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.

<pm|pn> = delta(m,n)=integral(0,2Pi,1/2Pi*e^(imt)*e^(-int)dt)

<pm|g(a,t)|pn>=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.