Example2.10. Consider the subgroup fe;d;fgofS3. Right-multiplying this subgroup by each element of S3yields the right cosets of this sub- group: fe;d;fge=fe;d;fg;fe;d;fga=fa;c;bg;fe;d;fgb=fb;a;cg fe;d;fgc=fc;b;ag;fe;d;fgd=fd;f;eg;fe;d;fgf=ff;e;dg Similarly, left-multiplying by each element of S3produces the left cosets of this subgroup: efe;d;fg=fe;d;fg;afe;d;fg=fa;b;cg;bfe;d;fg=fb;c;ag cfe;d;fg=fc;a;bg;dfe;d;fg=fd;f;eg;ffe;d;fg=ff;e;dg Thus, since the right and left cosets of fe;d;fgare the same, these elements form a self-conjugate subgroup of S3whose distinct cosets arefe;d;fgandfa;b;cg. Multiplying these subgroups together and neglecting duplicate elements yields fe;d;fgfe;d;fg=fe;d;fg;fe;d;fgfa;b;cg=fa;b;cg fa;b;cgfe;d;fg=fa;b;cg;fa;b;cgfa;b;cg=fe;d;fg Thequotient group (also called the factor group ) of a self-conjugate subgroup is the collection of cosets, each being considered an element.The order of the factor group is equal to the index of the self-conjugatesubgroup. With the notation used above, the quotient group is denotedbyG=H . Elements of Abstract Group Theory 31 Example 2.11. The cosets of the self-conjugate subgroup fe;d;fgof S3arefe;d;fgandfa;b;cg, so the order of the factor group is two. If we use the notation E=fe;d;fg;A=fa;b;cg (2.2) for the elements of the factor group, we can use the results of Example 2.8 to construct the multiplication table for this group (shown below)from which see that Eis the identity of the factor group, and Eand EA EEA AAE Aare their own inverses. Note that this multiplication table has the identical structure as the two-element group fe;agdiscussed in Sec. 2.4. 2.7 Summary In this chapter, we have covered only the most basic properties of groups. One of the remarkable aspects of this subject, already evidentin some of the discussion here, is that the four properties that deﬂne agroup, have such an enormous implication for the properties of groups,quite apart from their implications for physical applications, which willbe explored throughout this course. A comprehensive discussion of themathematical theory of groups, including many wider issues in bothpure and applied mathematics, may be found in the book by Gallian. 5 5J.A. Gallian, Contemporary Abstract Algebra 4th edn. (Houghton Mi†in, Boston, 1998). Chapter 3 Representations of Groups How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriateto the objects of reality? |Albert Einstein The structure of abstract groups developed in Chapter 2 forms the basis for the application of group theory to physical problems. Typi-cally in such applications, the group elements correspond to symmetryoperations which are carried out on spatial coordinates. When theseoperations are represented as linear transformations with respect to acoordinate system, the resulting matrices, together with the usual rulefor matrix multiplication, form a group that is equivalent to the groupof symmetry operations in a sense to be made precise later in this chap-ter. In essence, these matrices form what is called a representation of the symmetry group with each element corresponding to a particularmatrix. For applications to quantum mechanics, as we have seen in Sec- tion 1.2, the symmetry operations are performed on the Hamiltonian,whose invariance properties determine the symmetry group. The wave- functions, which do not all share the symmetry of the Hamiltonian,will be seen to determine the representations of the symmetry group inthe sense described above. These representations will, in turn, providea classiﬂcation scheme for the eigenfunctions of the Hamiltonian, in 33 34 Representations of Groups analogous fashion to the identiﬂcation of even and odd eigenfunctions in Section 1.2. The strength of the group-theoretic formalism that wewill develop is that this procedure can be carried out in a systematicfashion for a Hamiltonian having any symmetry without undue com-putational eﬁort. In this chapter, we will set out the basic deﬂnitions that enable us to construct a mathematical deﬂnition of what we mean by a representa-tion and discuss the basic types of representation. In the next chapterwe will develop a number of remarkable properties of representationsthat lie at the heart of applications of discrete group theory to quantummechanics. 3.1 Homomorphisms and Isomorphisms Consider two ﬂnite groups GandG0with elementsfe;a;b;:::gand fe0;a0;b0;:::gand which need not be of the same order. Suppose there is a mapping `between the elements of GandG0which preserves their composition rules, i.e., if a0=`(a) andb0=`(b), then `(ab)=`(a)`(b)=a0b0 If the order of the two groups is the same, then this mapping is said to be anisomorphism and the two groups are isomorphic to one another. This is denoted by G…G0. If the order of the two groups is not the same, then the mapping is a homomorphism and the two groups are said to be homomorphic . Thus, an isomorphism is a one-to-one correspondence between two groups, while a homomorphism is a many-to-one correspondence. An isomorphism preserves the structure of theoriginal group, but a homomorphism causes some of the structure of theoriginal group to be lost. Both properties are re°ected in the behaviorof multiplication tables under these mappings. Homomorphisms andisomorphisms are not limited to ﬂnite groups nor even to groups withdiscrete elements. Example 3.1. We saw in Sec. 2.2 that S 3is isomorphic to the planar symmetry operations of an equilateral triangle, since there is a one-to-one correspondence between the elements of the two groups and they Representations of Groups 35 have the same multiplication table. On the other hand, consider the correspondence between the elements of S3and the elements of the quotient group of S3discussed in Examples 2.9 and 2.11): fe;d;fg7!fEg;fa;b;cg7!fAg (3.1) i.e. the mapping `is deﬂned by `(e)=E`(d)=E`(f)=E `(a)=A`(b)=A`(c)=A(3.2) This is a homomorphism because three elements of S3correspond to a single element of the quotient group. To see that this mapping preservesmultiplication, we rearrange the multiplication table of S 3(Example 2.2) as follows: edfabc eedfabc ddfe cab ffed bca aabc edf bbcafed ccabdfe7!EA EEA AAE where the mapping of the multiplication table onto the elements fE;Ag is precisely that of a two-element group (cf. Example 2.9). This homo-morphism clearly causes some of the structure of the original group tobe lost. For example, S 3is non-Abelian group, but the two-element group is Abelian. 3.2 Representations Arepresentation of dimension nof an abstract group Gis a homomor- phism or isomorphism between the elements of Gand the group of nonsingular n£nmatrices (i.e. n£nmatrices with non-zero deter- minant) with complex entries and with ordinary matrix multiplicationas the composition law (Example 2.4). An isomorphic representation 36 Representations of Groups xy Figure 3.1: The coordinate system used to generate a two-dimensional rep- resentation of the symmetry group of the equilateral triangle. The origin ofthe coordinate system coincides with the geometric center of the triangle. is called a faithful representation and a homomorphic representation is called an unfaithful representation. According to this deﬂnition, if elements aandbofGare assigned matricesD(a) andD(b), thenD(a)D(b)=D(ab). The nonsingular nature of the matrices is required because inverses must be containedin the set (Example 2.4). Representations can also be comprised ofnumbers; the dimensionality of such representations is unity. Example3.2. Consider the following matrix representation of S 3based on the correspondence with planar symmetry operations of an equilat-eral triangle: e=ˆ10 01! ;a =1 2ˆ1¡p 3 ¡p 3¡1! ;b =1 2ˆ1p 3 p 3¡1! c=ˆ¡10 01! ;d=1 2ˆ¡1¡p 3 p 3¡1! ;f =1 2ˆ¡1p 3 ¡p 3¡1!(3.3) These matrices were generated by regarding each of the symmetry op- erations as a linear transformation in the coordinate system shown inFig. 3.1. Matrices a,b, andccorrespond to re°ections, so their deter- minant is¡1, while matrices dandfcorrespond to rotations, so their determinant is 1. These matrices form a faithful representation of S 3. Representations of Groups 37 Consider now the following mapping between the elements of S3 and the setf1;¡1g: fe;fg7!f 1g;fa;b;cg7!f¡ 1g (3.4) This is essentially a mapping between the elements of S3and the de- terminant of their matrix representation discussed above. Thus, theidentity matrix eand the rotations dandfhave determinants of 1, while the re°ections a,b, andchave determinants of ¡1. The physical interpretation of this homomorphism is therefore as a mapping fromthe individual elements of S 3to their parity, i.e., whether they change the orientation of the coordinate system ( ¡1) or not (1).1Since the determinant provides less information about a transformation than itsmatrix representation, it is clear that some information about the groupstructure of S 3is not preserved by this homomorphism. Finally, we note that the mapping of allelements to unity, fe;a;b;c;d;fg7! 1 is a representation of any group, though clearly an unfaithful one. This is called the identical representation . In the present case, the identical representation corresponds to a mapping from the group element tothe absolute value of the determinant. Since all of the transformationspreserve the lengths of vectors, any product of these transformationsdoes so as well. Representations of groups are important in quantum mechanics for several reasons. First, the eigenfunctions of a Hamiltonian will trans-form under the symmetry operations of that Hamiltonian according toa particular representation of that group. Second, quantum mechanicaloperators are often written in terms of their matrix elements, so it isconvenient to write symmetry operations in the same kind of matrixrepresentation. Moreover, the evaluation of these matrix elements maysometimes be simpliﬂed by identifying the appropriate selection rules 1In terms of the operations of S3, even parity corresponds to an even number of pairwise interchanges, while odd parity corresponds to an odd number of suchinterchanges. 38 Representations of Groups (Section 1.2). Finally, the algebra of matrices is generally simpler to carry out than abstract symmetry operations. Thus, in the next section,we discuss some of the important properties of matrix representationsof groups. 3.3 Reducible and Irreducible Represen- tations The deﬂnition of a representation provides for considerable °exibility in constructing matrix representations, which is manifested in severalways, but also indicates that representations are not unique. We con-sider some examples. Given a matrix representation n D(e);D(a);D(b);:::o of an abstract group with elements fe;a;b;:::g, we can obtain a new set of matrices which also form a representation by performing a transfor-mation known variously as a similarity ,equivalence ,o rcanonical trans- formation (cf. Sec. 2.6): n BD(e)B¡1;BD (a)B¡1;BD (b)B¡1;:::o (3.5) Such transformations arise quite naturally, for example, in carrying out a change of basis for a set of matrices. Thus, suppose one begins withthe matrix equation b=Aarelating two vectors aandbthrough a transformation A. If we now wish to express this equation in another basis which is obtained from the original basis by applying a transfor-mationB, we can write Bb=BAa=BAB ¡1Ba so in the new basis, our original equation becomes b0=A0a0 whereb0=Bb,a0=Ba, andA0=BAB¡1. A similarity transfor- mation can therefore be interpreted as a sequence of transformations Representations of Groups 39 involving ﬂrst a transformation to the original basis ( B¡1), then per- forming the transformation A, and ﬂnally transforming back to the new basis ( B). Referring back to the discussion in Section 2.6 on con- jugacy classes, we see that group elements in the same conjugacy classrepresent the same type of transformation (e.g., re°ection or rotation)which can be transformed into one another by particular symmetryoperations. Suppose we have representations of dimensions mandn. We can construct a representation of dimension m+nby forming block-diagonal matrices: ("D(e)0 0D0(e)# ;"D(a)0 0D0(a)# ;"D(b)0 0D0(b)# ;:::) (3.6) wherefD(e);D(a);D(b);:::gis ann-dimensional representation and fD0(e);D0(a);D0(b);:::ganm-dimensional representation of the group G, and the symbol 0 is an n£mor anm£nzero matrix, as required by its position in the supermatrix. Each of the m+n-dimensional matrices formed in this manner is called a directsum of then- andm-dimensional component matrices. The direct sum is denoted by \ '" to distinguish it from the ordinary addition of two matrices. Thus, we can write therepresentation in (3.6) as n D(e)'D0(e);D(a)'D0(a);D(b)'D0(b);¢¢¢o The two representations that form this direct sum can be either distinct or identical and, of course, the block-diagonal form can be continuedindeﬂnitely simply by incorporating additional representations in diag-onal blocks. However, in all such constructions, we are not actuallygenerating anything intrinsically new; we are simply reproducing theproperties of known representations. Thus, although representationsare a convenient way of associating matrices with group elements, thefreedom we have in constructing representations, exempliﬂed in (3.5)and (3.6), does not readily demonstrate that these matrices embodyany intrinsic characteristics of the group they represent. Accordingly,we now describe a way of classifying equivalent representations andthen introduce a reﬂnement of our deﬂnition of a representation. 40 Representations of Groups To overcome the problem of nonuniqueness posed by representations that are related by similarity transformations we consider the sum ofthe diagonal elements of an n£nmatrixA, called the trace ofAand by \tr": tr(A)=nX i=1Aii The utility of the trace stems from its invariance under similarity trans- formations, i.e., tr(A) = tr(BAB¡1) The importance of this invariance, the proof of which is discussed in Problem Set 4, is that, although there is an inﬂnite variety of represen-tations related by similarity transformations, each such representationhas the same set of traces associated with each of its elements. But working with the trace alone does not alleviate the nonunique- ness of representations posed by (3.6). To address this issue, we in-troduce the concept of an irreducible representation. Representations such as those in (3.6) are termed reducible because they are the direct sum of two (or more) representations. We could, of course, performa similarity transformation to obtain a representation that is not inblock form, but the representation so obtained is still deemed to bereducible because it was obtained from matrices which originally werein block form. Based on these considerations, we deﬂne reducible andirreducible representations as follows: Deﬁnition. If the same similarity transformation brings all of the ma- trices of a representation into the same block form (by which we meanmatrices of the same dimension in the same positions), then this repre-sentation is said to be reducible . Otherwise, the representation is said to beirreducible . Thus, irreducible representations cannot be expressed in terms of representations of lower dimensionality. One-dimensional representa-tions are, by deﬂnition, always irreducible. Determining the irreducible Representations of Groups 41 representations of groups is one of the central issues to be covered in the following chapters. Example 3.4. All of the representations of S3discussed in Example 3.2 are irreducible . This is clear for the identical representation and for the representation in (3.4), since they are composed of numbers. Butwe can use these representations to construct the following manifestlyreducible representation of S 3: e=ˆ10 01! ;a =ˆ10 0¡1! ;b =ˆ10 0¡1! c=ˆ10 0¡1! ;d=ˆ10 01! ;f =ˆ10 01! The representation in (3.3) is irreducible . There is no similarity transformation that will bring all of the matrices into block-diagonalform which, for the case here, means simple diagonalization. The easiestway to see this is from the point of view of the commutability of twomatrices. If two matrices can be brought into diagonal form by the samesimilarity transformation, then they commute. As diagonal matrices,they certainly commute, so they must also commute in their originalform. But a glance at the multiplication table for these matrices (recallthat they are a faithful representation of S 3) in Example 2.2 shows that they do not all commute. Hence, they cannot all be simultaneouslydiagonalized, so this representation is irreducible . 