Solutions of the Maxwell equations and photon

wave functions∗

Peter J. Mohr

National Institute of Standards and Technology, Gaithersburg, MD 20899-8420, USA

Abstract

Properties of six-component electromagnetic field solutions of a matrix form of the

Maxwell equations, analogous to the four-component solutions of the Dirac equation, are

described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the

electromagnetic fields, which may be interpreted as photon wave functions, are given both

for plane waves and for angular-momentum eigenstates. Rotationally invariant projection

operators are used to identify transverse or longitudinal electric and magnetic fields. For

plane waves, the velocity transformed transverse wave functions are also transverse, and the

velocity transformed longitudinal wave functions include both longitudinal and transverse

components. A suitable sum over these eigenfunctions provides a Green function for the

matrix Maxwell equation, which can be expressed in the same covariant form as the Green

function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic

transition current are calculated to illustrate applications of the Maxwell Green function.

1. Introduction

For quantum mechanics to provide a complete description of nature, it is necessary to

have a wave function for something as important as electromagnetic radiation or photons.

This has become increasingly relevant as the number of experiments on single photon production and detection, motivated by interest in the fields of quantum computation and

quantum cryptography, has grown rapidly over the past two decades [1]. The history of

theoretical efforts to define photon wave functions dates back to the early days of quantum

mechanics and is still unfolding. Overviews have been given in [2–4]. However, there is not

yet a consensus on the form a photon wave function should take or the properties it should

have. Further investigation of these questions is warranted, and possible answers are given

in this paper.

Quantum electrodynamics (QED) accurately describes the interaction of radiation with

free electrons and electrons bound in atoms, but as it is formulated in terms of an S matrix,

∗Annals of Physics 325, 607-663 (2010).
asymptotic states, and Feynman diagrams, it does not readily lend itself to the description of

the time evolution of radiation. In particular, interference effects or the space-time behavior

of a photon wave packet would be more naturally described in the framework of wave

mechanics with a wave function for a single photon.

There are a number of requirements that need to be be imposed on a formalism for

a quantum mechanical description of photons. First, the predicted behavior of radiation

should be consistent with the Maxwell equations. Time-dependent solutions of the Maxwell

equations provide the basis for both classical electromagnetic theory and QED, and it can

be expected that a photon wave function should also be based on solutions of the Maxwell

equations. This means that the wave function is simultaneously the solution of both of the

first-order Maxwell equations with time derivatives and not just a solution of a second-order

scalar wave equation.

A second requirement is that the wave functions obey the quantum mechanical principle

of linear superposition. Simply stated, this means that if two wave functions describe possible

states of radiation, then a linear combination of these wave functions also describes a possible

state. For example, a wave function for circularly polarized radiation can be written as a

linear combination of two wave functions for linearly polarized radiation, all of which must

be solutions of the same wave equation.

Another requirement is that the formalism be Lorentz invariant in order to properly

describe the space-time behavior of radiation. An approximation scheme like the reduction

of the Dirac equation to obtain the Schr ̈odinger equation for electron velocities that are

small compared to the speed of light is not an option for radiative photons.

Finally, it is necessary for the formalism to provide the tools for methods associated with

quantum mechanics. This includes a wave equation with a Hamiltonian that describes the

time development of states, wave functions that comprise a complete set of eigenfunctions of

the Hamiltonian, normalizable states with a probability distribution that corresponds to the

location of the photon, a law of conservation of probability, operators with expectation values

for observables, and wave packets that realistically describe the propagation of photons in

space and time.

To arrive at a wave equation that addresses these requirements, we examine an approach

in which the four-component matrix Dirac equation for a spin one-half electron is adapted

to a six-component form of the Maxwell equations for a spin-one photon. This version of

the Maxwell equations is a direct extension of the Dirac equation for the electron in which

two-by-two Pauli matrices are replaced by analogous three-by-three matrices. Since the

quantum mechanical properties of the Dirac equation, Hamiltonian, and wave functions are

well understood and tested experimentally, it is natural to consider the analogous Maxwell

equation, Hamiltonian, and wave functions as a quantum mechanical description of photons.

There are fundamental differences between the Dirac equation and the matrix Maxwell

equation, so the extension requires a detailed analysis. The most prominent difference is the

fact that there is a possible source term in the Maxwell equation which has no analog for

the Dirac equation [5]. Also, some properties of the three-by-three spin matrices differ from

those of the Pauli matrices, even though they have the same commutation relations.

Linear operators that are representations of the inhomogeneous Lorentz group can replace

2
the wave equation of a system for free electrons or transverse photons with no sources [6–8],

but the source terms and longitudinal solutions of the Maxwell equation fall outside this

framework. Taking the view that the Maxwell equation with a source is the most direct

contact with experiment, our approach is to start from the matrix Maxwell equation with

a source term and explicitly work out the Lorentz transformations of the solutions. It is

shown that the six-component equation is invariant under Lorentz transformations, as it

should be, but this is not self-evident, since the source term is essentially the three-vector

current density.

Next, six-component solutions are constructed and shown to be complete sets of orthogonal coordinate-space eigenfunctions of the Maxwell Hamiltonian, parameterized by physical

properties, such as linear momentum, angular momentum, and parity. These properties

are associated with operators that commute with the Hamiltonian. Complete sets of both

plane-wave solutions and angular-momentum eigenfunctions are given. Bilinear products of

normalizable linear combinations of these functions provide expressions for the probability

density and flux. The eigenfunctions are further classified according to whether they represent transverse or longitudinal states. These properties are associated with the electric

and magnetic fields, with the result that under a velocity boost, the transformed transverse

solutions are also transverse, unlike solutions corresponding to a transverse vector potential. Moreover, by summing over both transverse and longitudinal solutions, we obtain a

covariant Green function for the Maxwell equation, which is of the same form as the Green

function for the Dirac equation.

Solutions are obtained directly from the Maxwell equation, with no recourse to a vector

potential. This avoids problems such as extra polarization components and ambiguities

associated with gauge transformations [9–11]. Although integrals over a closed path of the

potential may be observables, as discussed in [12], such integrals can be expressed in terms

of the magnetic flux through the loop [13], so it is expected that the fields alone provide

a complete description of electrodynamics. It is also clear that photon wave functions

are closely aligned with electric and magnetic fields, and an approach that starts with

classical electrodynamics expressed in terms of fields only provides a natural framework for

the transition to the wave mechanics of photons. A possible advantage of using a vector

potential is that it is the solution of a scalar wave equation, which has a well-known Green

function. However, this advantage is offset by the fact that we provide a covariant Green

function for the Maxwell equation.

This paper is organized as follows. In Sec. 2 the vector Maxwell equations and the Dirac

equation are stated to define notation. The algebra of three-component spin matrices is

reviewed in Sec. 3, where both a spherical basis, which is the direct extension of the Pauli

matrices to three components, and a Cartesian basis with real components, are defined. The

Maxwell equations are written in terms of the spherical spin matrices and combined into the

Dirac equation form in Sec. 4. In Sec. 5, transverse and longitudinal projection operators are

defined and used to separate the Maxwell equations and solutions into the corresponding

disjoint sectors. Lorentz invariance is addressed in Sec. 6, where transformations of the

coordinates and derivatives, transformations of the Maxwell equation, and transformations of

the solutions are explicitly written. In Sec. 7, plane-wave solutions, which are eigenfunctions

3
of the momentum operator as well as the Hamiltonian, are given for both transverse and

longitudinal states, and the set of solutions is shown to be complete. The explicit action of

Lorentz transformations on the plane-wave solutions is described. Properties of normalizable

wave packets formed from the plane-wave solutions are illustrated. The angular-momentum

operator and the corresponding eigenfunctions are given and shown to be complete in Sec. 8.

In Sec. 9, the Maxwell Green function is written as an integral over the plane-wave solutions

in a form analogous to the Dirac Green function. As examples of applications of the Maxwell

Green function, formulas for radiation from a point dipole source and from a Dirac current

source are derived in Sec. 10. A summary of the main points of the paper is in Sec. 11 and

brief concluding remarks are made in Sec. 12.

The relation of the present study to earlier work is indicated in the sections where the

particular topics are discussed.

2. Three-vector Maxwell equations and the Dirac equation

The Maxwell equations in vacuum, in the International System of Units (SI), are

∇ · E =

ρ

ǫ0

, (1)

∇ × B −

1

c

2

∂E

∂t = μ0J, (2)

∇ × E +

∂B

∂t = 0, (3)

∇ · B = 0, (4)

where E and B are the electric and magnetic fields, ρ and J are the charge and current

densities, ǫ0 and μ0 are the electric and magnetic constants, and c = (ǫ0μ0)

−1/2

is the speed

of light. The continuity equation

∂ρ

∂t + ∇ · J = 0 (5)

follows from Eqs. (1) and (2).

The form of the Maxwell equations considered here is analogous to the Dirac equation

for the electron. The Dirac wave function φ(x) is a four-component column matrix that is

a function of the four-vector x. For a free electron, the Dirac equation is

( i ~γ

μ

∂μ − mec) φ(x) = 0, (6)

where ~ is the Planck constant divided by 2π, me is the mass of the electron, γ

μ

, μ = 0, 1, 2, 3,

are the 4 × 4 Dirac gamma matrices, given by

γ

0 =



I 0

0 −I



; γ

i =



0 σ

i

−σ

i 0



, i = 1, 2, 3, (7)

4
I and 0 are the 2 × 2 identity and zero matrices, σ

i

, i = 1, 2, 3, are the Pauli spin matrices

σ

1 =



0 1

1 0 

, σ2 =



0 −i

i 0 

, σ3 =



1 0

0 −1



, (8)

and the derivatives ∂μ are

∂0 =

∂

∂ct; ∂i =

∂

∂xi

, i = 1, 2, 3. (9)

We take the metric tensor g

μν to be

g

00 = 1; gii = −1, i = 1, 2, 3; gμν = 0, μ 6= ν. (10)

In terms of the spin matrices, the derivative term in Eq. (6) can be written as

γ

μ

∂μ =

 

I

∂

∂ct σ · ∇

−σ · ∇ −I

∂

∂ct

!

. (11)

The Pauli spin matrices act on two-component spin matrices in the electron wave function. Oppenheimer has suggested that since the Maxwell equations involve three-vectors,

three-component matrices should be considered for constructing a photon wave function [5].

Here we implement such an extension by replacing the Pauli spin matrices in the Dirac

equation by the analogous 3 × 3 matrices described in the next section.

3. Three-component spin matrices

As is well known, three-vectors and operations among them are interchangeable with

three-component matrices and matrix operations. In this section, formulas for these matrices

relevant to subsequent work are given. Some of these formulas have been given in [14]. It is

useful to define both Cartesian and spherical matrices to represent three-vectors.

The Cartesian matrix representing a vector a may be written as

ac =





a

1

a

2

a

3



 (12)

where a

1

, a2, a3 are the rectangular components of the vector a, and a spherical representation is denoted by

as = Mac, (13)

where M is a 3×3 unitary matrix specified in the following. The dot product of two vectors

is

a · b = a

†

cbc = a

†

sbs

, (14)

5
where † denotes the combined operations of matrix transposition and complex conjugation.

Explicit Hermitian τ matrices (τ

† = τ ), which are 3 × 3 versions of the Pauli matrices,

are obtained by taking τ

3

to be diagonal

τ

3 =





1 0 0

0 0 0

0 0 −1



 (15)

and applying appropriate rotation matrices to obtain τ

1

and τ

2

:

τ

1 = D(1)({0,

π

2

, 0}) τ

3 D(1)({0, −

π

2

, 0}) = 1

√

2





0 1 0

1 0 1

0 1 0



, (16)

τ

2 = D(1)({0, 0,

π

2

}) τ

1 D(1)({0, 0, −

π

2

}) = i

√

2





0 −1 0

1 0 −1

0 1 0



, (17)

where D(1)({α, β, γ}) is the j = 1 representation of the rotation group, parameterized by

the Euler angles α, β, γ [15]. In particular, D(1)({0,

π

2

, 0}) represents the rotation about the

2 axis by the angle π/2 and D(1)({0, 0,

π

2

}) represents the rotation about the 3 axis by the

angle π/2. The same rotations starting from σ

3

, with the j =

1

2

representation, reproduce

σ

1

and σ

2

. The τ matrices are related by



τ

i

, τ j



= iǫijk τ

k

, (18)

where ǫijk is the Levi-Civita symbol.1

The cross product of two vectors a and b can be written in terms of the scalar product

of the tau matrices with the vector a

τ · a = τ

i

a

i

(19)

acting on the spherical matrix for the vector b as

τ · a bs = i (a × b)

s

, (20)

provided the matrix M in Eq. (13) is suitably chosen. To determine M, we take the

Cartesian definition

(a × b)

i = ǫijkaj

b

k

(21)

1The tau matrices defined by Oppenheimer [5] are N†

τ

iN, where N =





1 0 0

0 i 0

0 0 −1



. [The minus sign

in τ

2

in Eq. (10) of that paper apparently is a typographical error, as indicated by inspection of Eq. (11).]

The matrices defined by Majorana [16] are M†τ

iM, where M is given in Eq. (23).

6
and write Eq. (20) as

τ · aM bc = iM(a × b)c. (22)

Imposing the requirement that this equation be valid for any vectors a and b fixes M, up

to a phase factor, to be

M =

1

√

2





−1 i 0

0 0 √

2

1 i 0



 , (23)

which yields

as =





−√

1

2

(a

1 − i a2

)

a

3

√

1

2

(a

1 + i a2

)



 . (24)

Consequences of Eq. (20) are

τ · a as = 0, (25)

τ · a bs + τ · b as = 0, (26)

a

†

s τ · b cs = i a · (b × c), (27)

(τ · a cs) · (τ · b ds) = (a × c) · (b × d)

= a · b c · d − a · d c · b. (28)

Equation (28) can be written as

c

†

s

(τ · a)

†

τ · b ds = c

†

s

(a · b − bs a

†

s

)ds (29)

for any vectors c and d, which yields the relation

(τ · a)

†

τ · b = a · b − bsa

†

s

, (30)

where it is understood that the first term on the right includes the 3 × 3 identity matrix as

a factor and the second term is also a 3 × 3 matrix. If ac has real components, then

(τ · a)

† = τ · a, (31)

(τ · a)

3 = a2

τ · a, (32)

where a

2 = a · a is the ordinary real vector scalar product.

Real Cartesian tau matrices τ ̃ may be defined so that

τ ̃ · a bc = (a × b)

c

. (33)

7
This relation follows from Eqs. (20) and (13) with the definition

τ ̃

i = −iM†

τ

i M, i = 1, 2, 3. (34)

These matrices are antisymmetric τ ̃

⊤ = −τ ̃, where ⊤ denotes matrix transposition, in

contrast to τ

† = τ . For vectors a and b with real Cartesian components, we have

τ ̃ · a τ ̃ · b = bca

⊤

c − a · b, (35)

(τ ̃ · a)

3 = −a2

(τ ̃ · a), (36)

(τ ̃ · a)

ij = −ǫijkak

. (37)

The matrix

τ ̃ · a =





0 −a

3 a2

a

3 0 −a1

−a

2 a1

0



 (38)

has the form of the lower right portion of the electromagnetic field-strength tensor F

μν, as

given in [17] for example.

4. Matrix Maxwell equation

In terms of the notation of the previous section, the matrix forms of the Maxwell equations in (2) and (3), for the source-free case (J = 0), are

i τ · ∇Bs +

1

c

∂Es

∂ct = 0, (39)

i τ · ∇Es − c

∂Bs

∂ct = 0. (40)

These equations may be written as two uncoupled equations



I

∂

∂ct + τ · ∇



(Es + i cBs) = 0, (41)



I

∂

∂ct − τ · ∇



(Es − i cBs) = 0, (42)

where I is the 3 × 3 identity matrix. In the Cartesian basis, Eqs. (41) and (42) are



I

∂

∂ct + iτ ̃ · ∇



(Ec + i cBc) = 0, (43)



I

∂

∂ct − iτ ̃ · ∇



(Ec − i cBc) = 0. (44)

In these expressions, it is evident that for real electric and magnetic fields, Eqs. (43) and

(44) are complex conjugates of each other and reduce to a single complex equation. It was

8
recognized in lectures by Riemann in the nineteenth century that this complex combination

of E and B is a solution of a single equation [18]. This fact was also discussed in [19, 20] and

is included in many works up to the present. Equations (43) and (44) may be interpreted as

Maxwell equations for right- and left-circularly polarized radiation, analogous to the Weyl

equations for right- and left-handed neutrino fields [8, 14].

However, in this paper, we consider the more restrictive case of complex electric and

magnetic fields that are simultaneously solutions of both Eqs (43) and (44), or equivalently

both Eqs. (39) and (40), for any polarization of radiation. The question of whether such

solutions can be found is answered by their explicit construction in subsequent sections of

the paper. To formulate this approach, we follow the Dirac equation and write

 

I

∂

∂ct τ · ∇

−τ · ∇ −I

∂

∂ct

! Es

i cBs

!

= 0, (45)

which is a restatement of Eqs. (39) and (40) in the form of the Dirac equation for an electron

wave function. It is a matrix equation with six components that may be viewed as a single

equation equivalent to Eqs. (39) and (40) for any polarization of the fields. Any complex

solution of Eq. (45) is a solution of both Eqs. (41) and (42). Similar wave functions have

been discussed in [21–23]. It should be noted that this formulation is different from the

six-component form considered by Oppenheimer in which the upper-three components and

lower-three components represent opposite helicity states [5].

If we define 6 × 6 gamma matrices by

γ

0 =



I 0

0 −I



; γ

i =



0 τ

i

−τ

i 0



, i = 1, 2, 3, (46)

where 0 is the 3 × 3 zero matrix, and write

Ψ(x) = 

Es(x)

i cBs(x)

!

, (47)

then Eq. (45) takes the covariant Dirac equation form

γ

μ

∂μΨ(x) = 0, (48)

which provides a concise expression for two of the Maxwell equations. We can also write

this as

Ψ(x)

←−∂ μγ

μ = 0, (49)

where Ψ(x) = Ψ†

(x) γ

0 and ←−∂ μ denotes differentiation of the function to the left. Although

these equations are simply algebraic rearrangements of the two Maxwell equations, the

resemblance to the Dirac equation and wave function is suggestive of a form that photon

wave functions might take.

9
It is of interest to note that for solutions of the Dirac equation for the hydrogen atom,

the lower two components are small and approach zero in the nonrelativistic limit, i.e., as

the velocity of the bound electron approaches zero. Similarly, for local electromagnetic fields

generated by moving charges, the magnetic field, given by the lower three components of Ψ,

also approaches zero in the limit as the velocity of the charges approaches zero.

To take source currents into account, Eq. (2) is written as

i τ · ∇Bs +

1

c

∂Es

∂ct = −μ0Js

, (50)

and a source term Ξ is defined to be

Ξ(x) = 

−μ0cJs(x)

0

!

, (51)

where 0 is a 3 × 1 matrix of zeros. This yields the expressions

γ

μ

∂μΨ(x) = Ξ(x) (52)

and

Ψ(x)

←−∂ μγ

μ = Ξ(x), (53)

either of which is referred to as the Maxwell equation here. The source term in Eq. (52)

or (53) represents a fundamental difference between the Dirac equation and the Maxwell

equation, as mentioned in Sec. 1 [5].

