1.1. LINGUISTIC PROLOGUE: NOTATION AND TERMINOLOGY 3
By the way, we will often write a scalar multiple of the identity operator, λI,
simply as λ. This commonly used bit of shorthand will cause no confusion if readers
will keep in mind that when they encounter a scalar where an operator or a matrix
seems to be needed, they should tacitly insert an I.
Vectors, tensors, and derivatives. In this book we reserve boldface type
almost exclusively to denote 3-dimensional vectors related to R3 as a model for the
physical space in which we live. (Exception: In §4.5 it is used to denote elements
of tensor products of Hilbert spaces.) Elements of
Rn
for general n are usually
denoted by lower-case italic letters (x, p, . . . ). We denote the n-tuple of partial
derivatives (∂1,...,∂n) on
Rn
by ∇, and we denote the Laplacian · =

∂j
2
by
∇2
(rather than Δ, for which we will have other uses).
When n = 4, a variant of this notation will be used for calculations arising
from relativistic mechanics. Four-dimensional space-time is taken to be
R4
with
coordinates
x0
= ct,
x1
= x,
x2
= y,
x3
= z
on
R4,
where t represents time, c the speed of light, and (x, y, z) a set of Cartesian
coordinates on physical space
R3.
Thus, a point x
R4
may be written as
(x0,
x)
when it is important to separate the space and time components. The Lorentz inner
product on
R4
is the bilinear form Λ defined by
Λ(x, y) =
x0y0

x1y1

x2y2

x3y3,
and
R4
equipped with this form is called Minkowski space. It is common in the
physics literature to denote the vector whose components are x0,...,x3 by
rather than x. This is the same sort of harmless abuse of language that is involved
in speaking of “the sequence an” or “the function f(x)”; we shall use it when it
seems convenient.
We shall generally use classical tensor notation for vectors and tensors associ-
ated to Minkowski space. In particular, the Lorentz form is defined by the matrix
(1.1) gμν =
gμν
= diag(1, −1, −1, −1).
We employ the Einstein summation convention: in any product of vectors and
tensors in which an index appears once as a subscript and once as a superscript,
that index is to be summed from 0 to 3. Thus, for example,
(1.2) Λ(x, y) = gμν
xμyν.
We shall also adopt the convention of using the matrix g to “raise or lower indices”:
=
gμνxν
,

=
gμνpν,
the practical effect of which is to change the sign of the last three components.
(Strictly speaking, vectors whose components are denoted by subscripts should
be construed as elements of the dual space
(R4)
, and the map

is the
isomorphism of
R4
with
(R4)
induced by the Lorentz form. But we shall not
attempt to distinguish between
R4
and its dual.) Formula (1.2) can then be written
as
Λ(x, y) =
xμyμ
=
xμyμ.
The Lorentz inner product of a vector x with itself is denoted by
x2
and its Euclidean
norm by |x|:
x2
=
xμxμ, |x|2
= x0
2
+ x1
2
+ x2
2
+
x3.2
Previous Page Next Page