Chapter 1

Introduction and Overview

1.1 Introduction

A famous quote from Richard Feynman goes “I think it is safe to say that no one
understands quantum mechanics.”[22]. In this book we’ll pursue one possible
route to such an understanding, emphasizing the deep connections of quan-
tum mechanics to fundamental ideas of modern mathematics. The strangeness
inherent in quantum theory that Feynman was referring to has two rather dif-
ferent sources. One of them is the striking disjunction and incommensurability
between the conceptual framework of the classical physics which governs our
everyday experience of the physical world, and the very different framework
which governs physical reality at the atomic scale. Familiarity with the pow-
erful formalisms of classical mechanics and electromagnetism provides deep un-
derstanding of the world at the distance scales familiar to us. Supplementing
these with the more modern (but still “classical” in the sense of “not quantum”)
subjects of special and general relativity extends our understanding into other
much less familiar regimes, while still leaving atomic physics a mystery.
Read in context though, Feynman was pointing to a second source of diffi-
culty, contrasting the mathematical formalism of quantum mechanics with that
of the theory of general relativity, a supposedly equally hard to understand
subject. General relativity can be a difficult subject to master, but its mathe-
matical and conceptual structure involves a fairly straightforward extension of
structures that characterize 19th century physics. The fundamental physical
laws (Einstein’s equations for general relativity) are expressed as partial dif-
ferential equations, a familiar if difficult mathematical subject. The state of a
system is determined by a set of fields satisfying these equations, and observable
quantities are functionals of these fields. The mathematics is largely that of the
usual calculus: differential equations and their real-valued solutions.
In quantum mechanics, the state of a system is best thought of as a different
sort of mathematical object: a vector in a complex vector space with a Hermitian
inner product, the so-called state space. Such a state space will sometimes be a