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 xμ

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”:

xμ =

gμνxν

,

pμ

=

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

xμ

→ xμ 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