outline a more respectable treatment. The chief di¬culty is the existence of the in-

tegrals de¬ning the operators s ’∇2 . This problem can be overcome by restricting

the functions • (r) in eqn (3.18) to good functions (Lighthill, 1964, Chap. 2), i.e.

in¬nitely-di¬erentiable functions that fall o¬ faster than any power of |r|. The Fourier

transform of a good function is also a good function, so all of the relevant integrals

exist, as long as s (|k|) does not grow exponentially at large |k|. The examples we need

±

are all of the form |k| , where ’1 ± 1, so eqns (3.18) and (3.21) are justi¬ed.

For physical applications the really important assumption is that all functions can be

approximated by good functions.

A generalized function is a linear functional, say G [•], de¬ned on the good

functions, i.e.

G [•] is a complex number and G [±• + βψ] = ±G [•] + βG [•] (A.83)

for any scalars ±, β and any good functions •, ψ. A familiar example is the delta

function. The rule

d3 rδ (r ’ R) • (r) = • (R) (A.84)

maps the function • (r) into the single number • (R). In this language, the transverse

delta function ∆⊥ (r ’ r ) is also a generalized function. An alternative terminology,

ij

often found in the mathematical literature, labels good functions as test functions and

generalized functions as distributions.

In quantum ¬eld theory, the notion of a generalized function is extended to linear

functionals sending good functions to operators, i.e. for each good function •,

X [•] is an operator and X [±• + βψ] = ±X [•] + βX [•] . (A.85)

Such functionals are called operator-valued generalized functions. For any density

operator ρ describing a physical state, X [•] de¬nes an ordinary (c-number) generalized

function Xρ [•] by

Xρ [•] = Tr (ρX [•]) . (A.86)

A.7 Improper functions

A.7.1 The Heaviside step function

The step function θ (x) is de¬ned by

1 for x > 0 ,

θ (x) = (A.87)

0 for x < 0 ,

and it has the useful representation

Improper functions

∞

ds e’isx

θ (x) = ’ lim , (A.88)

’0 2πi s + i

’∞

which is proved using contour integration.

A.7.2 The Dirac delta function

A Standard properties

(1) If the function f (x) has isolated, simple zeros at the points x1 , x2 , . . . then

1

δ x ’ xi .

δ (f (x)) = (A.89)

df

i dx x=xi

The multidimensional generalization of this rule is

1

δ x ’ xi ,

δ f (x) = (A.90)

‚f

det

i

‚x x=xi

where x = (x1 , x2 , . . . , xN ), f (x) = (f1 (x) , f2 (x) , . . . , fN (x)),

δ f (x) = δ (f1 (x)) · · · δ (fN (x)) ,

(A.91)

δ x ’ xi = δ x1 ’ xi · · · δ xN ’ xi ,

1 N

the Jacobian ‚f /‚x is the N — N matrix with components ‚fn /‚xm , and xi

satis¬es fn xi = 0, for n = 1, . . . , N .

(2) The derivative of the delta function is de¬ned by

∞

d df

δ (x ’ a) = ’

dxf (x) . (A.92)

dx dx

’∞ x=a

(3) By using contour integration methods one gets

1

1

= P ’ iπδ (x) , (A.93)

lim

’0 x + i x

where P is the principal part de¬ned by

∞ ’a ∞

f (x) f (x) f (x)

= lim + . (A.94)

P dx dx dx

x x x

a’0

’∞ ’∞ a

(4) The de¬nition of the Fourier transform yields

dtei(ω’ν)t = 2πδ (ω ’ ν) (A.95)

in one dimension, and

d3 rei(k’q)·r = (2π)3 δ (k ’ q) (A.96)

in three dimensions.

Mathematics

(5) The step function satis¬es

d

θ (x) = δ (x) . (A.97)

dx

(6) The end-point rule is

a

1

dxδ (x ’ a) f (x) = f (a) . (A.98)

2

’∞

(7) The three-dimensional delta function δ (r ’ r ) is de¬ned as

δ (r ’ r ) = δ (x ’ x ) δ (y ’ y ) δ (z ’ z ) , (A.99)

and is expressed in polar coordinates by

1

δ (r ’ r ) = δ (r ’ r ) δ (cos θ ’ cos θ ) δ (φ ’ φ ) . (A.100)

r2

B A special representation of the delta function

In many calculations, particularly in perturbation theory, one encounters functions of

the form

· (ωt)

ξ (ω, t) = , (A.101)

ω

which have the limit

lim ξ (ω, t) = ξ0 δ (ω) , (A.102)

t’∞

provided that the integral

∞

· (u)

ξ0 = du (A.103)

u

’∞

exists.