In this framework, an energy-momentum density operator is

p

μ =

ǫ0

2c

γ

μ

, (54)

which gives

Ψ cp0Ψ =

1

2



ǫ0|E|

2 +

1

μ0

|B|

2



= u, (55)

Ψ pΨ =

iǫ0

2



E

†

s τBs − B†s τEs



=

1

c

2μ0

Re E× B∗ = g. (56)

Eqs. (52) and (53) imply that

∂μΨ(x)γ

μΨ(x) = Ξ(x)Ψ(x) + Ψ(x) Ξ(x), (57)

which is a complex form of the Poynting theorem [see Eq. (14)]

∂u

∂t + ∇ · S = −Re E · J, (58)

where

S = c

2

g, (59)

which gives the conventional result if the fields and current are real [17].

10
5. Transverse and longitudinal fields

To make a Helmholtz decomposition of electromagnetic fields expressed in matrix form

into transverse and longitudinal components, we define 3 × 3 matrix transverse and longitudinal Hermitian projection operators Π T

s

(a) and Π L

s

(a) to be

Π T

s

(a) = (τ · a)

†

(τ · a)

a · a

, (60)

Π L

s

(a) = asa

†

s

a · a

. (61)

Based on identities in Sec. 3, these operators have the following properties:



Π T

s

(a)

2

= Π T

s

(a), (62)



Π L

s

(a)

2

= Π L

s

(a), (63)

Π T

s

(a) + Π L

s

(a) = I, (64)

Π T

s

(a) Π L

s

(a) = 0, (65)

Π T

s

(a) as = 0, (66)

Π L

s

(a) as = as. (67)

Acting on the matrix of an arbitrary vector b, the operators project the components perpendicular to and parallel to the argument a

Π T

s

(a) bs = bs −

a · b

a · a

as, (68)

Π L

s

(a) bs =

a · b

a · a

as. (69)

In addition to these algebraic relations, usefulness of the projection operators arises

from an extension to include differential and integral operations acting on coordinate-space

functions. Formally, we write

Π T

s

(∇) = (τ · ∇)

2

∇2

, (70)

Π L

s

(∇) = ∇s∇†

s

∇2

, (71)

which takes into account that fact that ∇ has real Cartesian components (in the sense that

they give real values when acting on a real function). The inverse Laplacian is defined by

the relation

1

∇2

f(x) = −

1

4π

Z

d x

′

1

|x − x′|

f(x

′

), (72)

which yields

∇2

1

∇2

f(x) = −

1

4π

Z

d x

′ ∇2

1

|x − x′|

f(x

′

) = f(x), (73)

11
based on

∇2

1

|x − x′|

= −4π δ(x − x

′

), (74)

where

δ(x − x

′

) = δ(x

1 − x′1

) δ(x

2 − x′2

) δ(x

3 − x′3

). (75)

Equation (73) indicates that the Laplacian operator follows analogs of the rules of algebra

in this context. For example, we have



Π T

s

(∇)

2

=



(τ · ∇)

2

∇2

2

=

∇2(τ · ∇)

2

∇2∇2

=

(τ · ∇)

2

∇2

= Π T

s

(∇), (76)

where either simply canceling the ∇2factors in the numerator and denominator or applying

the definition in Eq. (72) for the operators acting on a suitable function gives the same

result. Transverse and longitudinal components of the electric and magnetic fields and the

current density are identified by writing

Fs = F

T

s + F

L

s

, (77)

where

F

T

s = Π T

s

(∇)Fs, (78)

F

L

s = Π L

s

(∇)Fs, (79)

and Fs may be any of Es, Bs, or Js.

The separation of the Maxwell equations into two independent sets of equations which

involve either transverse components or longitudinal components takes the following form.

In terms of the spherical matrices, Eq. (1) is

∇†

sE

L

s =

ρ

ǫ0

, (80)

where the transverse component of the electric field is absent, because ∇†

sΠ Ts

(∇) = 0.

Equations (1) and (80) are equivalent, in the sense that each can be derived from the

other; Eq. (1) follows from Eq. (80) if the vanishing transverse component is added to

the latter equation. In the separated form, it is evident that the equation neither contains

information about or places any constraint on the transverse component ET

s

. The transverse

and longitudinal projection operators acting on Eq. (50), the matrix form of Eq. (2), yield

i τ · ∇BT

s +

1

c

∂ET

s

∂ct = −μ0J

T

s

, (81)

1

c

∂EL

s

∂ct = −μ0J

L

s

(82)

12
respectively, which take into account the commutation relation [Π T

s

(∇), τ · ∇] = 0 and the

fact that Π L

s

(∇) τ · ∇ = 0. Together, these equations are equivalent to Eq. (50) which can

be restored by writing the sum of Eq. (81) and Eq. (82) and adding the term that vanishes.

Evidently, this pair of equations is independent of BL

s

. Similarly, Eq. (3), or equivalently

Eq. (40), can be written as the pair

i τ · ∇E

T

s − c

∂BT

s

∂ct = 0, (83)

∂BL

s

∂ct = 0, (84)

which are independent of EL

s

. Equation (4) takes the form

∇†

sBLs = 0, (85)

independent of BT

s

. The transverse and longitudinal equations comprise two independent

sets.

Six-dimensional transverse and longitudinal projection operators are defined by

Π

T

(∇) = 

Π T

s

(∇) 0

0 Π T

s

(∇)

!

, (86)

Π

L

(∇) = 

Π L

s

(∇) 0

0 Π L

s

(∇)

!

, (87)

where 0 is the 3 × 3 matrix of zeros, Π T(∇) + Π L(∇) = I, and I is the 6 × 6 identity

matrix. The transverse equations are summarized by writing

γ

μ

∂μΨ

T

(x) = Ξ

T

(x), (88)

where

Ψ

T

(x) = Π

T

(∇)Ψ(x), (89)

Ξ

T

(x) = Π

T

(∇) Ξ(x). (90)

Equation (88) also follows directly from Eq. (52) and the fact that Π T(∇), γμ∂μ



= 0.

The longitudinal equations are Eqs. (80), (82), (84), and (85), together with the continuity

equation, Eq. (5), which can be expressed as

∂ρ

∂t + ∇†

sJ

L

s = 0. (91)

Since the continuity equation follows from Eqs. (80) and (82), it is not necessary to include

it in an independent set of equations; it is listed here only to show that it provides no

restriction on J

T

s

. Equations (84) and (85) are eliminated from consideration by taking

BL

s = 0. (92)

13
Constant fields are eliminated by the requirement that static fields vanish at infinite distances

for finite source distributions. However, a constant magnetic field may be approximated by

the field at the center of a current loop with a radius that is large compared to the extent

of the region of interest. Such a steady-state current density is transverse, as shown by

Eq. (82), and so the magnetic field, given by Eq. (81), is also transverse, which is consistent

with Eq. (92). A complete set of equations, equivalent to the set of Maxwell equations, is

provided by Eqs. (52), (80), and (92), and the transverse fields are completely described by

Eq. (88).

6. Lorentz transformations

Lorentz transformations of the matrix Maxwell equation are examined here in order

to confirm that this form of the Maxwell equations is Lorentz invariant. We adopt the

convention that transformations apply to the physical system rather than to the observer’s

coordinates.

To represent four-vector coordinates, the Cartesian matrices are extended to include a

time component x

0 = ct, so coordinate vectors take the form

x =





x

0

x

1

x

2

x

3



 =



ct

xc



. (93)

We employ the Cartesian basis for coordinate and momentum vectors and the spherical

basis for fields and currents, with a few exceptions that will be apparent. The notation x

represents either the four-coordinate argument of a function or a column matrix, depending on the context. It is sufficient for our purpose to consider only homogeneous Lorentz

transformations and to consider rotations and velocity transformations separately. These

transformations acting on four-vectors leave the scalar product

x · x = x

⊤g x = (ct)2 − x2

(94)

invariant, where g is the metric tensor given by

g =

 

1 0

0 −I

!

. (95)

A remark on notation is that a boldface 0 means either a 3 × 3, a 1 × 3, or a 3 × 1

rectangular array of zeros, as appropriate. We take the liberty of using an ordinary zero on

the right-hand side of equations to mean whatever sort of zero matches the left-hand side.

6.1. Rotation of coordinates

Rotations are parameterized by a vector u = θuˆ, where uˆ is a unit vector in the direction

of the axis of the rotation and θ is the angle of rotation. An infinitesimal rotation δθ uˆ

changes the point at position x to the point at position x

′

, where

x

′ = x + δθ uˆ × x + . . . , (96)

14
or

x

′

c = (I + δθ τ ̃ · uˆ) xc + . . . . (97)

For a finite rotation, the operation is exponentiated to give

x

′

c = eτ ̃·uxc = Rc(u) xc. (98)

Expansion of the exponential function in powers of θ, taking into account the fact that

(τ ̃ · uˆ)

3 = −τ ̃ · uˆ, yields

Rc(u) = I + (τ ̃ · uˆ)

2

(1 − cos θ) + τ ̃ · uˆ sin θ

= uˆcuˆ

⊤

c − (τ ̃ · uˆ)

2

cos θ + τ ̃ · uˆ sin θ. (99)

Evidently, R−1

c

(u) = Rc(−u) = R⊤

c

(u). It is confirmed that this operator has the appropriate action on a vector by calculating

x

′ = uˆ uˆ · x − uˆ × (uˆ × x) cos θ + uˆ × x sin θ. (100)

We use the notation

x

′ = R(u)x (101)

to represent the transformation in Eq. (100).

Rotations of a four-vector only change the spatial coordinates and are written as

x

′ = R(u) x =

 

ct

Rc(u) xc

!

, (102)

where

R(u) = 

1 0

0 Rc(u)

!

. (103)

The scalar product x · x is invariant under rotations, since x

2

is invariant.

The spatial coordinate rotation operator in the spherical basis, which follows from

Rs(u) = MRc(u)M†, (104)

is

Rs(u) = e−iτ·u = uˆsuˆ

†

s + (τ · uˆ)

2

cos θ − i τ · uˆ sin θ, (105)

and R−1

s

(u) = Rs(−u) = R†

s

(u). Starting from the geometrical constraint that the rotated

cross product of two vectors is the cross product of the rotated vectors, written as

Rs(u)(a × b)s = (a

′ × b′

)s, (106)

15
we have

Rs(u) τ · a bs = τ · a

′

b

′

s = τ · a

′Rs(u) bs

. (107)

Since this relation holds for any vector b, it yields

Rs(u) τ · a = τ · a

′Rs(u) (108)

and

Rs(u) τ · a R−1

s

(u) = τ · a

′

. (109)

A direct calculation provides the same result.

The relation between the rotated and the unrotated gradient operators is given by

∇′i =

∂

∂x′i =

∂xj

∂x′i

∂

∂xj

=

∂xj

∂x′i ∇j

, (110)

where, from Eq. (98), we have

∂xj

∂x′i = R−1

c ji(u) = Rc ij (u), (111)

so that

∇′

c = Rc(u) ∇c, (112)

and from Eq. (104),

∇′

s = Rs(u) ∇s

. (113)

Since the spherical gradient operator transforms as a spherical vector, we also have

Rs(u) τ · ∇R−1

s

(u) = τ · ∇′(114)

from Eq. (109). Equations (113) and (114) imply that transverse and longitudinal projection

operators transform according to

Π T

s

(∇′) = Rs(u) Π T

s

(∇)R−1

s

(u), (115)

Π L

s

(∇′) = Rs(u) Π L

s

(∇)R−1

s

(u). (116)

The action of the inverse Laplacian in terms of the rotated coordinates is the same as it is

for unrotated coordinates, which follows either because ∇′2 = ∇2from Eq. (113) or by the

definition in Eq. (72), taking into account the fact that the Jacobian for a rotation is unity.

16
6.2. Velocity transformation of coordinates

Velocity transformations are parameterized by a velocity vector v = c tanh ζ vˆ. If a

space-time point is given an infinitesimal velocity boost of δζ c vˆ, its spatial coordinate will

change to

x

′ = x + δζ ct vˆ + . . . , (117)

and its time coordinate must transform in such a way that the scalar product is invariant.

In particular, we require x

′

· x

′ = x · x, which yields

ct′ = ct + δζ vˆ · x + . . . . (118)

The complete infinitesimal transformation is



ct′

x

′

c



=



I + δζ 

0 vˆ

⊤

c

vˆc 0

 ct

xc



+ . . . . (119)

This may be written in terms of a 4 × 4 matrix valued function of the velocity direction:

K(vˆ) = 

0 vˆ

⊤

c

vˆc 0



, (120)

for which

K2(vˆ) = 

1 0

0 ˆvcvˆ

⊤

c



(121)

and K3(vˆ) = K(vˆ). For a finite velocity, the transformation is exponentiated to give

x

′ = eζK(vˆ)

x = V (v) x. (122)

Expansion in powers of ζ yields

V (v) = I + K2(vˆ)(cosh ζ − 1) + K(vˆ) sinh ζ

=

 

cosh ζ vˆ

⊤

c

sinh ζ

vˆc sinh ζ I + vˆcvˆ

⊤

c

(cosh ζ − 1) !

. (123)

The relations V

⊤(v) = V (v) and gV (v) = V−1

(v)g confirm the invariance of the scalar

product:

x

′

· x

′ = x⊤V⊤(v) g V (v) x = x · x. (124)

The transformation yields

ct′ = ct cosh ζ + vˆ · x sinh ζ, (125)

x

′ = x + vˆ vˆ · x(cosh ζ − 1) + ct vˆ sinh ζ. (126)

17
A point with x = 0 has the boosted velocity

x

′

t

′

= c tanh ζ vˆ = v. (127)

The spherical counterpart of the operator V (v), in the velocity transformation



ct′

x

′

s



= Vs(v)



ct

xs



, (128)

is

Vs(v) = 

1 0

0 M



V (v)



1 0

0 M†



=

 

cosh ζ vˆ

†

s

sinh ζ

vˆs sinh ζ I + vˆsvˆ

†

s

(cosh ζ − 1) !

. (129)

For the four-gradient operator, we have

∂

′

μ =

∂

∂x′μ =

∂xν

∂x′μ

∂

∂xν

=

∂xν

∂x′μ

∂ν , (130)

and from Eq. (122), which can be written as

x = V

−1

(v) x

′ = V (−v) x′

(131)

or

x

ν = Vνμ(−v) x′μ

, (132)

we also have

∂xν

∂x′μ = Vνμ(−v) = Vμν (−v), (133)

which yields

∂

′

μ = Vμν (−v) ∂ν. (134)

If a Cartesian gradient operator is defined as

∂c =

 ∂

∂ct

− ∇c

!

, (135)

then Eq. (134) gives

g ∂′

c = V (−v) g ∂c (136)

or

∂

′

c = V (v) ∂c, (137)

since g V (−v) g = V (v).

18
6.3. Parity and time reversal of coordinates

Lorentz transformations that leave the scalar product in Eq. (94) invariant include the

parity transformation P = g, time reversal T = −g, and total inversion PT = −I operations.

These transformations have the following defining effects on the coordinate vectors:

Px =



ct

−xc



, (138)

Tx =



−ct

xc



, (139)

PTx = −x. (140)

It is sufficient for the present purpose to consider only P and T. The coordinate derivatives

transform as

P∂c =

 ∂

∂ct

∇c

!

, (141)

T∂c =

 

−

∂

∂ct

− ∇c

!

. (142)

Comparison of Eqs. (95) and (103) shows that parity transformations commute with

rotations. On the other hand, for velocity transformations, the relation

PV (v) = V (−v)P (143)

applies as it should, because the space reflection of a point moving with a velocity v is a

point at the reflected position moving with a velocity −v. Similar conclusions follow for

time-reversal transformations.

6.4. Rotation of Ψ(x)

The result of a rotation, parameterized by the vector u, applied to the field Ψ(x) in

Eq. (52) is the field Ψ′(x) given by

Ψ

′

(x) = R(u)Ψ(R

−1

(u) x), (144)

where R(u) is a 6×6 matrix that gives the local transformation of the field Ψ(x) at any point

x. The inverse transformation of the argument on the right-hand-side takes into account the

fact that the transformed field at the point x originated from the field at the point that is

mapped into x by the transformation. Lorentz invariance is confirmed by showing that the

transformed field satisfies the same equation as the original field. We expect the current to

transform in the same way as Ψ and write

Ξ

′

(x) = R(u) Ξ(R

−1

(u) x). (145)

19
The objective is to show that

γ

μ

∂μΨ

′

(x) = Ξ

′

(x), (146)

for a suitable transformation R(u). In terms of the original field and source, Eq. (146) is

given by

γ

μ

∂μR(u)Ψ(R

−1

(u) x) = R(u) Ξ(R

−1

(u) x) (147)

or

γ

μ

∂

′

μR(u)Ψ(x) = R(u) Ξ(x), (148)

where the variable x has been replaced by x

′ = R(u) x. Thus Eq. (146) will follow if

R−1(u)γ

μ

∂

′

μR(u) = γ

μ

∂μ. (149)

We expect R(u) to be of the form

R(u) = 

Rs(u) 0

0 Rs(u)



, (150)

which yields

R−1(u)γ

μ

∂

′

μR(u) =





I

∂

∂ct R−1

s

(u) τ · ∇′Rs(u)

−R−1

s

(u) τ · ∇′Rs(u) −I

∂

∂ct



 , (151)

so Eq. (149) follows from

R−1

s

(u) τ · ∇′Rs(u) = τ · ∇, (152)

which, in turn, follows from Eq. (114). We conclude that, as expected, the solution and

source terms, transformed according to Eqs. (144) and (145), where R(u) is given in

Eq. (150), satisfy the same equation as the original solution and source terms. The sixdimensional rotation operator R(u) may be written as

R(u) = e−iS·u, (153)

where

S =



τ 0

0 τ



. (154)

Equations (144), (145), and (150) correspond to the separate equations

E

′

s

(x) = Rs(u) Es(R

−1

(u) x), (155)

B′

s

(x) = Rs(u) Bs(R

−1

(u) x), (156)

J

′

s

(x) = Rs(u) Js(R

−1

(u) x). (157)

20
It can be confirmed that Eqs. (1) and (4) in spherical form are invariant under rotations. In

particular, Eq. (1) for the rotated electric field and charge density ρ

′

(x) = ρ(R−1(u)x) is

∇†

s E

′

s

(x) = ρ

′

(x)

ǫ0

(158)

or

∇†

s Rs(u)Es(R

−1

(u) x) = ρ(R−1

(u) x)

ǫ0

. (159)

The substitution x → R(u) x gives

∇′†

s Rs(u)Es(x) = ρ(x)

ǫ0

, (160)

and Eq. (113) yields

∇†

s Es(x) = ρ(x)

ǫ0

. (161)

Hence, the transformed field and charge density satisfy Eq. (1) if the original field and charge

density do. Similarly,

∇†

s B′s

(x) = ∇†

s Bs(x) = 0. (162)

Thus, all of the Maxwell equations in matrix form are invariant under rotations.

The separation into transverse and longitudinal components of the electric and magnetic

fields is also invariant under rotations. This can be seen by considering the expression

Π s(∇)Fs(x), where Π s(∇) is either Π T

s

(∇) or Π L

s

(∇) and Fs(x) is any of Es(x), Bs(x),

or Js(x). We have

Π s(∇)F

′

s

(x) = Π s(∇) Rs(u)Fs(R

−1

(u) x) (163)

or

Π s(∇′)F

′

s

(x

′

) = Π s(∇′) Rs(u)Fs(x)

= Rs(u) Π s(∇)Fs(x), (164)

where the last line follows from either Eq. (115) or Eq. (116). This means that if the

original field is transverse or longitudinal, then the rotated field has the same character.

These results extend directly to the six-dimensional projection operators Π (∇), solution

Ψ(x), and source Ξ(x).