3.4 Unitary Representations Representations of groups are useful because of orthogonality theorems which we will prove in the next chapter. As background to that dis-cussion, we will prove in this section an important result about theunitarity of representations. But we ﬂrst review some general proper-ties of matrices. We begin by considering the transformation of an n£nmatrixA with entries A ij,i;j=1;2;:::;n , under the action of various opera- 42 Representations of Groups tions. The complex conjugate ofA, denoted by A⁄, has entries which are the complex conjugates of the corresponding entries of A: (A⁄)ij=(Aij)⁄(3.7) The transpose ofA, denoted by At, has its rows and columns inter- changed with respect to those of A: (At)ij=Aji (3.8) When applied to vectors, the transpose transforms a row vector into a column vector and vice versa . The transpose of a product of matrices A;B;C;::: is (ABC¢¢¢)t=¢¢¢CtBtAt(3.9) i.e., the order of matrix multiplication is reversed . This can be proven easily from the deﬂnition (3.8). Finally, the adjoint orHermitian conju- gate ofA, denoted by Ay, is the transposed conjugate of A, i.e. (Ay)ij=(Aji)⁄(3.10) In common with the transpose, the application of the Hermitian conju- gate to a product of matrices A;B;C;::: can be expressed as a product of Hermitian conjugates of the individual matrices, but with the orderreversed: (ABC¢¢¢) y=¢¢¢CyByAy(3.11) 3.4.1 Hermitian and Orthogonal Matrices A matrixAisHermitian if Ay=A (3.12) Hermitian matrices and Hermitian operators are familiar from quan- tum mechanics, where their properties of having real eigenvalues andorthogonal eigenvectors are of fundamental importance. A matrix Ais orthogonal if its transpose is its inverse: AtA=AAt=I (3.13) Representations of Groups 43 whereIis then£nunit matrix. In terms of matrix components, this condition reads nX k=1akiakj=nX k=1aikajk=–ij (3.14) where –ij=(0;ifi6=j; 1;ifi=j(3.15) is theKronecker delta . Thus, the rows of an orthogonal matrix are mutually orthogonal, as are the columns. The consequences of theorthogonality of a transformation matrix can be seen by examining theeﬁect of applying an orthogonal matrix Ato twon-dimensional vectors uandv, yielding vectors u 0andv0: u0=Au;v0=Av We now take the scalar, or ‘dot,’ product between u0andv0: (u0;v0)=(u0)tv0=(Au)tAv=utAtAv=utv=(u;v) (3.16) where we have used (3.11) and the fact that Ais orthogonal. This shows that the relative orientations and the lengths of vectors are pre-served by orthogonal transformations. Such transformations are eitherrigid rotations, which preserve the \handedness" (i.e., left or right)of a coordinate system, and are called proper rotations, or re°ections, which reverse the \handedness" of a coordinate system, and are called improper \rotations." 3.4.2 Unitary Matrices A third type of matrix, called unitary , has the property that AyA=AAy=I (3.17) By writing this condition in terms of matrix components, nX k=1a⁄ kiakj=nX k=1aika⁄ jk=–ij (3.18) 44 Representations of Groups we see that, in common with orthogonal matrices, the rows and columns of a unitary matrix are orthogonal, but with respect to a diﬁerent scalarproduct. For two vectors uandv, this scalar product is deﬂned as (u;v)=u yv (3.19) This generalizes the familiar dot product to complex vectors in ndi- mensions. We can now show, by proceeding as above, that unitarytransformations leave the scalar product invariant: (u 0;v0)=(u0)yv0=(Au)yAv=uyAyAv=uyv=(u;v) (3.20) The property of unitarity, when applied to operators, is of immense importance in quantum mechanics because it enables changes of basesto be performed while preserving the orthogonality of bases and, thus,the overlap between wavefunctions. In this sense, unitary matrices are associated with proper and improper \rotations," in analogy withorthogonal matrices. 3.4.3 Diagonalization of Hermitian Matrices⁄ LetHbe ann£nHermitian matrix. The eigenvalue equation for this matrix is Ha=‚a (3.21) By writing this equation as (H¡‚I)a= 0 (3.22) the eigenvalue equation in (3.21) has nontrivial solutions for aif and only if the determinant of the matrix of coe–cients in (3.22) vanishes: det(H¡‚I) = 0 (3.23) The expansion of the determinant leads to an nth-order polynomial in ‚whose solution yields the n(not necessarily distinct) eigenvalues of H:‚1;‚2;:::;‚n. We now show that the eigenvectors of Hwhich correspond to dis- tinct eigenvalues are orthogonal. Consider the eigenvalue equations for Representations of Groups 45 two eigenvectors aandbcorresponding to distinct eigenvalues ‚and „, respectively: Ha=‚a (3.24) Hb=„b (3.25) We now take the scalar product between band (3.24) and that between (3.25) anda: by(Ha)=‚bya (3.26) (byHy)a=„bya (3.27) Subtracting (3.27) from (3.26), and using the fact that His Hermitian yields (‚¡„)bya=byHa¡byHya= 0 (3.28) which, since ‚6=„, implies that bya= 0, i.e., that aandbare or- thogonal. If ‚and„are not distinct, we must use a Gram{Schmidt procedure to explicitly construct an orthogonal set of eigenvectors as-sociated with the degenerate eigenvalue. Thus, the eigenvectors of aHermitian matrix can always be chosen to form an orthogonal set. Consider the matrix Uwhose columns are the eigenvectors of H: U=(a 1;a2;:::;an) We can then write (3.21) in a form that subsumes allthe eigenvectors ofHas follows: HU=UD (3.29) whereDis the diagonal matrix whose entries are the eigenvalues of H: D=0 BBBBBB@‚10¢¢¢ 0 0‚2¢¢¢ 0 ............ 00¢¢¢‚ d1 CCCCCCA 46 Representations of Groups Since the rows of Uare composed of the (orthogonal) eigenvectors of H, it has the property that (cf. 3.18) UyU=UUy=I i.e.,U¡1=Uy,s oUisunitary . Hence, we can rewrite (3.29) as U¡1HU=UyHU=D We have proven the following theorem: Theorem 3.1. Any Hermitian matrix can be diagonalized by an ap- propriate unitary transformation. This theorem will be used in the next section to prove an important result concerning the existence of unitary group representations. 3.4.4 Transformation to Unitary Representations We have seen in Section 3.3 that there is considerable °exibility inconstructing group representations. In this section, we take a ﬂrst stepin restricting this freedom by showing that any representation can beexpressed entirely in terms of unitary matrices. Quite apart from theconvenient properties of unitary matrices discussed in Section 3.4.2,this theorem allows to think of group representations as proper andimproper complex \rotations." Theorem 3.2. Every representation can be brought into unitary form by a similarity transformation. Proof. LetfA 1;A2;:::;AjGjgbe ad-dimensional representation of a groupG, i.e., theAﬁare a set ofjGjd£dmatrices with nonvanishing determinants. From these matrices we form a matrix Hgiven by the sum H=jGjX ﬁ=1AﬁAy ﬁ Representations of Groups 47 This matrix is Hermitian because, using the property (3.11), Hy=X ﬁ(AﬁAy ﬁ)y=X ﬁAﬁAyﬁ=H According to Theorem 3.1, any Hermitian matrix can be diagonalized by some unitary transformation U. Denoting the diagonalized form of HbyD,w eh a v eD=UyHU, which enables us to write Das D=X ﬁUyAﬁAy ﬁU=X ﬁ(UyAﬁU)(UyAy ﬁU)=X ﬁ(UyAﬁU)(UyAﬁU)y By introducing the notation ~Aﬁ=UyAﬁU, we can write the last equa- tion in a more concise form as D=X ﬁ~Aﬁ~Ay ﬁ (3.30) The diagonal elements of Dare real, because Dkk=X ﬁX j(~Aﬁ)kj(~Ayﬁ)jk =X ﬁX j(~Aﬁ)kj(~Aﬁ)⁄ kj =X ﬁX jﬂﬂﬂ(~Aﬁ)kjﬂﬂﬂ2 fork=1;2;:::;d , and positive, because the summation over jincludes a diagonal element of the identity, which is a d£dunit matrix, and hence is equal to unity. Thus, the diagonal matrix D1=2, D1=2=0 BBBBBBB@D1=2 11 0¢¢¢ 0 0D1=2 22¢¢¢ 0 ............ 00¢¢¢D1=2 dd1 CCCCCCCA andD¡1=2, which is given by an analogous expression, both have posi- tive entries. 48 Representations of Groups We now form the matrices Bﬁ=D¡1=2~AﬁD1=2 from which we obtain the corresponding Hermitian conjugates: By ﬁ=(D¡1=2~AﬁD1=2)y=D1=2~AyﬁD¡1=2 We will now demonstrate that the Bﬁare unitary by ﬂrst showing that the product BﬁBy ﬁis equal to the identity matrix. The product BﬁBy ﬁ is given by BﬁBy ﬁ=‡ D¡1=2~AﬁD1=2·‡ D1=2~AyﬁD¡1=2· =D¡1=2~AﬁD~AyﬁD¡1=2 The deﬂnition of Din (3.30) and the associativity of matrix multipli- cation allow us to write this expression as BﬁBy ﬁ=D¡1=2X j~Aﬁ~Aﬂ~Ayﬂ~AyﬁD¡1=2 =D¡1=2X j(~Aﬁ~Aﬂ)(~Aﬁ~Aﬂ)yD¡1=2 Since theAﬁare a representation of G, then so are the ~Aﬁ(Problem 3, Problem Set 4). Hence, the product ~Aﬁ~Aﬂis another matrix ~A°in this representation. Moreover, according to the Rearrangement Theorem,the sum over all ﬂmeans that the set of ~A °obtained from these prod- ucts contains the matrix corresponding to each group element once andonly once. Thus, B ﬁBy ﬁ=D¡1=2X °~A°~Ay ° |{z} DD¡1=2=I whereIis thed£dunit matrix. This result can also be used to show thatBy ﬁBﬁ=I. Thus, the Bﬁ, which are obtained from the original representation by a similarity transformation, Bﬁ=D¡1=2U¡1AﬁUD1=2=(UD1=2)¡1Aﬁ(UD1=2) form a unitary representation of G. Hence, without any loss of gener- ality, we may always assume that a representation is unitary. Representations of Groups 49 3.5 Summary The main concepts introduced in this chapter are faithful and unfaithful representations, based on isomorphic and homomorphic mappings, re-spectively, reducible and irreducible representations, and the fact thatwe may conﬂne ourselves to unitary representations of groups. In thenext chapter we will focus on irreducible representations, both faithfuland unfaithful, since these cannot be decomposed into representationsof lower dimension and are, therefore, \intrinsic" to a symmetry group,since allreducible representations will be shown to be composed of direct sums of irreducible representations. Irreducible representationsoccupy a special place in group theory because they can be classiﬂed fora given symmetry group solely according to their traces and dimension. 50 Representations of Groups Chapter 4 Properties of Irreducible Representations Algebra is generous; she often gives more than is asked of her. |Jean d’Alembert We have seen in the preceding chapter that a reducible representa- tion can, through a similarity transformation, be brought into block-diagonal form wherein each block is an irreducible representation. Thus,irreducible representations are the basic components from which allrepresentations can be constructed. But the identiﬂcation of whether arepresentation is reducible or irreducible is a time-consuming task if itrelies solely on methods of linear algebra. 1In this chapter, we lay the foundation for a more systematic approach to this question by derivingthe fundamental theorem of representation theory, called the Great Or-thogonality Theorem. The utility of this theorem, and its central rolein the applications of group theory to physical problems, stem from thefact that it leads to simple criteria for determining irreducibility andprovides a direct way of identifying the number of inequivalent repre-sentations for a given group. This theorem is based on two lemmas ofSchur, which are the subjects of the ﬂrst two sections of this chapter. 1K. Hoﬁman and R. Kunze, Linear Algebra 2nd edn (Prentice{Hall, Englewood Cliﬁs, New Jersey, 1971), Ch. 6,7. 51 52 Properties of Irreducible Representations 4.1 Schur’s First Lemma Schur’s two lemmas are concerned with the properties of matrices that commute with all of the matrices of a irreducible representations. Theﬂrst lemma addresses the properties of matrices which commute witha given irreducible representation: Theorem4.1(Schur’sFirstLemma) . A non-zero matrix which com- mutes with all of the matrices of an irreducible representation is aconstant multiple of the unit matrix. Proof. LetfA 1;A2;:::;AjGjgbe the matrices of a d-dimensional irreducible representation of a group G, i.e., theAﬁared£dmatrices which cannot all be brought into block-diagonal form by the same sim-ilarity transformation. According to Theorem 3.2, we can take thesematrices to be unitary without any loss of generality. Suppose there isa matrixMthat commutes with all of the A ﬁ: MAﬁ=AﬁM (4.1) forﬁ=1;2;:::;jGj. By taking the adjoint of each of these equations, we obtain Ay ﬁMy=MyAy ﬁ: (4.2) Since theAﬁare unitary, Ay ﬁ=A¡1 ﬁ, so multiplying (4.2) from the left and right by Aﬁyields MyAﬁ=AﬁMy; (4.3) which demonstrates that, if Mcommutes with every matrix of a repre- sentation, then so does My. Therefore, given the commutation relations in (4.1) and (4.3) any linear combination of MandMyalso commute with these matrices: (aM+bMy)Aﬁ=Aﬁ(aM+bMy); whereaandbare any complex constants. In particular, the linear combinations H1=M+My;H 2=i(M¡My) Properties of Irreducible Representations 53 yield Hermitian matrices:Hi=Hy ifori=1;2. We will now show that a Hermitian matrix which commutes with all the matrices of anirreducible representation is a constant multiple of the unit matrix. Itthen follows that Mis also such a matrix, since M= 1 2(H1¡iH2) (4.4) The commutation between a general Hermitian matrix Hand the Aﬁis expressed as HAﬁ=AﬁH: (4.5) SinceHis Hermitian, there is a unitary matrix Uwhich transforms H into a diagonal matrix D(Theorem 3.1): D=U¡1HU: We now perform the same similarity transformation on (4.5): U¡1HAiU=U¡1HUU¡1AiU =U¡1AiHU=U¡1AiUU¡1HU By deﬂning ~Aﬁ=U¡1AﬁU, the transformed commutation relation (4.5) reads D~Aﬁ=~AﬁD: (4.6) Using the fact that Dis a diagonal matrix, i.e., that its matrix elements areDij=Dii–ij, where–ijis the Kronecker delta, the ( m;n)th matrix element of the left-hand side of this equation is (D~Aﬁ)mn=X kDmk(~Aﬁ)kn=X kDmm–mk(~Aﬁ)kn=Dmm(Aﬁ)mn: Similarly, the corresponding matrix element on the right-hand side is (~AﬁD)mn=X k(~Aﬁ)mkDkn=X k(~Aﬁ)mkDnn–kn=(~Aﬁ)mnDnn: Thus, after a simple rearrangement, the ( m;n)th matrix element of (4.6) is (~Aﬁ)mn(Dmm¡Dnn)=0: (4.7) 54 Properties of Irreducible Representations There are three cases that we must consider to understand the impli- cations of this equation. Case I. Suppose that all of the diagonal elements of Dare dis- tinct:Dmm6=Dnnifm6=n. Then, (4.7) implies that (~Aﬁ)mn=0;m6=n; i.e., the oﬁ-diagonal elements of ~Aﬁmust vanish, these are diagonal ma- trices and, therefore, according to the discussion in Section 3.3, theyform a reducible representation composed of done-dimensional rep- resentations. Since the ~A iare obtained from the Aiby a similarity transformation, the Aithemselves form a reducible representation. Case II. If all of the diagonal elements of Dare equal, i.e. Dmm= Dnnforallmandn, thenDis proportional to the unit matrix. The (~Aﬁ)mnare not required to vanish for anymandn. Thus, only this case is consistent with the requirement that the Aﬁform an irreducible representation. If Dis proportional to the unit matrix, then so is H=UDU¡1and, according to (4.4), the matrix Mis as well. Case III. Suppose that the ﬂrst pdiagonal entries of Dare equal, but the remaining entries are distinct from these and from each other:D 11=D22=¢¢¢=Dpp,Dmm6=Dnnotherwise. The ( ~Aﬁ)mnmust vanish for any pair of unequal diagonal entries. These correspond tothe cases where bothmandnlie in the range 1 ;2;:::;p and wherem andnare equal and both greater than p,s o allthe ~A iall have the following general form: ~Ai=ˆB10 0B2! ; whereB1is ap£pmatrix and B2i sa(p¡d)£(p¡d)diagonal matrix. Thus, the ~Aiare block diagonal matrices and are, therefore, reducible . We have shown that if a matrix that not a multiple of the unit matrix commutes with all of the matrices of a representation, thenthat representation is necessarily reducible (Cases I and III). Thus, if a non-zero matrix commutes with all of the matrices of an irreducible representation (Case III), that matrix must be a multiple of the unitmatrix. This proves Schur’s lemma. Properties of Irreducible Representations 55 4.2 Schur’s Second Lemma Schur’s ﬂrst lemma is concerned with the commutation of a matrix with a given irreducible representation. The second lemma generalizes thisto the case of commutation with two distinct irreducible representationswhich may have diﬁerent dimensionalities. Its statement is as follows: Theorem 4.2 (Schur’s Second Lemma). LetfA 1;A2;:::;AjGjgand fA0 1;A02;:::;A0 jGjgbe two irreducible representations of a group Gof dimensionalities dandd0, respectively. If there is a matrix Msuch that MAﬁ=A0 ﬁM forﬁ=1;2;:::;jGj, then ifd=d0, eitherM= 0 or the two represen- tations diﬁer by a similarity transformation. If d6=d0, thenM=0 . Proof. Given the commutation relation between Mand the two irreducible representations, MAﬁ=A0 ﬁM; (4.8) we begin by taking the adjoint: Ay ﬁMy=MyA0y ﬁ: (4.9) Since, according to Theorem 3.2, the Aﬁmay be assumed to be unitary, Ay ﬁ=A¡1 ﬁ, so (4.9) becomes A¡1 ﬁMy=MyA0¡1 ﬁ: (4.10) By multiplying this equation from the left by M, MA¡1 ﬁMy=MMyA0¡1 ﬁ; and utilizing the commutation relation (4.8) to write MA¡1 ﬁ=A0¡1 ﬁM; we obtain A0¡1 ﬁMMy=MMyA0¡1 ﬁ: 56 Properties of Irreducible Representations Thus, thed0£d0matrixMMycommutes with all the matrices of an irreducible representation. According to Schur’s First Lemma, MMy must therefore be a constant multiple of the unit matrix, MMy=cI; (4.11) wherecis a constant. We now consider individual cases. Case I.d=d0.I fc6= 0, Eq. (4.11) implies that2 M¡1=1 cMy: Thus, we can rearrange (4.8) as Aﬁ=M¡1A0 ﬁM; so our two representations are related by a similarity transformation and are, therefore, equivalent. Ifc= 0, thenMMy= 0. The (i;j)th matrix element of this product is (MMy)ij=X kMik(My)kj=X kMikM⁄ jk=0: By settingi=j, we obtain X kMikM⁄ ik=X kjMikj2=0; which implies that Mik= 0 for alliandk, i.e., that Mis the zero matrix. This completes the ﬂrst part of the proof. Case II.d6=d0. We taked<d0. ThenMis arectangular matrix withdcolumns and d0rows: M=0 BBBBBB@M11¢¢¢M1d M21¢¢¢M2d ......... M d01¢¢¢Md0d1 CCCCCCA: 2By multiplying (4.10) from the right byMand following analogous steps as above, one can show that MyM=cI, so that the matrix c¡1Myis both the left and right inverse of M. Properties of Irreducible Representations 57 We can make a d0£d0matrixNfromMby addingd0¡dcolumns of zeros: N=0 BBBBBB@M11¢¢¢M1d0¢¢¢ 0 M21¢¢¢M2d0¢¢¢ 0 .................. M d01¢¢¢Md0d0¢¢¢ 01 CCCCCCA·(M;0): Taking the adjoint of this matrix yields Ny=0 BBBBBBBBBBBBBBBBB@M11M⁄ 21¢¢¢M⁄ d01 M⁄ 12M⁄ 22¢¢¢M⁄ d02 ............ M ⁄ 1dM⁄ 2d¢¢¢M⁄ d0d 00¢¢¢ 0 ............ 00¢¢¢ 01 CCCCCCCCCCCCCCCCCA =ˆMy 0! : Note that this construction maintains the product MMy: NNy=(M;0)ˆMy 0! =MMy=cI: The determinant of Nis clearly zero. Thus, det(NNy) = det(N) det(Ny)=cd0=0 soc= 0, which means that MMy= 0. Proceeding as in Case I, we conclude that this implies M= 0. This completes the second part of the proof. 4.3 The Great Orthogonality Theorem Schur’s lemmas provide restrictions on the form of matrices which com-mute with all of the matrices of irreducible representations. But the 58 Properties of Irreducible Representations group property enables the construction of many matrices which sat- isfy the relations in Schur’s First and Second Lemmas. The interplaybetween these two facts provides the basis for proving the Great Or-thogonality Theorem. The statement of this theorem is as follows: Theorem4.3(GreatOrthogonalityTheorem). LetfA 1;A2;:::;AjGjg andfA0 1;A02;:::;A0 jGjgbe two inequivalent irreducible representations of a group Gwith elementsfg1;g2;:::;gjGjgand which have dimen- sionalitiesdandd0, respectively. The matrices AﬁandA0 ﬁin the two representations correspond to the element gﬁinG. Then X ﬁ(Aﬁ)⁄ ij(A0 ﬁ)i0j0=0 for all matrix elements. For the elements of a single unitary irreducible representation, we have X ﬁ(Aﬁ)⁄ ij(Aﬁ)i0j0=jGj d–i;i0–j;j0; wheredis the dimension of the representation. Proof. Consider the matrix M=X ﬁA0 ﬁXA¡1 ﬁ; (4.12) whereXis an arbitrary matrix with d0rows anddcolumns, so that M is ad0£d0matrix. We will show that for any matrix X,Msatisﬂes a commutation relation of the type discussed in Schur’s Lemmas. We now multiply Mfrom the left by the matrix A0 ﬂcorresponding to some matrix in the the \primed" representation: A0 ﬂM=X ﬁA0 ﬂA0ﬁXA¡1 ﬁ =X ﬁA0ﬂA0ﬁXA¡1 ﬁA¡1 ﬂAﬂ =X ﬁA0 ﬂA0ﬁX(AﬂAﬁ)¡1Aﬂ: (4.13) Properties of Irreducible Representations 59 Since theAﬁandA0 ﬁform representations of G, the products AﬁAﬂ andA0 ﬁA0ﬂyield matrices A°andA0 °, respectively, both corresponding to the same element in Gbecause representations preserve the group composition rule. Hence, by the Rearrangement Theorem (Theorem2.1), we can write the summation over ﬁon the right-hand side of this equation as X ﬁA0 ﬂA0ﬁX(AﬂAﬁ)¡1=X °A0 °XA¡1 °=M: Substituting this result into (4.13) yields A0 ﬂM=MAﬂ: (4.14) Depending on the nature of the two representations, this is precisely the situation addressed by Schur’s First and Second Lemmas. We considerthe cases of equivalent and inequivalent representations separately. Case I.d6=d 0or, ifd=d0, the representations are inequivalent (i.e., not related by a similarity transformation). Schur’s Second Lemmathen implies that Mmust be the zero matrix, i.e., that each matrix element of Mis zero. From the deﬂnition (4.12), we see that this requires M ii0=X ﬁX jj0(A0 ﬁ)ijXjj0(A¡1 ﬁ)j0i0=0: (4.15) By writing this sum as (note that because all sums are ﬂnite, their order can be changed at will) X jj0Xjj0•X ﬁ(Aﬁ)0 ij(A¡1 ﬁ)j0i0‚ =0; (4.16) we see that, since Xis arbitrary, each of its entries may be varied arbitrarily and independently without aﬁecting the vanishing of thesum. The only way to ensure this is to require that the coe–cients oftheX jj0vanish: X ﬁ(A0 ﬁ)ij(A¡1 ﬁ)j0i0=0: 60 Properties of Irreducible Representations For unitary representations, ( A¡1 ﬁ)j0i0=(Aﬁ)⁄ i0j0, so this equation re- duces to X ﬁ(A0 ﬁ)ij(Aﬁ)⁄ i0j0=0; which proves the ﬂrst part of the theorem. Case II.d=d0and the representations are equivalent. According to Schur’s First Lemma, M=cI, so, cI=X ﬁAﬁXA¡1 ﬁ: (4.17) Taking the trace of both sides of this equation, tr(cI)|{z} cd=t rµX ﬁAﬁXA¡1 ﬁ¶ =X ﬁtr(AﬁXA¡1 ﬁ)=X ﬁtr(X) |{z} jGjtr(X); yields an expression for c: c=jGj dtr(X): Substituting this into Eq. (4.17) and expressing the resulting equation in terms of matrix elements, yields X jj0Xjj0•X ﬁ(Aﬁ)ij(A¡1 ﬁ)j0i0‚ =jGj d–i;i0X jXjj; or, after a simple rearrangement, X jj0Xjj0"X ﬁ(Aﬁ)ij(A¡1 ﬁ)j0i0¡jGj d–i;i0–j;j0# =0: This equation must remain valid under any independent variation of the matrix elements of X. Thus, we must require that the coe–cient ofXjj0vanishes identically: X ﬁ(Aﬁ)ij(A¡1 ﬁ)j0i0=jGj d–i;i0–j;j0: Properties of Irreducible Representations 61 Since the representation is unitary, this is equivalent to X ﬁ(Aﬁ)ij(Aﬁ)⁄ i0j0=jGj d–i;i0–j;j0: This proves the second part of the theorem. 4.4 Some Immediate Consequences of the Great Orthogonality Theorem The Great Orthogonality Theorem establishes a relation between ma- trix elements of the irreducible representations of a group. Suppose wedenote the ﬁth matrix in the kth irreducible representation by A k ﬁand the (i;j)th element of this matrix by ( Ak ﬁ)ij. We can then combine the two statements of the Great Orthogonality Theorem as X ﬁ(Ak ﬁ)ij(Ak0 ﬁ)⁄ i0j0=jGj d–i;i0–j;j0–k;k0 (4.18) This expression helps us to understand the motivation for the name \Orthogonality Theorem" by inviting us to consider the matrix ele-ments of irreducible representations as entries in jGj-component vec- tors, i.e., vectors in a space of dimensionality jGj: V k ij=h (Ak 1)ij;(Ak 2)ij;:::; (Ak jGj)iji According to the statement of the Great Orthogonality Theorem, two such vectors are orthogonal if they diﬁer in any one of the indices i,j, ork, since (4.18) requires that Vk ij¢Vk0 i0j0=jGj d–i;i0–j;j0–k;k0 But, in ajGj-dimensional space there are at most jGjmutually or- thogonal vectors. To see the consequences of this, suppose we haveirreducible representations of dimensionalities d 1;d2;:::, where the dk 62 Properties of Irreducible Representations are positive integers. For the krepresentations, there are dkchoices for each of iandj, i.e., there are d2 kmatrix elements in each matrix of the representation. Summing over all irreducible representations, weobtain the inequality X kd2 k•jGj (4.19) Thus, the order of the group acts as an upper bound both for the number and the dimensionalities of the irreducible representations. Inparticular, a ﬂnite group can have only a ﬂnite number of irreducible representations. We will see later that the equality in (4.19) always holds. Example 4.1. For the group S 3, we have thatjGj= 6 and we have already identiﬂed two one-dimensional irreducible representations andone two-dimensional irreducible representation (Example 3.2). Thus,using (4.19), we have X kd2 k=12+12+22=6 so the Great Orthogonality Theorem tells us that there are no addi- tional distinct irreducible representations. For the two element group, we have found two one-dimensional rep- resentations,f1;1gandf1;¡1g(Example 3.3). According to the in- equality in (4.19), X kd2 k=1+1=2 so these are the only two irreducible representations of this group. 4.5 Summary The central result of this chapter is the statement and proof of the Great Orthogonality Theorem. Essentially all of the applications inthe next several chapters are consequences of this theorem. The impor-tant advance provided this theorem is that it provides an orthogonality Properties of Irreducible Representations 63 relation between the entries of the matrices of the irreducible repre- sentations of a group. While this can be used to test whether a givenrepresentation is reducible or irreducible (Problem Set 6), its main rolewill be in a somewhat \reduced" form, such as that used in Sec. 4.4 toplace bounds on the number of irreducible representations of a ﬂnitegroup. One of the most important aspects of the Great OrthogonalityTheorem for applications to physical problems is in the constructionof \character tables," i.e., tables of traces of matrices of an irreduciblerepresentation. This is taken up in the next chapter. 64 Properties of Irreducible Representations 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. Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems . . . These are the problem of the inﬂnitesimal, the inﬂnite, and continuity . . . |Bertrand Russell The groups we have considered so far have been, in all but a few cases, discrete and ﬂnite. Most of the central theorems for these groups andtheir representations have relied on carrying out sums over the groupelements, often in conjunction with the Rearrangement Theorem (The-orem 2.1). These results provide the basis for the application of groupsand representations to physical problems through the construction andmanipulation of character tables and the associated computations thatrequire direct sums, direct products, orthogonality and decomposition. But the notion of symmetry transformations that are based on con- tinuous quantities also occur naturally in physical applications. For example, the Hamiltonian of a system with spherical symmetry (e.g.,atoms and, in particular, the hydrogen atom) is invariant under allthree-dimensional rotations. To address the consequences of this in-variance within the framework of group theory necessitates confrontingseveral issues that arise from the continuum of rotation angles. Theseinclude deﬂning what we mean by a \multiplication table," determin-ing how summations over group elements are carried out, and deriving 107 108 Continuous Groups, Lie Groups, and Lie Algebras the appropriate re-statement of the Rearrangement Theorem to enable the Great Orthogonality Theorem and its consequences to be obtainedfor continuous groups. More generally, the existence of a continuumof group elements, when combined with the requirement of analyticity,introduces new structures associated with constructing diﬁerentials andintegrals of group elements. In eﬁect, this represents an amalgamationof group theory and analysis, so such groups are the natural objectsfor describing the symmetry of analytic structures such as diﬁerentialequations and those that arise in diﬁerential geometry. In fact, theintroduction of analytic groups by Sophus Lie late in the 19th centurywas motivated by the search for symmetries of diﬁerential equations. In this chapter we begin our discussion about the modiﬂcations to our development of groups and representations that are necessitated byhaving a continuum of elements. We begin in the ﬂrst section with thedeﬂnition of a continuous group and specialize to the most commontype of continuous group, the Lie group. We then introduce the ideaof an inﬂnitesimal generator of a transformation, from which everyelement can be obtained by repeated application. These generatorsembody much of the structure of the group and, because there are aﬂnite number of these entities, are simpler to work with than the fullgroup. This leads naturally to the Lie algebra associated with a Liegroup. All of these concepts are illustrated with the groups of properrotations in two and three dimensions. The representation of thesegroups, their character tables, and basis functions will be discussed inthe next chapter. 7.1 Continuous Groups Consider a set of elements Rthat depend on a number of real continuous parameters, R(a)·R(a1;a2;:::;ar). These elements are said to form acontinuous group if they fulﬂll the requirements of a group (Section 2.1) and if there is some notion of ‘proximity’ or ‘continuity’ imposedon the elements of the group in the sense that a small change in oneof the factors of a product produces a correspondingly small change intheir product. If the group elements depend on rparameters, this is called anr-parameter continuous group. Continuous Groups, Lie Groups, and Lie Algebras 109 In general terms, the requirements that a continuous set of elements form a group are the same as those for discrete elements, namely, clo-sure under multiplication, associativity, the existence of a unit, andan inverse for every element. Consider ﬂrst the multiplication of twoelementsR(a) andR(b) to yield the product R(c): R(c)=R(a)R(b): Thencmust be a continuous real function fofaandb: c=f(a;b): This deﬂnes the structure of the group in the same way as the multi- plication table does for discrete groups. The associativity of the com-position law, R(a) h R(b)R(c)i |{z} R[f(b;c)]=[R(a)R(b)i |{z} R[f(b;c)]R(c); requires that f[a;f(b;c)] =f[f(a;b);c]: The existence of an identity element, which we denote by R(a0), R(a0)R(a)=R(a)R(a0)=R(a); is expressed in terms of fas f(a0;a)=f(a;a0)=a: The inverse of each element R(a), denoted by R(a0), produces R(a0)R(a)=R(a)R(a0)=R(a0): Therefore, f(a0;a)=f(a;a0)=a0: Iffis an analytic function, i.e., a function with a convergent Tay- lor series expansion within the domain deﬂned by the parameters, the 110 Continuous Groups, Lie Groups, and Lie Algebras resulting group is called an r-parameter Liegroup , named after Sophus Lie, a Norwegian mathematician who provided the foundations for suchgroups. Our interest in physical applications centers around transformations ond-dimensional spaces. Examples include Euclidean spaces, where the variables are spatial coordinates, Minkowski spaces, where the variablesare space-time coordinates, and spaces associated with internal degreesof freedom, such as spin or isospin. In all cases, these are mappings ofthe space onto itself and have the general form x 0 i=fi(x1;x2;:::;xd;a1;a2;:::;ar);i =1;2;:::;d: If thefiare analytic, then this deﬂnes an r-parameter Lie group of transformations. Example 7.1 Consider the one-dimensional transformations x0=ax (7.1) whereais an non-zero real number. This transformation corresponds to stretching the real line by a factor a. The product of two such operations, x00=ax0andx0=bxis x00=ax0=abx: By writing x00=cx, we have that c=ab; (7.2) so the multiplication of two transformations is described by an analytic function that yields another transformation of the form in (7.1). Thisoperation is clearly associative, as well as Abelian, since the producttransformation corresponds to the multiplication of real numbers. Thisproduct can also be used to determine the inverse of these transfor-mations. By setting c= 1 in (7.2), so that x 00=x, the inverse of (7.1) is seen to correspond to the transformation with a0=a¡1, which explains the requirement that a6= 0. Finally, the identity is deter- mined from x0=x, which clearly corresponds to the transformation Continuous Groups, Lie Groups, and Lie Algebras 111 witha= 1. Hence, the transformations deﬂned in (7.1) form a one- parameter Abelian Lie group. Example 7.2 Now consider the one-dimensional transformations x0=a1x+a2; (7.3) where again a1is an non-zero real number. These transformations cor- responds to the stretching of the real line by a factor a1,a si nt h e preceding Example, and a translation by a2. The product of two oper- ations is x00=a1x0+a2=a1(b1x+b2)+a2=a1b1x+a1b2+a2: By writing x00=c1x+c2, we have that c1=a1b1;c 2=a1b2+a2; so the multiplication of two transformations is described by an analytic function and yields another transformation of the form in (7.1). How-ever, although this multiplication is associative, it is not Abelian, ascan be seen from the fact that the indices do not enter symmetricallyinc 2. By setting, c1=c2= 1, the inverse of (7.3) is the transformation x0=x a1¡a2 a1: The identity is again determined from x0=x, which requires that a1= 1 anda2= 0. Hence, the transformations in (7.3) form a two- parameter (non-Abelian) Lie group. 7.2 Linear Transformation Groups An important class of transformations is the group of linear transfor-mations in ddimensions. These can be represented by d£dmatrices. For example, the most general such transformation in two dimensionsisx 0=Axor, in matrix form, ˆx0 y0! =ˆa11a12 a21a22!ˆx y! ; (7.4) 112 Continuous Groups, Lie Groups, and Lie Algebras where det(A)=a11a22¡a12a216= 0 (Example 2.4). With no further restriction, and with the composition of two elements given by the usualrules of matrix multiplication, these matrices form a four-parameter Liegroup. This Lie group is called the generallineargroup in two dimensions and is denoted by GL(2,R), where the ‘R’ signiﬂes that the entriesare real; the corresponding group with complex entries is denoted byGL(2,C). In ndimensions, these transformation groups are denoted by GL(n,R), or, with complex entries, by GL( n,C). 7.2.1 Orthogonal Groups Many transformations in physical applications are required to preserve length in the appropriate space. If that space is ordinary Euclideann-dimensional space, the restriction that lengths be preserved means that x 02 1+x02 2+¢¢¢+x02 n=x2 1+x2 2+¢¢¢+x2 n: (7.5) The corresponding groups, which are subgroups of the general linear group, are called orthogonal , and are denoted by O( n). Consider the orthogonal group in two-dimensions, i.e., O(2), where the coordinates are xandy. By substituting the general transformation (7.4) into (7.5), we require that x02+y02=(a11x+a12y)2+(a21x+a22y)2 =(a2 11+a2 21)x2+2 (a11a12+a21a22)xy+(a2 12+a2 22)y2: For the right-hand side of this equation to be equal to x2+y2forallx andy, we must set a2 11+a2 21=1;a 11a12+a21a22=0;a2 12+a2 22=1: Thus, we have three conditions imposed on four parameters, leaving one free parameter. These conditions can be used to establish the followingrelation: (a 11a22¡a12a21)2=1: Continuous Groups, Lie Groups, and Lie Algebras 113 Recognizing the quantity in parentheses as the determinant of the trans- formation, this condition implies that det(A)=§1: If det(A) = 1, then the parity of the coordinate system is not changed by the transformation; this corresponds to a proper rotation. If det( A)= ¡1, then the parity of the coordinate system is changed by the transfor- mation; this corresponds to an improper rotation. As we have already seen, both types of transformations are important in physical applica-tions, but we will ﬂrst examine the proper rotations in two-dimensions.This group is called the special orthogonal group in two dimensions and is denoted by SO(2), where \special" signiﬂes the restriction to properrotations. The parametrization of this group that we will use is R(’)= ˆcos’¡sin’ sin’ cos’! ; (7.6) where’, the single parameter in this Lie group, is the rotation angle of the transformation. As can easily be checked using the trigonometricidentities for the sum of two angles, R(’ 1+’2)=R(’1)R(’2); (7.7) so this group is clearly Abelian. 