QuantumPhysics.dvi

3. The Jacobi identity [X 1 ,[X 2 ,X 3 ]] + [X 2 ,[X 3 ,X 1 ]] + [X 3 ,[X 1 ,X 2 ]] = 0 holds;

A representationDof a Lie algebraGis a map fromGintoN×Nmatrices such that

for allX 1 ,X 2 ∈G, we have,

1’ D([X 1 ,X 2 ]) = [D(X 1 ),D(X 2 )];

2’ D(λ 1 X 1 +λ 2 X 2 ) =λ 1 D(X 1 ) +λ 2 D(X 2 ), and it follows thatD(0) = 0;

3’ The Jacobi identity is automatically obeyed on matrices.

8.5 Direct sum and reducibility of representations

Consider two representationsρ(1)andρ(2)of a groupG, or any two representationsD(1)and

D(2)of a Lie algebraG, withρ(i),D(i)of dimensionNi. We form the direct sum representa-

tions as follows,

(

ρ(1)⊕ρ(2)

)

(g) =







ρ(1)(g) 0

0 ρ(2)(g)







(

D(1)⊕D(2)

)

(X) =







D(1)(X) 0

0 D(2)(X)





 (8.13)

It is immediate to see thatρ(1)⊕ρ(2)is a representation ofG, and thatD(1)⊕D(2)is a

representation ofG, both of dimensionN=N 1 +N 2.

A representationρofG(similarlyDofG) isreducibleifρcan be written as the direct

sum of two representations ofG,

ρ=ρ(1)⊕ρ(2) dimρ(1),dimρ(2) 6 = 0 (8.14)

A representationρofGisirreducibleif it is not reducible.

8.6 The irreducible representations of angular momentum

During our study of the quantum system of angular momentum, we have already identified all

the irreducible representations ofSO(3). They are labeled by the total angular momentum

j, such that the eigenvalue ofJ^2 is ̄h^2 j(j+1). We shall denote these representations byD(j).

Since they are representations, they satisfy the same algebra astheJ,

[

D(j)(Ja),D(j)(Jb)

]

=i ̄h

∑^3

c=1

εabcD(j)(Jc) dimD(j)= 2j+ 1 (8.15)