21
6.5. Velocity transformation of Ψ(x)

The result of the velocity transformation, by a velocity v, applied to the field Ψ(x) in

Eq. (52) is the function Ψ′(x) given by

Ψ

′

(x) = V(v)Ψ(V

−1

(v) x), (165)

where V(u) is a 6 × 6 matrix that gives the local transformation of the field Ψ(x) at any

point. The inverse transformation of the argument on the right-hand-side plays the same

role as for rotations. Our objective is to establish the covariance of Eq. (52) by showing that

if Ψ(x) is a solution of that equation with a source Ξ(x), then

γ

μ

∂μΨ

′

(x) = Ξ

′

(x), (166)

where Ξ

′

(x) is a suitably transformed source term. Equation (166) can be written as

γ

μ

∂μV(v)Ψ(V

−1

(v) x) = Ξ

′

(x) (167)

or

γ

μ

∂

′

μV(v)Ψ(x) = Ξ

′

(V (v) x), (168)

where the variable x has been replaced by x

′ = V (v) x. The 6 × 6 matrix V(v) is based on

the conventional local velocity transformation of the electric and magnetic fields as discussed

in Appendix A. Here we write it as

V(v) = eζK·vˆ, (169)

where

K =



0 τ

τ 0



. (170)

Expansion of the exponential function in Eq. (169) in powers of ζ yields

V(v) = I + (K · vˆ)

2

(cosh ζ − 1) + (K · vˆ) sinh ζ

=



I + (τ · vˆ)

2

(cosh ζ − 1) τ · vˆ sinh ζ

τ · vˆ sinh ζ I + (τ · vˆ)

2

(cosh ζ − 1) 

, (171)

where (K · vˆ)

3 = K · vˆ is taken into account, so that

V(v)Ψ =

 

E′

s

i cB′

s

!

=

 

Es + (τ · vˆ)

2Es(cosh ζ − 1) + i τ · vˆ cBs sinh ζ

i [cBs + (τ · vˆ)

2

cBs(cosh ζ − 1) − i τ · v Eˆ s sinh ζ ]

!

. (172)

In Eq. (168), we have

γ

μ

∂

′

μ =

 

I

∂

∂ct′ τ · ∇′

−τ · ∇′ −I

∂

∂ct′

!

, (173)

22
where, from Eqs. (135) and (137),

∂

∂ct′

= cosh ζ

∂

∂ct − sinh ζ vˆ · ∇, (174)

∇′ = ∇ + (cosh ζ − 1)vˆ vˆ · ∇ − sinh ζ vˆ

∂

∂ct. (175)

Multiplication of Eq. (171) by Eq. (173) yields the identity

γ

μ

∂

′

μ V(v) = 

I + vˆsvˆ

†

s

(cosh ζ − 1) 0

0 I + vˆsvˆ

†

s

(cosh ζ − 1) 

γ

μ

∂μ

+



− sinh ζ vˆs∇†

s 0

0 sinh ζ vˆs∇†

s



(176)

and hence

γ

μ

∂

′

μV(v)Ψ(x) = 

I + vˆsvˆ

†

s

(cosh ζ − 1) 0

0 I + vˆsvˆ

†

s

(cosh ζ − 1) 

Ξ(x)

+



− sinh ζ vˆs ∇ · E(x)

i sinh ζ vˆs c ∇ · B(x)



=



−μ0c [Js(x) + vˆsvˆ · J(x)(cosh ζ − 1) + sinh ζ vˆscρ(x)]

0



=



−μ0cJ

′

s

(x)

0



, (177)

where J

′

s

(x) is the velocity transformed three-vector source current, and

Ξ

′

(x) = 

−μ0cJ

′

s

(V

−1

(v) x)

0

!

. (178)

The result in Eq. (177) takes into account the additional two Maxwell equations in (1) and

(4), besides Eqs. (2) and (3) used to construct Eq. (52). It also requires the conventional

result that the current



cρ(x)

Js(x)



(179)

transforms as a four-vector under the velocity transformation given by Eq. (129). Equation (177) establishes the validity of Eq. (166), provided the transformed three-vector source

current is the three-vector component of the transformed four-vector current.

The covariance of Eq. (52), even though the charge density does not appear in the source

term, is linked to the fact that the current density satisfies the continuity equation. Since

23
the continuity equation follows from the Maxwell equations, it cannot be expected that

consistent solutions may be found for an arbitrary four-vector current density. However, for

valid sources, information about the charge density may be obtained from the three-vector

current density and the continuity equation. For example, if the electric field is specified

at a particular time, then the charge density at that time is known from Eq. (1) and may

be determined at any other time from knowledge of the three-vector current density by

use of the continuity equation. Thus the time evolution of the electromagnetic fields can

be described relativistically with no reference to the charge density. The Maxwell Green

function, which provides the solutions of Eq. (52), is discussed in Sec. 9.

Since V

†γ0V = γ0 and V†γ0ηV = γ0η, where

η =



0 I

I 0



, (180)

the invariance of the quantities

ΨΨ = |E|

2 − c2

|B|

2

, (181)

ΨηΨ = 2 i c Re E · B (182)

is evident.

6.6. Parity and time-reversal transformations of Ψ(x)

We expect the fields Ψ(x) to transform under a parity change according to

Ψ

′

(x) = PΨ( P

−1 x), (183)

where P is a 6 × 6 matrix, and we assume that the current transforms in the same way, so

that

Ξ

′

(x) = PΞ( P

−1 x). (184)

We can obtain

γ

μ

∂μΨ

′

(x) = Ξ

′

(x), (185)

by finding a suitable matrix P. In terms of the original field and source, Eq. (185) is given

by

γ

μ

∂μPΨ(P

−1x) = PΞ(P−1x) (186)

or

γ

μ

∂

′

μPΨ(x) = PΞ(x), (187)

where ∂

′

μ

is the parity transformed derivative given by Eq. (141). Equation (185) will follow

if

P

−1

γ

μ

∂

′

μP = γ

μ

∂μ, (188)

24
which corresponds to

P

−1

γ

0P = γ0

, (189)

P

−1

γ

i P = −γi

, i = 1, 2, 3. (190)

Solutions of these equations are provided by

P = ±



I 0

0 −I



. (191)

The minus sign corresponds to the conventional choice of how classical electric and

magnetic fields transform under a parity change, that is, the current and electric fields

change sign and the magnetic fields do not [17].

To examine time-reversal invariance, we first consider Ψ(x) as a field which is real in

the Cartesian basis. In this case, the conventional use of an anti-unitary operator for time

reversal is unnecessary, and the same can be expected to be true for fields expressed in the

spherical basis. We write

Ψ

′

(x) = T Ψ( T

−1 x) (192)

and

Ξ

′

(x) = −T Ξ( T

−1 x), (193)

where T is a suitable 6 × 6 matrix. The minus sign for the current provides the result

that the electric field does not change sign under time reversal and the current does. The

objective is to find a matrix T such that

γ

μ

∂μΨ

′

(x) = Ξ

′

(x) (194)

or

γ

μ

∂μT Ψ(T

−1x) = −T Ξ(T−1x), (195)

and so

γ

μ

∂

′

μT Ψ(x) = −T Ξ(x), (196)

where ∂

′

μ

is given by Eq. (142). Such a matrix satisfies

T

−1

γ

μ

∂

′

μT = −γ

μ

∂μ (197)

or

T

−1

γ

0

T = γ

0

, (198)

T

−1

γ

i

T = −γ

i

, i = 1, 2, 3, (199)

25
with a solution provided by

T =



I 0

0 −I



. (200)

The matrices P and T commute with the rotation matrix, as they should, and we have

the result that the matrix form of the Maxwell equations is invariant under parity and

time-reversal transformations.

For quantum-mechanical time reversal, the time-reversal operator is anti-unitary and

includes complex conjugation, or Hermitian conjugation in the case of a matrix solution,

which has the effect of interchanging initial and final states. For an example where such an

interchange corresponds to observable consequences in QED, see [24, 25]. We thus write

Ψ

′

(x) = TΨ( T

−1 x) = Ψ†

( T

−1 x) U−1

, (201)

Ξ

′

(x) = −TΞ( T

−1 x) = −Ξ†

( T

−1 x) U−1

, (202)

where T = C U is the product of the Hermitian conjugation operator C, which has the action

CΨ(x) = Ψ†(x) and a unitary matrix U. The objective is to find a U such that

Ψ

′

(x)γ

0←−∂ μγμ = Ξ′

(x)γ

0

(203)

if Ψ(x) is a solution of Eq. (53). Equation (203) can be written as

Ψ

†

(x) U

−1

γ

0←−∂′

μγ

μ = −Ξ†

(x) U

−1

γ

0

, (204)

where ∂

′

μ

is given by Eq. (142). Equation (204) follows from Eq. (53) provided

γ

0 U−1

γ

0←−∂′

μγ

μ

γ

0 Uγ0 = −

←−∂ μγ

μ

, (205)

which has as a solution

U = T (206)

and

T = C T . (207)

7. Plane-wave eigenfunctions

Following the analogy with the Dirac equation, the Hamiltonian for the Maxwell equation

is

H = c α · p = −i ~c α · ∇, (208)

where α

i = γ0 γi

, and the wave functions for the photon may be identified as the complete

set of eigenfunctions of H. The solutions considered here are coordinate-space plane waves

26
characterized by a wave vector k and a polarization vector ǫˆλ; both positive- and negativeenergy solutions, as well as zero-energy solutions are included to form a complete set. These

solutions are also eigenfunctions of the momentum operator

P = Ip = −i ~ I ∇, (209)

which commutes with H.

The plane-wave solutions are not normalizable, because their modulus squared is independent of x and the integral over all space does not exist. As a result, the solutions include

an arbitrary multiplicative factor, that could be a function of k. Here a factor is chosen to

provide the simple result in Eq. (225).

7.1. Transverse plane-wave photons

We first consider transverse photons, i.,e., photons for which the electric and magnetic

fields are perpendicular to the wave vector. The polarization vector is a unit vector proportional to the electric or magnetic fields, represented by a three component, possibly

complex, vector in the spherical basis. As such, the polarization vector does not transform

as the spatial component of a four-vector under velocity transformations.

Two polarization vectors, both in the plane perpendicular to kˆ, are denoted by

ǫˆλ(kˆ) ; λ = 1, 2. (210)

They have the orthonormality properties

ǫˆ

†

λ2

(kˆ) ǫˆλ1(kˆ) = δλ2,λ1, (211)

kˆ†

s

ǫˆλ(kˆ) = 0, (212)

and the completeness property

X

2

λ=1

ǫˆλ(kˆ) ǫˆ

†

λ

(kˆ) = I − kˆs kˆ†

s = (τ · kˆ)

2 = Π T

s

(kˆ). (213)

From Eq. (212), we also have

(τ · kˆ)

2

ǫˆλ(kˆ) = ǫˆλ(kˆ). (214)

The polarization vectors can represent linear polarization, circular polarization, or any combination by a suitable choice of ǫˆλ(kˆ). For example, for k in the eˆ

3 direction, linear polarization vectors in the eˆ

1 and eˆ2 directions are

ǫˆ1(eˆ

3

) =





−√

1

2

0

√

1

2



 ; ǫˆ2(eˆ

3

) =





√

i

2

0

√

i

2



 , (215)

27
according to Eq. (24). Similarly, circular polarization vectors are (see Sec. 8.2)

ǫˆ1(eˆ

3

) =





1

0

0



 ; ǫˆ2(eˆ

3

) =





0

0

1



 . (216)

These polarization vectors can be transformed to the vectors corresponding to any direction

of k with the rotation operator in Eq. (105). (See also Sec. 7.5.)

Positive (+) and negative (−) energy transverse photon wave functions are given by

ψ

(±)

k,λ (x) = 1

p

2(2π)

3



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



e

±ik·x

. (217)

Although constructed geometrically to be transverse by the choice of polarization vectors,

these wave functions are also transverse in the sense defined in Sec. 5. We have (see Appendix

B for more detail)

Π

T

(∇) ψ

(±)

k,λ (x) = Π

T

(kˆ) ψ

(±)

k,λ (x) = ψ

(±)

k,λ (x), (218)

Π

L

(∇) ψ

(±)

k,λ (x) = Π

L

(kˆ) ψ

(±)

k,λ (x) = 0, (219)

where Π T and Π L are defined in Eqs. (86) and (87). The wave functions in Eq. (217) are

eigenfunctions of the Hamiltonian in Eq. (208) with eigenvalues ±~c |k|. In particular,

H ψ

(±)

k,λ (x) = ±~c α · k ψ

(±)

k,λ (x), (220)

and



0 τ · k

τ · k 0

  ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



= |k|



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



, (221)

so that

H ψ

(±)

k,λ (x) = ±~c |k| ψ

(±)

k,λ (x). (222)

Also,

P ψ

(±)

k,λ (x) = ±~ k ψ

(±)

k,λ (x). (223)

The wave functions have the expected property

ψ

(±)

k,λ(x) ψ

(±)

k,λ (x) = 0, (224)

since the electric and magnetic field strengths are equal for a transverse photon. Normalization and orthogonality relations are

Z

dx ψ

(±)†

k2,λ2

(x) ψ

(±)

k1,λ1

(x) = δλ2λ1δ(k2 − k1), (225)

Z

dx ψ

(±)†

k2,λ2

(x) ψ

(∓)

k1,λ1

(x) = 0. (226)

28
The latter relation follows from a cancellation of terms between the upper-three and lowerthree components of the wave function:

Z

dx ψ

(±)†

k2,λ2

(x) ψ

(∓)

k1,λ1

(x) = 1

2(2π)

3

Z

dx ǫˆ

†

λ2

(kˆ2)

h

I + τ · kˆ2 τ · kˆ1

i

ǫˆλ1(kˆ1) e∓i(k2+k1)·x

=

1

2

ǫˆ

†

λ2

(kˆ2)

h

I + τ · kˆ2 τ · kˆ1

i

ǫˆλ1(kˆ1)δ(k2 + k1) = 0. (227)

The transverse wave functions constitute a complete set of such functions. The completeness is established by writing

X

2

λ=1

Z

dk ψ

(±)

k,λ (x2)ψ

(±)†

k,λ (x1)

=

X

2

λ=1

Z

dk

1

2(2π)

3



ǫˆλ(kˆ) ǫˆ

†

λ

(kˆ) ǫˆλ(kˆ) ǫˆ

†

λ

(kˆ)τ · kˆ

τ · kˆ ǫˆλ(kˆ) ǫˆ

†

λ

(kˆ) τ · kˆ ǫˆλ(kˆ) ǫˆ

†

λ

(kˆ)τ · kˆ



e

±ik·(x2−x1)

=

Z

dk

1

2(2π)

3



(τ · kˆ)

2 τ · kˆ

τ · kˆ (τ · kˆ)

2



e

±ik·(x2−x1)

. (228)

To evaluate the integrals for the sum over positive and negative energy solutions, we use κ

to represent either a plus sign or a minus sign and write

X

κ→±

Z

dk (τ · kˆ)

2

e

κik·(x2−x1) = 2(2π)3 Π T

s

(∇2) δ(x2 − x1) (229)

and

X

κ→±

Z

dk τ · kˆ e

κik·(x2−x1) = 0, (230)

which yields the transverse completeness relation

X

κ→±

X

2

λ=1

Z

dk ψ

(κ)

k,λ(x2)ψ

(κ)†

k,λ (x1) = Π

T

(∇2) δ(x2 − x1). (231)

7.2. Longitudinal plane-wave photons

The transverse wave functions alone do not provide a complete description of electromagnetic fields. For example, the field of a point charge q at rest at the origin, given by

Es(x) = q

4πǫ0

xs

|x|

3

= −

q

4πǫ0

∇s

1

|x|

, (232)

is purely longitudinal, because

Π L

s

(∇) Es(x) = Es(x), (233)

Π T

s

(∇) Es(x) = 0. (234)

29
The longitudinal photons are represented by a third polarization state, labeled λ = 0, with

the polarization vector taken to be

ǫˆ0(kˆ) = kˆs. (235)

If k is in the eˆ

3 direction, the longitudinal polarization vector is (up to a phase factor)

ǫˆ0(eˆ

3

) =





0

1

0



 . (236)

Longitudinal wave functions are

ψ

(+)

k,0

(x) = 1

p

(2π)

3



ǫˆ0(kˆ)

0



e

ik·x

(237)

or

ψ

(−)

k,0

(x) = 1

p

(2π)

3



0

ǫˆ0(kˆ)



e

−ik·x

. (238)

This polarization state has the property that

Π

L

(∇) ψ

(±)

k,0

(x) = Π

L

(kˆ) ψ

(±)

k,0

(x) = ψ

(±)

k,0

(x), (239)

Π

T

(∇) ψ

(±)

k,0

(x) = Π

T

(kˆ) ψ

(±)

k,0

(x) = 0. (240)

The wave function has an energy eigenvalue of zero,

H ψ

(±)

k,0

(x) = ±~c α · k ψ

(±)

k,0

(x) = 0, (241)

because τ · k ǫˆ0(kˆ) = 0. However,

P ψ

(±)

k,0

(x) = ±~ k ψ

(±)

k,0

(x). (242)

Normalization and orthogonality of the λ = 0 wave functions, as well as the transverse

wave functions, are given by Eqs. (225) and (226), where λ1 and λ2 may take on any of the

values 0, 1, or 2.

Completeness relations are given by

Z

dk ψ

(+)

k,0

(x2)ψ

(+)†

k,0

(x1) = Zdk

1

(2π)

3



ǫˆ0(kˆ) ǫˆ

†

0

(kˆ) 0

0 0 

e

ik·(x2−x1)

(243)

and

Z

dk ψ

(−)

k,0

(x2)ψ

(−)†

k,0

(x1) = Zdk

1

(2π)

3



0 0

0 ˆǫ0(kˆ) ǫˆ

†

0

(kˆ)



e

−ik·(x2−x1)

, (244)

30
where

Z

dk ǫˆ0(kˆ) ǫˆ

†

0

(kˆ)e±ik·(x2−x1) = (2π)

3 Π L

s

(∇2) δ(x2 − x1), (245)

which yields

X

κ→±

Z

dk ψ

(κ)

k,0

(x2)ψ

(κ)†

k,0

(x1) = Π

L

(∇2) δ(x2 − x1). (246)

For the example of a point charge at the origin, we have

Ψp(x) = q

4πǫ0|x|

3



xs

0



, (247)

which may be written as (see Appendix C for some detail)

Ψp(x) = Π

L

(∇)Ψp(x) = Zdx1Π

L

(∇) δ(x − x1)Ψp(x1)

=

X

κ→±

Z

dk ψ

(κ)

k,0

(x)

Z

dx1 ψ

(κ)†

k,0

(x1)Ψp(x1)

= −

i q

p

(2π)

3 ǫ0

Z

dk

1

|k|

ψ

(+)

k,0

(x). (248)

7.3. Full orthogonality and completeness of the plane wave solutions

The full orthogonality relations are

Z

dx ψ

(κ2)†

k2,λ2

(x) ψ

(κ1)

k1,λ1

(x) = δκ2κ1δλ2λ1δ(k2 − k1), (249)

where the factor δκ2κ1is 1 if κ2 and κ1 represent the same sign and is 0 for opposite signs,

and λ2 and λ1 may be any of 0,1,2. The combined result of the transverse and longitudinal

completeness relations, Eqs. (231) and (246), is

X

κ→±

X

2

λ=0

Z

dk ψ

(κ)

k,λ(x2)ψ

(κ)†

k,λ (x1) = I δ(x2 − x1), (250)

where Π T(∇) + Π L(∇) = I.

