CHAPTER 1

Prologue

Men can do nothing without the make-believe of a beginning. Even

Science, the strict measurer, is obliged to start with a make-believe

unit, and must fix on a point in the stars’ unceasing journey when his

sidereal clock shall pretend time is at Nought. His less accurate grand-

mother Poetry has always been understood to start in the middle, but

on reflection it appears that her proceeding is not very different from

his; since Science, too, reckons backwards as well as forwards, divides

his unit into billions, and with his clock-finger at Nought really sets

off in medias res. No retrospect will take us to the true beginning;

and whether our prologue be in heaven or on earth, it is but a fraction

of the all-presupposing fact with which our story sets out.

—George Eliot, Daniel Deronda (epigraph of Chapter 1)

The purpose of this preliminary chapter is to present some notation and termi-

nology, discuss a few matters on which mathematicians’ and physics usages differ,

and provide some basic definitions and formulas from physics and Lie theory that

will be needed throughout the book.

1.1. Linguistic prologue: notation and terminology

Hilbert space and its operators. Quantum mechanicians and mathematical

analysts both spend a lot of time in Hilbert space, but they have different ways

of speaking about it. In this book we generally follow physicists’ conventions, so

it will be well to explain how to translate from one dialect to the other. (The

physicists’ dialect is largely due to Dirac, and its ubiquity is a testimony to the

profound influence that his book [22] had on several generations of physicists.)

To begin with, complex conjugates are denoted by asterisks rather than over-

lines:

x − iy = (x +

iy)∗

rather than x + iy.

The inner product on a Hilbert space is denoted by ·|· and is taken to be linear

in the second variable and conjugate-linear in the first:

u|av + bw = a u|v + b u|w , v|u = u|v

∗.

The Hermitian adjoint (conjugate transpose) of a matrix or an operator is denoted

by a dagger rather than an asterisk:

if A = (xjk + iyjk), then (xkj − iykj) =

A†

rather than

A∗.

(More about adjoints below.) These shifted uses of

∗

and

†

may seem annoying at

first, but one gets used to them — and there is another consideration. The overline

is employed in physics not for conjugation but to denote the “Dirac adjoint” of a

Dirac spinor, as we shall explain in §4.1. This usage turns up suﬃciently frequently

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http://dx.doi.org/10.1090/surv/149/01