of norm fixed to the value 1, which fixes the amplitude ofc, leaving a remaining
ambiguity which is a phaseeiθ. By the above principles this phase will not
contribute to the calculated probabilities of measurements. We will however
not take the point of view that this phase information can just be ignored. It
plays an important role in the mathematical structure, and the relative phase
of two different states certainly does affect measurement probabilities.
1.3 Unitary group representations
The mathematical framework of quantum mechanics is closely related to what
mathematicians describe as the theory of “unitary group representations”. We
will be examining this notion in great detail and working through many examples
in coming chapters, but here is a quick summary of the general theory.
1.3.1 Lie groups
A fundamental notion that appears throughout different fields of mathematics
is that of a group:
Definition(Group).A groupGis a set with an associative multiplication, such
that the set contains an identity element, as well as the multiplicative inverse
of each element.
If the set has a finite number of elements, this is called a “finite group”.
The theory of these and their use in quantum mechanics is a well-developed
subject, but one we mostly will bypass in favor of the study of “Lie groups”,
which have an infinite number of elements. The elements of a Lie group make
up a geometrical space of some dimension, and choosing local coordinates on the
space, the group operations are given by differentiable functions. Most of the Lie
groups we will consider are “matrix groups”, meaning subgroups of the group
ofnbyninvertible matrices (with real or complex matrix entries). The group
multiplication in this case is matrix multiplication. An example we will consider
in great detail is the group of all rotations about a point in three dimensional
space, in which case such rotations can be identified with 3 by 3 matrices, with
composition of rotations corresponding to multiplication of matrices.
Digression.A standard definition of a Lie group is as a smooth manifold, with
group laws given by smooth (infinitely differentiable) maps. More generally, one
might consider topological manifolds and continuous maps, but this gives nothing
new (by the solution to Hilbert’s Fifth problem). Most of the finite dimensional
Lie groups of interest are matrix Lie groups, which can be defined as closed
subgroups of the group of invertible matrices of some fixed dimension. One
particular group of importance in quantum mechanics (the metaplectic group,
see chapter 20) is not a matrix group, so the more general definition is needed
to include this case.