Chapter 9 Unitary Groups and SU(N)

1 Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describingthe 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.}

2 Since, as we will show in Section , SU(2) islocally isomorphic to SO(3), we can de ne a total spinSin an abstractthree-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 this Chapter , we introduce Unitary Groups and their irreduciblerepresentations in a similar manner to which we developed SO(3). Webegin by de ning unitarity in terms of the invariance of an appropriatequantity and proceed to discuss the construction of irreducible repre-sentations of these Groups inNdimensions. Higher-dimensional irre-ducible representations will be obtained with the aid of Young tableaux,which is a diagrammatic technique for determining the dimensionali-147148 Unitary Groups and SU(N) ties and the basis functions of irreducible representations derived fromdirect SU(2)As with orthogonal matrices, the Unitary Groups can be de ned in termsof quantities which are left invariant.

3 Consider a general complex trans-formation in two dimensions,x0=Axwhich, in matrix form, reads: x0y0!= abcd! xy!wherea,b,c, anddare complex, so there are eight free determinant of this matrix is nonzero to permit the constructionof Unitary TransformationsSuppose we require the quantityjxj2+jyj2to be an invariant of such atransformation. Then,jx0j2+jy0j2=jax+byj2+jcx+dyj2=(ax+b y)(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 necessitatessetting the following conditions on the matrix elements:jaj2+jcj2=1;jbj2+jdj2=1;ab +cd =0 These four conditions (the last equation provides two conditions be-cause it involves complex quantities) means that the original eight freeUnitary Groups and SU(N) 149parameters are reduced to four.

4 These conditions are the same as thoseobtained by requiring theAyA= 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 theN-dimensional Special Unitary TransformationsIf, in addition to the conditions above, we require that the determinantof the transformation is unity, the transformation matrix must have theform x0y0!= ab b a ! xy!;jaj2+jbj2= 1( )There are now three free parameters and the group of these matrices isdenoted by SU(2) where, as in our discussion of orthogonal Groups , the`S' signi es `special' because of the requirement of a unit Relation between SU(2) and SO(3) Pauli MatricesIf the matrix elements of the general Unitary matrix in ( ) are ex-pressed in terms of their real and imaginary parts, we can decomposethis matrix into the components of a \basis.

5 " Thus, witha=ar+iaiandb=br+ibi,wehaveU= ar+iaibr+ibi br+ibiar iai!=ar 1001!+iai 100 1!+br 01 10!|{z}ibr 0 ii0!+ibi 0110!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 x= 100 1!; y= 0 ii0!; z= 0110!These three (Hermitian) matrices are known as thePauli matrices. Theysatisfy the following multiplication rules: 2i=I(i=x;y;z) i j= j i=i"ijk k(fi;j;kg=x;y;z)( )whereIis the 2 2 unit matrix. These multiplication rules can beused to obtain a concise expression for the product of two matriceswritten asa andb , wherea=(ax;ay;az),b=(bx;by;bz), and =( x; y; z):(a )(b )=(a b)I+i(a b) ( ) In nitesimal GeneratorsMoreover, if we de ne matricesXi= 12i i, fori=1;2;3, then the sec-ond of the multiplication rules in ( ) yield the following commutationrelations:[Xi;Xj]="ijkXkThese are identical to commutators of the in nitesimal generators ofSO(3) in ( ).

6 Thus, locally at least, there is an isomorphism betweenSO(3) and SU(2). Motivated by the discussion in Section , considerthe matrixU= exp ( 12i'n )where'nis the axis-angle representation of a rotation (Section ( ).Since the exponential of a matrix is de ned by its Taylor series expan-sion, we haveU=1Xk=0( i)nn!(12')n(n )nUnitary Groups and SU(N) 151 From Equation ( ), (n )2=I,soU=I1Xk=0( 1)n(2n)!(12')2n i(n )1Xk=0( 1)n(2n+ 1)!(12')2n+1= cos (12')I i(n ) sin (12')=264cos (12') inzsin (12') (ny+inx) sin (12')(ny inx) sin (12')cos (12')+inzsin (12')375( )This matrix is manifestly of the Unitary form in ( ) with unit deter-minant. The Pauli matrices are, therefore, the in nitesimal generatorsof SU(2) and form a representation of its Lie Local and Global Mappings between SU(2)and SO(3)The matrix in ( ) is parametrized in the same way as rotations inSO(3), namely, in terms of a rotation angle'and a rotation , although the mapping between SU(2) and SO(3) islocal lyan iso-morphism, since their algebras are isomorphic,globallythis relationshipis a homomorphism.)

7 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 particularU(0;n)=I, butU(2 ;n)= I, so bothof these elements are associated with the identity of SO(3). Moreover,these elements form an invariant subgroup of SU(2) (Section ) whichis isomorphic to the group Z2=f1; 1gunder ordinary general, using the trigonometric identities,cosh12('+2 )i= cos (12')sinh12('+2 )i= sin (12')we nd thatU('+2 ;n)= U(';n)152 Unitary Groups and SU(N) Thus, if we form the cosets of the subgroupfU(0;n);U(2 ;n)g,weobtainnU(0;n);U(2 ;n)oU(';n)=nU(';n);U('+2 ;n)oThus, the factor group SU(2)/Z2is isomorphic to SO(3):SU(2)=Z2= SO(3)In fact, this double-valuedness extends to characters as well.

8 Takingthe trace of the matrix in ) yields2 cos (12')If we compare this expression with that for (`)(') for SO(3) with`=12,we nd (1=2)(')=sin'sin (12')= 2 cos (12')so the two-dimensional (irreducible) representation of SU(2) generatedby 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 ofsingle-valuednessof the spherical harmonics, so the double-valued correspondence between SU(2) and SO(3) results in this half-integer Irreducible Representations of SU(2)When we constructed the irreducible representations of SO(2) andSO(3), we used as basis functions obtained from the coordinatesfx;ygandfx;y;zg, respectively, and to obtain higher-order irreducible rep-resentations from direct products.

9 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 Groups and SU(N) Basis StatesBy associating the Pauli matrices with angular momentum operatorsthroughJi=12 h i, we choose as our basis states the vectorsu1= 10!;u2= 01!There are several physical interpretations of these states. For exam-ple, they can represent the two possible energy eigenstates of a spin-12particle, such an electron or proton. Another possibility is thatu1andu2represent the isospin eigenstates of an isospin-12particle, such as aproton or a neutron.

10 The fact that the proton and neutron are not ex-actly degenerate means that isospin symmetry is only an approximatesymmetry. A third interpretation ofu1andu2is as \up" and \down"quarks which make up nucleons. We will discuss further re nements ofthe quark model in the context of SU(N) later in this Multiparticle Systems and Direct ProductsWhen using basis states of SU(2) to construct multiparticle statesthrough direct products, we must respect the indistinguishability ofthe particles. Thus,measurableproperties of a quantum system can-not depend on the labelling of the particles, thoughwavefunctions, ofcourse, need not obey this invariance. Consider a two-particle system,with particle `1' in stateiand particle `2' in statej.