For a unitary representation, the matricesπ(g) take values in a subgroup
U(n)⊂GL(n,C). In our review of linear algebra (chapter 4) we will see that
U(n) can be characterized as the group ofnbyncomplex matricesUsuch that
U−^1 =U†whereU†is the conjugate-transpose ofU. Note that we’ll be using the notation
“†” to mean the “adjoint” or conjugate-transpose matrix. This notation is pretty
universal in physics, whereas mathematicians prefer to use “∗” instead of “†”.
1.4 Representations and quantum mechanics
The fundamental relationship between quantum mechanics and representation
theory is that whenever we have a physical quantum system with a groupG
acting on it, the space of statesHwill carry a unitary representation ofG(at
least up to a phase factor ambiguity). For physicists working with quantum
mechanics, this implies that representation theory provides information about
quantum mechanical state spaces whenGacts on the system. For mathemati-
cians studying representation theory, this means that physics is a very fruitful
source of unitary representations to study: any physical system with a groupG
acting on it will provide one.
For a representationπand group elementsgthat are close to the identity,
exponentiation can be used to writeπ(g)∈GL(n,C) as
π(g) =eAwhereAis also a matrix, close to the zero matrix. We will study this situation
in much more detail and work extensively with examples, showing in particular
that ifπ(g) is unitary (i.e., in the subgroupU(n)⊂GL(n,C)), thenAwill be
skew-adjoint:
A†=−A
whereA†is the conjugate-transpose matrix. DefiningB=iA, we find thatB
is self-adjoint
B†=B
We thus see that, at least in the case of finite dimensionalH, the unitary rep-
resentationπofGonHcoming from an action ofGof our physical system gives
us not just unitary matricesπ(g), but also corresponding self-adjoint operators
BonH. Lie group actions thus provide us with a class of quantum mechanical
observables, with the self-adjointness property of these operators corresponding
to the unitarity of the representation on state space. It is a remarkable fact that
for many physical systems the class of observables that arise in this way include
the ones of most physical interest.
In the following chapters we’ll see many examples of this phenomenon. A
fundamental example that we will study in detail is that of action by translation
in time. Here the group isG=R(with the additive group law) and we get