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some situations in which more care is required. For these contingencies we brie¬‚y
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


F (r, ω) e’iωt .
F (r, t) = (B.16)

’∞

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)


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

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