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-
x iy = (x +
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) =
rather than
(More about adjoints below.) These shifted uses of


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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