7.4. Time dependence of the wave functions

The time dependence of the photon wave functions is given by2

ψ

(κ)

k,λ(x) = ψ

(κ)

k,λ(x) e−κiωt

, (251)

2The notation f(x) = f(x)





t=0 is employed throughout this paper.

31
where ω is determined by the equation

γ

μ

∂μ ψ

(κ)

k,λ(x) = γ

0



I

∂

∂ct + α · ∇



ψ

(κ)

k,λ(x) = 0. (252)

For transverse photons

ω = c|k|, (253)

and for longitudinal photons

ω = 0. (254)

The complete exponential factor is thus

e

−κik·x

, (255)

where

k =



|k|

kc



or 

0

kc



, (256)

depending on whether the photon is transverse or longitudinal. This corresponds to the

eigenvalue equation

H ψ

(κ)

k,λ(x) = κ~ω ψ(κ)

k,λ(x) (257)

and a time dependence given by

ψ

(κ)

k,λ(x) = e−iHt/~ψ

(κ)

k,λ(x). (258)

It is also of interest to consider the effect of a hypothetical photon mass mγ on the

longitudinal photon wave function. Such a modification, with an infinitesimal mass, resolves

an ambiguity in the construction of the Green function as discussed in Sec. 9. Following the

form of the Dirac equation in Eq. (6), we have

(i ~γ

μ

∂μ − mγc) ψ

(κ)

k,0

(x) = 0 (259)

or



~κω

c

γ

0 − mγc I



ψ

(κ)

k,0

(x) = 0, (260)

which yields

~ω = mγc

2

, (261)

since

γ

0ψ

(κ)

k,0

(x) = κ ψ(κ)

k,0

(x). (262)

32
7.5. Rotation of the wave functions

The rotations of the wave functions follow from the discussion of Sec. 6.4, with an

additional consideration of the vector k. On physical grounds, a rotation parameterized by

the vector u of the state of a photon means rotation of the vector k into the vector k

′

,

according to

k

′ = R(u)k, (263)

where R(u) is defined by Eq. (101). Similarly, the polarization vector is transformed by the

spherical rotation operator

ǫˆλ(kˆ′) = Rs(u) ǫˆλ(kˆ) (264)

and

τ · kˆ′ǫˆλ(kˆ′) = Rs(u) τ · k Rˆ −1

s

(u)Rs(u) ǫˆλ(kˆ), (265)

where the rotation angle θ, axis direction uˆ, kˆ, and kˆ′ are related by

uˆ sin θ = kˆ × kˆ′. (266)

For applications, the rotation operator can be expressed as a function of kˆ and kˆ′rather

than u. From

Rs(u) = I − (τ · uˆ)

2

(1 − cos θ) − i τ · uˆ sin θ, (267)

one has for transverse polarization, λ = 1, 2,3

ǫˆλ(kˆ′) = (τ · kˆ′

)

2 + τ · kˆ′ τ · kˆ

1 + kˆ′· kˆ

ǫˆλ(kˆ), (268)

τ · kˆ′ǫˆλ(kˆ′) = (τ · kˆ′

)

2 + τ · kˆ′ τ · kˆ

1 + kˆ′· kˆ

τ · kˆ ǫˆλ(kˆ), (269)

and for longitudinal polarization, λ = 0,

ǫˆ0(kˆ′) = kˆ′

skˆ†s ǫˆ0(kˆ). (270)

We thus have for a rotation of the wave functions characterized by the vector u

ψ

(κ)

k′, λ(x) = R(u) ψ

(κ)

k,λ(R

−1

(u) x), (271)

3The identity τ · k × k′ = i(k′

s k

†

s − ks k

′†

s

) is useful here.

33
where k

′

· x = k · R−1(u) x in the exponent of the wave function. This yields the result

γ

μ

∂μψ

(κ)

k′, λ(x) = 0, (272)

according to the discussion in Sec. 6.4.

The expected transformation in Eq. (271) can be confirmed by an explicit calculation

based on the completeness of the wave functions. For a rotation of a transverse wave function,

we write

R(u) ψ

(κ)

k,λ(R

−1

(u) x) = Zdx1 δ(x − x1) R(u) ψ

(κ)

k,λ(R−1

(u) x1) e−κiωt

=

X

κ1→±

X

2

λ1=1

Z

dk1

Z

dx1 ψ

(κ1)

k1,λ1

(x)ψ

(κ1)†

k1,λ1

(x1)R(u) ψ

(κ)

k,λ(R−1

(u) x1) e−κiωt, (273)

where λ = 1 or 2, and from Eq. (164) and the subsequent remarks, it follows that the rotated

wave function is also transverse, so λ1 is restricted to 1 or 2. The evaluation requires the

matrix element

Z

dx1 ψ

(κ1)†

k1,λ1

(x1)R(u) ψ

(κ)

k,λ(R−1

(u) x1)

=

1

2(2π)

3

Z

dx1

h

ǫˆ

†

λ1

(kˆ1) Rs(u) ǫˆλ(kˆ) + ǫˆ

†

λ1

(kˆ1) τ · kˆ1 Rs(u) τ · kˆ ǫˆλ(kˆ)

i

×e

−κ1ik1·x1 eκik·R−1(u)x1

=

1

2

δκ1κ

h

ǫˆ

†

λ1

(kˆ′) Rs(u) ǫˆλ(kˆ) + ǫˆ

†

λ1

(kˆ′) τ · kˆ′ Rs(u) τ · kˆ ǫˆλ(kˆ)

i

δ(k1 − k

′

)

= δκ1κ δλ1λ δ(k1 − k

′

). (274)

In the exponent in Eq. (274), we have k · R−1(u)x1 = R(u)k · x1 = k

′

· x1. The factor

δκ1κ results from the requirement that k

′ → k as the rotation angle θ → 0, i.e., that k does

not change sign for an infinitesimal rotation. The factor δλ1λ follows from Eq. (264) and

the discussion that follows it, together with the orthonormality of the polarization vectors.

Substitution of Eq. (274) into (273) yields

R(u) ψ

(κ)

k,λ(R

−1

(u) x) = ψ

(κ)

k′,λ(x), (275)

in accord with the general arguments leading to Eq. (271).

For a rotation of a longitudinal wave function, we have

R(u) ψ

(κ)

k,0

(R

−1

(u) x) = Zdx1 δ(x − x1) R(u) ψ

(κ)

k,0

(R−1(u) x1)

=

X

κ1→±

Z

dk1

Z

dx1 ψ

(κ1)

k1,0

(x)ψ

(κ1)†

k1,0

(x1)R(u) ψ

(κ)

k,0

(R−1(u) x1) (276)

34
where only longitudinal wave functions contribute, and the evaluation requires the matrix

element

Z

dx1 ψ

(κ1)†

k1,0

(x1)R(u) ψ

(κ)

k,0

(R−1(u) x1)

=

1

(2π)

3

Z

dx1δκ1κ ǫˆ

†

0

(kˆ1) Rs(u) ǫˆ0(kˆ) e−κ1ik1·x1e

κik·R−1(u)x1

= δκ1κ ǫˆ

†

0

(kˆ′) Rs(u) ǫˆ0(kˆ) δ(k1 − k

′

)

= δκ1κ δ(k1 − k

′

). (277)

Substitution of Eq. (277) into (276) yields

R(u) ψ

(κ)

k,0

(R

−1

(u) x) = ψ

(κ)

k′,0

(x), (278)

in agreement with Eq. (271).

7.6. Velocity transformation of the wave functions

7.6.1. Transverse wave functions

Under the velocity transformation of a transverse photon by a velocity v, the four-vector

k =



|k|

kc



(279)

is transformed to

k

′ = V (v) k =





|k|



cosh ζ + vˆ · kˆ sinh ζ



kc + |k|vˆc

h

sinh ζ + vˆ · kˆ(cosh ζ − 1)i



 , (280)

and the wave function is expected to transform according to

ξ ψ′ (κ)

k′, λ(x) = V(v) ψ

(κ)

k,λ(V

−1

(v) x). (281)

The prime on the transformed function indicates that it is a function of modified polarization

vectors, which in general are not simply rotated vectors corresponding to kˆ → kˆ′. The

transformed transverse wave function is proportional to

V(v)



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



=





n

I + (τ · vˆ)

2

(cosh ζ − 1) + τ · v τ ˆ · kˆ sinh ζ

o

ǫˆλ(kˆ)

n

τ · vˆ sinh ζ + [I + (τ · vˆ)

2

(cosh ζ − 1)] τ · kˆ

o

ǫˆλ(kˆ)





= ξ



ǫˆ

′

λ

(kˆ′)

τ · kˆ′ ǫˆ

′

λ

(kˆ′)



, (282)

where the fact that transformed wave function can be written in the form given at the

right-hand end is based on the three identities:







n

I + (τ · vˆ)

2

(cosh ζ − 1) + τ · v τ ˆ · kˆ sinh ζ

o

ǫˆλ(kˆ)







= cosh ζ + vˆ · kˆ sinh ζ, (283)

35
k

′

s

†

n

I + (τ · vˆ)

2

(cosh ζ − 1) + τ · v τ ˆ · kˆ sinh ζ

o

ǫˆλ(kˆ) = 0, (284)

τ · vˆ sinh ζ +



I + (τ · vˆ)

2

(cosh ζ − 1)τ · kˆ

= τ · kˆ′

n

I + (τ · vˆ)

2

(cosh ζ − 1) + τ · v τ ˆ · kˆ sinh ζ

o

, (285)

where

kˆ′ =

kˆ + vˆ

h

sinh ζ + vˆ · kˆ(cosh ζ − 1)i

cosh ζ + vˆ · kˆ sinh ζ

. (286)

Equation (283) determines the scalar multiplicative factor in Eq. (281) to be

ξ = cosh ζ + vˆ · kˆ sinh ζ. (287)

According to Eq. (284), the transformed polarization vector is in the plane perpendicular

to k

′

, that is, the transformed transverse polarization vector is also transverse. This is in

contrast to the vector potential, which does not maintain transversality under a velocity

transformation, so the Coulomb, or radiation, gauge condition is not preserved. The difference is just the fact that the vector potential transforms as a vector, so that the angle

between the space component of the vector potential and the space component of the wave

vector is not necessarily preserved under a velocity transformation, whereas the polarization

vector transforms as the electric field, i.e., as a component of a second-rank tensor, and

Eq. (284) shows that for this case the transversality is preserved. Equation (285) shows

that the lower components of the transformed wave function, as given in Eq. (282), can be

written as τ · kˆ′times the upper components in the same expression. This together with

the relation k

′

· x = k · V

−1

(v) x in the exponent of the wave function insures that the

transformed wave function is a solution of the source-free Maxwell equation. It also follows

from Eq. (282) that if λ2 6= λ1, then

ǫˆ

′ †

λ2

(kˆ′) ǫˆ

′

λ1

(kˆ′) = 0, (288)

so the orthogonality of the polarization vectors is preserved by the velocity transformation.

Equation (281) can be be obtained by an explicit calculation based on the completeness

of the wave functions, as for rotations. A velocity transformation of a transverse wave

function is given by

V(v) ψ

(κ)

k,λ(V

−1

(v) x)

=

Z

dx1 δ(x − x1) V(v) ψ

(κ)

k,λ(V

−1

(v) x1)

=

X

κ1→±

X

2

λ1=0

Z

dk1

Z

dx1 ψ

′ (κ1)

k1,λ1

(x)ψ

′ (κ1)†

k1,λ1

(x1)V(v) ψ

(κ)

k,λ(V

−1

(v) x1), (289)

36
where λ = 1 or 2, t1 = t, and the primes on the wave functions indicate that the velocity

transformed polarization vectors provide the basis vectors for λ1 = 1, 2. For λ1 = 0,

Z

dx1 ψ

′ (+)†

k1,0

(x1)V(v) ψ

(κ)

k,λ(V

−1

(v) x1)

=

1

√

2(2π)

3

Z

dx1



ǫˆ

′†

0

(kˆ1) 0



V(v)



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



e

−ik1·x1

e

−κik·V −1(v)x1

=

1

√

2

δ+κ ξ ǫˆ

′†

0

(kˆ′) ǫˆ

′

λ

(kˆ′) δ(k1 − k

′

) e−ik

′0

ct = 0, (290)

Z

dx1 ψ

′ (−)†

k1,0

(x1)V(v) ψ

(κ)

k,λ(V

−1

(v) x1)

=

1

√

2(2π)

3

Z

dx1



0 ˆǫ

′†

0

(kˆ1)



V(v)



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



e

ik1·x1e−κik·V −1(v)x1

=

1

√

2

δ−κ ξ ǫˆ

′†

0

(kˆ′) τ · kˆ′ǫˆ

′

λ

(kˆ′) δ(k1 − k

′

) eik

′0

ct = 0. (291)

In the exponent in Eqs. (290) and (291), k ·V

−1

(v)x1 = V (v)k ·x1 = k

′

·x1. As for rotations,

the factors δ+κ and δ−κ result from the requirement that k

′ → k as the velocity |v| → 0,

and the last equalities follow from Eqs. (284) and (285). For λ1 = 1 or 2, we have

Z

dx1 ψ

′(κ1)†

k1,λ1

(x1)V(v) ψ

(κ)

k,λ(V

−1

(v) x1)

=

1

2(2π)

3

Z

dx1



ǫˆ

′†

λ1

(kˆ1) ǫˆ

′†

λ1

(kˆ1) τ · kˆ1



V(v)



ǫˆλ(kˆ)

τ · kˆ ǫˆλ(kˆ)



×e

−κ1ik1·x1 e−κik·V

−1

(v)x1

= δκ1κ δλ1λ ξ δ(k1 − k

′

) e−κik

′0

ct

. (292)

Substitution of Eq. (292) into (289) yields

V(v) ψ

(κ)

k,λ(V

−1

(v) x) = ξ ψ′ (κ)

k′,λ (x), (293)

where

ξ ǫˆ

′

λ

(kˆ′) = nI + (τ · vˆ)

2

(cosh ζ − 1) + τ · v τ ˆ · kˆ sinh ζ

o

ǫˆλ(kˆ), (294)

in accord with Eq. (281).

7.6.2. Longitudinal wave functions

For a longitudinal solution, the four-vector k in the invariant phase factor k · x, given by

k =



0

kc



(295)

37
from Eqs. (254)-(256), has a zero time component. However, the transformed phase, given

by k · V

−1

(v) x = V (v)k · x = k

′

· x, where

k

′ = V (v) k =



vˆ · k sinh ζ

kc + vˆc vˆ · k(cosh ζ − 1) 

, (296)

does have time dependence. Thus the wave function is expected to transform according to

X

κ′→±

X

3

λ′=0

ξ

κ

′κ

λ′ 0

(t) ψ

′ (κ

′

)

k′, λ′ (x) = V(v) ψ

(κ)

k,0

(V

−1

(v) x), (297)

where the coefficients include the extra time dependence introduced by the velocity transformation. The sum over polarization states is necessary, because unlike the case of the

transverse wave function, the transformed longitudinal wave function is mixture of longitudinal and transverse components. This is expected on physical grounds, since moving

charges cause radiative atomic transitions. On the other hand, the transformed space-like

wave vector does not the match the wave vector of either the longitudinal or transverse basis

functions. Since the solutions are classified according to their three-wave-vector, there is a

residual time dependence in the expansion that is included in the transformation coefficients.

To be explicit, for λ 6= 0, we consider specific polarization vectors. Let ǫˆ

′

1

(kˆ′) be a

linear polarization vector in the plane of kˆ′and vˆ and ǫˆ

′

2

(kˆ′) be a linear polarization vector

perpendicular to kˆ′ and vˆ. These conditions, together with k

′ †

s

ǫˆ

′

1

(kˆ′) = 0, yield (up to phase

factors)

ǫˆ

′

1

(kˆ′) = vˆ · kˆ′ kˆ′

s − vˆs q

1 − (vˆ · kˆ′)

2

, (298)

ǫˆ

′

2

(kˆ′) = τ · vˆ kˆ′

q s

1 − (vˆ · kˆ′)

2

, (299)

where

ǫˆ

′

2

(kˆ′) = τ · kˆ′ǫˆ

′

1

(kˆ′), (300)

ǫˆ

′

1

(kˆ′) = τ · kˆ′ǫˆ

′

2

(kˆ′). (301)

The transformed upper- and lower-component longitudinal wave functions follow from the

expressions (for t = 0 and x = 0)

V(v)



ǫˆ0(kˆ)

0



=



[I + (τ · vˆ)

2

(cosh ζ − 1)] ǫˆ0(kˆ)

τ · vˆ sinh ζ ǫˆ0(kˆ)



= ξ

++

00 (0) 

ǫˆ

′

0

(kˆ′)

0



+

ξ

++

10 (0)

√

2



ǫˆ

′

1

(kˆ′)

τ · kˆ′ ǫˆ

′

1

(kˆ′)



+

ξ

−+

10 (0)

√

2



ǫˆ

′

1

(kˆ′)

−τ · kˆ′ ǫˆ

′

1

(kˆ′)



, (302)

38
and

V(v)



0

ǫˆ0(kˆ)



=



τ · vˆ sinh ζ ǫˆ0(kˆ)

[I + (τ · vˆ)

2

(cosh ζ − 1)] ǫˆ0(kˆ)



= ξ

−−

00 (0) 

0

ǫˆ

′

0

(kˆ′)



+

ξ

−−

20 (0)

√

2



ǫˆ

′

2

(kˆ′)

τ · kˆ′ ǫˆ

′

2

(kˆ′)



+

ξ

+−

20 (0)

√

2



−ǫˆ

′

2

(kˆ′)

τ · kˆ′ ǫˆ

′

2

(kˆ′)



, (303)

based on the relations

kˆ =

kˆ′ − vˆ vˆ · kˆ′(1 − sech ζ)

q

1 − (vˆ · kˆ′)

2 tanh2

ζ

, (304)

kˆ′

skˆ′†s



I + (τ · vˆ)

2

(cosh ζ − 1)kˆs = cosh ζ

q

1 − (vˆ · kˆ′)

2

tanh2ζ kˆ′

s

, (305)

kˆ′

skˆ′†s τ · vˆ sinh ζ kˆs = 0, (306)

(I − kˆ′

skˆ′†s

)



I + (τ · vˆ)

2

(cosh ζ − 1)kˆs =

sinh ζ tanh ζ vˆ · kˆ′(vˆ · kˆ′kˆ′

s − vˆs)

q

1 − (vˆ · kˆ′)

2

tanh2ζ

, (307)

(I − kˆ′

skˆ′†s

) τ · vˆ sinh ζ kˆs =

sinh ζ τ · kˆ′(vˆ · kˆ′kˆ′

s − vˆs)

q

1 − (vˆ · kˆ′)

2

tanh2ζ

, (308)

with coefficients given by

ξ

++

00 (0) = ξ

−−

00 (0) = cosh ζ

q

1 − (vˆ · kˆ′)

2

tanh2ζ , (309)

ξ

++

10 (0) = ξ

−−

20 (0) = sinh ζ

√

2

q

1 − (vˆ · kˆ′)

2

s

1 + vˆ · kˆ′tanh ζ

1 − vˆ · kˆ′ tanh ζ

, (310)

ξ

−+

10 (0) = ξ

+−

20 (0) = −

sinh ζ

√

2

q

1 − (vˆ · kˆ′)

2

s

1 − vˆ · kˆ′tanh ζ

1 + vˆ · kˆ′tanh ζ

. (311)

