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where dµ (») is an operator-valued measure de¬ned by the following statement:
for each subset ∆ of the real line,

P (∆) = dµ (») (A.26)


’1
is the projection operator onto the subspace of vectors ψ such that (» ’ A) ψ <∞
for all » ∈ ∆ (Riesz and Sz.-Nagy, 1955, Chap. VIII, Sec. 120).
/
A linear operator U is unitary if it preserves inner products, i.e.

(U ψ, U φ) = (ψ, φ) (A.27)

for any pair of vectors ψ, φ in the Hilbert space. A necessary and su¬cient condition
for unitarity is that the operator is norm preserving, i.e.

(U ψ, U ψ) = (ψ, ψ) for all ψ if and only if U is unitary . (A.28)

The spectral resolution for a unitary operator with a pure point spectrum is

eiθn Pn , θn real ,
U= (A.29)
n

and for a continuous spectrum

eiθ dµ (θ) , θ real .
U= (A.30)

A linear operator N is said to be a normal operator if

N, N † = 0 . (A.31)
The hermitian and unitary operators are both normal. The hermitian operators N1 =
N + N † /2 and N2 = N ’ N † /2i satisfy N = N1 + iN2 and [N1 , N2 ] = 0. Normal
operators therefore have the spectral resolutions

N= (xn P1n + iyn P2n ) , [P1n , P2m ] = 0 (A.32)
n
¼ Mathematics

for a point spectrum, and

N= x dµ1 (x) + i y dµ2 (y) , dµ1 (x) , dµ2 (y) = 0 (A.33)
∆1 ∆1

for a continuous spectrum.

A.3.4 Matrices
A linear operator X acting on an N -dimensional Hilbert space, with basis f (1) , . . . ,
f (N ) , is represented by the N — N matrix

Xmn = f (m) , Xf (n) . (A.34)

The operator and its matrix are both called X. The matrix for the product XY of
two operators is the matrix product
N
(XY )mn = Xmk Ykn . (A.35)
k=1

The determinant of X is de¬ned as
X1n1 · · · XN nN ,
det (X) = (A.36)
n1 ···nN
n1 ···nN

where the generalized alternating tensor is
§
⎪1 n1 · · · nN is an even permutation of 12 · · · N ,

’1 n1 · · · nN is an odd permutation of 12 · · · N , (A.37)
n1 ···nN =

©
n1 ···nN 0 otherwise .
The trace of X is
N
Tr X = Xnn . (A.38)
n=1

The transpose matrix X T is de¬ned by Xnm = Xmn . The adjoint matrix X †
T
† —
is the complex conjugate of the transpose: Xnm = Xmn . A matrix X is symmetric if
X = X T , self-adjoint or hermitian if X † = X, and unitary if X † X = XX † = I, where I
is the N — N identity matrix. Unitary transformations preserve the inner product. The
hermitian and unitary matrices both belong to the larger class of normal matrices
de¬ned by X † X = XX † .
A matrix X is positive de¬nite if all of its eigenvalues are real and non-negative.
This immediately implies that the determinant and trace of the matrix are both non-
negative. An equivalent de¬nition is that X is positive de¬nite if
φ† Xφ 0 (A.39)
for all vectors φ. For a positive-de¬nite matrix X, there is a matrix Y such that
X = Y Y †.
The normal matrices have the following important properties (Mac Lane and Birk-
ho¬, 1967, Sec. XI-10).
½
Fourier transforms

Theorem A.1 (i) If f is an eigenvector of the normal matrix Z with eigenvalue z,
then f is an eigenvector of Z † with eigenvalue z — , i.e. Zf = zf ’ Z † f = z — f .
(ii) Every normal matrix has a complete, orthonormal set of eigenvectors.

Thus hermitian matrices have real eigenvalues and unitary matrices have eigenvalues
of modulus 1.

