SO(n) = orthogonal matrix with determinant 1. SO==special orthogonal Dim(SO(n))=n(n-1)/2 For X a defining rep of SO(3): (Xab)ij=delta(i,a)delta(j,b)-delta(j,a)delta(i,b) =>Commutation relations: [Xab,Xgd]=delta(b,g)Xad-delta(a,d)Xbg-delta(a,g)Xbg+delta(b,d)Xad Xab=eabgLg for eabg=antisymmetric tensor. SO(4) is special. It is the only SO(n) which decomposes. SO(4) ~= SO(3) x SO(3) Decomposition: Xij=-Xji=SO(3) sub algebra Define Lk as: Xij=i/2*fijkLk Xi4=-X4i Define Ki as:-iXij=Ki Define J(1)=1/2(L+K) (SU(2) group) Define J(2)=1/2(L-K) (Another SU(2) group) Note that [J(1)i,J(2)i]=0 => D(j1,j2) labeling ok. For arbitrary D(j1,j2) j1+j2=integer => rep exists. Example Of SO(4) symmetry in nature: SO(4) is equivalent to the symmetries of a planet orbiting a star or electron around a hydrogen atom (classically) Define: M=1/2m(PxL - LxP)-kX/r where * X= vector of current direction * P=current momentum * L=current angular momentum * m = reduced mass of system Note that L*M=0 and [M,H]=0 S0(2) = rotational symmetry of a circle. G={g(theta)|0<=theta<2Pi} g(th1)g(th2)=g(th1+th2) if th1+th2<2Pi, g(th1+th2-2Pi) if th1+th2=>2Pi Defining representation: g(th)= |cos(th) -sin(th)| |sin(th) cos(th)| =D(th) Transpose(D)D=Id.=> D real and unitary. This is an abelian group so its irreps must be 1-d. M^DM=D~= |e^ith 0 | |0 e^-ith| with M=1/sqrt(2)* |1 -i| |-i 1| Irreps of SO(2): Dm(th)=e^imth for m any integer. D=D1+D-1 D0 is the trivial rep. for M!=0 g(th)-> Dm(th) is |m| to 1 mapping. only D1 and D-1 are faithful. Integral(G,dg)=1. Here dg=dth/2Pi. Generator: D(dth)= |1 -dth| |dth 1 | +O(dth^2) =Id-idthJ+O(dth^2) where J= | 0 -i | | i 0 | Note that adjoint(J)=J. characters are orthogonal. Integral(G,Xm*(g)Xn(g)dg)=delta(m,n) <=> Integral(0,2Pi,e^imth * e^-inth dth/2Pi)=delta(m,n) Completeness: f(g)=Sum(-infinity,infinity,ame^-imth) f(th)=Sum(m,cme^-imth) whre cm=Integral(0,2Pi,e^imth * f(th)dth/2Pi) This is basically the Fourier series. SO(1,1)= |cosh t sinh t| |sinh t cosh t| -infinity Fourier transform The conjugate rep. to a rep in SO(m) is found by filling in the missing spaces in a young diagram. For example the conjugate to: ### # is: ## ###