space of functions known as wavefunctions. While these may, like classical fields,
satisfy a differential equation, one non-classical feature is that wavefunctions are
complex-valued. What’s completely different about quantum mechanics is the
treatment of observable quantities, which correspond to self-adjoint linear op-
erators on the state space. When such operators don’t commute, our intuitions
about how physics should work are violated, as we can no longer simultaneously
assign numerical values to the observables.
During the earliest days of quantum mechanics, the mathematician Hermann
Weyl quickly recognized that the mathematical structures being used were ones
he was quite familiar with from his work in the field of representation theory.
From the point of view that takes representation theory as a central theme
in mathematics, the framework of quantum mechanics looks perfectly natural.
Weyl soon wrote a book expounding such ideas [100], but this got a mixed reac-
tion from physicists unhappy with the penetration of unfamiliar mathematical
structures into their subject (with some of them characterizing the situation as
the “Gruppenpest”, the group theory plague). One goal of this book will be to
try and make some of this mathematics as accessible as possible, boiling down
part of Weyl’s exposition to its essentials while updating it in the light of many
decades of progress towards better understanding of the subject.
Weyl’s insight that quantization of a classical system crucially involves un-
derstanding the Lie groups that act on the classical phase space and the uni-
tary representations of these groups has been vindicated by later developments
which dramatically expanded the scope of these ideas. The use of representa-
tion theory to exploit the symmetries of a problem has become a powerful tool
that has found uses in many areas of science, not just quantum mechanics. I
hope that readers whose main interest is physics will learn to appreciate some
of such mathematical structures that lie behind the calculations of standard
textbooks, helping them understand how to effectively exploit them in other
contexts. Those whose main interest is mathematics will hopefully gain some
understanding of fundamental physics, at the same time as seeing some crucial
examples of groups and representations. These should provide a good ground-
ing for appreciating more abstract presentations of the subject that are part
of the standard mathematical curriculum. Anyone curious about the relation
of fundamental physics to mathematics, and what Eugene Wigner described as
“The Unreasonable Effectiveness of Mathematics in the Natural Sciences”[101]
should benefit from an exposure to this remarkable story at the intersection of
the two subjects.
The following sections give an overview of the fundamental ideas behind
much of the material to follow. In this sketchy and abstract form they will
likely seem rather mystifying to those meeting them for the first time. As we
work through basic examples in coming chapters, a better understanding of the
overall picture described here should start to emerge.
lily
(lily)
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