A.7.3 Integral kernels

The de¬nition of a generalized function as a linear rule assigning a complex number

to each good function can be extended to a linear rule that maps a good function, e.g.

f (t), to another good function g (t). The linear nature of the rule means that it can

always be expressed in the form

g (t) = dt W (t, t ) f (t ) . (A.104)

For a ¬xed value of t, W (t, t ) de¬nes a generalized function of t which is called an

integral kernel. This de¬nition is easily extended to functions of several variables,

e.g. f (r). The delta function, the Heaviside step function, etc. are examples of integral

kernels. An integral kernel is positive de¬nite if

dt f — (t) W (t, t ) f (t )

dt 0 (A.105)

for every good function f (t).

Probability and random variables

A.8 Probability and random variables

A.8.1 Axioms of probability

The abstract de¬nition of probability starts with a set „¦ of events and a probability

function P that assigns a numerical value to every subset of „¦. In principle, „¦ could be

any set, but in practice it is usually a subset of RN or CN , or a subset of the integers.

The essential properties of probabilities are contained in the axioms (Gardiner, 1985,

Chap. 2):

0 for all S ‚ „¦;

(1) P (S)

(2) P („¦) = 1;

(3) if S1 , S2 , . . . is a discrete (countable) collection of nonoverlapping sets, i.e.

Si © Sj = … for i = j , (A.106)

then

P (S1 ∪ S2 ∪ · · · ) = P (Sj ) . (A.107)

j

The familiar features 0 P (S) 1, P (…) = 0, and P (S ) = 1 ’ P (S), where S

is the complement of S, are immediate consequences of the axioms. If „¦ is a discrete

(countable) set, then one writes P (x) = P ({x}), where {x} is the set consisting of

the single element x. If „¦ is a continuous (uncountable) set, then it is customary to

introduce a probability density p (x) so that

P (S) = dx p (x) , (A.108)

S

where dx is the natural volume element on „¦.

If „¦ = Rn , the probability density is a function of n variables: p (x1 , x2 , . . . , xn ).

The marginal distribution of xj is then de¬ned as

dx1 · · · dxj+1 · · ·

pj (xj ) = dxj’1 dxn p (x1 , x2 , . . . , xn ) . (A.109)

The joint probability for two sets S and T is P (S © T ); this is the probability

that an event in S is also in T . This is more often expressed with the notation

P (S, T ) = P (S © T ) , (A.110)

which is used in the text. The conditional probability for S given T is

P (S © T )

P (S, T )

P (S | T ) = = ; (A.111)

P (T ) P (T )

this is the probability that x ∈ S, given that x ∈ T .

¼ Mathematics

The compound probability rule is just eqn (A.111) rewritten as

P (S, T ) = P (S | T ) P (T ) . (A.112)

This can be generalized to joint probabilities for more than two outcomes by applying

it several times, e.g.

P (S, T, R) = P (S | T, R) P (T, R)

= P (S | T, R) P (T | R) P (R) . (A.113)

Dividing both sides by P (R) yields the useful rule

P (S, T | R) = P (S | T, R) P (T | R) . (A.114)

Two sets of events S and T are said to be independent or statistically inde-

pendent if the joint probability is the product of the individual probabilities:

P (S, T ) = P (S) P (T ) . (A.115)

A.8.2 Random variables

A random variable X is a function X (x) de¬ned on the event space „¦. The function

can take on values in „¦ or in some other set. For example, if „¦ = R, then X (t) could

be a complex number or an integer. The average value of a random variable is

X= dx p (x) X (x) . (A.116)

If the function X does take on values in „¦, and is one“one, i.e. X (x1 ) = X (x2 ) implies

x1 = x2 , then the distinction between X (x) and x is often ignored.