Terms that do not appear in Eq. (302) or (303) make no contribution

ξ

+−

00 (0) = ξ

−+

00 (0) = ξ

−−

10 (0) = ξ

+−

10 (0) = ξ

++

20 (0) = ξ

−+

20 (0) = 0. (312)

An explicit calculation of Eq. (297) is made by evaluating

V(v) ψ

(κ)

k,0

(V

−1

(v) x) = Zdx1 δ(x − x1) V(v) ψ

(κ)

k,0

(V

−1

(v) x1)

=

X

κ1→±

X

2

λ1=0

Z

dk1

Z

dx1 ψ

′ (κ1)

k1,λ1

(x)ψ

′ (κ1)†

k1,λ1

(x1)V(v) ψ

(κ)

k,0

(V

−1

(v) x1). (313)

39
For λ1 = 0,

Z

dx1 ψ

′ (+)†

k1,0

(x1)V(v) ψ

(+)

k,0

(V

−1

(v) x1)

=

1

(2π)

3

Z

dx1



ǫˆ

′†

0

(kˆ1) 0



V(v)



ǫˆ0(kˆ)

0



e

−ik1·x1

e

−ik·V −1(v)x1

= ξ

++

00 (0) δ(k1 − k

′

) e−ik

′0

ct

, (314)

Z

dx1 ψ

′ (−)†

k1,0

(x1)V(v) ψ

(−)

k,0

(V

−1

(v) x1)

=

1

(2π)

3

Z

dx1



0 ˆǫ

′†

0

(kˆ1)



V(v)



0

ǫˆ0(kˆ)



e

ik1·x1 eik·V −1(v)x1

= ξ

−−

00 (0) δ(k1 − k

′

) eik

′0

ct

, (315)

where

kˆ′ =

kˆ + vˆ vˆ · kˆ(cosh ζ − 1)

q

1 + (vˆ · kˆ)

2

sinh2ζ

, (316)

and k

′0

is the time component associated with the transformed space-like vector k

k

′0 = vˆ · k sinh ζ = vˆ · k′

tanh ζ. (317)

For λ1 = 1 or 2,

Z

dx1 ψ

′ (κ1)†

k1,λ1

(x1)V(v) ψ

(+)

k,0

(V

−1

(v) x1)

=

1

√

2(2π)

3

Z

dx1



ǫˆ

′†

λ1

(kˆ1) ǫˆ

′†

λ1

(kˆ1) τ · kˆ1



V(v)



ǫˆ0(kˆ)

0



×e

−κ1ik1·x1 e−ik·V −1(v)x1

= δκ1+δλ11 ξ

++

10 (0) δ(k1 − k

′

) e−ik

′0

ct + δκ1−δλ11 ξ−+

10 (0) δ(k1 + k

′

) e−ik

′0

ct

, (318)

Z

dx1 ψ

′ (κ1)†

k1,λ1

(x1)V(v) ψ

(−)

k,0

(V

−1

(v) x1)

=

1

√

2(2π)

3

Z

dx1



ǫˆ

′ †

λ1

(kˆ1) ǫˆ

′ †

λ1

(kˆ1) τ · kˆ1



V(v)



0

ǫˆ0(kˆ)



×e

−κ1ik1·x1eik·V −1(v)x1

= δκ1−δλ12 ξ

−−

20 (0) δ(k1 − k

′

) eik

′0

ct + δκ1+δλ12 ξ

+−

20 (0) δ(k1 + k

′

) eik

′0

ct

. (319)

40
Here both signs of κ1 are included, because there is no continuity condition on the transverse

solutions, which are absent in the limit of small velocity transformations. These results yield

Eq. (297) with the non-zero coefficients given by

ξ

++

00 (t) = ξ

++

00 (0) e−ivˆ·k

′

tanh ζ ct

, (320)

ξ

−−

00 (t) = ξ

−−

00 (0) eivˆ·k

′

tanh ζ ct

, (321)

ξ

++

10 (t) = ξ

++

10 (0) ei(|k

′

|−vˆ·k

′

tanh ζ) ct

, (322)

ξ

−+

10 (t) = ξ

−+

10 (0) e−i(|k

′

|+vˆ·k

′

tanh ζ) ct

, (323)

ξ

+−

20 (t) = ξ

+−

20 (0) ei(|k

′

|+vˆ·k

′

tanh ζ) ct

, (324)

ξ

−−

20 (t) = ξ

−−

20 (0) e−i(|k

′

|−vˆ·k

′

tanh ζ) ct

. (325)

These transformed longitudinal solutions can be used, for example, to describe the fields of

a moving charge by means of the expansion in Eq. (248).

7.7. Standing-wave parity eigenfunctions

The parity operator P changes the sign of the coordinates and includes multiplication

by the matrix P = −γ

0

, so that the transformed wave function is also a solution of the

Maxwell equations, as discussed in Sec. 6.6. We thus have

Pψ

(κ)

k,λ (x) = −γ

0ψ

(κ)

k,λ(−x). (326)

With this definition, the parity operator commutes with the Hamiltonian in Eq. (208)

PH = HP, (327)

so we may identify eigenstates of both parity and energy. Since

P2ψ

(κ)

k,λ(x) = ψ

(κ)

k,λ(x), (328)

the parity eigenvalues are ±1.

Transverse parity and energy eigenstates are

ψ

(κ,+)

k,λ (x) = 1

p

(2π)

3



κi ǫˆλ(kˆ) sin k · x

τ · kˆ ǫˆλ(kˆ) cos k · x



, (329)

ψ

(κ,−)

k,λ (x) = 1

p

(2π)

3



ǫˆλ(kˆ) cos k · x

κi τ · kˆ ǫˆλ(kˆ) sin k · x



, (330)

where

Pψ

(κ,±)

k,λ (x) = ±ψ

(κ,±)

k,λ (x), (331)

Hψ

(κ,±)

k,λ (x) = κ~c|k|ψ

(κ,±)

k,λ (x), (332)

41
and λ = 1, 2. These states are linear combinations of plane-wave states that form standing

plane waves. Orthogonality relations are calculated with the aid of the integrals

Z

dx cos k2 · x cos k1 · x = 4π

3

[δ(k2 − k1) + δ(k2 + k1)] , (333)

Z

dx sin k2 · x sin k1 · x = 4π

3

[δ(k2 − k1) − δ(k2 + k1)] , (334)

Z

dx cos k2 · x sin k1 · x = 0, (335)

which show that the states that differ only by the sign of k are not orthogonal. One of the

overlapping states may be removed by including only states with k such that k · k0 > 0,

where k0 is a fixed direction in space. With this restriction on k2 and k1, the orthonormality

of the states is given by

Z

dx ψ

(κ2,π2)†

k2,λ2

(x) ψ

(κ1,π1)

k1,λ1

(x) = δκ2κ1δπ2π1δλ2λ1δ(k2 − k1). (336)

Completeness of these eigenfunctions, including the restriction on k provided by a factor

θ(k · k0), where the theta function is defined as

θ(x) =







1 for x > 0

1

2

for x = 0

0 for x < 0

, (337)

follows from

X

κ,π→±

X

2

λ=1

Z

dk θ(k · k0) ψ

(κ,π)

k,λ (x2) ψ

(κ,π)†

k,λ (x1)

=

2

(2π)

3

Z

dk θ(k · k0)Π

T

(kˆ)(cos k · x2 cos k · x1 + sin k · x2 sin k · x1)

= Π

T

(∇2) δ(x2 − x1). (338)

Longitudinal parity and energy eigenstates are given by

ψ

(+,+)

k,0

(x) = 1

√

4π

3



ǫˆ0(kˆ) sin k · x

0



, (339)

ψ

(+,−)

k,0

(x) = 1

√

4π

3



ǫˆ0(kˆ) cos k · x

0



, (340)

ψ

(−,+)

k,0

(x) = 1

√

4π

3



0

ǫˆ0(kˆ) cos k · x



, (341)

ψ

(−,−)

k,0

(x) = 1

√

4π

3



0

ǫˆ0(kˆ) sin k · x



, (342)

42
where

Pψ

(κ,±)

k,0

(x) = ±ψ

(κ,±)

k,0

(x), (343)

Hψ

(κ,±)

k,0

(x) = 0. (344)

As for the transverse eigenfunctions, there is overlap between states with opposite signs of

k, so the same condition on k may be applied here, that is θ(k · k0) > 0, to eliminate the

redundancy. With this condition on k2 and k1, the orthonormality relation is

Z

dx ψ

(κ2,π2)†

k2,0

(x) ψ

(κ1,π1)

k1,0

(x) = δκ2κ1δπ2π1δ(k2 − k1).

(345)

In addition, the longitudinal parity eigenfunctions are orthogonal to the transverse parity

eigenfunctions. Completeness of the longitudinal parity eigenfunctions is given by

X

κ,π→±

Z

dk θ(k · k0) ψ

(κ,π)

k,0

(x2) ψ

(κ,π)†

k,0

(x1)

=

2

(2π)

3

Z

dk θ(k · k0)Π

L

(kˆ)(cos k · x2 cos k · x1 + sin k · x2 sin k · x1)

= Π

L

(∇2) δ(x2 − x1). (346)

The combined completeness relation is thus

X

κ,π→±

X

2

λ=0

Z

dk θ(k · k0) ψ

(κ,π)

k,λ (x2) ψ

(κ,π)†

k,λ (x1) = Iδ(x2 − x1). (347)

7.8. Wave packets

The plane wave solutions considered in this section are not normalizable as ordinary

functions. Rather, integrals over products of solutions should be interpreted in the sense

of distributions or generalized functions, like the delta function [26]. That is, they provide

a well-defined value for an integral when they are included in the integrand together with

a suitable weight, or test function. However, the plane waves can provide the basis for

an expansion of a normalizable wave packet as a sum and integral over a complete set of

solutions of the Maxwell equation. If f

(κ)

λ

(k) is a suitable function, we write

Ψf (x) = X

κ λ

Z

dk f

(κ)

λ

(k) ψ

(κ)

k,λ(x), (348)

and Ψf is a solution of the Maxwell equation

γ

μ

∂μΨf (x) = X

κ λ

Z

dk f

(κ)

λ

(k) γ

μ

∂μ ψ

(κ)

k,λ(x) = 0. (349)

43
Further, Ψf will be normalized if f

(κ)

λ

is, because

Z

dx Ψ

†

f

(x)Ψf (x) = X

κ λ

X

κ′ λ′

Z

dk

Z

dk

′

f

(κ)∗

λ

(k)

Z

dx ψ

(κ)†

k,λ (x) ψ

(κ

′

)

k′,λ′(x)f

(κ

′

)

λ′ (k

′

)

=

X

κ λ

Z

dk





f

(κ)

λ

(k)







2

= 1. (350)

In view of the orthonormality (in the generalized sense) of the plane-wave solutions, Eq. (348)

may be inverted to give

f

(κ)

λ

(k) = Zdx ψ

(κ)†

k,λ (x)Ψf (x), (351)

where we have specified Ψf (x) = Ψf (x)|

t=0, and the time dependence of the wave function

is given by Eq. (348).

An example is a normalized photon wave packet which at t = 0 has (approximately) a

wave vector k0, a transverse polarization vector ǫˆ1(kˆ0), and a Gaussian envelope of length

a and width b centered at x = 0:

Ψf (x) = 1

a

1

2 b



2

π

3

1

4



ǫˆ1(kˆ0)

τ · kˆ0 ǫˆ1(kˆ0)



e

ik0·x

e

−(x2

k

/a2+x2

⊥/b2

)

, (352)

where xk = x· kˆ0 kˆ0 and x⊥ = x−xk. For a and b large compared to |k0|

−1

, the packet has

a functional form that resembles a positive-energy transverse plane wave. From Eq. (351),

we have

f

(κ)

λ

(k) = a

1

2 b

2



2

π

3

1

4

F

(κ)

λ

(kˆ) e−

h

(k0−κkk)

2

a

2/4+k2

⊥b

2/4

i

, (353)

where kk = k · kˆ0 kˆ0, k⊥ = k − kk, for λ = 1, 2,

F

(κ)

λ

(kˆ) = 1

2

ǫˆ

†

λ

(kˆ)



I + τ · k τ ˆ · kˆ0



ǫˆ1(kˆ0), (354)

and

F

(+)

0

(kˆ) = 1

√

2

ǫˆ

†

0

(kˆ) ǫˆ1(kˆ0), (355)

F

(−)

0

(kˆ) = 1

√

2

ǫˆ

†

0

(kˆ) τ · kˆ0ǫˆ1(kˆ0). (356)

The wave packet has small longitudinal components, because F

(κ)

0

(kˆ) is not necessarily

zero unless kˆ = ±kˆ0. It has negative-energy components, but they are also suppressed,

particularly as a, b → ∞, because for κ → −, the exponential factors in Eq. (353) favor k =

44
−k0, and for λ = 1, 2, F

(κ)

λ

(−kˆ0) = 0 as compared to F

(κ)

λ

(kˆ0) = δλ 1. Thus a cancellation

between the upper-three and lower-three components of the wave function suppresses the

contribution of negative-energy eigenstates to the wave packet.

The expectation value of the Hamiltonian H, Eq. (208), is

hΨf | H | Ψf i = −i ~c

Z

dxΨ

†

f

(x) α · ∇Ψf (x) = ~c|k0| = ~ω0, (357)

and the expectation value of the momentum P, Eq. (209), is

hΨf | P | Ψf i = ~k0. (358)

The initial probability density Q(x) is

Q(x) = Ψ

†

f

(x)Ψf (x) = 2

ab2



2

π

3

1

2

e

−2(x2

k

/a2+x2

⊥/b2

)

, (359)

with

Z

dx Q(x) = 1, (360)

and the initial energy density E(x) is

E(x) = Ψ

†

f

(x) H Ψf (x) = ~ω0Q(x) −

i

2

~ckˆ0 · ∇Q(x). (361)

The real part of the energy density is proportional to the probability density for the photon,

and the imaginary term, which vanishes upon integration to arrive at the expectation value

in Eq. (357), reflects the change in the initial probability density at the point x due to the

motion of the wave packet. At a fixed point in the path of the wave packet, the probability

density increases as the packet approaches and decreases after the maximum of the wave

packet has passed by. The time-dependent probability density is

Q(x) = Ψ

†

f

(x)Ψf (x) =



e

−i H t/~Ψf (x)





2

, (362)

and the change at t = 0 is

∂Q(x)

∂t







t=0

= −

i

~

Ψ

†

f

(x)



H − ←−H

†



Ψf (x)

=

2

~

ImΨ

†

f

(x)HΨf (x) = −ckˆ0 · ∇Q(x). (363)

The gradient produces a vector that points toward the maximum of the wave packet, so

that on the forward side of the packet kˆ0 · ∇Q(x) is negative and the probability density is

increasing, as expected. Eq. (363) also shows that the wave packet is initially moving with

velocity c in the direction of kˆ0.

45
The time dependence of the wave packet, Eq. (352) at t = 0, is given exactly by Eq. (348).

However, approximations may be made in order to obtain a more transparent expression.

In view of the exponential factors in Eq. (353), the assumption that a, b ≫ |k0|

−1

implies

k ≈ κk0, and

F

(κ)

λ

(kˆ) ≈ F

(κ)

λ

(κkˆ0) = δλ1δκ+, (364)

so that from Eq. (348), Ψf → Ψ′

f

, where

Ψ

′

f

(x) = a

1

2 b

8π

3

2



2

π

3

1

4

Z

dk



ǫˆ1(kˆ)

τ · kˆ ǫˆ1(kˆ)



e

−i(|k|ct−k·x)

e

−

h

(k0−kk)

2

a

2/4+k2

⊥b

2/4

i

. (365)

This is a normalized positive energy wave function with polarization ǫˆ1 that is an exact solution of the Maxwell equation γ

μ∂μΨ′

f

(x) = 0. Further simplifications are the replacements

kˆ → kˆ0 in the polarization vector matrix and |k| → k · kˆ0 in the exponent, which yield

Ψ′

f → Ψ′′f

, with

Ψ

′′

f

(x) = a

1

2 b

8π

3

2



2

π

3

1

4



ǫˆ1(kˆ0)

τ · kˆ0 ǫˆ1(kˆ0)

 Z

dk e

−i(k·kˆ0ct−k·x)

e

−

h

(k0−kk)

2

a

2/4+k2

⊥b

2/4

i

=

1

a

1

2 b



2

π

3

1

4



ǫˆ1(kˆ0)

τ · kˆ0 ǫˆ1(kˆ0)



e

−i(ω0t−k0·x)

e

−

h

(ct−kˆ0·x)

2

/a2+x2

⊥

/b2

i

, (366)

which is an approximate wave function with a normalized Gaussian probability distribution

Q

′′(x) = 2

ab2



2

π

3

1

2

e

−2

h

(ct−kˆ0·x)

2

/a2+x2

⊥/b2

i

, (367)

that moves with velocity c in the kˆ0 direction.

7.9. Conservation of probability

The formulation of the Poynting theorem in Sec. 4 can be reinterpreted here to demonstrate conservation of probability. We define the probability density four-vector to be

q

μ

(x) = Ψ(x)γ

μΨ(x), (368)

where in the previous section Q(x) = q

0

(x). For the source-free case, Ξ(x) = 0, Eq. (57) is

∂μΨ(x)γ

μΨ(x) = 0, (369)

which is the statement of conservation of probability

∂

∂t q

0

(x) + c ∇ · q(x) = 0. (370)

46
Applying this relation to plane-wave states gives consistent, although trivial, results. We

have

q

0

(x) = ψ

(κ)†

k,λ (x) ψ

(κ)

k,λ(x) = (2π)

−3

, (371)

which reflects the fact that the probability distribution for a plane wave is uniform throughout space and not normalizable. For transverse plane waves, λ = 1, 2, the probability flux

vector is

q(x) = ψ

(κ)†

k,λ (x) α ψ

(κ)

k,λ(x) = (2π)

−3kˆ, (372)

and for longitudinal plane waves q(x) = 0.

For the wave packet in Eq. (352), at t = 0

q(x) = kˆ0 q

0

(x), (373)

so Eq. (363) is essentially the conservation of probability equation evaluated at t = 0.

Conservation of probability is valid for any solution of the homogeneous Maxwell equation

from the definition of the probability density operator. However, it also happens to be valid

for the wave packet represented by Ψ′′

f

, which is not an exact solution of the wave equation.

The wave equation is not satisfied by Ψ′′

f

, because there is a small extra term resulting from

the gradient operator acting on the perpendicular coordinate x⊥. On the other hand, q(x)

for Ψ′′

f

is proportional to kˆ0 and kˆ0 · ∇ x⊥ = 0, so there is no corresponding extra term in

∇ · q(x).

8. Angular momentum eigenfunctions

Radiation emitted in atomic transitions is characterized by its angular momentum and

parity. In this section, wave functions that are eigenfunctions of energy, angular momentum,

and parity are given; they are also classified according to whether they are transverse or

longitudinal. The three-component angular-momentum matrices given here are to some

extent parallels of the three-vector functions of [27].

8.1. Angular momentum

The spatial angular-momentum operator is given by

L = x × p = −i ~ x × ∇, (374)

and following the example of the Dirac equation, the total angular momentum is given as a

3 × 3 matrix by [5]

J = L + ~τ , (375)

where it is understood that the first term on the right side is a 3 × 3 matrix with L for

diagonal elements and zeros for the rest.4(In order to adhere to convention, we denote

4

In some works, where electric and magnetic fields or the vector potential are three-vector valued fields,

the spin operator is represented by a cross product. For example, Corben and Schwinger [27] write JzΦ =

LzΦ+i ez ×Φ, where Φ is a vector potential, and in Edmonds [28], spin is represented symbolically as i eˆ×.

