QuantumPhysics.dvi

Examples of groups includeZ,+;Q,+;R,+;C,+; Q^0 ,×; R^0 ,×;C^0 ,×; as well as the group

of allm×nmatrices under addition, and the group of all invertiblen×nmatrices under

multiplication, a group denotedGl(n). These groups are all infinite groups, namely having

a infinite set of elements. The quintessential example of a finite group is the groupSn of

permutations acting on any set ofndistinct elements. The crystallographic groups are other

finite groups.

Arepresentationρof dimensionN of the groupGis a mapρfromGinto the group of

N×N invertible matrices,Gl(N),

1’ Group multiplication carries over to multiplication of matrices

ρ(g 1 ∗g 2 ) =ρ(g 1 )ρ(g 2 ) for allg 1 ,g 2 ∈G;

2’ Associativity is automatic for matrices;

3’ The image ofeis the unit matrix,ρ(e) =I;

4’ The image of the inverse is the inverse matrix,ρ(g−^1 ) =

(

ρ(g)

)− 1

In other words, a representation of a groupGgives a representation of the elements ofG

and of the group multiplication law∗in terms ofN×N matrices.

One distinguishes the following special types of representations,

• Thetrivial representationρ(g) =I for allg∈G;

• Afaithful representationis such that the mapρis injective;

• Areal representationis such thatρ(g) is a realN×N matrix for allg∈G;

• Acomplex representationis such thatρ(g) is complex for at least oneg∈G;

• Aunitary representationis such thatρ(g)†ρ(g) =Ifor allg∈G.

Representations allow us to represent the action of an abstract group concretely on a linear

vector space, such as a Hilbert space in quantum mechanics.

8.4 General Lie Algebras and their Representations

A Lie algebraGis a linear vector space, endowed with a bilinear pairing, usually denoted as

the commutator. The defining properties are as follows; for allX 1 ,X 2 ,X 3 ∈G, we have,

1. The commutator is antisymmetric, [X 1 ,X 2 ] =−[X 2 ,X 1 ], and belongs toG;

2. Bilinearity [λ 1 X 1 +λ 2 X 2 ,X 3 ] =λ 1 [X 1 ,X 3 ] +λ 2 [X 2 ,X 3 ] forλ 1 ,λ 2 ∈C;