7.3 Inﬂnitesimal Generators A construction of immense utility in the study of Lie groups, which was introduced and extensively studied by Lie, is the inﬁnitesimalgenerator . The idea behind this is that instead of having to consider the group asa whole, for many purposes it is su–cient to consider an inﬂnitesimaltransformation around the identity. Any ﬂnite transformation can thenbe constructed by the repeated application, or \integration," of thisinﬂnitesimal transformation. 114 Continuous Groups, Lie Groups, and Lie Algebras 7.3.1 Matrix Form of Generators For SO(2), we ﬂrst expand R(’) in a Taylor series around the identity (’= 0): R(’)=R(0) +dR d’ﬂﬂﬂﬂ ’=0’+1 2d2R d’2ﬂﬂﬂﬂ ’=0’2+¢¢¢: (7.8) The coe–cients in this series can be determined directly from (7.6), but a more elegant solution may be found by ﬂrst diﬁerentiating (7.7) withrespect to’ 1, d d’1R(’1+’2)=dR(’1) d’1R(’2); (7.9) then setting ’1= 0. Using the chain rule, the left-hand side of this equation is •dR(’1+’2) d(’1+’2)d(’1+’2) d’1‚ﬂﬂﬂﬂ ’1=0=dR(’2) d’2; so Eq. (7.9) becomes dR(’) d’=XR(’); (7.10) where dR(’1) d’1ﬂﬂﬂﬂ ’1=0=ˆ0¡1 10! ·X: (7.11) Equations (7.10) and (7.11) allow us to determine all of the expan- sions coe–cients in (7.9). By setting ’= 0 in (7.10) and observing thatR(0) =I, whereIis the 2£2 unit matrix, I=ˆ10 01! ; we obtain dR(’) d’ﬂﬂﬂﬂ ’=0=X: (7.12) Continuous Groups, Lie Groups, and Lie Algebras 115 To determine the higher-order derivatives of R, we diﬁerentiate (7.10) ntimes, and set ’=0 : dnR(’) d’nﬂﬂﬂﬂ ’=0=Xdn¡1R(’) d’n¡1ﬂﬂﬂﬂ ’=0: This yields, in conjunction with (7.12), dnR(’) d’nﬂﬂﬂﬂ ’=0=Xn: Substituting this expression into the Taylor series in (7.8) allows us to write R(’)=I+X’+1 2X2’2+¢¢¢ =1X n=01 n!(X’)n ·e’X; whereX0=Iand the exponential of a matrix is deﬂned by the Taylor series expansion of the exponential. Thus, every rotation by a ﬂniteangle can be obtain from the exponentiation of the matrix X, which is called the inﬁnitesimal generator of rotations. Since X2=I,i ti sa straightforward matter to show directly from the Taylor series of theexponential (Problem 4, Problem Set 9) that e ’X=Icos’+Xsin’=ˆcos’¡sin’ sin’ cos’! : 7.3.2 Operator Form of Generators An alternative way of representing inﬂnitesimal generators through which connections with quantum mechanics can be directly made isin terms of diﬁerential operators. To derive the operator associatedwith inﬂnitesimal rotations, we expand (7.6) to ﬂrst order in d ’to obtain the transformation x 0=xcos’¡ysin’=x¡yd’; y0=xsin’+ycos’=xd’+y: 116 Continuous Groups, Lie Groups, and Lie Algebras An arbitrary diﬁerentiable function F(x;y) then transforms as F(x0;y0)=F(x¡yd’;xd’+y): Retaining terms to ﬂrst order in d ’on the right-hand side of this equa- tion yields F(x0;y0)=F(x;y)+µ ¡y@F @x+x@F @y¶ d’: SinceFis an arbitrary function, we can associate inﬂnitesimal rotations with the operator X=x@ @y¡y@ @x: As we will see in the next section, this operator is proportional to the z-component of the angular momentum operator. The group SO(2) is simple enough that the full beneﬂts of an in- ﬂnitesimal generator are not readily apparent. We will see in the nextsection, where we discuss SO(3), that the inﬂnitesimal generators em-body much of the structure of the full group. 7.4 SO(3) The orthogonal group in three dimensions is comprised of the trans-formations that leave the quantity x 2+y2+z2invariant. The group GL(3,R) has 9 parameters, but the invariance of the length produces sixindependent conditions, leaving three free parameters, so O(3) forms athree-parameter Lie group. If we restrict ourselves to transformationswith unit determinant, we obtain the group of proper rotations in threedimensions, SO(3). There are three common ways to parametrize these rotations: †Successive rotations about three mutually orthogonal ﬂxed axes. †Successive about the z-axis, about the newy-axis, and then about thenewz-axis. These are called Euler angles . Continuous Groups, Lie Groups, and Lie Algebras 117 †The axis-angle representation, deﬂned in terms of an axis whose direction is speciﬂed by a unit vector (two parameters) and arotation about that axis (one parameter). In this section, we will use the ﬂrst of these parametrizations to demonstrate some of the properties of SO(3). In the next chapter,where we will develop the orthogonality relations for this group, theaxis-angle representation will prove more convenient. 7.4.1 Rotation Matrices Consider ﬂrst rotations about the z-axis by an angle ’3: R3(’3)=0 BB@cos’3¡sin’3 0 sin’3 cos’3 0 00 11 CCA: The corresponding inﬂnitesimal generator is calculated as in (7.11): X3=dR3 d’3ﬂﬂﬂﬂ ’3=0=0 BB@0¡10 1000001 CCA: These results are essentially identical to those for SO(2). However, for SO(3), we have rotations about two other axes to consider. Forrotations about the x-axes by an angle ’ 1, the rotation matrix is R1(’1)=0 BB@10 0 0 cos’1¡sin’1 0 sin’1 cos’11 CCA and the corresponding generator is X1=dR1 d’1ﬂﬂﬂﬂ ’1=0=0 BB@000 00¡1 0101 CCA 118 Continuous Groups, Lie Groups, and Lie Algebras Finally, for rotations about the y-axis by an angle ’2,w eh a v e R2(’2)=0 BB@cos’20 sin’2 01 0 ¡sin’20 cos’21 CCA and the generator is X2=dR2 d’2ﬂﬂﬂﬂ ’2=0=0 BB@001 000 ¡1001 CCA As can be easily veriﬂed, the matrices Ri(’i) do not commute, nor do theXi. However, the Xihave an additional useful property, namely closure under commutation. As an example, consider the productsX 1X2andX2X1: X1X2=0 BB@000 00¡1 0101 CCA0 BB@001 000 ¡1001 CCA=0 BB@000 1000001 CCA X2X1=0 BB@001 000 ¡1001 CCA0 BB@000 00¡1 0101 CCA=0 BB@010 0000001 CCA Thus, the commutator of X1andX2, denoted by [ X1;X2] is given by [X1;X2]·X1X2¡X2X1=0 BB@0¡10 1000001 CCA=X3 Similarly, we have [X2;X3]=X1; [X3;X1]=X2 The commutation relations among all of the Xican be succinctly sum- marized by introducing the anti-symmetric symbol "ijk, which takes the Continuous Groups, Lie Groups, and Lie Algebras 119 value 1 for a symmetric permutation of distinct i,j, andk, the value ¡1 for an antisymmetric permutation, and is zero otherwise (i.e., if two or more ofi,jandkare equal). We can then write [Xi;Xj]="ijkXk (7.13) We will discuss the physical interpretation of these generators once we obtain their operator form in the next section. 7.4.2 Operators for Inﬂnitesimal Rotations As was the case in Section 7.3, an alternative to the matrix represen-tation of inﬂnitesimal generators is in terms of diﬁerential operators.Proceeding as in that section, we ﬂrst write the general rotation as anexpansion to ﬂrst order in each of the ’ iabout the identity. This yields the transformation matrix 0 BB@x0 y0 z01 CCA=0 BB@1¡’3’2 ’3 1¡’1 ¡’2’1 11 CCA0 BB@x y z1 CCA Substituting this coordinate transformation into a diﬁerentiable func- tionF(x;y;z ), F(x0;y0;z0)=F(x¡’3y+’2z;y+’3x¡’1z;z¡’2x+’1y) and expanding the right-hand side to ﬂrst order in the ’iyields the following expression: F(x0;y0;z0)=F(x;y;z ) +µ@F @zy¡@F @yz¶ ’1+µ@F @xz¡@F @zx¶ ’2+µ@F @yx¡@F @xy¶ ’3 SinceFis an arbitrary diﬁerentiable function, we can identify the gen- eratorsXiof rotations about the coordinate axes from the coe–cients of the’i, i.e., with the diﬁerential operators X1=y@ @z¡z@ @y 120 Continuous Groups, Lie Groups, and Lie Algebras X2=z@ @x¡x@ @z(7.14) X3=x@ @y¡y@ @x Notice that X3is the operator obtained for SO(2) in Section 7.3. We can now assign a physical interpretation to these operators by compar-ing them with the vectors components of the angular operators in thecoordinate representation, obtained from the deﬂnition L=r£p=r£(¡i„hr) Carrying out the cross-product yields the standard expressions L 1=¡i„hµ y@ @z¡z@ @y¶ L2=¡i„hµ z@ @x¡x@ @z¶ (7.15) L3=¡i„hµ x@ @y¡y@ @x¶ for thex,y, andzcomponents of L, respectively. Thus, Li=¡i„hXi, fori=1;2;3, and (7.13) becomes [Li;Lj]=i„h"ijkLk which are the usual angular momentum commutation relations. There- fore, we can associate the vector components of the angular momentumoperator with the generators of inﬂnitesimal rotations about the cor-responding axes. An analogous association exists between the vectorcomponents of the coordinate representation of the linear momentumoperator and diﬁerential translation operations along the correspondingdirections. 7.4.3 The Algebra of Inﬂnitesimal Generators The commutation relations in (7.13) deﬂne a \product" of two gener-ators which yields the third generator. Thus, the set of generators is Continuous Groups, Lie Groups, and Lie Algebras 121 closed under this operation. Triple products, which determine whether or not this composition law is associative, can be written in a conciseform using only the deﬂnition of the commutator, i.e., in the form ofan identity, without any explicit reference to the quantities involved.Beginning with the triple product h A;[B;C]i =A[B;C]¡[B;C]A =ABC¡ACB¡BCA +CBA We now add and subtract the quantities BAC andCAB on the right- hand side of this equation and rearrange the resulting expression intocommutators to obtain h A;[B;C]i =ABC¡ACB¡BCA +CBA +BAC¡BAC +CAB¡CAB =¡C(AB¡BA)+(AB¡BA)C +B(AC¡CA)¡(AC¡CA)B =¡h [A;B];Ci +h [C;A];Bi A simple rearrangement yields the Jacobi identity : h A;[B;C]i +h B;[C;A]i +h C;[A;B]i =0 Notice that this identity has been obtained using only the deﬂnition of the commutator. For the inﬂnitesimal generators of the rotation group, with the com- mutator in (7.13), each of the terms in the Jacobi identity vanishes.Thus, h A;[B;C]i =h [A;B];C]i so the product of these generators is associative. In the more general case, however, products of quantities deﬂned in terms of a commutatorare not associative. The Liealgebra associated with the Lie group from 122 Continuous Groups, Lie Groups, and Lie Algebras which the generators are obtained consists of quantities A;B;C;::: deﬂned by A=3X k=1akXk;B =3X k=1bkXk;C =3X k=1ckXk; etc: where theak;bk;ck;:::are real coe–cients and from which linear com- binationsﬁA+ﬂBwith realﬁandﬂcan be formed. The product is given by [A;B]=¡[B;A] and the Jacobi identity is, of course, satisﬂed. The formal deﬂnition of a Lie algebra, which is an abstraction of the properties just discussed, is as follows. Deﬁnition. ALie algebra is a vector space Lover some ﬂeld F1 (typically the real or complex numbers) together with a binary opera- tion [¢;¢]:L£L!L, called the Lie bracket , which has the following properties: 1.Bilinearity . [ax+by;z]=a[x;z]+b[y;z] [z;ax +by]=a[z;x]+b[z;y] for allaandbinFandx,y, andzinL. 2.Jacobi identity . h [x;y];zi +h [z;x];y]+h [y;z];xi =0 for allx,y, andzinL. 1A ﬂeld is an algebraic system of elements in which the operations of addition, subtraction, multiplication, and division (except by zero) may be performed withoutleaving the system (closure) and the associative, commutative, and distributiverules, familiar from the arithmetic of ordinary numbers, hold. Examples of ﬂeldsare the rational numbers, the real numbers, and the complex numbers. The smallestﬂeld has only two elements: f0;1g. The concept of a ﬂeld is useful for deﬂning vectors and matrices, whose components can be elements of any ﬂeld. Continuous Groups, Lie Groups, and Lie Algebras 123 3.Antisymmetry . [x;y]=¡[y;x] for allxandyinL. 7.5 Summary In this chapter, we have described the properties of Lie groups in terms of speciﬂc examples, especially SO(2) and SO(3). With thisbackground, we can generalize our discussion to any Lie group. Anr-parameter Lie group of transformations on an n-dimensional space is x 0 i=fi(x1;x2;:::;xn;a1;a2;:::;ar) wherei=1;2;:::;n . If only one of the rparameters aiis changed from zero, while all the other parameters are held ﬂxed, we obtain theinﬂnitesimal transformations X iassociated with this Lie group. These can be expressed as diﬁerential operators by examining the eﬁect ofthese inﬂnitesimal coordinate transformations on an arbitrary diﬁeren-tiable function F: dF= nX j=1@F @xjdxj =nX j=1@F @xjµrX i=1@fj @aiﬂﬂﬂﬂ a=0dai¶ =rX i=1daiµnX j=1@fj @aiﬂﬂﬂﬂ a=0@ @xj¶ F We identify the diﬁerential operators Xias the coe–cient of d aiin this diﬁerential: Xi=nX j=1@fj @aiﬂﬂﬂﬂ a=0@ @xj forr=1;2;:::;r . These operators satisfy commutation relations of the form [Xi;Xj]=ck ijXk 124 Continuous Groups, Lie Groups, and Lie Algebras where theck ijare called structure constants and are a property of the group. The commutator satisﬂes the Jacobi identity, h Xi;[Xj;Xk]i +h Xj;[Xk;Xi]i +h Xk;[Xi;Xj]i =0 which places a constraint on the structure constants. The commutator and the Jacobi identity, together with the ability to form real linearcombinations of the X iendows these generators with the structure of an algebra, called the Lie algebra associated with the Lie group. Chapter 6 Groups and Representations in Quantum Mechanics The universe is an enormous direct product of representations of sym- metry groups. |Steven Weinberg1 This chapter is devoted to applying the mathematical theory of groupsand representations which we have developed in the preceding chaptersto the quantum mechanical description of physical systems. The powerof applying group theory to quantum mechanics is that it provides aframework for making exact statements about a physical system with a knowledge only of the symmetry operations which leave its Hamil-tonian invariant, the so-called \group of the Hamiltonian." Moreover,when we apply the machinery of groups to quantum mechanics, we ﬂndthat representations|and irreducible representations in particular|arise quite naturally, as do related concepts such as the importance ofunitarity of representations and the connection between the symmetryof a physical system and the degeneracy of its eigenstates. We willfollow the general sequence of the discussion in Sections 1.2 and 1.3, 1Steven Weinberg, Sheldon Glashow, and Abdus Salam were awarded the 1979 Nobel Prize in Physics for their incorporation of the weak and electromagneticinteractions into a single theory. 83 84 Groups and Representations in Quantum Mechanics beginning with the group of the Hamiltonian, using this to establish the symmetry properties of the eigenfunctions, and concluding with a dis-cussion of selection rules, which demonstrates the power and economyof using character tables. As a demonstration of the usefulness of theseconstructions, we will prove Bloch’s theorem, the fundamental princi-ple behind the properties of wavefunctions in periodic systems such as electrons and phonons (the quanta of lattice vibrations) in single crys-tals. The application of group theory to selection rules necessitates theintroduction of the \direct product" of matrices and groups, thoughhere, too, quantum mechanics provides a motivation for this concept. 6.1 The Group of the Hamiltonian Recall the deﬂnition of a similarity transformation introduced in Section3.3. Two matrices, or operators, AandBare related by a similarity transformation generated by a matrix (or operator) Rif B=RAR ¡1: The quantity Bis therefore the expression of Aunder the transforma- tionR. Consider now a Hamiltonian Hand its transformation by an operationR RHR¡1: The Hamiltonian is said to be invariant underRif H=RHR¡1; (6.1) or, equivalently, RH=HR: (6.2) Thus the order in whichHand theRare applied is immaterial, so H andRcommute: [H;R] = 0. In this case, Ris said to be a symmetry operation of the Hamiltonian. Consider set of all symmetry operations of the Hamiltonian, which we will denote by fRﬁg. We now show that these operations form a Groups and Representations in Quantum Mechanics 85 group. To demonstrate closure, we observe that if RﬁandRﬂare two operations which satisfy (6.1), then RﬁHR¡1 ﬁ=Rﬁ(RﬂHR¡1 ﬂ)R¡1 ﬁ=(RﬁRﬂ)H(RﬁRﬂ)¡1=H: Thus, the product RﬁRﬂ=R°is also a symmetry operation of the Hamiltonian. Associativity is clearly obeyed since these operationsrepresent transformations of coordinates and other variables of theHamiltonian. 2The unit element Ecorresponds to performing no op- eration at all and the inverse R¡1 ﬁof a symmetry operation Rﬁis the application of the reverse operation to \undo" the original transfor-mation. Thus, the set fR ﬁgforms a group, called the group of the Hamiltonian . 