E(n)=Euclidean group in n dimensions = symmetries of n-dimensional space = rotations and translations properties: * E(n) is Non compact, non-abelian. * Unitary irreps are infinite dimensional * T(n) = translation group in n dimensions a normal subgroup * E(n) = semi direct product(SO(n),T(n)) xi=Rijxj+ai for R in SO(n). g=T(a)R for R in SO(n) and a a translation vector. Matrix representation: | |a1| | R |: | | |: | | |an| |________|__| |0.....0 | 1| =n+1 dimensional defining rep. generators of translations: matrix all 0 except in the aith position. generators of rotations: same as SO(n) For E(2) [P1,P2]=0, [M,P1]=P2 [M,P2]=-1P1 R^T(a)R=T(R^a) Unitary reps: D(a,I)=D(T(a))=e^(ip*a) for a an arbitrary vector and p another vector. D(a,I)f(p)=e^(ip*a)f(p) D(0,R)f(p)=f(R^p) =>D(a,R)=D(a,I)D(0,R)f(p)=e^(iR^p*a)f(R^p) This rep is unitary w.r.t. the inner product of quantum mechanics. Instead of having this continously infinite dimensional space, take the fourier series of of f(p,t). => f(p,t)-> |pm> where p in reals and m in Z. = delta(m,n)=integral(0,2Pi,1/2Pi*e^(imt)*e^(-int)dt) =D(p)mn(a,t)=i^(n-m)e^(imp)Jn-m(pa)e^(in(t-p)) where Jn(z)=bessel function=integral(0,2Pi,1/2Pi*e^(izSin(p) - np)dp) The properties of the group give rise to several orthogonality relations on the bessel functions.