value for this observable, the integerqj. A general state will be a linear super-
position of state vectors from differentHqjand there will not be a well-defined
numerical value for the observableQon such a state.
From the action ofQonH, the representation can be recovered. The action
of the groupU(1) onHis given by multiplying byiand exponentiating, to get
π(eiθ) =eiQθ=
eiq^1 θ 0 ··· 0
0 eiq^2 θ ··· 0
··· ···
0 0 ··· eiqnθ
∈U(n)⊂GL(n,C)The standard physics terminology is that “Q is the generator of theU(1) action
by unitary transformations on the state spaceH”.
The general abstract mathematical point of view (which we will discuss in
much more detail in chapter 5) is that a representationπis a map between
manifolds, from the Lie groupU(1) to the Lie groupGL(n,C), that takes the
identity ofU(1) to the identity ofGL(n,C). As such it has a differentialπ′,
which is a linear map from the tangent space at the identity ofU(1) (which
here isiR) to the tangent space at the identity ofGL(n,C) (which is the space
M(n,C) ofnbyncomplex matrices). The tangent space at the identity of a Lie
group is called a “Lie algebra”. In later chapters we will study many different
examples of such Lie algebras and such mapsπ′, with the linear mapπ′often
determining the representationπ.
In theU(1) case, the relation between the differential ofπand the operator
Qis
π′:iθ∈iR→π′(iθ) =iQθ
The following drawing illustrates the situation: