E(n)=Euclidean group in n dimensions = symmetries of n-dimensional space = rotations and translations
xi=Rijxj+ai for R in SO(n).
g=T(a)R for R in SO(n) and a a translation vector.
| |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
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)
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.
<pm|pn> = delta(m,n)=integral(0,2Pi,1/2Pi*e^(imt)*e^(-int)dt)
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.