a unitary representation ofRon the space of statesH. The corresponding
self-adjoint operator is the Hamiltonian operatorH (divided by~) and the
representation is given by
t∈R→π(t) =e−
~iHtwhich one can check is a group homomorphism from the additive groupRto a
group of unitary operators. This unitary representation gives the dynamics of
the theory, with the Schr ̈odinger equation 1.1 just the statement that−~iH∆t
is the skew-adjoint operator that gets exponentiated to give the unitary trans-
formation that moves statesψ(t) ahead in time by an amount ∆t.
One way to construct quantum mechanical state spacesHis as “wavefunc-
tions”, meaning complex-valued functions on space-time. Given any group ac-
tion on space-time, we get a representationπon the state spaceHof such
wavefunctions by the construction of equation 1.3. Many of the representations
of interest will however not come from this construction, and we will begin our
study of the subject in the next few chapters with such examples, which are
simpler because they are finite dimensional. In later chapters we will turn to
representations induced from group actions on space-time, which will be infinite
dimensional.
1.5 Groups and symmetries
The subject we are considering is often described as the study of “symmetry
groups”, since the groups may occur as groups of elements acting by transfor-
mations of a spaceMpreserving some particular structure (thus, a “symmetry
transformation”). We would like to emphasize though that it is not necessary
that the transformations under consideration preserve any particular structure.
In the applications to physics, the term “symmetry” is best restricted to the
case of groups acting on a physical system in a way that preserves the equations
of motion (for example, by leaving the Hamiltonian function unchanged in the
case of a classical mechanical system). For the case of groups of such symme-
try transformations, the use of the representation theory of the group to derive
implications for the behavior of a quantum mechanical system is an important
application of the theory. We will see however that the role of representation
theory in quantum mechanics is quite a bit deeper than this, with the overall
structure of the theory determined by group actions that are not symmetries
(in the sense of not preserving the Hamiltonian).
1.6 For further reading
We will be approaching the subject of quantum theory from a different direc-
tion than the conventional one, starting with the role of symmetry and with the
simplest possible finite dimensional quantum systems, systems which are purely