6.2 Eigenfunctions and Representations There are a number of consequences of the discussion in the preced- ing section for the representations of the group of the Hamiltonian.Consider an eigenfunction ’of a Hamiltonian Hcorresponding to the eigenvalueE: H’=E’: We now apply a symmetry operation R ﬁto both sides of this equation, RﬁH’=ERﬁ’; and use (6.2) to write RﬁH’=HRﬁ’: Thus, we have H(Rﬁ’)=E(Rﬁ’): If the eigenvalue is nondegenerate, then Rﬁ’diﬁers from ’by at most a phase factor: Rﬁ’=ei`ﬁ’: 2The associativity of linear operations is discussed by Wigner in Group Theory (Academic, New York, 1959), p. 5. 86 Groups and Representations in Quantum Mechanics The application of a second operation Rﬂthen produces Rﬂ(Rﬁ’)=ei`ﬂei`ﬁ’: (6.3) The left-hand side of this equation can also be written as (RﬂRﬁ)’=ei`ﬂﬁ’; (6.4) Equating the right-hand sides of Eqs. (6.3) and (6.4), yields ei`ﬂﬁ=ei`ﬂei`ﬁ; i.e., these phases preserve the multiplication table of the symmetry op- erations. Thus, the repeated application of all of the Rﬁto’generates aone-dimensional representation of the group of the Hamiltonian. The other case to consider occurs if the application of all of the symmetry operations to ’produces‘distinct eigenfunctions. These eigenfunctions are said to be ‘-fold degenerate. If these are the only eigenfunctions which have energy E, this is said to be a normal degen- eracy . If, however, there are other degenerate eigenfunctions which are not captured by this procedure, this is said to be an accidental degen- eracy . The term \accidental" refers to the fact that the degeneracy is not due to symmetry. But an \accidental" degeneracy can also occurbecause a symmetry is \hidden," i.e., not immediately apparent, so thegroup of the Hamiltonian is not complete. One well-known example ofthis is the level degeneracy of the hydrogen atom. For a normal degeneracy, there are orthonormal eigenfunctions ’ i, i=1;2;:::;‘ which, upon application of one of the symmetry op- erationsRﬁare transformed into linear combinations of one another. Thus, if we denote by ’the‘-dimensional row vector ’=(’1;’2;:::;’‘); we can write Rﬁ’=’¡(Rﬁ); where ¡(Rﬁ)i sa n‘£‘matrix. In terms of components, this equation reads Rﬁ’i=‘X k=1’k[¡(Rﬁ)]ki (6.5) Groups and Representations in Quantum Mechanics 87 The successive application of operations RﬁandRﬂthen yields RﬂRﬁ`i=Rﬂ‘X k=1’k[¡(Rﬁ)]ki=‘X k=1(Rﬂ’k)[¡(Rﬁ)]ki: The operation Rﬂ’kcan be written as in (6.5): Rﬂ’k=‘X j=1’j[¡(Rﬂ)]jk: Thus, RﬂRﬁ`i=‘X k=1‘X j=1’j[¡(Rﬂ)]jk[¡(Rﬁ)]ki =‘X j=1’j‰‘X k=1[¡(Rﬂ)]jk[¡(Rﬁ)]ki¾ : (6.6) Alternatively, we can write RﬂRﬁ’i=‘X j=1’j[¡(RﬂRﬁ)]ji: (6.7) By comparing (6.6) and (6.7) and using the orthonormality of the wave- functions, we conclude that ¡(RﬂRﬁ)=¡ (Rﬂ)¡(Rﬁ); so the ¡(Ri) form an‘-dimensional representation of the group of the Hamiltonian. Since the eigenfunctions can be made orthonormal, thisrepresentation can always be taken to be unitary (Problem Set 8). We will now show that this representation is also irreducible . We ﬂrst consider the eﬁect of replacing the ’ iby a linear combination of these functions,ˆ=’U. Then the eﬁect of operating with Ron theˆis Rˆ=R’U=’¡U=ˆU¡1¡U; i.e., the representation with the transformed wavefunctions is related by a similarity transformation to that with the original eigenfunctions, 88 Groups and Representations in Quantum Mechanics i.e., the two representations are equivalent . Suppose that this represen- tation is reducible. Then there is a unitary transformation of the ’j such that there are two or more subsets of the ˆithat transform only among one another under the symmetry operations of the Hamiltonian.This implies that the application of the R ito any eigenfunction gen- erates eigenfunctions only in the same subset. The degeneracy of theeigenfunctions in the other subset is therefore accidental , in contradic- tion to our original assertion that the degeneracy is normal . Hence, the representation obtained for a normal degeneracy is irreducible and the corresponding eigenfunctions are said to generate , or form a basis for this representation. We can summarize the results of this section as follows: †To each eigenvalue of a Hamiltonian there corresponds a unique irreducible representation of the group of that Hamiltonian. †The degeneracy of an eigenvalue is the dimensionality of this ir- reducible representation. Thus, the dimensionalities of the irre-ducible representations of a group are the possible degeneraciesof Hamiltonians with that symmetry group. †Group theory provides \good quantum numbers," i.e., labels cor- responding irreducible representations to which eigenfunctionsbelong. †Although these statements have been shown for ﬂnite groups, they are also valid for continuous groups. 6.3 Group Theory in Quantum Mechanics The fact that eigenfunctions corresponding to an ‘-fold degenerate eigenvalue form a basis for an ‘-dimensional irreducible representation of the group of the Hamiltonian is one of the fundamental principlesbehind the application of group theory to quantum mechanics. In thissection, we brie°y describe the two main types of such applications,namely, where group theory is used to obtain exact results, and where it is used in conjunction with perturbation theory to obtain approximate results. Groups and Representations in Quantum Mechanics 89 6.3.1 Exact Results One of the most elegant applications of group theory to quantum me- chanics involves using the group of the Hamiltonian to determine the(normal) degeneracies of the eigenstates, which are just the dimen-sions of the irreducible representations. Because such a classiﬂcation isderived from the symmetry properties of the Hamiltonian, it can be ac-complished without having to solve the Schr˜ odinger equation. Among the most historically important of such applications is the classiﬂca-tion of atomic spectral lines. The atomic Hamiltonian is comprisedof the sum of the kinetic energies of the electrons and their Coulombinteractions, so an exact solution is impractical, even for few-electronatoms such as He. Nevertheless, the spherical symmetry of the Hamil-tonian enables the identiﬂcation of the irreducible representations ofatomic states from which are derived the angular momentum additionrules and multiplet structures. This will be explored further when wediscuss continuous groups. Another exact result is Bloch’s theorem,which is the basis for many aspects of condensed matter physics. Thistheorem uses the translational invariance of perfect periodic crystalsto determine the form of the eigenfunctions. As discussed in the nextsection, Bloch’s theorem can be reduced to a statement about the (one-dimensional) irreducible representations and basis functions of cyclicgroups. The lowering of the symmetry of a Hamiltonian by a perturbation can also be examined with group theory. In particular, the question ofwhether the allowed degeneracies are aﬁected by such a perturbationcan be addressed by examining the irreducible representations of thegroups of the original and perturbed Hamiltonians. Group theory canaddress not only whether degeneracies can change (from the irreduciblerepresentations of the two groups), but how irreducible representationsof the original group are related to those of the perturbed group. Typi-cally, when the symmetry of a system is lowered, the dimensionalities ofthe irreducible representations can also be lowered, resulting in a \split-ting" of the original irreducible representations into lower-dimensionalirreducible representations of the group of the perturbed system. Finally, on a somewhat more practical level, group theory can be used to construct symmetrized linear combinations of basis functions 90 Groups and Representations in Quantum Mechanics to diagonalize a Hamiltonian. Examples where this arises is the low- ering of the symmetry of a system by a perturbation, where the basisfunctions are the eigenfunctions of the original Hamiltonian, the bond-ing within molecules, where the basis functions are localized aroundthe atomic sites within the molecule, and vibrations in molecules andsolids, where the basis functions describe the displacements of atoms.These applications are discussed by Tinkham. 3 6.3.2 Approximate Results The most common application of group theory in approximate calcula-tions involves the calculation of matrix elements in perturbation theory.A typical example is involved adding to a Hamiltonian H 0and pertur- bationH0due to an electromagnetic ﬂeld which causes transitions be- tween the eigenstates of the original Hamiltonian. The transition rateWis calculated from ﬂrst-order time-dependent perturbation theory, with the result known as Fermi’s Golden Rule : 4 W=2… „h%ifj(ijH0jf)j2; where%ifis called the \joint density of states," which is a measure of the number of initial and ﬂnal states which are available for the excitation,and (ijH 0jf) is a matrix element of H0between the initial and ﬂnal states. The application of group theory to this problem, which is thesubject of Section 6.6, involves determining when this matrix elementvanishes by reasons of symmetry. 6.4 Bloch’s Theorem⁄ Bloch’s theorem is of central importance to many aspects of electrons, phonons, and other excitations in crystalline solids. One of the mainresults of this theorem, namely, the form of the eigenfunctions, can bederived solely from group theory. We will work in one spatial dimension, 3M. Tinkham, Group Theory and Quantum Mechanics (McGraw{Hill, New York, 1964) 4L.I. Schiﬁ, Quantum Mechanics 2nd edn (McGraw{Hill, New York, 1955) Groups and Representations in Quantum Mechanics 91 but the discussion can be extended easily to higher dimensions. We consider a one-dimensional crystal where the distance between nearestneighbors is aand the number of repeat units is N(a large number for a macroscopic solid). Since this system is ﬂnite, it has no translationalsymmetry. However, by imposing a type of boundary condition knownasperiodic , whereby the Nth unit is identiﬂed with the ﬂrst unit| eﬁectively forming a circle from this solid|we now have Ndiscrete symmetries. The Schr˜ odinger equation for a particle of mass mmoving in the periodic potential of this system is • ¡„h2 2md2 dx2+V(x)‚ ’=E’; whereV(x+a)=V(x). 6.4.1 The Group of the Hamiltonian The translation of an eigenfunction by awill be denoted by Ra: Ra’(x)=’(x+a): The basic properties of translations originate with the observation that a translation through na, Rna’(x)=’(x+na); can be written as the n-fold product of Ra: Rn a’(x)=RaRa¢¢¢Ra|{z} nfactors’(x)=’(x+na): Moreover, because of the periodic boundary conditions, we identify the Nth unit with the ﬂrst, so RN a=R0; which means that no translation is carried out at all. Thus, the col- lection of all the translations can be written as the powers of a singleelement,R a: fRa;R2 a;:::;RN a=Eg; (6.8) 92 Groups and Representations in Quantum Mechanics whereEis the identity. This shows that the group of the Hamiltonian is a cyclic group of order N. In particular, since cyclic groups are Abelian, there are None-dimensional irreducible representations of this group, i.e., each eigenvalue is nondegenerate and labelled by one of theseirreducible representations. 6.4.2 Character Table and Irreducible Represen- tations Having identiﬂed the algebraic structure of the group of the Hamilto- nian, we now construct the character table. Since RN a=E, and since all irreducible representations are one-dimensional, the character for Ra in each of these representations, ´(n)(Ra) must obey this product: h ´(n)(Ra)iN=1: The solutions to this equation are the Nth roots of unity (cf. Problem 3, Problem Set 5): ´(n)(Ra)=e2…in=N;n =0;1;2;:::N¡1: The character table is constructed by choosing one of these values for each irreducible representation and then determining the remaining en-tries from the multiplication table of the group (since each irreduciblerepresentation is one-dimensional). The resulting character table is: fEgfRagfR2 ag ¢¢¢ fRN¡1 ag ¡1 11 1 ¢¢¢ 1 ¡2 1!!2¢¢¢!N¡1 ¡3 1!2!4¢¢¢!2(N¡1) .................. ¡N 1!N¡1!2N¡2¢¢¢!(N¡1)2 where!=e2…i=N. If we denote the eigenfunction corresponding to the nth irreducible representation by ’n, then applying Rayields Ra’n(x)=!n¡1’n(x)=’n(x+a): Groups and Representations in Quantum Mechanics 93 Since the characters of this group are pure phases, the moduli of the eigenfunctions are periodic functions: j’n(x+a)j2=j’n(x)j2: Thus, the most general form of the ’nis ’n(x)=ei`n(x)un(x); (6.9) where`n(x) is a phase function, which we will determine below, and theunhave the periodicity of the lattice: un(x+a)=un(x). By com- bining this form of the wavefunction with the transformation properties required by the character table, we can write Rm a’n(x)=!m(n¡1)’n(x)=!m(n¡1)ei`n(x)un(x): Alternatively, by applying the same translation operation directly to (6.9) yields Rm a’n(x)=’n(x+ma)=ei`n(x+ma)un(x): By equating these two ways of writing Rm a’n(x), we ﬂnd that their phase changes must be equal. This, in turn, requires that the phasefunction satisﬂes ` n(x+ma)=`n(x)+2…m(n¡1) N: (6.10) Thus,`nis alinear function of mand, therefore, also of x+ma, since `nis a function of only a single variable: `n(x)=Ax+B; whereAandBare constants to be determined. Upon substitution of this expression into both sides of (6.10), A(x+ma)+B=Ax+B+2…m(n¡1) N; and cancelling common factors, we obtain `n(x)=knx+B; 94 Groups and Representations in Quantum Mechanics where kn=2…(n¡1) Na=2…(n¡1) L andL=Nais the size system. The wavefunction in (6.9) thereby reduces to ’n(x)=eiknxun(x); where we have absorbed the constant phase due to Binto the deﬂnition ofun(x). This is called a Bloch function : a function un(x) with the periodicity of the lattice modulated by a plane wave.5This is one of the two main results of Bloch’s theorem, the other being the existenceof energy gaps, which is beyond the scope of the discussion here. 6.5 Direct Products The direct product provides a way of enlarging the number of elementsin a group while retaining the group properties. Direct products oc-cur in several contexts. For example, if a Hamiltonian or Lagrangiancontains diﬁerent types of coordinates, such as spatial coordinates fordiﬁerent particles, or spatial and spin coordinates, then the symmetryoperations on the diﬁerent coordinates commute with each other. Ifthere is a coupling between such degrees of freedom, such as particleinteractions or a spin-orbit interaction, then the direct product is re-quired to determine the appropriate irreducible representations of theresulting eigenstates. In this section, we develop the group theory as-sociated with direct products and their representations. We will thenapply these concepts to selection rules in the following section. 5A related issue which can be addressed by group theory is the nature of the quantity „hkn. Although it has units of momentum, it does not represent a true momentum, but is called the \crystal momentum." The true momentum „ hkla- bels the irreducible representations of the translation group, which is a continuousgroup and will be discussed in the next chapter. The discrete translations of a pe-riodic potential form a subgroup of the full translation group, so the correspondingirreducible representations cannot be labelled by momentum. Groups and Representations in Quantum Mechanics 95 6.5.1 Direct Product of Groups Consider two groups Ga=fe;a2;:::;ajGajg;Gb=fe;b2;:::;bjGbjg; such that all elements in Gacommute with all elements in Gb: aibj=bjai; fori=1;2;:::;jGajandj=1;2;:::;jGbj. We have deﬂned a1=eand b1=e. Thedirectproduct ofGaandGb, denoted by Ga›Gb, is the set containing all elements aibj: Ga›Gb=fe;a2;:::;ajGaj;b2;:::;bjGbj;:::;aibj;:::g:(6.11) As shown in Problem 3 of Problem Set 8, the direct product is a group of orderjGajjGbj. Example 6.1. Consider the symmetry operations on an equilateral triangle that has a thickness , i.e., the triangle has become a \wedge." Thus, in addition to the original symmetry operations of the planarequilateral triangle, there is now also a re°ection plane ¾ h. There are now six vertices, which are labelled as in Example 2.1, except thatwe now distinguish between points which lie above, f1 +;2+;3+g, and below,f1¡;2¡;3¡g, the re°ection plane. The original six operations do not transform points above and below the re°ection plane into oneanother. The re°ection plane, on the other hand, only transforms cor- responding points above and below the plane into one another. Hence,the 6 operations of a planar triangle commute with¾ h. The symmetry group of the equilateral wedge consists of the original 6 operations of a planar triangle, the horizontal re°ection plane, andtheir products. Since the set with elements fE;¾ hgforms a group (and each element commutes with the symmetry operations of an equilateraltriangle), the appropriate group for the wedge is thereby obtained bytaking the direct product fE;¾ v;1;¾v;2;¾v;3;C3;C2 3g›fE;¾hg: The 12 elements of this group are fE;¾v;1;¾v;2;¾v;3;C3;C2 3;¾h;¾h¾v;1;¾h¾v;2;¾h¾v;3;¾hC3;¾hC2 3g: 96 Groups and Representations in Quantum Mechanics 6.5.2 Direct Product of Matrices The determination of the irreducible representations and the character table of a direct product group does not require a separate new calcu-lation of the type discussed in the preceding chapter. Instead, we canutilize the irreducible representations of the two groups used to formthe direct to obtain these quantities. To carry out these operationsnecessitates introducing the direct product of matrices. The direct product Cof two matrices AandB, written as A›B= C, is deﬂned in terms of matrix elements by a ijbkl=cik;jl: (6.12) Note that the row and column labels of the matrix elements of Care composite labels: the row label, ik, is obtained from the row labels of the matrix elements of AandBand the column label, jl, is obtained from the corresponding column labels. The matrices need not have thesame dimension and, in fact, need not even be square. However, sincewe will apply direct products to construct group representations, wewill conﬂne our discussion to square matrices. In this case, if Ais an n£nmatrix and Bis anm£mmatrix,Cis anmn£mnmatrix. Example 6.2. For matrices AandBgiven by A= ˆa11a12 a21a22! ;B =0 BB@b11b12b13 b21b22b23 b31b32b331 CCA; the direct product C=A›Bis A›B=0 BBBBBBBBBBBBBB@a11b11a11b12a11b13a12b11a12b12a12b13 a11b21a11b22a11b23a12b21a12b22a12b23 a11b31a11b32a11b33a12b31a12b32a12b33 a21b11a21b12a21b13a22b11a22b12a22b13 a21b21a21b22a21b23a22b21a22b22a22b23 a21b31a21b32a21b33a22b31a22b32a22b331 CCCCCCCCCCCCCCA : Groups and Representations in Quantum Mechanics 97 Another way of writing the direct product that more clearly displays its structure is A›B=ˆa11Ba 12B a21Ba 22B! : The notion of a direct product arises quite naturally in quantum mechanics if we consider the transformation properties of a product oftwo eigenfunctions. Suppose we have two eigenfunctions ’ iand’i0of a HamiltonianHwhich is invariant under some group of operations. As in Section 6.2, the action of these operations on the eigenfunctionsofHis R’ i=‘X j=1’j¡ji(R); R’i0=‘0X j0=1’j0¡j0i0(R): The question we now ask is: how does the product’i’i0transform under the symmetry operations of the Hamiltonian? Given the transformationproperties of ’ iand’i0noted above, we ﬂrst observe that R(’i’i0)=R(’i)R(’i0): In other words, since Rrepresents a coordinate transformation, its ac- tion on any function of the coordinates is to transform each occurrenceof the coordinates. Thus, R(’ i’i0)=‘X j=1‘0X j0=1’j’j0¡ji(R)¡j0i0(R) =‘X j=1‘0X j0=1’j’j0¡jj0;ii0(R); so’i’i0transforms as the direct product of the irreducible representa- tions associated with ’iand’i0. 