A.4 Fourier transforms
A.4.1 Continuous transforms
In the mathematical literature it is conventional to denote the Fourier (integral)
transform of a function f (x) of a single, real variable by

dxf (x) e’ikx ,
f (k) = (A.40)
’∞

so that the inverse Fourier transform is

dk
f (k) eikx .
f (x) = (A.41)

’∞

The virtue of this notation is that it reminds us that the two functions are, generally,
drastically di¬erent, e.g. if f (x) = 1, then f (k) = 2πδ (k) .
On the other hand, the is a typographical nuisance in any discussion involving
many uses of the Fourier transform. For this reason, we will sacri¬ce precision for
convenience. In our convention, the Fourier transform is indicated by the same letter,
and the distinction between the functions is maintained by paying attention to the
arguments.
The Fourier transform pair is accordingly written as

dxf (x) e’ikx ,
f (k) = (A.42)
’∞

dk
f (k) eikx .
f (x) = (A.43)

’∞

This is analogous to the familiar idea that the meaning of a vector V is independent
of the coordinate system used, despite the fact that the components (Vx , Vy , Vz ) of
V are changed by transforming to a new coordinate system. From this point of view,
the functions f (x) and f (k) are simply di¬erent representations of the same physical
quantity. Confusion is readily avoided by paying attention to the physical signi¬cance
of the arguments, e.g. x denotes a point in position space, while k denotes a point
in the reciprocal space or k-space.
If the position-space function f (x) is real, then the Fourier transform satis¬es

f — (k) = [f (k)] = f (’k) . (A.44)

When the position variable x is replaced by the time t, it is customary in physics to
use the opposite sign convention:
¾ Mathematics


dxf (x) eiωt ,
f (ω) = (A.45)
’∞


f (k) e’iωt .
f (t) = (A.46)

’∞

Fourier transforms of functions of several variables, typically f (r), are de¬ned
similarly:

d3 rf (r) e’ik·r ,
f (k) = (A.47)

d3 k
(k) eik·r ,
f (r) = 3f (A.48)
(2π)
where the integrals are over position space and reciprocal space (k-space) respectively.
If f (r) is real then
f — (k) = f (’k) . (A.49)
Combining these conventions for a space“time function f (r, t) yields the transform
pair

dtf (r, t) e’i(k·r’ωt) ,
d3 r
f (k, ω) = (A.50)

d3 k dω
f (k, ω) ei(k·r’ωt) .
f (r, t) = (A.51)
3 2π
(2π)
The last result is simply the plane-wave expansion of f (r, t). If f (r, t) is real, then
the Fourier transform satis¬es
f — (k, ω) = f (’k, ’ω) . (A.52)
Two related and important results on Fourier transforms”which we quote for the
one- and three-dimensional cases”are Parseval™s theorem:
dω —
dtf — (t) g (t) = f (ω) g (ω) , (A.53)

d3 k —

3
d rf (r) g (r) = 3 f (k) g (k) , (A.54)
(2π)
and the convolution theorem:
dt f (t ’ t ) g (t )
h (t) = if and only if h (ω) = f (ω) g (ω) , (A.55)


f (ω ’ ω ) g (ω )
h (ω) = if and only if h (t) = f (t) g (t) , (A.56)

d3 r f (r ’ r ) g (r )
h (r) = if and only if h (k) = f (k) g (k) , (A.57)

d3 k
(k ’ k ) g (k ) if and only if h (r) = f (r) g (r) .
h (k) = 3f (A.58)
(2π)
These results are readily derived by using the delta function identities (A.95) and
(A.96).
¿
Fourier transforms

A.4.2 Fourier series
It is often useful to simplify the mathematics of the one-dimensional continuous trans-
form by considering the functions to be de¬ned on a ¬nite interval (’L/2, L/2) and
imposing periodic boundary conditions. The basis vectors are still of the form
uk (x) = C exp (ikx), but the periodicity condition, uk (’L/2) = uk (L/2), restricts k
to the discrete values
2πn
(n = 0, ±1, ±2, . . .) .
k= (A.59)
L

Normalization requires C = 1/ L, so the transform is
L/2
1
dxf (x) e’ikx ,
fk = √ (A.60)
L ’L/2

and the inverse transform f (x) is
1
f (x) = √ fk eikx . (A.61)
L k

The continuous transform is recovered in the limit L ’ ∞ by ¬rst using eqn (A.60)
to conclude that √
Lfk ’ f (k) as L ’ ∞ , (A.62)
and writing the inverse transform as

1
Lfk eikx .
f (x) = (A.63)
L
k

The di¬erence between neighboring k-values is ∆k = 2π/L, so this equation can be
recast as
∆k √ dk
Lfk eikx ’ f (k) eikx .
f (x) = (A.64)
2π 2π
k