47
both the current three-vector and the angular-momentum matrix by J. In either case, the

meaning should be clear from the context.) The extension to a 6 × 6 matrix is

J =



J 0

0 J



= x × P + ~S (376)

and we have

[H, J ] = 0, (377)

so eigenfunctions of both energy and angular momentum may be constructed. The vanishing

of the commutator follows from the relations

[τ · ∇, L] = −i~τ × ∇, (378)

[τ · ∇, τ ] = iτ × ∇, (379)

[τ · ∇, J] = 0. (380)

It is of interest to see that J commutes with H only for the (relative) combination of L and

τ given in Eq. (375). To obtain eigenstates of the square of the total angular momentum

J

2

and the third component of angular momentum J

3

, given by

J

2 =



J

2 0

0 J

2



(381)

and

J

3 =



J

3 0

0 J

3



, (382)

we construct matrix spherical harmonics that are analogous to conventional vector spherical

harmonics and are three-component extensions of the Dirac two-component spin-angularmomentum eigenfunctions. Orthonormal basis matrices are given by

ǫˆ

(1) =





1

0

0



 , ǫˆ

(0) =





0

1

0



 , ǫˆ

(−1) =





0

0

1



 , (383)

and they satisfy the eigenvalue equations

τ

2

ǫˆ

(ν) = 2 ǫˆ(ν)

, (384)

τ

3

ǫˆ

(ν) = ν ǫˆ(ν)

, (385)

where the 2 may be regarded as s(s + 1), with s = 1 as the spin eigenvalue. The matrix

spherical harmonics are

Y

m

jl (xˆ) = X

ν

(l m − ν 1 ν|l 1 j m)Y

m−ν

l

(xˆ) ǫˆ

(ν)

, (386)

48
with vector addition coefficients and spherical harmonics in the notation of [28]. The spherical harmonics satisfy the eigenvalue equations

L

2Ym

l

(xˆ) = ~

2

l(l + 1) Y

m

l

(xˆ), (387)

L

3Ym

l

(xˆ) = ~ m Y m

l

(xˆ), (388)

and by their construction, the matrix spherical harmonics are eigenfunctions of the total

angular momentum:

J

2Ym

jl (xˆ) = ~

2

j(j + 1) Y

m

jl (xˆ), (389)

J

3Ym

jl (xˆ) = ~ m Y

m

jl (xˆ). (390)

Explicit expressions in terms of spherical harmonics are

Y

m

jj (xˆ) =





−

q(j+m)(j+1−m)

2j(j+1) Y

m−1

j

(xˆ)

√ m

j(j+1)

Y

m

j

(xˆ)

q(j−m)(j+1+m)

2j(j+1) Y

m+1

j

(xˆ)





, (391)

Y

m

jj+1(xˆ) =





q(j+1−m)(j+2−m)

(2j+2)(2j+3) Y

m−1

j+1 (xˆ)

−

q(j+1−m)(j+1+m)

(j+1)(2j+3) Y

m

j+1(xˆ)

q(j+2+m)(j+1+m)

(2j+2)(2j+3) Y

m+1

j+1 (xˆ)





, (392)

Y

m

jj−1

(xˆ) =





q(j−1+m)(j+m)

(2j−1)2j

Y

m−1

j−1

(xˆ)

q(j−m)(j+m)

(2j−1)j

Y

m

j−1

(xˆ)

q(j−1−m)(j−m)

(2j−1)2j

Y

m+1

j−1

(xˆ)





. (393)

These functions are orthonormal

Z

dΩ Y

m2†

j2l2

(xˆ) Y

m1

j1l1

(xˆ) = δj2j1δl2l1δm2m1, (394)

which follows from the relations

Z

dΩ Y

m2∗

l2

(xˆ) Y

m1

l1

(xˆ) = δl2l1δm2m1, (395)

ǫˆ

(ν2)†

ǫˆ

(ν1) = δν2ν1

, (396)

X

ν

(l 1 j2 m|l m − ν 1 ν)(l m − ν 1 ν|l 1 j1 m) = δj2j1, (397)

49
and they are complete

X

jlm

Y

m

jl (xˆ2)Y

m†

jl (xˆ1) = I δ(cos θ2 − cos θ1) δ(φ2 − φ1), (398)

based on the relations

X

j

(l m − ν2 1 ν2|l 1 j m)(l 1 j m|l m − ν1 1 ν1) = δν2ν1, (399)

X

ν

ǫˆ

(ν)

ǫˆ

(ν)† = I, (400)

X

lm

Y

m

l

(xˆ2)Y

m∗

l

(xˆ1) = δ(cos θ2 − cos θ1) δ(φ2 − φ1), (401)

where θi, φi are the spherical coordinates of xˆi.

An alternative set of matrix angular-momentum eigenfunctions is

Xjm

1

(xˆ) = 1

~

p

j(j + 1)

LsY

m

j

(xˆ), (402)

Xjm

2

(xˆ) = 1

~

p

j(j + 1)

τ · xLˆ sY

m

j

(xˆ), (403)

Xjm

3

(xˆ) = xˆsY

m

j

(xˆ). (404)

For j = 0, X00

1

(xˆ) = X00

2

(xˆ) = 0. From a comparison of Eqs. (391)-(393) to Eqs. (402)-

(404), one has

X

jm

1

(xˆ) = Y

m

jj (xˆ), (405)

X

jm

2

(xˆ) = −

s

j

2j + 1

Y

m

jj+1(xˆ) −

s

j + 1

2j + 1

Y

m

jj−1

(xˆ), (406)

X

jm

3

(xˆ) = −

s

j + 1

2j + 1

Y

m

jj+1(xˆ) + s

j

2j + 1

Y

m

jj−1

(xˆ). (407)

In view of the relations in Eqs. (405)-(407), the functions Xjm

i

(xˆ) are eigenfunctions of J

2

and J

3 with

J

2Xjm

i

(xˆ) = ~

2

j(j + 1) Xjm

i

(xˆ), (408)

J

3Xjm

i

(xˆ) = ~ m Xjm

i

(xˆ). (409)

This can be confirmed directly from the definitions in Eqs. (402)-(404) with the aid of the

commutation relations



L

i

, Lj



= i~ǫijkL

k

, (410)



L

i

, xj



= i~ǫijkx

k

(411)

50
and the tau matrix identities in Sec. 3, which provide the operator identities

J

iLs = LsLi

, (412)



J

i

, τ · xˆ



= 0, (413)

J

ixˆs = xˆsLi

. (414)

These functions are orthonormal

Z

dΩ Xj2m2†

i2

(xˆ) Xj1m1

i1

(xˆ) = δi2i1δj2j1δm2m1, (415)

and they are complete

X

ijm

Xjm

i

(xˆ2) Xjm†

i

(xˆ1) = I δ(cos θ2 − cos θ1) δ(φ2 − φ1), (416)

where the latter fact may be seen from the completeness of the matrix spherical harmonics

and the relation

X

i

Xjm

i

(xˆ2) Xjm†

i

(xˆ1) = X

l

Y

m

jl (xˆ2)Y

m†

jl (xˆ1). (417)

The parity of the eigenfunctions is given by

Xjm

1

(−xˆ) = (−1)jXjm

1

(xˆ), (418)

Xjm

2

(−xˆ) = (−1)j+1Xjm

2

(xˆ), (419)

X

jm

3

(−xˆ) = (−1)j+1X

jm

3

(xˆ), (420)

which follows from Y

m

j

(−xˆ) = (−1)jY

m

j

(xˆ).

8.2. Helicity eigenstates

Special cases of the transverse plane-wave solutions in Eq. (217) are circularly polarized

states with the polarization vectors in Eq. (216). They can be grouped with the longitudinal

plane-wave states in Eqs. (237) and (238) with the polarization vector in Eq. (236). These

polarization vectors are summarized here as

ǫˆ1(eˆ

3

)=





1

0

0



 ; ǫˆ0(eˆ

3

)=





0

1

0



 ; ǫˆ−1(eˆ

3

)=





0

0

1



 ,

where we have changed the label for λ from 2 to −1 for this section.

The states have a well-defined helicity; they are eigenfunctions of the operator for the

projection of angular momentum in the direction of the wave vector J · kˆ [5, 29]. In view

of the relations

L · kˆ e

±ik·x = 0, (421)

τ · kˆ ǫˆλ(kˆ) = λ ǫˆλ(kˆ) (422)

for the polarizations considered here, we have

J · kˆ ψ

(±)

k,λ (x) = λ~ ψ

(±)

k,λ (x) (423)

for these states.

51
8.3. Transverse spherical photons

Transverse spherical wave functions are given by

ψ

T(κ,+)

ω,jm (x) = 1

√

2





fω,j (r)Xjm

1

(xˆ)

−κi

c

ω

τ · ∇fω,j (r)X

jm

1

(xˆ)



 , (424)

ψ

T(κ,−)

ω,jm (x) = 1

√

2





c

ω

τ · ∇fω,j (r)Xjm

1

(xˆ)

κifω,j (r)Xjm

1

(xˆ)



 , (425)

where r = |x| and j ≥ 1. They are transverse because ∇†

s Ls = ∇†s τ · ∇ = 0, so that

Π

T

(∇) ψ

T(κ,π)

ω,jm (x) = ψ

T(κ,π)

ω,jm (x), (426)

Π

L

(∇) ψ

T(κ,π)

ω,jm (x) = 0. (427)

The wave functions are eigenfunctions of angular momentum with eigenvalues given by [see

Eq. (380)]

J

2 ψ

T(κ,π)

ω,jm (x) = ~

2

j(j + 1) ψ

T(κ,π)

ω,jm (x), (428)

J

3 ψ

T(κ,π)

ω,jm (x) = ~ m ψT(κ,π)ω,jm (x), (429)

and they are eigenfunctions of H, with eigenvalue κ~ω

H ψ

T(κ,π)

ω,jm (x) = −i ~c α · ∇ψ

T(κ,π)

ω,jm (x) = κ~ω ψT(κ,π)ω,jm (x), (430)

provided



∇2 +

ω

2

c

2



fω,j (r)Xjm

1

(xˆ) = 0, (431)

which is true if



1

r

∂

2

∂r2

r −

j(j + 1)

r

2

+

ω

2

c

2



fω,j (r) = 0. (432)

Solutions of Eq. (432) are spherical Bessel functions given by [30]

fω,j (r) ∝







jj (ωr/c)

h

(1)

j

(ωr/c)

. (433)

We employ the normalized solution

fω,j (r) = ω

c

r

2

πc

jj (ωr/c) (434)

52
for the wave functions; any other linear combination of spherical Bessel functions (with

j ≥ 1) is not integrable as r → 0. The parity of the wave functions is

P ψ

T(κ,+)

ω,jm (x) = (−1)j+1 ψ

T(κ,+)

ω,jm (x), (435)

P ψ

T(κ,−)

ω,jm (x) = (−1)j ψ

T(κ,−)

ω,jm (x). (436)

This provides the conventional parity and angular-momentum attributes for electric and

magnetic multipole radiation. Namely, ψ

T(κ,+)

ω,jm (x) is magnetic 2j-pole or Mj radiation and

ψ

T(κ,−)

ω,jm (x) is electric 2j-pole or Ej radiation.

Alternative forms for the lower three components in Eq. (424) or the upper three components in Eq. (425) are obtained by writing (see Appendix D)

τ · ∇ =

1

r

∂

∂r r τ · xˆ +

1

~r



Ls xˆ

†

s + xˆs L

†

s



, (437)

which yields

τ · ∇ fω,j (r)Xjm

1

(xˆ) = 1

r

∂

∂r r fω,j (r)Xjm

2

(xˆ) −

p

j(j + 1)

r

fω,j (r)Xjm

3

(xˆ). (438)

Relations among spherical Bessel functions provide

1

r

∂

∂r r fω,j (r) = ωc

1

2j + 1

[(j + 1)fω,j−1(r) − j fω,j+1(r)], (439)

1

r

fω,j (r) = ω

c

1

2j + 1

[fω,j−1(r) + fω,j+1(r)], (440)

which together with Eqs. (406) and (407) yield the second alternative form

c

ω

τ · ∇ fω,j (r)Xjm

1

(xˆ) = −

s

j + 1

2j + 1

fω,j−1(r) Y

m

jj−1

(xˆ) + s

j

2j + 1

fω,j+1(r) Y

m

jj+1(xˆ). (441)

This latter form is useful in calculating the wave function orthonormality and completeness

relations.

An analogous longitudinal function is obtained by writing [Eq. (D.6)]

∇s = xˆs

∂

∂r −

1

~r

τ · x Lˆ s (442)

and

ω

c

F

jm

ω

(x) = ∇s fω,j (r) Y

m

j

(xˆ)

=

∂

∂r fω,j (r)X

jm

3

(xˆ) −

p

j(j + 1)

r

fω,j (r)X

jm

2

(xˆ), (443)

53
so that

τ · ∇ F

jm

ω

(x) = 0. (444)

From the additional relation

∂

∂r fω,j (r) = ωc

1

2j + 1

[j fω,j−1(r) − (j + 1)fω,j+1(r)], (445)

together with Eqs. (406) and (407), one has

F

jm

ω

(x) = s

j

2j + 1

fω,j−1(r) Y

m

jj−1

(xˆ) + s

j + 1

2j + 1

fω,j+1(r) Y

m

jj+1(xˆ). (446)

The orthonormality of the transverse wave functions is given by

Z

dx ψ

T(κ2,π2)†

ω2,j2m2

(x)ψ

T(κ1,π1)

ω1,j1m1

(x) = δκ2κ1δπ2π1δj2j1δm2m1δ(ω2 − ω1), (447)

which takes into account the integral

Z ∞

0

dr r2fω2,j (r) fω1,j (r) = δ(ω2 − ω1). (448)

The completeness relation for the transverse wave functions is

Z ∞

0

dω

X

κπjm

ψ

T(κ,π)

ω,jm (x2)ψ

T(κ,π)†

ω,jm (x1) = Π

T

(∇2) δ(x2 − x1), (449)

which is shown in some detail by writing

X

κπ

ψ

T(κ,π)

ω,jm (x2)ψ

T(κ,π)†

ω,jm (x1) = 

S

jm

ω

(x2, x1) 0

0 S

jm

ω

(x2, x1)



, (450)

where

S

jm

ω

(x2, x1) = fω,j (r2)Xjm

1

(xˆ2)fω,j (r1)Xjm†

1

(xˆ1)

+

c

2

ω2



τ · ∇2fω,j (r2)Xjm

1

(xˆ2)

τ · ∇1fω,j (r1)Xjm

1

(xˆ1)

†

, (451)

and

S

jm

ω

(x2, x1) = Π T

s

(∇2)

h

S

jm

ω

(x2, x1) + F

jm

ω

(x2)F

jm†

ω

(x1)

i

= Π T

s

(∇2)

h

fω,j (r2)Y

m

jj (xˆ2)fω,j (r1)Y

m†

jj (xˆ1)

+fω,j−1(r2) Y

m

jj−1

(xˆ2)fω,j−1(r1) Y

m†

jj−1

(xˆ1)

+fω,j+1(r2) Y

m

jj+1(xˆ2)fω,j+1(r1) Y

m†

jj+1(xˆ1)

i

, (452)

54
which gives

Z ∞

0

dω

X

jm

S

jm

ω

(x2, x1) = Π T

s

(∇2)

1

r2r1

δ(r2 − r1)

×

X

jm

h

Y

m

jj (xˆ2)Y

m†

jj (xˆ1) + Y

m

jj−1

(xˆ2)Y

m†

jj−1

(xˆ1) + Y

m

jj+1(xˆ2)Y

m†

jj+1(xˆ1)

i

= Π T

s

(∇2) δ(x2 − x1), (453)

based on

Z ∞

0

dω fω,j (r2) fω,j (r1) = 1

r2r1

δ(r2 − r1). (454)

8.4. Longitudinal spherical photons

Longitudinal spherical wave functions are

ψ

L(+)

k,jm(x) = 1

k

 

0

∇s gk,j(r)Y

m

j

(xˆ)

!

, (455)

ψ

L(−)

k,jm(x) = 1

k

 ∇s gk,j(r)Y

m

j

(xˆ)

0

!

. (456)

It follows from the identity τ · ∇ ∇s = 0 that they are longitudinal

Π

L

(∇) ψ

L(π)

k,jm(x) = ψ

L(π)

k,jm(x), (457)

Π

T

(∇) ψ

L(π)

k,jm(x) = 0 (458)

and that they are eigenfunctions of H, with eigenvalue 0

H ψ

L(π)

k,jm(x) = 0, (459)

with no condition on g. However, in order to have a complete set of longitudinal wave

functions, a set of functions, indexed by the parameter k is specified here. The form of the

plane-wave longitudinal solutions in Eqs. (237) and (238) and of the expansion of a plane

wave in spherical waves suggest the choice

gk,j(r) = k

r

2

π

jj (kr) (460)

for the radial wave function, where k is a free parameter. This set of functions provides an

infinite orthonormal set of degenerate (ω = 0) basis functions for each j.

From the form of ∇sin Eq. (442) and the fact that Xjm

2

(xˆ) and Xjm

3

(xˆ) are eigenfunctions of angular momentum, it follows that the longitudinal spherical wave functions are

also eigenfunctions of angular momentum

J

2 ψ

L(π)

k,jm(x) = ~

2

j(j + 1) ψ

L(π)

k,jm(x), (461)

J

3 ψ

L(π)

k,jm(x) = ~ m ψL(π)k,jm(x). (462)

55
They have parity given by

P ψ

L(+)

k,jm(x) = (−1)j+1 ψ

L(+)

k,jm(x), (463)

P ψ

L(−)

k,jm(x) = (−1)j ψ

L(−)

k,jm(x). (464)

With spherical Bessel functions for the radial wave functions, we have, following Eqs. (442)

to (446),

∇s

k

gk,j(r)Y

m

j

(xˆ) = s

j

2j + 1

gk,j−1(r) Y

m

jj−1

(xˆ) + s

j + 1

2j + 1

gk,j+1(r) Y

m

jj+1(xˆ). (465)

This identity facilitates calculation of the orthonormality relation for the longitudinal wave

functions, which is

Z

dx ψ

L(π2)†

k2,j2m2

(x)ψ

L(π1)

k1,j1m1

(x) = δπ2π1δj2j1δm2m1δ(k2 − k1), (466)

with

Z ∞

0

dr r2gk2,j (r) gk1,j (r) = δ(k2 − k1). (467)

The completeness relation for the longitudinal wave functions is

Z ∞

0

dk

X

πjm

ψ

L(π)

k,jm(x2)ψ

L(π)†

k,jm (x1) = Π

L

(∇2) δ(x2 − x1), (468)

which follows from

X

π

ψ

L(π)

k,jm(x2)ψ

L(π)†

k,jm (x1) = 

T

jm

k

(x2, x1) 0

0 T

jm

k

(x2, x1)



, (469)

where

T

jm

k

(x2, x1) = −

∇2 s∇†

1 s

∇2

2

gk,j (r2)gk,j(r1)Y

m

j

(xˆ2)Y

m∗

j

(xˆ1), (470)

and

∇2gk,j (r)Y

m

j

(xˆ) = −k

2

gk,j(r)Y

m

j

(xˆ). (471)

Thus from

Z ∞

0

dk gk,j(r2)gk,j (r1) = 1

r2r1

δ(r2 − r1), (472)

56
we have

Z ∞

0

dk

X

jm

T

jm

k

(x2, x1) = Π L

s

(∇2)δ(x2 − x1), (473)

which provides Eq. (468). That result, together with Eq. (449) gives the full completeness

relation.