98 Groups and Representations in Quantum Mechanics 6.5.3 Representations of Direct Product Groups Determining the representations of direct products and the construction of their character tables are based on the following theorem: Theorem6.1. The direct product of the representations of two groups is a representation of the direct product of these groups. Proof. A typical product of elements in the direct product group in (6.11) is (apbq)(ap0bq0)=(apap0)(bqbq0)=arbr0: A representation of the direct product group must preserve the mul- tiplication table. We will use the notation that the matrix A(apbq) corresponds to the element apbq. Thus, we must require that A(apbq)A(ap0bq0)=A(arbr0): By using the deﬂnition of the direct product of two matrices in Equation (6.12), we can write this equation in terms of matrix elements as h A(apbq)A(ap0bq0)i ik;jl=X m;nA(apbq)ik;mn|{z} A(ap)imA(bq)knA(ap0bq0)mn;jl|{z} (A(ap0)mjA(bq0)nl =µX mA(ap)imA(ap0)mj¶ |{z} A(apap0)ijµX nA(bq)]knA(bq0)nl¶ |{z} A(bpbp0)kl =A(ar)ijA(br0)kl =A(arbr0)ik;jl: Thus, the direct product of the representations preserves the multipli- cation table of the direct product group and, hence, is a representationof this group. In fact, as shown in Problem 4 of Problem Set 8, the direct product of irreducible representations of two groups is an irreducible represen-tation of the direct product of those groups. An additional convenient Groups and Representations in Quantum Mechanics 99 feature of direct product groups is that the characters of its represen- tations can be computed directly from the characters of the represen-tations of the two groups forming the direct product. This statementis based on the following theorem: Theorem6.2. If´(a p) and´(bq) are the characters of representations of two groups GaandGb, the characters ´(apbq) of the representation formed from the matrix direct product of these representations is ´(apbq)=´(ap)´(bq): Proof. From the deﬂnition of the direct product, a representation of the direct product group is A(apbq)ij;kl=A(ap)ikA(bq)jl: Taking the trace of both sides of this expression yields X i;jA(apbq)ij;ij |{z} ´(apbq)=µX iA(ap)ii¶ |{z} ´(ap)µX jA(bq)jl¶ |{z} ´(bq): Thus, ´(apbq)=´(ap)´(bq); which proves the theorem. Since the characters are associated with a given class, the characters for the classes of the direct product are computed from the charactersof the classes of the original groups whose elements contribute to eachclass of the direct product. Moreover, the number of classes in the directproduct group is the product of the numbers of classes in the originalgroups. This can be seen immediately from the equivalence classes inthe direct product group. Using the fact that elements belonging tothe diﬁerent groups commute, (a ibj)¡1(akbl)(aibj)¡1=(a¡1 iakai)(b¡1 jblbj): 100 Groups and Representations in Quantum Mechanics Thus, equivalence classes in the direct product group must be formed from elements in equivalence classes in the original groups. Example 6.3. Consider the direct product group of the equilateral wedge in Example 6.1. The classes of S3are (Example 2.9), in the notation of Example 5.5, fEg;f¾v;1;¾v;2;¾v;3g;fC3;C2 3g; and the classes of the group fE;¾hgare fEg;f¾hg: There are, therefore, six classes in the direct product group, which are obtained by taking the products of elements in the original classes, asdiscussed above: fEg;fC 3;C2 3g;f¾v;1;¾v;2;¾v;3g; f¾hg;¾hfC3;C2 3g;¾hf¾v;1;¾v;2;¾v;3g: The structure of the character table of the direct product group can now be determined quite easily. We denote the character for the ﬁth class of the jth irreducible representation of group Gaby´j ﬁ(ap). Simi- larly, we denote the the character for the ﬂth class of the lth irreducible representation of group Gaby´l ﬂ(bq). Since the direct products of irre- ducible representations of GaandGbare irreducible representations of Ga›Gb(Problem 4, Problem Set 8), and since the classes of Ga›Gb are formed from products of classes of GaandGb, the character table of the direct product group has the form ´jl ﬁﬂ(apbq)=´j ﬁ(ap)´l ﬂ(bq): In other words, with the character tables regarded as square matrices, the character table of the direct product group Ga£Gbis constructed as a direct product of the character tables of GaandGb! Groups and Representations in Quantum Mechanics 101 Example 6.4. For the direct product group in Example 6.3, the char- acter tables of the original groups are E 3¾v2C3 A111 1 A21¡11 E 20¡1 and E¾h A111 A21¡1 The character table of the direct product group is, therefore, the direct product of these tables (cf. Example 6.1): E 3¾v2C3¾h3¾h¾v2¾hC3 A+ 111 1 11 1 A+ 21¡11 1¡11 E+20¡1 20¡1 A¡ 111 1¡1¡1¡1 A¡ 21¡11¡11¡1 E¡20¡1¡20 1 where the superscript on the irreducible representation refers to the parity under re°ection through ¾h. 6.6 Selection Rules One common application of direct products and their representations is in the determination of selection rules. In this section, we will applythe techniques developed in this chapter to determine the conditionswhere symmetry requires that a matrix element vanishes. 102 Groups and Representations in Quantum Mechanics 6.6.1 Matrix Elements As discussed in Section 6.3.2, the determination of selection rules is based on using group theory to ascertain when the matrix element Mif=(ijH0jf)=Z ’i(x)⁄H0’f(x)dx (6.13) vanishes by reasons of symmetry. In this matrix element, the initial state transforms according to an irreducible representation ¡(i)and the ﬂnal state transforms according to an irreducible representation¡ (f). It only remains to determine the transformation properties of H0. We do this by applying each of the operations of the group of the original Hamiltonian H0to the perturbation H0. If we retain only the distinct results of these operations we obtain, by construction, arepresentation of the group of the Hamiltonian, which we denote by ¡ 0. This representation may be either reducible or irreducible, dependingonH 0and on the symmetry of H0.I fH0has the same symmetry as H0, then this procedure generates the identical representation. At the otherextreme, ifH 0hasnone of the symmetry properties of H0, then this procedure generates a reducible representation whose dimensionality isequal to the order of the group. We now consider the symmetry properties under transformation of the productH(x)’ f(x). From the discussion in Section 6.5.2, we con- clude that this quantity transforms as the direct product ¡0›¡(f). Since quantities that transform to diﬁerent irreducible irreducible rep-resentations are orthogonal (Problem 6, Problem Set 8), the matrixelement (6.13) vanishes if this direct product is either not equal to ¡ (i) or, if it is reducible, does not include ¡(i)in its decomposition. We can summarize this result in the following theorem: Theorem 6.3. The matrix element (ijH0jf)=Z ’i(x)⁄H0’f(x)dx vanishes if the irreducible representation ¡(i)corresponding to ’iis not included in the direct product ¡0›¡(f)of the representations ¡0and ¡(f)corresponding toH0and’f, respectively. Groups and Representations in Quantum Mechanics 103 It is important to note that this selection rule only provides a con- dition that guarantees that the matrix element will vanish. It does not guarantee that the matrix element will not vanish even if the conditionsof the theorem are fulﬂlled. 6.6.2 Dipole Selection Rules As a scenario which illustrates the power of group theoretical methods,suppose thatH 0transforms as a vector, i.e., as ( x;y;z ). This situation arises when the transitions described by Fermi’s Golden Rule (Section6.3.2) are caused by an electromagnetic ﬂeld. The form of H 0in the presence of an electromagnetic potential Ais obtained by making the replacement6 p!p¡eA for the momentum in the Hamiltonian. For weak ﬂelds, this leads to a perturbation of the form H0=e mp¢A (6.14) Since the electromagnetic is typically uniform, we can write the matrix elementMifas Mif»(ijpjf)¢A so the transformation properties of p=(px;py;pz), which are clearly those of a vector , determine the selection rules for electromagnetic tran- sitions. These are called the dipole selection rules. The examination of many properties of materials rely on the evaluation of dipole matrixelements. Example 6.6. Suppose the group of the Hamiltonian corresponds to the symmetry operations of an equilateral triangle, i.e., C 3v, the char- acter table for which is (Example 5.5) 6H. Goldstein, Classical Mechanics (Addison{Wesley, Reading, MA, 1950) 104 Groups and Representations in Quantum Mechanics C3vE 3¾v2C3 A111 1 A21¡11 E 20¡1 To determine the dipole selection rules for this system, we must ﬂrst determine the transformation properties of a vector r=(x;y;z ). We take thex- andy-axes in the plane of the equilateral triangle and the z-axis normal to this plane to form a right-handed coordinate system. Applying each symmetry operation to rproduces a reducible repre- sentation because these operations are either rotations or re°ectionsthrough vertical planes. Thus, the zcoordinate is invariant under ev- ery symmetry operation of this group which, together with the fact thatan (x;y) basis generates the two-dimensional irreducible representation E, yields ¡ 0=A1'E We must now calculate the characters associated with the direct products of between ¡0and each irreducible representation to determine the allowed ﬂnal states given the transformation properties of the initialstates. The characters for these direct products are shown below ¡ 0=A1'E C3vE 3¾v2C3 A1 11 1 A2 1¡11 E 20¡1 A1›¡031 0 A2›¡03¡10 E›¡060 0 Using the decomposition theorem, we ﬂnd A1›¡0=A1'E A2›¡0=A2'E E›¡0=A1'A2'2E Groups and Representations in Quantum Mechanics 105 Thus, if the initial state transforms as the identical representation A1, the matrix element vanishes if the ﬂnal state transforms as A2. If the initial state transforms as the \parity" representation A2, the matrix element vanishes if the ﬂnal state transforms as A1. Finally, there is no symmetry restriction if the initial state transforms as the \coordinate"representation E. 6.7 Summary This chapter has demonstrated how the mathematics of groups and their representations are used in quantum mechanics and, indeed, howmany of the structures introduced in the preceding chapters appearquite naturally in this context. Apart from exact results, such asBloch’s theorem, we have focussed on the derivation of selection rulesinduced by perturbations, and derived the principles behind dipole se-lection rules. A detailed discussion of other applications of discretegroups to quantum mechanical problems is described in the book byTinkham. Many of the proofs concerning the relation between quan-tum mechanics and representations of the group of the Hamiltonian arediscussed by Wigner. 106 Groups and Representations in Quantum Mechanics Chapter 5 Characters and Character Tables In great mathematics there is a very high degree of unexpectedness, com- bined with inevitability and economy. |Godfrey H. Hardy1 In the preceding chapter, we proved the Great Orthogonality Theorem,which is a statement about the orthogonality between the matrix ele-ments corresponding to diﬁerent irreducible representations of a group.For many applications of group theory, however, the full matrix rep-resentations of a group are not required, but only the traces withinclasses of group elements|called \characters." A typical applicationinvolves determining whether a given representation is reducible or irre-ducible and, if it is reducible, to identify the irreducible representationscontained within that representation. In this chapter, we develop the mathematical machinery that is used to assemble the characters of the irreducible representations of a groupin what are called \character tables." The compilation of character ta-bles requires two types of input: the order of the group and the numberof classes it contains. These quantities provide stringent restrictions 1G.H. Hardy, A Mathematician’s Apology (Cambridge University Press, London, 1941) 65 66 Characters and Character Tables on the number of irreducible representations and their dimensionali- ties. Moreover, orthogonality relations derived from the Great Orthog-onality Theorem will be shown to provide constraints on characters ofdiﬁerent irreducible representations, which considerably simpliﬂes theconstruction of character tables. 5.1 Orthogonality Relations The Great Orthogonality Theorem, X ﬁ(Ak ﬁ)ij(Ak0 ﬁ)⁄ i0j0=jGj dk–i;i0–j;j0–k;k0 (5.1) is a relationship between the matrix elements of the irreducible repre- sentations of a group G. In this section, we show how this statement can be manipulated into an expression solely in terms of the traces ofthe matrices in these representations. This will open the way to estab-lishing a sum rule between the number of irreducible representationsand the number of classes in a group. We begin by setting j=iandj 0=i0in (5.1), X ﬁ(Ak ﬁ)ii(Ak0 ﬁ)⁄ i0i0=jGj dk–i;i0–k;k0; (5.2) where we have used the fact that –i;i0–i;i0=–i;i0. Summing over iandi0 on the left-hand side of this equation yields X i;i0X ﬁ(Ak ﬁ)ii(Ak0 ﬁ)i0i0=X ﬁ"X i(Ak ﬁ)ii# |{z} tr(Ak ﬁ)"X i0(Ak0 ﬁ)⁄ i0i0# |{z} tr(Ak0 ﬁ)⁄ =X ﬁtr(Ak ﬁ)tr(Ak0 ﬁ)⁄; and, by summing over iandi0on the right-hand side of (5.2), we obtain jGj dk–k;k0X iX i0–i;i0=jGj dk–k;k0X i1 |{z} dk=jGj–k;k0: Characters and Character Tables 67 We have thereby reduced the Great Orthogonality Theorem to X ﬁtr(Ak ﬁ)tr(Ak0⁄ ﬁ)=jGj–k;k0: (5.3) This expression can be written in a more useful form by observing that matrices corresponding to elements in the same conjugacy classhave the same trace. To see this, recall the deﬂnition in Section 2.6of the conjugacy of two elements aandbgroupG. There must be an element ginGsuch thata=gbg ¡1. Any representation fAﬁg, reducible or irreducible, must preserve this relation: Aa=AgAbAg¡1: This representation must also have the property that Ag¡1=A¡1 g.T h u s (Problem 2, Problem Set 4), tr(Aa) = tr(AgAbA¡1 g) = tr(A¡1 gAgAb) = tr(Ab): We can now introduce the notation ´k ﬁfor the trace corresponding to allof the elements of the ﬁth class of the kth irreducible representation. This is called the character of the class. If there are nﬁelements in this class, then we can write the relation (5.3) in terms of characters as asum over conjugacy classes CX ﬁ=1nﬁ´k ﬁ´k0⁄ ﬁ=jGj–k;k0; (5.4) whereCis the number of conjugacy classes. In arriving at this relation, we have proven the following theorem: Theorem 5.1 (Orthogonality Theorem for Characters). The char- acters of the irreducible representations of a group obey the relation X ﬁnﬁ´k ﬁ´k0⁄ ﬁ=jGj–k;k0: This orthogonality theorem can be used to deduce a relationship between the number classes of a group and the number of irreduciblerepresentations. By rearranging (5.4) as X ﬁ•µnﬁ jGj¶1=2 ´k ﬁ‚•µnﬁ jGj¶1=2 ´k0⁄ ﬁ‚ =–k;k0 68 Characters and Character Tables and introducing the vectors e´k=jGj¡1=2(pn1´k 1;pn2´k 2;:::;pnC´k C); we can write the orthogonality relation for characters as e´k¢e´k0=–k;k0: Thee´kreside in a space whose dimension is the number of classes Cin the group. Thus, the maximum number of a set of mutually orthogonalvectors in this space is C. But these vectors are labelled by an index k corresponding to the irreducible representations of the group. Hence,the number of irreducible representations must be less than or equal tothe number of classes . It is also possible 2to obtain an orthogonality relation with the roles of the irreducible representations and classes reversed in comparison tothat in Theorem 5.1: X k´k ﬁ´k⁄ ﬂ=jGj nﬁ–ﬁ;ﬂ: (5.5) By following analogous reasoning as above, we can deduce that this or- thogonality relation implies that the number of irreducible representa- tions must be greater than or equal to the number of classes. Combined with the statement of Theorem 5.1, we have the following theorem: Theorem5.2. The number of irreducible representations of a group is equal to the number of conjugacy classes of that group. Example 5.1. For Abelian subgroups each element is in a class by itself (Problem 6, Problem Set 3). Thus, the number of classes is equalto the order of the group, so, according to Theorem 5.2, the numberof irreducible representations must also equal the order of the group.When combined with the restriction imposed by Eqn. (4.19), which wecan now write as jGjX k=1d2 k=jGj; 2M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, 1989, New York) pp. 106{110. Characters and Character Tables 69 we have an alternative way (cf. Problem 4, Problem Set 5) of seeing that all of the the irreducible representations of an Abelian group areone-dimensional, i.e., d k= 1, fork=1;2;:::;jGj. Example 5.2. For the group S3, there are three classes: feg,fa;b;cg, andfd;fg(Example 2.9). Thus, there are three irreducible represen- tations which, as we have seen, consist of two one-dimensional repre-sentations and one two-dimensional representation. 