Open Journal of Microphysics, 2011, 1, 41-52
doi:10.4236/ojm.2011.13008 Published Online November 2011 (http://www.SciRP.org/journal/ojm)
The Photon Wave Function
Joseph Cugnon
AGO Department, University of Liège, allée du 6 Août 17, bât. B5, B-4000 Liège 1, Belgium
Received September 13, 2011; revised October 15, 2011; accepted October 23, 2011
Abstract
The properties of a wave equation for a six-component wave function of a photon are re-analyzed. It is
shown that the wave equation presents all the properties required by quantum mechanics, except for the ones
that are linked with the definition of the position operator. The situation is contrasted with the three-
component formulation based on the Riemann-Silberstein wave function. The inconsistency of the latter with
the principles of quantum mechanics is shown to arise from the usual interpretation of the wave function.
Finally, the Lorentz invariance of the six-component wave equation is demonstrated explicitly for Lorentz
boosts and space inversion.
Keywords: Photon, Wave Function, Quantum Mechanics, Lorentz Invariance
1. Introduction
The wave function of a photon is a topic that has for long
been ignored since the physicists have been primarily
interested in emission and absorption processes, for
which solid theories, such as the Glauber theory, exist.
But in the last two decades, an increased interest has
been focused on the description of a single photon by a
wave function. Two difficulties have shown up. The first
one deals with the problem of the proper wave equation.
The second one is linked with the definition of the posi-
tion operator. Largely irrespective of the proper wave
equation, it seems difficult to define a position operator
as usual, which, at the same time, satisfies the commuta-
tion relations with the total angular momentum dictated
by Poincaré symmetry. It should however be mentioned
that acceptable wave functions have been shown to de-
scribe a photon with good localization properties.
In this paper, we focus on the first question mentioned
above, namely the wave equation. It has been realized
during the last years, that the wave equation should be
similar, or at least consistent, with the Maxwell equa-
tions. But the proper form of the wave equation is still
controversial. Furthermore, for some choices, the wave
function does not fulfill basic principles of quantum me-
chanics. Here, we study a particular wave equation and
show it to be consistent with the requirements of quan-
tum mechanics and to present good symmetry properties.
It nevertheless presents the same problems as regarding
the definition of the position operator.
2. The Wave Equation
2.1. Introduction
The systematic search for a relativistic wave equation for
the photon has been undertaken by several authors and in
particular in References [1-7], to restrict oneself to the
recent years. Without entering into details, the wave
equation should be linear in the time derivative and as-
sume the canonical form
i
H
t



 (1)
where the Hamiltonian H is linear in the momentum p
and is such that plane wave solutions are consistent with
the relativistic dispersion relation. More precisely, H
should be a square root of the operator p2c2 in the space
of the components, like the Dirac Hamiltonian is a square
root of the operator p2c2 + m2c4 in the space of the Dirac
spinors. On the other hand, the number of components of

is not fixed a priori. It is reasonable to admit that it
should be chosen as low as possible, for the sake of
simplicity.
As a guide to determine the minimum number of com-
ponents, it is instructive to look at other simple exam-
ples. For massive boson particles with spin zero,

has
two components and H can be taken as

J. CUGNON 42
22
22
2
22
2
2
pp
mc
m
H=
pp
mc
mm

 






2m



(2)
This is the well-known two-component formulation of
the Klein-Gordon equation [8], linear in the time deri-
vative. The m = 0 limit is not very much instructive,
since it is singular. On the other hand, this equation
suggests that an equation for a spin one boson should
have more than two components. For a massive spin 1/2
particle, 

has four components and H has the Dirac
form
2.
D
H
cmc   αp (3)
Although it applies to fermions, this equation strongly
suggests that the Hamiltonian H of Equation (1) should
be taken as a linear and homogeneous form in the mo-
mentum p. In the m = 0 limit, the Dirac equation reduces
to the Weyl equation, which shows that the appropriate
number of components for massless spin 1/2 particles is
two. So, again, the minimum number of components for
the photon wave function seems to be three. This choice
have been made by several authors. The form of H is
then practically uniquely determined and the equation is:
ψiic
t
 
pψ (4)
The Hamiltonian
H
=ic

p
is hermitian because the
operator is anti-hermitian. This can be
seen from the following property:
ip
 
k
kk
k
i
==p
x





kk
ψHψHψ



(5)
where the matrices Hk are given explicitly by
123
00 00 01010
00 1,000,100
01 010000 0
=








HHH
(6)
See Appendix A for details. In the following, we will
use the short-hand notation

ii
i
=pH pH (7)
the presence of indicating automatically 3 × 3 ma-
trices in the space of the components.
H
The wave Equation (4) is compatible with the criteria
enunciated above, namely that the square of H is equal to
p2c2. This however requires some restriction on the wave
functions. Indeed, the square of the Hamiltonian matrix
is given, with the help Equation (64) of Appendix A, by


2
222 2
1
ii
i
icicpHpcc






p
Hpp (8)
where 1 is the 3 × 3 unit matrix and where the second
term is the direct product of
p
by itself. This operator
reduces to the p2c2 operator, if the wave functions are
restricted to divergenceless functions, satisfying
=0

