with Lie bracket the matrix commutator (which is zero here). Such a Lie algebra
can be identified with the Lie algebraR(with trivial Lie bracket).
We will sometimes find this way of expressing elements ofRas matrices
useful, but will often instead label elements of the group by scalarsa, and use
the additive group law. The same scalarsaare also used to label elements of
the Lie algebra, with the exponential map from the Lie algebra to the Lie group
now just the identity map. Recall that the Lie algebra of a Lie group can be
thought of as the tangent space to the group at the identity. For examples of
Lie groups likeRthat are linear spaces, the space and its tangent space can be
identified, and this is what we are doing here.
The irreducible representations of the groupRare the following:
Theorem 10.1.Irreducible representations ofRare labeled byc∈Cand given
by
πc(a) =eca
Such representations are unitary (inU(1)) whencis purely imaginary.
The proof of this theorem is the same as for theG=U(1) case (theorem 2.3),
dropping the final part of the argument, which shows that periodicity (U(1) is
justRwithaanda+N 2 πidentified) requirescto beitimes an integer.
The representations ofRthat we are interested in are on spaces of wave-
functions, and thus infinite dimensional. The simplest case is the representation
induced on functions onRby the action ofRon itself by translation. Here
a∈Racts onq∈R(whereqis a coordinate onR) by
q→a·q=q+aand the induced representationπon functions (see equation 1.3) is
π(g)f(q) =f(g−^1 ·q)which for this case will be
π(a)f(q) =f(q−a) (10.1)To get the Lie algebra version of this representation, the above can be dif-
ferentiated, finding
π′(a) =−ad
dq(10.2)
In the other direction, knowing the Lie algebra representation, exponentiation
give
π(a)f=eπ′(a)
f=e−a
dqd
f(q) =f(q)−a
df
dq+
a^2
2!d^2 f
dq^2+···=f(q−a)which is just Taylor’s formula.^1
(^1) This requires restricting attention to a specific class of functions for which the Taylor
series converges to the function.