5.2 The Decomposition Theorem One of the main uses of characters is in the decomposition of a givenreducible representation into its constituent irreducible representations.The procedure by which this is accomplished is analogous to projectinga vector onto a set of complete orthogonal basis vectors. The theoremwhich provides the foundation for carrying this out with characters isthe following: Theorem 5.3 (Decomposition Theorem). The character ´ ﬁfor the ﬁth class of any representation can be written uniquely in terms of the corresponding characters of the irreducible representations of the groupas ´ ﬁ=X kak´k ﬁ; where ak=1 jGjX ﬁnﬁ´k⁄ ﬁ´ﬁ: Proof. For a reducible representation, the same similarity transfor- mation brings all of the matrices into the same block-diagonal form. Inthis form, the matrix A ﬁcan be written as the direct sum of matrices Ak jof irreducible representations: Aﬁ=Ak1 ﬁ'Ak2 ﬁ'¢¢¢'Akn ﬁ; 70 Characters and Character Tables whereﬁ=1;2;:::;jGjandk1;k2;:::knlabel irreducible representa- tions. Given this, and the fact that similarity transformations leavethe trace invariant, we can write the character ´ iof this reducible rep- resentation corresponding to the ith class as ´ﬁ=X kak´k ﬁ; (5.6) where the akmust be nonnegative integers . We now multiply both sides of this equation by nﬁ´k0⁄ ﬁ, sum overﬁ, and use the orthogonality relation (5.4): X ﬁnﬁ´k0⁄ ﬁ´ﬁ=X kakX ﬁnﬁ´kﬁ´k0⁄ ﬁ |{z} jGj–k;k0=jGjak0 Thus, ak0=1 jGjX ﬁnﬁ´k0⁄ ﬁ´ﬁ; (5.7) soak0is the projection of the reducible representation onto the k0th irreducible representation. Note that, because the number of irreduciblerepresentations equals the number of classes, the orthogonal vectors ofcharacters span the space whose dimensionality is the number of classes,so this decomposition is unique. The Decomposition Theorem reduces the task of determining the ir- reducible representations contained within a reducible representation toone of vector algebra. Unless a particular application requires the ma-trix forms of the representations, there is no need to block-diagonalizea representation to identify its irreducible components. We can follow a procedure analogous to that used to prove the De- composition Theorem to derive a simple criterion to identify whethera representation is reducible or irreducible. We begin with the decom-position (5.6) and take its complex conjugate: ´ ⁄ ﬁ=X k0ak0´k0⁄ ﬁ; (5.8) Characters and Character Tables 71 where we have used the fact that the akare integers, so a⁄ k=ak.W e now take the product of (5.6) and (5.8), multiply by nﬁ, sum over ﬁ, and invoke (5.4): X ﬁnﬁ´ﬁ´⁄ ﬁ=X k;k0akak0X inﬁ´kﬁ´k0⁄ ﬁ |{z} jGj–k;k0=jGjX ka2 k: Thus, X ﬁnﬁj´2 ﬁj=jGjX ka2 k: (5.9) If the representation in question is irreducible, then all of the akare zero, except for the one corresponding to that irreducible representation,which is equal to unity. If the representation is reducible, then therewill be at least two of the a kwhich are positive integers. We can summarize these observations with a simple criterion for reducibility.If the representation is irreducible, then X ﬁnﬁj´ﬁj2=jGj; (5.10) and if the representation is reducible, X ﬁnﬁj´ﬁj2>jGj: (5.11) Example 5.3. Consider the following representation of S3: e=ˆ10 01! ;a =1 2ˆ1¡p 3 ¡p 3¡1! ;b =1 2ˆ1p 3 p 3¡1! ; c=ˆ¡10 01! ;d=1 2ˆ¡1p 3 ¡p 3¡1! ;f =1 2ˆ¡1¡p 3 p 3¡1! : There are three classes of this group, feg,fa;b;cg, andfd;fg,s ow e haven1=1 ,n2= 3, andn3= 2, respectively. The corresponding characters are ´1=2;´ 2=0;´ 3=¡1: 72 Characters and Character Tables Forming the sum in (5.9), we obtain 3X ﬁ=1nﬁj´ﬁj2=( 1£4 )+( 3£0 )+( 2£1 )=6; which is equal to the order of the group. Therefore, this representation isirreducible , as we have already demonstrated in Example 3.4 and in Problem 1, Problem Set 6. Example 5.4. Another representation of S3is e=d=f=ˆ10 01! ;a =b=c=1 2ˆ¡1¡p 3 ¡p 31! : The characters corresponding to the three classes are now ´1=2;´ 2=0;´ 3=2: Forming the sum in (5.9), we ﬂnd 3X i=1n1j´ij2=( 1£4 )+( 3£0 )+( 2£4 )=1 2; which is greater than the order of the group, so this representation is reducible (cf. Problem 2, Problem Set 6). To determine the irreducible constituents of this representation, we use the decomposition theorem.There are three irreducible representations of S 3: the one-dimensional identical representation, with characters ´1 1=1;´1 2=1;´1 3=1; the one-dimensional \parity" representation, with characters ´2 1=1;´2 2=¡1;´2 3=1; and the two-dimensional \coordinate" representation discussed above in Example 5.3, with characters ´3 1=2;´3 2=0;´3 3=¡1: Characters and Character Tables 73 We now calculate the akusing the expression in Equation (5.7). These determine the \projections" of the characters of the reducible represen-tation onto the characters of the irreducible representation. We obtain a 1=1 6h (1£1£2 )+( 3£1£0 )+( 2£1£2)i =1; a2=1 6h (1£1£2 )+( 3£¡1£0 )+( 2£1£2)i =1; a3=1 6h (1£2£2 )+( 3£0£0 )+( 2£¡1£2)i =0: Thus, this reducible representation is composed of the identical repre- sentation and the \parity" representation, with no contribution fromthe \coordinate" representation. The block-diagonal form of this rep-resentation is, therefore, e=d=f= ˆ10 01! ;a =b=c=1 2ˆ10 0¡1! ; which is the result obtained in Problem 5, Problem Set 5 by applying matrix methods. 5.3 The Regular Representation Our construction of irreducible representations has thus far proceededin an essentially ad hoc fashion, relying in large part on physical argu- ments. We have not yet developed a systematic procedure for construct-ing all of the irreducible representations of a group. In this section, weintroduce a method, based on what is called the \regular" represen-tation, which enables us to accomplish this. However, our purposefor introducing such a methodology is not the determination of irre-ducible representations as such, since even for relatively simple groups,the approach we describe would present a computationally demandingprocess, but as a theoretical tool for proving a theorem. Moreover, wewill ﬂnd that, for applications of group theory to quantum mechanics,the irreducible representations of the group of operations that leaveHamiltonian invariant will emerge naturally without having to rely onany auxiliary constructions. 74 Characters and Character Tables Theregular representation is a reducible representation that is ob- tained directly from the multiplication table of a group. As we will showbelow, this representation contains every irreducible representation ofa group at least once. The construction of the regular representationis based on arranging the multiplication table of a group so that theunit element appears along the main diagonal of the table. Within suchan arrangement the columns (or rows) of the table are labelled by thegroup elements, arranged in any order, and the corresponding order ofthe inverses labels the rows (or columns). As an example, consider the multiplication table for S 3(Section 2.4) arranged in the way just described: eabcdf e=e¡1eabcdf a=a¡1aedfbc b=b¡1bfedca c=c¡1cdfeab f=d¡1fbcaed d=f¡1dcabfe The matrices of the regular representation are obtained by regarding the multiplication table as an jGj£jGjarray from which the matrix representation for each group element is assembled by putting a ‘1’where that element appears in the multiplication table and zero else-where. For example, the matrices corresponding to the unit eand the elementaare e!0 BBBBBBBBBBBBB@100000 0100000010000001000000100000011 CCCCCCCCCCCCCA ;a!0 BBBBBBBBBBBBB@010000 1000000000010000100001000010001 CCCCCCCCCCCCCA with analogous matrices for the other group elements. Characters and Character Tables 75 Our ﬂrst task is to show that the regular ‘representation’ is indeed a representation of the group. First of all, it is clear that the mappingwe have described is one-to-one. For any two elements g 1andg2of this group, we denote the matrices in the regular representation thatcorrespond to these elements as A reg(g1) andAreg(g2). Thus, to show that these matrices form a representation of S3, we need to verify that Areg(g1g2)=Areg(g1)Areg(g2); i.e., that the multiplication table is preserved by this representation. We consider this relation expressed in terms of matrix elements: h Areg(g1g2)i ij=X kh Areg(g1)i ikh Areg(g2)i kj: (5.12) From the way the regular representation has been constructed, the ith row index of these matrix elements can be labelled the inverse of theith group element g ¡1 iand thejth column can be labelled by the jth group element gj: h Areg(g1g2)i ij=h Areg(g1g2)i g¡1 i;gj=8 < :1;ifg¡1 igj=g1g2; 0; otherwise h Areg(g1)i ik=h Areg(g1)i g¡1 i;gk=8 < :1;ifg¡1 igk=g1; 0; otherwise h Areg(g2)i kj=h Areg(g2)i g¡1 k;gj=8 < :1;ifg¡1 kgj=g2; 0; otherwise Therefore, in the sum over kin (5.12), we have nonzero entries only when g1g2=(g¡1 igk)(g¡1 kgj)=g¡1 igj; which gives precisely the nonzero matrix elements of Areg(g1g2). Hence, the matrices Areg(g1) preserve the group multiplication table and thereby form a faithful representation of the group. 76 Characters and Character Tables Our main purpose in introducing the regular representation is to prove the following theorem: Theorem 5.4. The dimensionalities dkof the irreducible representa- tions of a group are related to the order jGjof the group by X kd2 k=jGj: This theorem shows that the inequality (4.19), which was deduced di- rectly from the Great Orthogonality Theorem is, in fact, an equality. Proof. We ﬂrst show, using Eqn. (5.9), that the regular represen- tation is reducible. To evaluate the sums on the left-hand side of thisequation, we note that, from the construction of the regular repre-sentation, the characters ´ reg;ivanish for every class except for that corresponding to the unit element. Denoting this character by ´reg;e, we see that its value must be equal to the order of the group: ´reg;e=jGj: Thus, X ﬁnﬁj´ﬁj2=´2 reg;e=jGj2; which, forjGj>1i sgreater thanjGj. Thus, for groups other than the single-element group feg, the regular representation is reducible. We will now use the Decomposition Theorem to identify the irre- ducible constituents of the regular representation. Thus, the characters´ reg;ﬁfor theﬁth class in the regular representation can be written as ´reg;ﬁ=X kak´k ﬁ: According to the Decomposition Theorem, the akare given by ak=1 jGjX ﬁnﬁ´k⁄ ﬁ´reg;ﬁ: Characters and Character Tables 77 We again use the fact that ´reg;e=jGj, with all other characters vanish- ing. The corresponding value of ´k eis determined by taking the trace of the identity matrix whose dimensionality is that of the kth irreducible representation: ´k e=dk. Therefore, the Decomposition Theorem yields ak=1 jGj£dk£jGj=dk; i.e., thekth irreducible representation appears dktimes in the regular representation: each one-dimensional irreducible representation appearsonce, each two-dimensional irreducible representation appears twice,and so on. Since the dimensionality of the regular representation is jGj, and sincea kis the number of times the kth irreducible representation appears in the regular representation, we have the constraint X kakdk=jGj; i.e., X kd2 k=jGj: This sum rule, and that equating the number of classes to the num- ber of irreducible representations (Theorem 5.2), relate a property ofthe abstract group (its order and the number of classes) to a propertyof the irreducible representations (their number and dimensionality).The application of these rules and the orthogonality theorems for char-acters is the basis for constructing character tables. This is describedin the next section. 5.4 Character Tables Character tables are central to many applications of group theory tophysical problems, especially those involving the decomposition of re-ducible representations into their irreducible components. Many text-books on group theory contain compilations of character tables for the 78 Characters and Character Tables most common groups. In this section, we will describe the construction of character tables for S3. We will utilize two types of information: sum rules for the number and dimensionalities of the irreducible represen-tations, and orthogonality relations for the characters. Additionally,the group multiplication table can be used to establish relationshipsfor one-dimensional representations. By convention, characters tablesare displayed with the columns labelled by the classes and the rows bythe irreducible representations. The ﬂrst step in the construction of this character table is to note that, sincejS 3j= 6 and there are three classes (Example 2.9), there are 3 irreducible representations whose dimensionalities must satisfy d2 1+d2 2+d2 3=6: The unique solution of this equation (with only positive integers) is d1=1 ,d2= 1, andd3= 2, so there are two one-dimensional irreducible representations and one two-dimensional irreducible representation. In the character table for any group, several entries can be made im- mediately. The identical representation, where all elements are equal tounity, is always a one-dimensional irreducible representation. Similarly,the characters corresponding to the unit element are equal to the di-mensionality of that representation, since they are calculated from thetrace of the identity matrix with that dimensionality. Thus, denotingbyﬁ,ﬂ,°, and–quantities that are to be determined, the character table forS 3is: S3fegfa;b;cgfd;fg ¡111 1 ¡21ﬁﬂ ¡32°– where the ¡ iare a standard label for the irreducible representations. The remaining entries are determined from the orthogonality re- lations for characters and, for one-dimensional irreducible representa-tions, from the multiplication table of the group. The orthogonalityrelation in Theorem 5.1, which is an orthogonality relation for the rows of a character table, yield 1+3ﬁ+2ﬂ=0; (5.13) Characters and Character Tables 79 1+3ﬁ2+2ﬂ2=6: (5.14) The group multiplication table requires that a2=e; b2=e; c2=e; d2=f: Since the one-dimensional representations must obey the multiplication table, these products imply that ﬁ2=1;ﬂ2=ﬂ: Substituting these relations into (5.14), yields 4 + 2 ﬂ= 6, i.e., ﬂ=1 Upon substitution of this value into (5.13), we obtain 3 + 3 ﬁ= 0, i.e., ﬁ=¡1 From the orthogonality relation (5.5), which is an orthogonality relation between the columns of a character table, we obtain 1+ﬁ+2°=0 1+ﬂ+2–=0 Substituting the values obtained for ﬁandﬂinto these equations yields °=0;– =¡1 The complete character table for S3is therefore given by S3fegfa;b;cgfd;fg ¡111 1 ¡21¡11 ¡320¡1 When character tables are compiled for the most common groups, a notation is used which re°ects the fact that the group elements cor-respond to transformations on physical objects. The notation for theclasses ofS 3are as follows: 80 Characters and Character Tables †feg!E. The identity. †fa;b;cg! 3¾v. Re°ection through vertical planes, where ‘verti- cal’ refers to the fact that these planes contain the axis of highestrotational symmetry, in this case, the z-axis. The ‘3’ refers to there being three elements in this class. †fd;fg! 2C 3. Rotation by2 3…radians, with the ‘2’ again referring to the there being two elements in this class. The notation C2 3 is for rotations by4 3…radians, so the ‘class’ notation is meant only to indicate the type of operation. In general, Cnrefers to rotations through 2 …=nradians. Several notations are used for irreducible representations. One of the most common is to use Afor one-dimensional representations, E for two-dimensional representations, and Tfor three-dimensional rep- resentations, with subscripts used to distinguish multiple occurrencesof irreducible representations of the same dimensionality. The notation¡ is often used to indicate a generic (usually irreducible) representa-tion, with subscripts and superscripts employed to distinguish betweendiﬁerent representations. With the ﬂrst of these conventions, the char-acter table for S 3, which is known as the group C3vwhen interpreted as the planar symmetry operations of an equilateral triangle, is C3vE 3¾v2C3 A111 1 A21¡11 E 20¡1 5.5 Summary This chapter has been devoted to characters and character tables. The utility of characters in applications stems from the following: Characters and Character Tables 81 1. The character is a property of the class of an element. 2. Characters are unaﬁected by similarity transformations, so equiv- alent representations|reducible or irreducible|have the samecharacters. 3. As shown in Equations (5.10) and (5.11), the characters of a repre- sentation indicate, through a straightforward calculation, whetherthat representation is reducible or irreducible. 4. Characters of irreducible representations obey orthogonality the- orems which, when interpreted in the context of character ta-bles, correspond to the orthogonality relations of their rows andcolumns. 5. According to the Decomposition Theorem, once the character ta- ble of a group is known, the characters of any representation canbe decomposed into its irreducible components. 82 Characters and Character Tables Chapter 9 Unitary Groups and SU(N)⁄ The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which haverotational symmetry in three dimensions. But there are many systemsfor which operations on classical coordinates must be supplementedby operations on \internal" degrees of freedom which have no classicalanalogue. For example, the Stern{Gerlach experiment showed thatelectrons are endowed with an internal degree of freedom called \spin"which has the properties of an angular momentum. The two spin statesare therefore inconsistent with the dimensionalities of the irreduciblerepresentations of SO(3), so another group|SU(2)|must be used todescribe these states. Since, as we will show in Section 9.2, SU(2) islocally isomorphic to SO(3), we can deﬂne a total spin Sin an abstract three-dimensional space, analogous to the total angular momentum inreal space. In particle physics, unitary symmetry was used to describethe approximate symmetry (called isospin) of neutrons and protonsand, more recently, to describe particle spectra within the frameworkof the quark model. In this chapter, we introduce unitary groups and their irreducible representations in a similar manner to which we developed SO(3). Webegin by deﬂning unitarity in terms of the invariance of an appropriatequantity and proceed to discuss the construction of irreducible repre-sentations of these groups in Ndimensions. Higher-dimensional irre- ducible representations will be obtained with the aid of Young tableaux,which is a diagrammatic technique for determining the dimensionali- 147 148 Unitary Groups and SU(N) ties and the basis functions of irreducible representations derived from direct products. 9.1 SU(2) As with orthogonal matrices, the unitary groups can be deﬂned in termsof quantities which are left invariant. Consider a general complex trans-formation in two dimensions, x 0=Axwhich, in matrix form, reads: ˆx0 y0! =ˆab cd!ˆx y! wherea,b,c, anddare complex, so there are eight free parameters. The determinant of this matrix is nonzero to permit the constructionof inverses. 9.1.1 Unitary Transformations Suppose we require the quantity jxj2+jyj2to be an invariant of such a transformation. Then, jx0j2+jy0j2=jax+byj2+jcx+dyj2 =(ax+by)(a⁄x⁄+b⁄y⁄)+(cx+dy)(c⁄x⁄+d⁄y⁄) =(jaj2+jcj2)jxj2+(ab⁄+cd⁄)xy⁄+(a⁄b+c⁄d)x⁄y +(jbj2+jdj2)jyj2 =jxj2+jyj2 Sincexandyare independent variables, this invariance necessitates setting the following conditions on the matrix elements: jaj2+jcj2=1;jbj2+jdj2=1;a b⁄+cd⁄=0 These four conditions (the last equation provides two conditions be- cause it involves complex quantities) means that the original eight free Unitary Groups and SU(N) 149 parameters are reduced to four. These conditions are the same as those obtained by requiring the AyA= 1, so the determinant of the result- ing matrix has modulus unity. These transformations are analogousto orthogonal transformations of real coordinates and, indeed, orthog-onal transformations are also unitary. The group comprised of unitarymatrices is denoted by U(2) and by U(N) for the N-dimensional case. 9.1.2 Special Unitary Transformations If, in addition to the conditions above, we require that the determinant of the transformation is unity, the transformation matrix must have theform ˆx0 y0! =ˆab ¡b⁄a⁄!ˆx y! ;jaj2+jbj2= 1 (9.1) There are now three free parameters and the group of these matrices is denoted by SU(2) where, as in our discussion of orthogonal groups, the‘S’ signiﬂes ‘special’ because of the requirement of a unit determinant. 9.2 Relation between SU(2) and SO(3) 9.2.1 Pauli Matrices If the matrix elements of the general unitary matrix in (9.1) are ex-pressed in terms of their real and imaginary parts, we can decomposethis matrix into the components of a \basis." Thus, with a=a r+iai andb=br+ibi,w eh a v e U=ˆar+iaibr+ibi ¡br+ibiar¡iai! =arˆ10 01! +iaiˆ10 0¡1! +brˆ01 ¡10! |{z} ibrˆ0¡i i0!+ibiˆ01 10! 150 Unitary Groups and SU(N) Thus, any 2£2 unitary matrix can be represented as a linear combi- nation of the unit matrix and the matrices ¾x=ˆ10 0¡1! ;¾y=ˆ0¡i i0! ;¾z=ˆ01 10! These three (Hermitian) matrices are known as the Paulimatrices . They satisfy the following multiplication rules: ¾2 i=I (i=x;y;z ) ¾i¾j=¡¾j¾i=i"ijk¾k (fi;j;kg=x;y;z )(9.2) whereIis the 2£2 unit matrix. These multiplication rules can be used to obtain a concise expression for the product of two matriceswritten asa¢¾andb¢¾, wherea=(a x;ay;az),b=(bx;by;bz), and ¾=(¾x;¾y;¾z): (a¢¾)(b¢¾)=(a¢b)I+i(a£b)¢¾ (9.3) 9.2.2 Inﬂnitesimal Generators Moreover, if we deﬂne matrices Xi=¡1 2i¾i, fori=1;2;3, then the sec- ond of the multiplication rules in (9.2) yield the following commutationrelations: [X i;Xj]="ijkXk These are identical to commutators of the inﬂnitesimal generators of SO(3) in (7.13). Thus, locally at least, there is an isomorphism betweenSO(3) and SU(2). Motivated by the discussion in Section 7.3, considerthe matrix U= exp (¡ 1 2i’n¢¾) where’nis the axis-angle representation of a rotation (Section (8.3). Since the exponential of a matrix is deﬂned by its Taylor series expan-sion, we have U=1X k=0(¡i)n n!(1 2’)n(n¢¾)n Unitary Groups and SU(N) 151 From Equation (9.3), ( n¢¾)2=I,s o U=I1X k=0(¡1)n (2n)!(1 2’)2n¡i(n¢¾)1X k=0(¡1)n (2n+ 1)!(1 2’)2n+1 = cos (1 2’)I¡i(n¢¾) sin (1 2’) =2 64cos (1 2’)¡inzsin (1 2’)¡(ny+inx) sin (1 2’) (ny¡inx) sin (1 2’) cos (1 2’)+inzsin (1 2’)3 75 (9.4) This matrix is manifestly of the unitary form in (9.1) with unit deter- minant. The Pauli matrices are, therefore, the inﬂnitesimal generatorsof SU(2) and form a representation of its Lie algebra. 9.2.3 Local and Global Mappings between SU(2) and SO(3) The matrix in (9.4) is parametrized in the same way as rotations in SO(3), namely, in terms of a rotation angle ’and a rotation axis n. But, although the mapping between SU(2) and SO(3) is locally an iso- morphism, since their algebras are isomorphic, globally this relationship is a homomorphism. The reason for this stems from the periodicity ofthe two groups: SO(3) has a periodicity of 2 …, while SU(2) has a peri- odicity of 4 …. In particular U(0;n)=I, butU(2…;n)=¡I, so both of these elements are associated with the identity of SO(3). Moreover,these elements form an invariant subgroup of SU(2) (Section 2.4) whichis isomorphic to the group Z 2=f1;¡1gunder ordinary multiplication. In general, using the trigonometric identities, cosh 1 2(’+2…)i =¡cos (1 2’) sinh 1 2(’+2…)i =¡sin (1 2’) we ﬂnd that U(’+2…;n)=¡U(’;n) 152 Unitary Groups and SU(N) Thus, if we form the cosets of the subgroup fU(0;n);U(2…;n)g,w e obtain n U(0;n);U(2…;n)o U(’;n)=n U(’;n);U(’+2…;n)o Thus, the factor group SU(2)/Z 2is isomorphic to SO(3): SU(2)=Z2= SO(3) In fact, this double-valuedness extends to characters as well. Taking the trace of the matrix in 9.4) yields 2 cos (1 2’) If we compare this expression with that for ´(‘)(’) for SO(3) with ‘=1 2, we ﬂnd ´(1=2)(’)=sin’ sin (1 2’)= 2 cos (1 2’) so the two-dimensional (irreducible) representation of SU(2) generated by the Pauli matrices corresponds to a representation of SO(3) with ahalf-integer index. The integer values of ‘can be traced to the require- ment of single-valuedness of the spherical harmonics, so the double- valued correspondence between SU(2) and SO(3) results in this half-integer index. 9.3 Irreducible Representations of SU(2) When we constructed the irreducible representations of SO(2) andSO(3), we used as basis functions obtained from the coordinates fx;yg andfx;y;zg, respectively, and to obtain higher-order irreducible rep- resentations from direct products. The basic procedure is much thesame for unitary groups, except that we can no longer rely on basisstates expressed in terms of coordinates. In this section, we carry outthe required calculations for SU(2) and then generalize the method forSU(N) in the next section. Unitary Groups and SU(N) 153 9.3.1 Basis States By associating the Pauli matrices with angular momentum operators throughJi=1 2„h¾i, we choose as our basis states the vectors u1=ˆ1 0! ;u 2=ˆ01! There are several physical interpretations of these states. For exam- ple, they can represent the two possible energy eigenstates of a spin-1 2 particle, such an electron or proton. Another possibility is that u1and u2represent the isospin eigenstates of an isospin-1 2particle, such as a proton or a neutron. The fact that the proton and neutron are not ex-actly degenerate means that isospin symmetry is only an approximatesymmetry. A third interpretation of u 1andu2is as \up" and \down" quarks which make up nucleons. We will discuss further reﬂnements ofthe quark model in the context of SU(N) later in this chapter. 9.3.2 Multiparticle Systems and Direct Products When using basis states of SU(2) to construct multiparticle statesthrough direct products, we must respect the indistinguishability ofthe particles. Thus, measurable properties of a quantum system can- not depend on the labelling of the particles, though wavefunctions, of course, need not obey this invariance. Consider a two-particle system,with particle ‘1’ in state iand particle ‘2’ in state j. The corresponding wavefunction is ˆ i;j(1;2). We require that jˆi;j(1;2)j2=jˆi;j(2;1)j2 which implies that ˆi;j(2;1 )=eiµˆi;j(1;2) for some phase angle µ. Since a two-fold exchange restores the original labelling, ˆi;j(1;2 )=eiµˆi;j(2;1 )=e2iµˆi;j(1;2) 154 Unitary Groups and SU(N) we must have that e2iµ= 1, or that µ=0o rµ=…. In the ﬂrst case, the wavefunction is symmetric under the interchange of particles, ˆi;j(2;1) =ˆi;j(1;2) while in the latter case, the wavefunction is antisymmetric under the interchange of particles, ˆi;j(2;1) =¡ˆi;j(1;2) Consider now a two-particle system each of which occupy one of the states of SU(2). The basis of these two-particle states is comprised offu 1u1;u1u2;u2u1;u2u2g, where we have adopted the convention that the order of the states corresponds to the order of the particle coordinates,e.g.,u 1u1·u1(1)u1(2). But not all of these states are symmetric or antisymmetric under the interchange of particles. Hence, we constructthe new basis n u1u1;u1u2+u2u1;u2u2|{z} symmetric;u1u2¡u2u1|{z} antisymmetrico (9.5) We can compare this result with that obtained from the two-fold direct product representation of SU(2): ´(1=2)(’)´(1=2)(’)=h 2 cos (1 2)i2 =‡ ei’=2+e¡i’=2·2 =‡ ei’=2+1+e¡i’=2· +1 =´(1)(’)+´(0)(’) we see that the three symmetric wavefunctions for a basis for the ‘=1 irreducible representation of SO(3) and the antisymmetric wavefunc- tions transforms as the identical representation ( ‘= 0) of SO(3). If we think of these as spin-1 2particles, the symmetric state corresponds to a total spinS= 1, while the antisymmetric state corresponds to S=0 . We could proceed in this way to construct states for larger numbers ofparticles, but in the next section we introduce a technique which is farmore e–cient and which can be applied to other SU(N) groups, wherethe direct method described in this section becomes cumbersome. Unitary Groups and SU(N) 155 9.3.3 Young Tableaux Determining the dimensionalities of the irreducible representations of direct products of basis states of SU(N) is a problem which is encoun-tered in several applications in physics and group theory. Youngtableaux provide a diagrammatic method for carrying this out in a straightfor-ward manner. In this section, we repeat the calculation in the precedingsection to illustrate the method, and in the next section, we describethe general procedure for applying Young tableaux to SU(N). The basic unit of a Young tableau is a ‘box’, shown below which denotes a basis state. If there is no entry in the box, then thistableau represents any state. An entry, signiﬂed by a number denotesone of the basis states in some reference order. Thus, for SU(2), wehave u 1=1 u2=2 The utility of Young tableaux centers around the construction of direct products. For the two-fold direct products of SU(2) in (9.5),there are two types of states, symmetric and antisymmetric. The Youngtableau for a generic two-particle symmetric state is and the two-particle antisymmetric state is The Young tableaux for three symmetric states in (9.5) are 156 Unitary Groups and SU(N) 11 12 22 and that for the antisymmetric state is 1 2 In the framework of Young tableaux, the two-fold direct product is written as £ = + The three-fold direct product illustrates the conventions used in theconstruction of Young tableaux and their labelling. The generic tableauxare ££ = + + The rules for constructing the \standard" arrangement of Young tableauxare as follows †The rows are constructed from left to right †The columns are constructed from top to bottom †No row is longer than any row above it †No column is longer than any column to the left of it Thus, with these conventions, a typical tableau is shown below: Unitary Groups and SU(N) 157 The states for the three-fold direct product are as follows. There are four symmetric states: 111 112 122 222 which correspond to a four-dimensional irreducible representation, andtwo \mixed" states: 11 212 2 which correspond to a two-dimensional irreducible representation. Thereare no totally antisymmetric three-particle states because we have onlytwo distinct basis states. Thus, the rules for entering states into Youngtableaux are: †The numbers within rows are nondecreasing from left to right. †The numbers within columns are increasing from top to bottom. The two sets of rules for constructing Young tableaux of generic states and identifying particular states enables the calculation of thedimensionalities in a straightforward manner, often by identifying ap-propriate combinatorial rules. 9.4 Young Tableaux for SU(N) The groups SU(N) have acquired an importance in particle physicsbecause of the quark model. This necessitates calculating direct prod-ucts of basis states to determine the characteristics of particle spectra. 158 Unitary Groups and SU(N) This, in turn, requires that we adapt the methodology of the Young tableaux developed in the preceding section to SU(N), which turns outto be straightforward given the rules stated in the preceding section.There is no change to the construction of the generic tableaux; the onlychanges are in the labelling of the tableaux. Consider, for example thecase of a two-fold direct product of SU(3). There are six symmetricstates 11 12 13 22 23 33 and three antisymmetric states 1 21 32 3 As is evident from these constructions, the number of states asso- ciated with a tableau of a particular topology increases sharply withthe number of basis states. The rules in the preceding section allowthe number of such symmetric and antisymmetric states to be calcu-lated for SU(N). There 1 2N(N+ 1) symmetric states and1 2N(N¡1) antisymmetric states. The only other modiﬂcation to our discussion of SU(2) is that for larger numbers of basis states, tableaux which make no contributionto SU(2), may make a contribution to SU(N). Consider, for example,the antisymmetric three-particle state. This state vanishes for SU(2)because there are only two basis states, but for SU(3), we have 1 2 3 In fact, this is a direct consequence of the rule for labelling Young tableaux, and we see that, for SU(N), any column with more than N boxes makes no contribution. Unitary Groups and SU(N) 159 9.5 Summary In this chapter, we have extended our discussion of orthogonal groups to unitary groups. These groups play an especially important role inquantum mechanics because of their property of conserving probabilitydensity. We have constructed direct products of basis states, which arerequired in a number of applications of these groups. The use of Youngtableaux was shown to be an especially convenient way to determinethe dimensionalities of higher-dimensional irreducible representationsof unitary groups and their basis functions.