As an illustration of a role of the spherical functions, we revisit the example of a point

charge at the origin, for which

Ψp(x) = −

q

4πǫ0

 

∇s

1

r

0

!

. (474)

In view of the integral

Z ∞

0

dk

1

k

gk,0(r) = r

π

2

1

r

, (475)

one has

Ψp(x) = −

q

√

2 πǫ0

Z ∞

0

dk ψL(−)

k,00 (x), (476)

which is the analog, for spherical solutions, of Eq. (248) for plane wave solutions.

9. Maxwell Green function

A solution of the Maxwell equation for the electric and magnetic fields Ψ(x) given a

specified current source Ξ(x), as they are related in Eq. (52)

γμ∂

μΨ(x) = Ξ(x),

can be found with the aid of the 6 × 6 matrix Maxwell Green function DM(x2 − x1), given

by

DM(x2 − x1) = X

2

λ=0

Z

dk ψ

(+)

k,λ (x2)ψ

(+)

k,λ(x1) θ(t2 − t1)

−

X

2

λ=0

Z

dk ψ

(−)

k,λ (x2)ψ

(−)

k,λ(x1) θ(t1 − t2),

(477)

where ψ

(±)

k,λ (x) is given by Eq. (217), (237) or (238), and Eq. (251). In view of the relations

γμ∂

μ

2 ψ

(±)

k,λ (x2) = 0, (478)

γμ∂

μ

2

θ(t2 − t1) = γ0 δ(ct2 − ct1), (479)

γμ∂

μ

2

θ(t1 − t2) = −γ0 δ(ct2 − ct1) (480)

57
and the completeness of the wave functions, we have

γμ∂

μ

2 DM(x2 − x1) = I δ(x2 − x1) (481)

and

DM(x2 − x1) γμ

←−∂

μ

1 = −I δ(x2 − x1), (482)

where

δ(x2 − x1) = δ(ct2 − ct1) δ(x2 − x1). (483)

In terms of the Green function, a solution for the electric and magnetic fields is

Ψ(x2) = Zd

4

x1 DM(x2 − x1) Ξ(x1), (484)

as is confirmed by the application of γμ∂

μ

2

to both sides. In Eq. (484) d4x1 = c dt1dx1. A

separation into transverse or longitudinal solutions may be made by restricting the sum over

polarizations to λ = 1, 2 for a transverse solution or λ = 0 for a longitudinal solution.

The Maxwell Green function also may be written as an integral over the four-vector k

of the plane-wave solutions. For this it is useful to make the separation into transverse and

longitudinal components. For the transverse part we have

D

T

M(x2 − x1) = X

2

λ=1

Z

dk ψ

(+)

k,λ (x2)ψ

(+)

k,λ(x1) e−iω(t2−t1)

θ(t2 − t1)

−

X

2

λ=1

Z

dk ψ

(−)

k,λ (x2)ψ

(−)

k,λ(x1) eiω(t2−t1)

θ(t1 − t2), (485)

and we employ the identities

e

−iω(t2−t1)

θ(t2 − t1) = i

2π

Z ∞

−∞

dk0

e

−ik0(ct2−ct1)

k0(1 + iδ) − ω/c, (486)

−e

iω(t2−t1)

θ(t1 − t2) = i

2π

Z ∞

−∞

dk0

e

−ik0(ct2−ct1)

k0(1 + iδ) + ω/c, (487)

where the limit δ → 0

+ for the integral is understood. We also have

α · k ψ

(±)

k,λ (x2) = |k| ψ

(±)

k,λ (x2) = ω

c

ψ

(±)

k,λ (x2). (488)

Together these relations yield [see Eq. (228)]

D

T

M(x2 − x1) = i

2π

X

2

λ=1

Z

d

4

k



e

−ik0(ct2−ct1)

k0(1 + iδ) − α · k

ψ

(+)

k,λ (x2)ψ

(+)

k,λ(x1)

58
+

e

−ik0(ct2−ct1)

k0(1 + iδ) + α · k

ψ

(−)

k,λ (x2)ψ

(−)

k,λ(x1)



=

i

(2π)

4

Z

d

4

k e

−ik0(ct2−ct1)

e

ik·(x2−x1)

γ

0k0(1 + iδ) − γ · k



(τ · kˆ)

2 0

0 (τ · kˆ)

2



=

i

(2π)

4

Π

T

(∇2)

Z

CF

d

4

k

e

−ik·(x2−x1)

γ

μkμ

, (489)

where d4k = dk0 dk, and CF indicates that the contour of integration over k0 is the Feynman

contour, which passes from −∞ below the negative real axis, through 0, and above the

positive real axis to +∞; this is equivalent to including the factor (1 + iδ) multiplying k0 in

the denominator and integrating along the real axis.

For applications, it is useful to consider an alternative form for the transverse Green

function. Taking into account the relation

Π

T

(kˆ)

1

γ

0k0(1 + iδ) − γ · k

= Π

T

(kˆ)

γ

0k0 − γ · k

k

2

0 − k2 + iδ

=



k0 (τ · kˆ)

2 −τ · k

τ · k −k0 (τ · kˆ)

2



1

k

2 + iδ

, (490)

we have, from Eq. (489),

D

T

M(x2 − x1) = i

(2π)

4

Z ∞

−∞

dk0 e

−ik0(ct2−ct1)



k0 Π T

s

(∇) i τ · ∇

−i τ · ∇ −k0 Π T

s

(∇)

 Z

dk

e

ik·r

k

2 + iδ

=

1

8π

2

i

Z ∞

−∞

dk0 e

−ik0(ct2−ct1)



k0 Π T

s

(∇) i τ · ∇

−i τ · ∇ −k0 Π T

s

(∇)



e

i(k

2

0+iδ)

1/2

|r|

|r|

→

1

8π

2

i

Z ∞

−∞

dk0 e

−ik0(ct2−ct1)

×



k0(τ · rˆ)

2 −(k2

0 + iδ)

1/2 τ · rˆ

(k

2

0 + iδ)

1/2 τ · rˆ −k0(τ · rˆ)2



e

i(k

2

0+iδ)

1/2

|r|

|r|

, (491)

where r = x2 − x1, the gradient ∇ is with respect to r, and the branch of the square

root in the exponent is determined by the condition Im(k

2

0 + iδ)

1/2 > 0, which specifies that

(k

2

0 + iδ)

1/2 → |k0| for real values of k0. In the last line of Eq. (491), higher-order terms in

(k0 |r|)

−1 are not included, but the exact expression follows from the formulas in Appendix

E.

For the longitudinal Green function, we write

D

L

M(x2 − x1) = Z

dk ψ

(+)

k,0

(x2)ψ

(+)

k,0

(x1) e−ǫ(ct2−ct1)θ(t2 − t1)

−

Z

dk ψ

(−)

k,0

(x2)ψ

(−)

k,0

(x1) eǫ(ct2−ct1)θ(t1 − t2), (492)

where damping factors with ǫ > 0 are added so that the Green function falls off for large

time differences. In addition, we assume that the longitudinal wave functions are solutions

59
of the Maxwell equation with an infinitesimal mass mǫ

included, as given in Eq. (259), in

order to be able to use the Feynman contour to specify the path of integration over k0 in

relation to the poles of the integrand. We employ the identities

e

−imǫc

2

(t2−t1)/~

e

−ǫ(ct2−ct1)

θ(t2 − t1) = i

2π

Z ∞

−∞

dk0

e

−ik0(ct2−ct1)

k0 + iǫ − mǫc/~

, (493)

−e

imǫc

2

(t2−t1)/~

e

ǫ(ct2−ct1)

θ(t1 − t2) = i

2π

Z ∞

−∞

dk0

e

−ik0(ct2−ct1)

k0 − iǫ + mǫc/~

(494)

and

γ

0 ψ

(κ)

k,0

(x2) = κ ψ(κ)

k,0

(x2), (495)

α · k ψ

(κ)

k,0

(x2) = 0 (496)

to obtain

D

L

M(x2 − x1) = i

2π

Z

d

4

k



e

−ik0(ct2−ct1)

k0 + iǫ − mǫc/~

ψ

(+)

k,0

(x2)ψ

(+)

k,0

(x1)

+

e

−ik0(ct2−ct1)

k0 − iǫ + mǫc/~

ψ

(−)

k,0

(x2)ψ

(−)

k,0

(x1)



=

i

2π

Z

CF

d

4

k

e

−ik0(ct2−ct1)

k0 − γ

0mǫc/~

X

κ→±

ψ

(κ)

k,0

(x2)ψ

(κ)

k,0

(x1)

=

i

(2π)

4

Π

L

(∇2)

Z

CF

d

4

k

e

−ik·(x2−x1)

γ

μkμ − mǫc/~

. (497)

Here the limit ǫ → 0 would be undefined without the mass term. A concise alternative

expression for the longitudinal Green function is obtained by the substitution of the partial

completeness relations that follow from Eqs. (243) and (244) into Eq. (492):

D

L

M(x2 − x1) = Π

L

(∇2) δ(x2 − x1)



I θ(t2 − t1) 0

0 I θ(t1 − t2)



. (498)

The transverse and longitudinal Green functions in Eqs. (489) and (497) differ only by

the type of projection operator and the infinitesimal mass term in Eq. (497). However, such

a mass term in the last line of Eqs. (489) would not change the relation of the path of

integration over k0 to the location of the poles of the integrand, so it could also be included

in that expression. In particular, the poles in Eq. (490) at k0 = ± (k

2 − iδ)

1/2 would move to

k0 = ± [k

2 + (mǫc/~)2 − iδ]

1/2

. These poles lie on curves in the second and fourth quadrants

of the complex k0 plane, whereas the Feynman contour passes through the first and third

quadrants. Thus, we may write DT

M(x2 − x1) + DLM(x2 − x1) = DM(x2 − x1), with

DM(x2 − x1) = i

(2π)

4

Z

CF

d

4

k

e

−ik·(x2−x1)

γ

μkμ − mǫc/~

. (499)

60
This result is a covariant Green function for the Maxwell equation which is of the same form

as the well-known Green function for the Dirac equation. A formal coordinate-representation

is

DM(x2 − x1) = 1

γμ∂

μ

2

δ(x2 − x1). (500)

The fields given by Eq. (484) represent a particular solution of the Maxwell equation.

Any solution of Eq. (484) for Ξ(x1) = 0, such as the field of a static charge distribution, may

be added to the particular solution, and the sum will be a solution with the same source

function. In fact, even if the three-vector current density vanishes in the distant past and

future, there could be a static charge distribution that persists, with or without a net total

charge, for which the fields would be non-zero indefinitely. To deal with this case, we obtain

an expression that takes into account the possible fields in the past and future by writing

the time derivative

∂

∂(ct1)

Z

dx1 DM(x2 − x1)γ0Ψ(x1) = Zdx1 DM(x2 − x1)γ0

←−∂

0

1Ψ(x1)

+

Z

dx1 DM(x2 − x1)γ0∂

0

1 Ψ(x1), (501)

and for fields that vanish for large space-like distances, we write

Z

dx1 ∇1 · DM(x2 − x1)γΨ(x1) = Zdx1 DM(x2 − x1)γ ·

←−∇1Ψ(x1)

+

Z

dx1 DM(x2 − x1)γ · ∇1Ψ(x1) = 0, (502)

where the result is zero, because it may be written as an integral over the bounding surface,

by the Gauss-Ostrogradsky theorem. The sum of Eqs. (501) and (502) is

∂

∂(ct1)

Z

dx1 DM(x2 − x1)γ0Ψ(x1)

=

Z

dx1 DM(x2 − x1)γμ

←−∂

μ

1 Ψ(x1) + Z

dx1 DM(x2 − x1)γμ∂

μ

1 Ψ(x1)

= −

Z

dx1 δ(x2 − x1)Ψ(x1) + Zdx1 DM(x2 − x1) Ξ(x1). (503)

Integration of Eq. (503) over t1 from ti to tf, where ti < t2 < tf, yields

Ψ(x2) = Zdx1 DM(x2 − x1)γ0Ψ(x1)







t1=ti

−

Z

dx1 DM(x2 − x1)γ0Ψ(x1)







t1=tf

+c

Z tf

ti

dt1

Z

dx1 DM(x2 − x1) Ξ(x1) (504)

61
or

Ψ(x2) = Zdx1

X

2

λ=0

Z

dk



ψ

(+)

k,λ (x2)ψ

(+)†

k,λ (x1)Ψ(x1)







t1=ti

+ ψ

(−)

k,λ (x2)ψ

(−)†

k,λ (x1)Ψ(x1)







t1=tf



+c

Z tf

ti

dt1

Z

dx1 DM(x2 − x1) Ξ(x1). (505)

As a consistency check of this expression, we note that it properly reduces to the expected

result for the field of a constant charge distribution. In this case, the current source term

vanishes, the initial and final fields are the same, and they are purely longitudinal. As a

result, only longitudinal functions with no time dependence will contribute to the sum over

states, which is just the longitudinal completeness relation, and Eq. (505) reduces to the

proper identity.

10. Applications of the Maxwell Green function

The Maxwell Green function is used here to calculate the radiation fields of a point

dipole source as an example of an application. Only the large distance transverse fields are

considered, and they are given by

Ψd(x2) = Zd

4

x1 D

T

M(x2 − x1) Ξd(x1), (506)

with the source term

Ξd(x) = 

−μ0c js(x) e−iωdt

0



. (507)

The classical source for dipole radiation is a charge q with position

xd(t) = x0 cos ωdt (508)

which produces a current density

jcl(x) = q δ(x − xd(t)) x ̇ d(t)

≈ −ωd d δ(x) sinωdt, (509)

where d = q x0. This is the real part of

j(x) = −i ωd d δ(x)e−iωdt, (510)

which is the source current for the radiation [17]. For the transverse Maxwell Green function,

we use the expression on the last line of Eq. (491). Integration over t1 yields a factor

2πδ(k0c − ωd), and evaluation of the integration over k0 follows. The result is

c

Z

dt1 D

T

M(x2 − x1) Ξd(x1)

= −

μ0c k

4πi



(τ · rˆ)

2 −τ · rˆ

τ · rˆ −(τ · rˆ)

2

  js(x1)

0



e

ik|r|

|r|

e

−iωdt2 + . . . , (511)

62
where k = ωd/c. Since the source is point-like at the origin x1 = 0, r = x2, and

Ψd(x) = k

2

4πǫ0



(τ · xˆ)

2 ds

τ · x dˆ s



e

ik|x|

|x|

e

−iωdt + . . . . (512)

The time-average differential radiated power, based on Eqs. (54) to (59) with a factor 1/2

from the time averaging [17] is

dId

dΩ

=

1

2

x

2 xˆ · S(x) = cǫ0

4

x

2 Ψd(x)γ · xˆΨd(x)

=

ck4

32 π

2ǫ0

d

†

s

(τ · xˆ)

2 ds =

ck4

32 π

2ǫ0



d

2 − (xˆ · d)2



, (513)

which is the well-known result.

A more realistic example is the radiation produced by a Dirac transition current, which

is given by

ΞD(x) = 

−μ0c j

if

s

(x)

0



=

 

2e

ǫ0

φ

†

f

(x) αs φi(x) e−iωif t

0

!

, (514)

where φi and φf are the initial and final hydrogen atom Dirac wave functions, here α is the

4 × 4 Dirac matrix, and

ωif =

Ei − Ef

~

(515)

is the frequency corresponding to the energy difference of the transition. The factor of 2

multiplying the matrix element accounts for the difference between a classical dipole moment

and the quantum mechanical dipole moment operator in Eq. (520). (See the footnote on p.

407 of [17].) We have

c

Z

dt1 D

T

M(x2 − x1) ΞD(x1)

=

ik

4πǫ0c



(τ · rˆ)

2 −τ · rˆ

τ · rˆ −(τ · rˆ)

2

  j

if

s

(x1)

0



e

ik|r|

|r|

e

−iωif t2 + . . . , (516)

where k = ωif /c. For distances far from the source atom, |x2| ≫ |x1|, kˆ ≈ rˆ ≈ xˆ2, and in

the exponent k|r| = k|x2| − k · x1 + . . . , which yields

ΨD(x2) = Zd

4x1 DT

M(x2 − x1) ΞD(x1)

=

ik

4πǫ0c

Z

dx1



(τ · kˆ)

2

j

if

s

(x1)

τ · k j ˆ if

s

(x1)



e

−ik·x1

e

ik|x2|

|x2|

e

−iωif t2 + . . . . (517)

63
The average radiated power is

dID

dΩ

=

1

2

x

2

2 xˆ2 · S(x2) = cǫ0

4

x

2

2 Ψ D(x2)γ · kˆΨD(x2)

=

k

2

32 π

2ǫ0c

Z

dx1 j

if†

s

(x1) eik·x1(τ · kˆ)

2

Z

dx

′

1

j

if

s

(x

′

1

) e−ik·x′

1

= ~ωif

αkc

2π

X

2

λ=1

Z

dx φ

†

i

(x) ǫˆλ(kˆ) · α e

ik·xφf (x)

Z

dx

′ φ

†

f

(x

′

) ǫˆλ(kˆ) · α e

−ik·x′

φi(x

′

),

(518)

where α = e

2/4πǫ0~c is the fine-structure constant. The radiated power integrated over

directions of the vector kˆ may be interpreted as ~ωifAif , where Aif is the radiative transition

rate for i → f, that is, the probability that the atom providing the source current makes a

transition from state i to state f in one second. This gives

Aif =

αkc

2π

Z

dΩk

X

2

λ=1

D

i







ǫˆλ(kˆ) · α e

ik·x





 f

EDf







ǫˆλ(kˆ) · α e

−ik·x







i

E

, (519)

which is the same as the relativistic radiative transition rate given by QED (see Appendix

F). In the dipole approximation eik·x → 1, hi |α| fi = i k hi |x| fi, which follows from the

identity [H, x] = [cα · p, x] = −i~cα, where H is the Dirac Hamiltonian, and integration

over kˆ yields the familiar result

Aif →

4αω3

if

3c

2

|hf |x|ii|2. (520)

11. Summary

In Eq. (52), two of the Maxwell equations, Eqs. (2) and (3), are written in the form of

the Dirac equation without a mass, but with the addition of a source term Ξ(x):

γ

μ

∂μΨ(x) = Ξ(x),

where the gamma matrices are 6 × 6 versions of the Dirac gamma matrices in Eq. (46), and

Ψ(x) = 

Es(x)

i cBs(x)

!

, Ξ(x) = 

−μ0cJs(x)

0

!

from Eqs. (47) and (51). The source-free version of this equation, with Ξ(x) = 0, can be

written as a Schr ̈odinger-like equation from Eq. (45) or (252)

i~

∂

∂tΨ(x) = HΨ(x), (521)

64
where the Hamiltonian, Eq. (208),

H = −i ~c α · ∇

is the analog of the Dirac Hamiltonian for the electron. The factors of ~ are not essential

here, but they are introduced to provide the conventional units of frequency and energy. As

with the Dirac wave functions, where all four components are necessary to describe an electron bound in an atom relativistically, all six of the components of the photon wave function

apparently are necessary to properly account for the space-time properties of electromagnetic fields. There are three polarization degrees of freedom, two for radiation and one for

electrostatic interactions, and relativistic covariance requires twice that many components.

Alternatively stated, six complex functions are necessary to describe the six components of

the electric and magnetic fields, and they are coupled by the Maxwell equation and Lorentz

transformations.