Appendix B

Classical electrodynamics

B.1 Maxwell™s equations

In SI units the microscopic form of Maxwell™s equations is

ρ

∇·E = , (B.1)

0

‚E

∇ — B = µ0 j + , (B.2)

0

‚t

‚B

∇—E =’ , (B.3)

‚t

∇· B = 0. (B.4)

The homogeneous equations (B.3) and (B.4) are identically satis¬ed by introducing

the scalar potential • and the vector potential A (r) and setting

B = ∇ — A,

(B.5)

‚A

E=’ ’ ∇• .

‚t

A further consequence of this representation is that eqn (B.1) becomes the Poisson

equation

ρ

∇2 • = ’ , (B.6)

0

which has the Coulomb potential as its solution.

The vector and scalar potentials A and • are not unique. The same electric and

magnetic ¬elds are produced by the new potentials A and • de¬ned by a gauge

transformation,

A ’ A = A + ∇χ , (B.7)

• ’ • = • ’ ‚χ/‚t , (B.8)

where χ (r, t) is any di¬erentiable, real function. This is called gauge invariance.

This property can be exploited to choose the gauge that is most convenient for the

problem at hand. For example, it is always possible to perform a gauge transformation

such that the new potentials satisfy ∇ · A = 0 and • = ¦, where ¦ is a solution of

eqn (B.6). This is called the Coulomb gauge, since • = ¦ is the Coulomb potential,

or the radiation gauge, since the vector potential is transverse (Jackson, 1999, Sec.

6.3).

¾ Classical electrodynamics

The ¬‚ow of energy in the ¬eld is described by the continuity equation (Poynting™s

theorem),

‚u (r, t)

+ ∇ · S (r, t) = 0 , (B.9)

‚t

where

12

0

u (r, t) = E 2 (r, t) + B (r, t) (B.10)

2 2µ0

is the electromagnetic energy density, and the Poynting vector

1

E —B (B.11)

S=

µ0

is the energy ¬‚ux.

B.2 Electrodynamics in the frequency domain

It is often useful to describe the ¬eld in terms of its frequency and/or wavevector con-

tent. Let F (r,t) be a real function representing any of the components of E, B, or A .

Under the conventions established in Appendix A.4, the four-dimensional (frequency

and wavevector) Fourier transform of F (r, t) is

dte’i(k·r’ωt) F (r, t) ,

d3 r

F (k, ω) = (B.12)

and the inverse transform is

∞

d3 k dω

F (k, ω) ei(k·r’ωt) .

F (r, t) = (B.13)

3 2π

(2π) ’∞

According to eqn (A.52) the reality of F (r, t) imposes the conditions

F — (k, ω) = F (’k, ’ω) . (B.14)

For many applications it is also useful to consider the temporal Fourier transform at

a ¬xed position r:

∞

dteiωt F (r, t) ,

F (r, ω) = (B.15)

’∞

with the inverse transform

∞

dω

F (r, ω) e’iωt .

F (r, t) = (B.16)

2π

’∞

The function F (r, ω) satis¬es

F — (r, ω) = F (r, ’ω) . (B.17)

2

The quantity F (+) (r, ω) is called the power spectrum of F ; it can be used to

de¬ne an average frequency, ω0 , by

¿

Wave equations

∞ dω 2

d3 r F (+) (r, ω) ω

’∞ 2π

ω0 = ω = . (B.18)

∞ dω 2

d3 r F (+) (r, ω)

’∞ 2π

The frequency spread of the ¬eld is characterized by the rms deviation ∆ω”the

frequency or spectral width”de¬ned by

∞ dω 2 2

(r, ω) (ω ’

d3 r (+)

’∞ 2π F ω0 )

2 2

(∆ω) = (ω ’ ω0 ) = . (B.19)

∞ 2

d3 r ’∞ dω F (+) (r, ω)

2π

The average wavevector k0 and deviation ∆k are similarly de¬ned by

∞ dω 2

d3 k

F (+) (k, ω) k

’∞ 2π

(2π)3

k0 = k = , (B.20)

∞ dω 2

d3 k

F (+) (k, ω)

’∞ 2π

(2π)3

∞ dω 2

d3 k 2

(k, ω) (k ’

(+)

’∞ 2π F k0 )

(2π)3

2 2

(∆k) = (k ’ k0 ) = . (B.21)

∞ dω 2

d3 k

F (+) (k, ω)

(2π)3 ’∞ 2π

B.3 Wave equations

The microscopic Maxwell equations (B.1)“(B.4) can be replaced by two second-order

wave equations for E and B:

1 ‚2 ‚j 1