In Cartesian coordinates the three-dimensional discrete transform is de¬ned on a
rectangular parallelepiped with dimensions Lx , Ly , Lz . The one-dimensional results
then imply
1
d3 rf (r) e’ik·r ,
fk = √ (A.65)
VV
where the k-vector is restricted to
2πny 2πnz
2πnx
ux + uy + uz , (A.66)
k=
Lx Ly Lz
and V = Lx Ly Lz . The inverse transform is
1
f (r) = √ fk eik·r , (A.67)
V k

and the integral transform is recovered by
Mathematics


V fk ’ f (k) as V ’ ∞ . (A.68)

The sum and integral over k are related by

d3 k
1
’ , (A.69)
(2π)3
V
k

which in turn implies
3
V δk,k ’ (2π) δ (k ’ k ) . (A.70)

A.5 Laplace transforms
Another useful idea”which is closely related to the one-dimensional Fourier trans-
form”is the Laplace transform de¬ned by

dt e’ζt f (t) .
f (ζ) = (A.71)
0

In this case, we will use the standard mathematical notation f (ζ), since we do not use
Laplace transforms as frequently as Fourier transforms. The inverse transform is
ζ0 +i∞
dζ ζt
f (t) = e f (ζ) . (A.72)
2πi
ζ0 ’i∞

The line (ζ0 ’ i∞, ζ0 + i∞) in the complex ζ-plane must lie to the right of any poles
in the transform function f (ζ).
The identity
df
(ζ) = ζ f (ζ) ’ f (0) (A.73)
dt
is useful in treating initial value problems for sets of linear, di¬erential equations. Thus
to solve the equations
dfn
= Vnm fm , (A.74)
dt m

with a constant matrix V , and initial data fn (0), one takes the Laplace transform to
get
ζ fn (ζ) ’ Vnm fm (ζ) = fn (0) . (A.75)
m

This set of algebraic equations can be solved to express fn (ζ) in terms of fn (0).
Inverting the Laplace transform yields the solution in the time domain.
The convolution theorem for Laplace transforms is
t ζ0 +i∞

dt g (t ’ t ) f (t ) = g (ζ) f (ζ) eζt , (A.76)
2πi
ζ0 ’i∞
0


where the integration contour is to the right of any poles of both g (ζ) and f (ζ).
Functional analysis

An important point for applications to physics is that poles in the Laplace trans-
form correspond to exponential time dependence. For example, the function f (t) =
exp (zt) has the transform
1
f (ζ) = . (A.77)
ζ ’z
More generally, consider a function f (ζ) with N simple poles in ζ:
1
f (ζ) = , (A.78)
(ζ ’ z1 ) · · · (ζ ’ zN )
where the complex numbers z1 , . . . , zN are all distinct. The inverse transform is
ζ0 +i∞
eζt

f (t) = , (A.79)
2πi (ζ ’ z1 ) · · · (ζ ’ zN )
ζ0 ’i∞

where ζ0 > max[Re z1 , . . . , Re zN ]. The contour can be closed by a large semicircle in
the left half plane, and for N > 1 the contribution from the semicircle can be neglected.
The integral is therefore given by the sum of the residues,
N
1
ezn t
f (t) = , (A.80)
zn ’ zj
n=1 j=n

which explicitly exhibits f (t) as a sum of exponentials.

A.6 Functional analysis
A.6.1 Linear functionals
In normal usage, a function, e.g. f (x), is a rule assigning a unique value to each value
of its argument. The argument is typically a point in some ¬nite-dimensional space,
e.g. the real numbers R, the complex numbers C, three-dimensional space R3 , etc. The
values of the function are also points in a ¬nite-dimensional space. For example, the
classical electric ¬eld is represented by a function E (r) that assigns a vector”a point
in R3 ”to each position r in R3 .
A rule, X, assigning a value to each point f in an in¬nite-dimensional space M
(which is usually a space of functions) is called a functional and written as X [f ].
The square brackets surrounding the argument are intended to distinguish functionals
from functions of a ¬nite number of variables.
If M is a vector space, e.g. a Hilbert space, then a functional Y [f ] that obeys

Y [±f + βg] = ±Y [f ] + βY [g] , (A.81)

for all scalars ±, β and all functions f, g ∈ M, is called a linear functional. The
family, M , of linear functionals on M is called the dual space of M. The dual space
is also a vector space, with linear combinations of its elements de¬ned by

(±X + βY ) [f ] = ±X [f ] + βY [f ] (A.82)

for all f ∈ M.
Mathematics

A.6.2 Generalized functions
In Section 3.1.2 the de¬nition (3.18) and the rule (3.21) are presented with the cavalier
disregard for mathematical niceties that is customary in physics. There are however

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