According to Eq. (521), as in Eq. (258), the time dependence of the solution is given by

Ψ(x) = e−iHt/~Ψ(x). (522)

The time-independent solutions may be expanded in eigenfunctions of the Hamiltonian with

eigenvalues En given by

HΨn(x) = EnΨn(x), (523)

where n is a set of parameters that characterize the state represented by the wave function.

For each eigenfunction, one has

Ψn(x) = e−iEnt/~Ψn(x). (524)

The eigenfunctions are orthonormal

Z

dx Ψ

†

n2

(x)Ψn1(x) = δn2n1, (525)

and they are complete

X

n

Ψn(x2)Ψ

†

n

(x1) = δ(x2 − x1). (526)

The state index n includes continuous variables, so Eq. (525) has delta functions in those

variables on the right-hand side, and the summation symbol in Eq. (526) includes integration

over those variables.

States considered in detail in this paper are propagating plane waves in Secs. 7.1 and

7.2, standing plane waves in Sec. 7.7, and angular-momentum eigenstates in Secs. 8.3 and

8.4. The propagating plane-wave states are eigenfunctions of the momentum operator in

Eq. (209)

P = −i ~ I ∇.

65
This operator commutes with the Hamiltonian, [H, P] = 0, and eigenstates of both energy

and momentum are given in Eqs. (217), (237), and (238). These plane-wave states are further

characterized by polarization vectors given in Eq. (210) or (235). Linear combinations of

the traveling plane waves, combined to give standing-wave parity eigenfunctions, are in

Eqs. (329), (330), and (339) to (342). The angular-momentum operator, Eq. (376), is

J = x × P + ~S,

where the spin matrix, Eq. (154), is

S =



τ 0

0 τ



,

and τ is given by Eqs. (15) to (17). One has [H, J ] = 0 in Eq. (377), and simultaneous

eigenfunctions of energy, angular momentum squared J

2

, third component of angular momentum J

3

, and parity are given in Eq (424), (425), (455), and (456). All three sets of

eigenfunctions listed above are shown to be orthogonal and complete, as in Eqs. (525) and

(526).

The eigenfunctions considered here are not normalizable wave functions. However, they

provide basis functions for the expansion of a normalizable wave packet, as discussed in

Sec. 7.8. For the sum

Ψf (x) = X

n

fnΨn(x), (527)

from the orthonormality of the eigenfunctions one has

fn =

Z

dxΨ

†

n

(x)Ψf (x) (528)

and

Z

dx Ψ

†

f

(x)Ψf (x) = X

n

|fn|

2 = 1 (529)

for suitably chosen fn. For the example of a Gaussian wave packet in Eq. (352), the expectation value of the Hamiltonian, in Eq. (357), is

hΨf | H | Ψf i = ~ω0,

where ω0 = c|k0| is the frequency corresponding to the wave vector k0 of the wave packet. It

is clear that Eq. (357) applies in more generality than just to the wave packet in Eq. (352). If

the Gaussian shape function were replaced by any normalized real function, the expectation

value of the Hamiltonian would still be exactly ~ω0. For the wave packet in Eq. (352), the

expectation expectation value of the momentum operator, Eq. (358), is

hΨf | P | Ψf i = ~k0,

66
and the expectation value of the projection of the angular momentum in the direction of the

wave vector is

D

Ψf





 J · kˆ0







Ψf

E

= ~ ǫˆ

†

1

(kˆ0) τ · kˆ0 ǫˆ1(kˆ0), (530)

where the result depends on the polarization state represented by ǫˆ1(kˆ0). For circular

polarization, Eq. (216), the expectation value is ±~, while for linear polarization, Eq. (215),

it is 0. The real part of the energy density is ~ω0 times the probability density

Re Ψ

†

f

(x)HΨf (x) = ~ω0Ψ

†

f

(x)Ψf (x) (531)

for the wave packet.

For any wave packet, represented by Ψf , the photon probability density four-vector

defined in Eq. (368) is

q

μ

f

(x) = Ψf (x)γ

μΨf (x), (532)

and the differential conservation of probability is given by Eq. (370)

∂

∂t q

0

f

(x) + c ∇ · qf (x) = 0. (533)

This is valid for any solution of the homogeneous Maxwell equation. By integrating over

a closed volume and converting the divergence to an integral of the normal component of

the vector over the surface, one obtains a statement of conservation of probability for the

volume. If the surface of the volume is taken to infinity in all directions, where the wave

function vanishes, this expression shows the time independence of the normalization of the

wave function

∂

∂t Z

dx Ψ

†

f

(x)Ψf (x) = 0. (534)

Of course, this does not give meaningful results for a plane wave, because in this case, the

probability density is constant over space and is not normalizable.

For electromagnetic fields and photons, Lorentz invariance is a necessary consideration.

In Secs. 6.4 and 6.5 it is shown that

γ

μ

∂μΨ

′

(x) = Ξ

′

(x),

where the primes indicate that the field and source have been transformed by either a rotation

or a velocity boost. For rotations represented by the vector u = θuˆ, the transformed field,

in Eq. (144), is

Ψ

′

(x) = R(u)Ψ(R

−1

(u) x),

where R(u) is the coordinate rotation operator in Eq. (103) and

R(u) = e−iS·u

67
in Eq. (153). The source term transforms in the same way. For velocity transformations,

corresponding to the velocity v = c tanh ζ vˆ, Eq. (165) is

Ψ

′

(x) = V(v)Ψ(V

−1

(v) x),

where V (v) is the coordinate velocity transformation operator in Eq. (123) and

V(v) = eζK·vˆ

in Eq. (169), where

K =



0 τ

τ 0



in Eq. (170). The transformation of the source term under a velocity boost is noteworthy.

In the presence of a non-zero source, the Maxwell equation is invariant, but the left- and

right-hand sides do not transform separately. As shown in Sec. 6.5, the derivatives acting

on the fields produce terms that combine with the original source term in such a way as to

produce the velocity transformed source term, even though it is the three-vector current. In

the absence of sources, the transformation reduces to a more conventional form.

In the case of photon wave functions, sources are taken to be absent and the wave

functions are solutions of the homogeneous Maxwell equation. The Lorentz transformations

of the plane-wave functions are explicitly shown in Secs. 7.5 and 7.6. As with the Dirac

equation for an electron, the eigenvalues in Eq. (523) may be either positive or negative,

and here they also may be zero, for both plane-wave and spherical-wave eigenfunctions. The

negative eigenvalues, which are associated with relativistic invariance, are necessary in order

to have a complete set of solutions satisfying Eq. (526). It is relevant to note that for the

six-component wave packet in Eq. (352), there is an interaction between the upper-three

components and lower-three components, evident in Eq. (354), that suppresses the role of

the negative energy states.

In Sec. 5, Eqs. (86) and (87), orthogonal transverse and longitudinal projection operators

are defined:

Π

T

(∇) = 

Π T

s

(∇) 0

0 Π T

s

(∇)

!

, Π

L

(∇) = 

Π L

s

(∇) 0

0 Π L

s

(∇)

!

,

where from Eqs. (70) and (71)

Π T

s

(∇) = (τ · ∇)

2

∇2

, Π L

s

(∇) = ∇s∇†

s

∇2

.

These operators commute with the Hamiltonian, the momentum operator, and the angularmomentum operator

[H, Π ] = [P, Π ] = [J , Π ] = 0, (535)

68
where Π represents either projection operator, Π

T(∇) or ΠL

(∇), so all the eigenstates

considered in this paper are classified as being either transverse or longitudinal, with

Π

T

(∇)Ψ

T

n

(x) = Ψ

T

n

(x), (536)

Π

L

(∇)Ψ

L

n

(x) = Ψ

L

n

(x), (537)

respectively. The transverse states describe radiation and have non-zero eigenvalues in

Eq. (523), while the longitudinal states correspond to electrostatic interactions, with an

eigenvalue of zero. An exception is that the longitudinal states may have a non-zero eigenvalue if a hypothetical mass term is considered, as discussed in Sec. 7.4. The projection

operators commute with rotations, but not, in general, with velocity boosts. However, as

shown in Sec. 7.6.1, the velocity transformed transverse plane-wave states are also transverse.

This corresponds to the fact that radiation may be treated relativistically independent of

electrostatic interactions. On the other hand, as shown in Sec. 7.6.2, the velocity transformed

longitudinal states have both longitudinal and transverse components, corresponding to the

fact that moving charges may excite radiative transitions. Both transverse and longitudinal

states are necessary in order to have a complete set, as in Eq. (526).

A solution of the inhomogeneous Maxwell equation may be obtained with the Maxwell

Green function, as discussed in Sec. 9. The Green function satisfies the equation

γμ∂

μ

2 DM(x2 − x1) = I δ(x2 − x1)

in Eq. (481), and a solution of the Maxwell equation is given by

Ψ(x2) = Zd

4x1 DM(x2 − x1) Ξ(x1)

in Eq. (484). It is shown, by summing over the complete set of plane-wave solutions, that

the Green function is

DM(x2 − x1) = i

(2π)

4

Z

CF

d

4

k

e

−ik·(x2−x1)

γ

μkμ − mǫc/~

in Eq. (499), which is the same form as the Dirac Green function, except that here it is a

6 × 6 matrix instead of a 4 × 4 matrix. In this equation, CF is the Feynman contour and

the infinitesimal mass is included to resolve an ambiguity in the longitudinal contribution.

In Sec. 10, applications of the Maxwell Green function are made, including a calculation of

radiation from a Dirac-electron current source. In this example the six-component Maxwell

formalism couples radiation to the Dirac current relativistically with a result that is the

same as the result of a calculation that starts from Feynman-gauge QED.

12. Conclusion

We conclude that the criteria for properties of a single-photon wave function proposed

in the introduction are met by the formalism described in the subsequent sections. In

69
particular, the example of a photon wave packet provides a normalizable solution of the wave

equation whose properties can be verified by explicit calculations. It yields the unanticipated

result that for virtually any probability distribution, under rather mild assumptions about

the form of the wave packet, the expectation value of the Hamiltonian is exactly h H i = ~ω0,

where ω0 is the frequency associated with the wave vector of the packet.

Appendix A. Velocity transformation of electromagnetic fields

The velocity transformation of electromagnetic fields is derived here without invoking

potentials for completeness. With the aid of the identity ∇⊤

c τ ̃ · cB = (∇ × cB)

⊤

c

, Eqs. (1)

and (2) may be written as

∇⊤

c Ec = μ0c

2

ρ, (A.1)

∂E⊤

c

∂ct − ∇⊤

c τ ̃ · cB = −μ0cJ

⊤

c

(A.2)

or

∂

⊤

c

gF = μ0J

⊤

, (A.3)

where

F =

1

c

 

0 −E⊤

c

Ec τ ̃ · cB

!

(A.4)

is the field tensor [see Eq. (38)] and

J =



cρ

Jc



. (A.5)

Since V (v) g V (v) = g, Eq. (A.3) is equivalent to

∂

⊤

c V (v) g V (v) F(x)V (v) = μ0J

⊤(x)V (v) (A.6)

or

∂

⊤

c

g V (v) F



V

−1

(v)x



V (v) = μ0J

⊤



V

−1

(v)x



V (v). (A.7)

Assuming the current transforms as a four-vector,

J

′

(x) = V (v) J



V

−1

(v)x



, (A.8)

Eq. (A.3) will be invariant if the field tensor transforms according to

F

′

(x) = V (v) F



V

−1

(v)x



V (v). (A.9)

70
Direct calculation yields5

F

′

(x

′

) = 1

c

 

0 −E′⊤

c

(x)

E′

c

(x) τ ̃ · cB′(x)

!

, (A.10)

where x

′ = V (v) x and

E

′

c = Ec cosh ζ − vˆc vˆ · E (cosh ζ − 1) − τ ̃ · vˆ cBc sinh ζ, (A.11)

cB ′

c = cBc cosh ζ − vˆc vˆ · cB (cosh ζ − 1) + τ ̃ · v Eˆ c sinh ζ. (A.12)

These relations are equivalent (up to the velocity sign convention) to the electric and magnetic field transformations in [17], and in the spherical basis they are

E

′

s = Es + (τ · vˆ)

2Es(cosh ζ − 1) + i τ · vˆ cBs sinh ζ, (A.13)

cB′

s = cBs + (τ · vˆ)

2

cBs(cosh ζ − 1) − i τ · v Eˆ s sinh ζ. (A.14)

The Cartesian transformation in Eqs. (A.11) and (A.12) can be written as

Ψ

′

c

(x

′

) = e−ζK ̃ ·vˆΨc(x), (A.15)

where

K ̃ =



0  ̃τ

−τ ̃ 0



(A.16)

and

Ψc(x) = 

Ec(x)

cBc(x)



. (A.17)

Similarly, the Cartesian form of the rotation transformation, corresponding to the operator

in Eq. (153), is

Ψ

′

c

(x

′

) = eS ̃·uΨc(x), (A.18)

where

S ̃ =



τ ̃ 0

0  ̃τ



. (A.19)

The components of the matrices in Eqs. (A.16) and (A.19) have the commutation relations

h

S ̃i, S ̃j

i

= ǫijk S ̃k, (A.20)

h

S ̃i, K ̃j

i

= ǫijk K ̃k, (A.21)

h

K ̃i, K ̃j

i

= −ǫijk S ̃k, (A.22)

characteristic of the Lie algebra of Lorentz transformations.

5The identity ǫijk = ǫljkvˆivˆl + ǫilkvˆjvˆl + ǫijlvˆkvˆl may be useful here.

71
Appendix B. Inverse Laplacian

For cases where the integral definition of the inverse Laplacian converges poorly, we use a

generalized definition that includes a damping factor to resolve ambiguity in the intermediate

steps of the calculation. From the equation



∇2 − ǫ

2

 e

−ǫ |x−x′|

|x − x′|

= −4πδ(x − x

′

), (B.1)

one has

1

∇2 − ǫ

2

δ(x − x

′

) = −

1

4π

e

−ǫ |x−x′|

|x − x′|

. (B.2)

Multiplication by f(x

′

) and integration over x

′ yields

1

∇2 − ǫ

2

f(x) = −

1

4π

Z

dx

′

e

−ǫ |x−x′|

|x − x′|

f(x

′

). (B.3)

We thus have, for example,

1

∇2 − ǫ

2

e

ik·x = −

1

k2 + ǫ

2

e

ik·x → −

1

k2

e

ik·x

, (B.4)

by direct calculation of the integral.

Appendix C. Coulomb matrix element

The calculation of the Coulomb field matrix element in Eq. (248) requires evaluation of

the integral

Z

dx

kˆ · x

|x|

3

e

−ik·x = limǫ→0

Z

dx e

−ǫ|x| kˆ · x

|x|

3

e

−ik·x

→ − i

4π

|k|

, (C.1)

which is defined with the convergence factor e−ǫ|x|. The inverse transformation requires the

integral

Z

dk

kˆ

|k|

e

ik·x = limǫ→0

Z

dk e

−ǫ|k| kˆ

|k|

e

ik·x

→ i

2π

2x

|x|

3

, (C.2)

which confirms the result that

−

i q

p

(2π)

3 ǫ0

Z

dk

1

|k|

ψ

(+)

k,0

(x) = q

4πǫ0|x|

3



xs

0



.

(C.3)

72
Appendix D. Separation of the transverse and longitudinal gradient operators

The transverse gradient operator τ · ∇ is separated into radial and angular parts by

writing

τ · ∇ = τ · x τ ˆ · x τ ˆ · ∇ + xˆs xˆ

†

s τ · ∇

=

∂

∂r τ · xˆ − τ · xˆ ∇s xˆ

†

s +

1

~r

xˆsL

†

s

(D.1)

where the line over the gradient operator indicates that it does not act on the unit vector

directly to the right. That term is

τ · xˆ ∇s xˆ

†

s = τ · xˆ



∇sxˆ

†

s +

1

r

xˆsxˆ

†

s −

1

r

I



= −

1

~r

Lsxˆ

†

s −

1

r

τ · xˆ. (D.2)

Thus

τ · ∇ =

1

r

∂

∂r r τ · xˆ +

1

~r



Ls xˆ

†

s + xˆs L

†

s



. (D.3)

Acting on Ls, the transverse gradient operator yields

τ · ∇ Ls =

1

r

∂

∂r r τ · x Lˆ s −

1

~r

xˆs L

2

, (D.4)

where L†

sLs = −L2

.

For the longitudinal gradient operator ∇s, the identity

∇ = xˆ

∂

∂r −

i

~r

xˆ × L (D.5)

provides

∇s = xˆs

∂

∂r −

1

~r

τ · x Lˆ s. (D.6)

Appendix E. Exact transverse Green function

The exact transverse Maxwell Green function follows from Eq. (491) together with the

relations

τ · ∇

e

−w|r|

|r|

= −w τ · rˆ

e

−w|r|

|r|



1 +

1

w|r|



(E.1)

and

1

∇2

e

−w|r|

|r|

= −

1

4π

Z

dx

1

|r − x|

e

−w|x|

|x|

= −

1

w2|r|



1 − e

−w|r|



, (E.2)

73
which yield

∇i∇j

∇2

e

−w|r|

|r|

=

r

i

r

j

|r|

2

e

−w|r|

|r|

+



3

r

i

r

j

|r|

2

− δij e

−w|r|

w|r|

2

+

e

−w|r|

w2|r|

3

−

1

w2|r|

3



(E.3)

and

Π T

s

(∇)

e

−w|r|

|r|

= (τ · rˆ)

2

e

−w|r|

|r|

+



(τ · rˆ)

2 − 2 rˆs rˆ†

s





e

−w|r|

w|r|

2

+

e

−w|r|

w2|r|

3

−

1

w2|r|

3



. (E.4)

Appendix F. Radiative decay in quantum electrodynamics

In QED, the radiative decay rate of an excited state may be obtained from the imaginary

part of the radiative correction to the energy level of that state

~

X

f

Aif = −2 Im(∆Ei), (F.1)

where the sum is over all states with a lower unperturbed energy. This gives a correction

to the level that, roughly speaking, results in an exponentially damped time dependence for

the population of the state:

|e

−i ∆E t/~

|

2

= e−

P

f Aif t

. (F.2)

For one-photon decays, the rate is included in the second-order self-energy correction to the

level. An expression derived from Feynman-gauge QED that includes some of the real part

and all of the imaginary part of this level shift in hydrogen-like atoms is given by [31]

∆Ei = −

α~

2

c

2

4π

2

Z

~ck<Ei

dk

1

k



δjl −

k

jkl

k2

 α

j

e

ik·x

1

H − Ei + ~ck − iδ

α

l

e

−ik·x



= −

α~

2

c

2

4π

2

Z

~ck<Ei

dk

1

k

X

2

λ=1

X

f

D

i







ǫˆλ(kˆ) · α e

ik·x





 f

E

×

1

Ef − Ei + ~ck − iδ

D

f







ǫˆλ(kˆ) · α e

−ik·x







i

E

, (F.3)

where H is the Dirac Hamiltonian. The integrand is real, except for the imaginary infinitesimal in the denominator, for which

Im 1

Ef − Ei + ~ck − iδ

→ πδ(Ef − Ei + ~ck) (F.4)

and hence

X

f

Aif =

X

f

αkc

2π

Z

dΩk

X

2

λ=1

D

i







ǫˆλ(kˆ) · α e

ik·x





 f

EDf







ǫˆλ(kˆ) · α e

−ik·x







i

E

, (F.5)

with the restriction 0 < Ef < Ei on the sum over f. The contribution to the decay rate

from each final state f coincides with Eq. (519) for the transition rate Aif .

74
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