ψ

or, equivalently, =ψ0.p
Equation (4) corresponds to the Maxwell equations, as
motivated and discussed below. However, this is only
true for a special and criticizable choice of the wave
function, as explained later.
2.2. An Equation with Good Quantum
Mechan i cal Featu re s
The proposed equation involves a six component wave
function that we will write as
1
2
, 




 (9)
1

, 2

both having three components and being
divergenceless. The wave equation is given by:
11
22
0
0
c
i
c t
  


  
 
 

pH
pH 



, (10)
where all elements of the matrix on the right-hand side
are 3 × 3 matrices. The non-vanishing 3 × 3 matrices are
anti-hermitian, but the big 6 × 6 matrix, which is nothing
but the Hamiltonian, is hermitian. It is hermitian in the
space of the components and in the space of the nor-
malizable functions 1

and 2, due to the presence of
the momentum operator. Equation (10) has also been
proposed by Wang et al. [9,10], Bialynicki-Birula [3]
and others1. Here we study in some detail the quantum
and symmetry properties of this wave equation.

2.3. Quantum Properties
We are going to show that Equation (10) has good quan-
tum mechanical properties, except for the ones that are
linked with the position operator.
1) Hilbert space. Like for Equation (4), it is necessary
to restrict the functions 1

and 2 to be divergence-
less in order to ensure H2 = p2c2. This is easily verified
using Equation (8). The total configurational Hilbert
space is thus the direct product of two similar Hilbert
spaces built on normalizable divergenceless functions in
configurational space. These are perfect Hilbert spaces.
A possible (limiting) basis is provided by familiar plane
waves with transverse polarization.

1E. Majorana seems to have proposed this choice for the first time
[11].

J. CUGNON43
2) Probability density and current. It is easy to check,
by the usual method, that one can define a probability
density
2
12
 
2
(11)
and a probability current


*
12
2 c j

 (12)
which satisfy the continuity equation. Details are given
in Appendix B.
3) Phase of the wave function. If

given by Equa-
tion (9) is a solution of Equation (10), then
i
ψ'=ψe

(13)
where

is a real constant, is also a solution. This re-
sults from the linearity of the equation. Transformation
(13) leaves the density and current of probability invari-
ant, as it should. It is of interest to notice that the wave
equation (10) is real, in the sense that, besides the func-
tions 1 and 2, it involves real operators and real
coefficients. Indeed, owing to Equation (5), it can be
written as
 
12
21
1
1
=
ct
=
ct






(14)
The other remarkable property, which is shared by the
Weyl equation, is that has disappeared. 
4) Plane wave solutions and energy eigenvalues. Let
us consider solutions of Equation (10) of the type

i.ωt
ik.x
ψ=ae =aekr

, (15)
where is a six-component column vector,
1
2
a=


a
a
with both 1 and 2 a orthogonal to . Inserting form
(15) into Equation (10), one gets the eigenvalue problem
expressed by
a k

0.
ca
c








kH
kH (16)
The eigenvalues are

= 0, kc, −kc, with k=k,
each doubly degenerate. A set of eigenvectors can be
constructed as follows. Let us define
0
L==
k

kk, (17)
1
ε any real unit vector orthogonal to k
0
11
0, 1=εkεε
1

1
L
2
,
2







2
(18)
and
0
2
=εkε. (19)
The eigenvectors can taken as:
0: ,,
L
LL
aa 
 
 
 

 

εε
εε
(20)
1
21
:,ω=kc aa
 

 

 

εε
εε
(21)
1
21
:,ω=kc aa
 
 
 

 

εε
εε
. (22)
This is easily verified using Equation (10). The eigen-
vectors for ω = 0 are just formal solutions of Equation
(16). If one requires these eigenvectors to be orthogonal
to , there is simply no eigenvector. This result corre-
sponds to the fact that the photon cannot have a longitu-
dinal polarization. The positive eigenvalues correspond
to two independent transverse polarizations. It is easy to
check that the various solutions are orthogonal to each
other. The negative eigenvalues correspond to propaga-
tion in the opposite direction. In general, negative eigen-
values are associated with antiparticles. The fact that
negative eigenvalues solutions simply duplicate the posi-
tive ones correspond to the fact the photon is identical to
its antiparticle or, equivalently, has no antiparticle.
k
Alternatively, one can consider circular polarization.
A set of eigenvectors is then provided, for the ω = ±kc
eigenvalues, by
:,
:,
ω=kc a=a
ii
ω=kc a=a
ii




 

 

 
 

 

 


εε
εε
εε
εε
(23)
with
1


εεε (24)
The orthogonality between different solutions i and
j is to be understood in the sense of the following sca-
lar product
a
a


*.
ijij ij
a,a=aa =δ.
5) Spin and helicity operators. The spin operator can
be taken as
0.
0
i
=i



H SH (25)
We remind that the Hk’s are antihermitian operators.
Actually, the operators iHk form the adjoint representa-
tion of the SU(2) algebra (see Equation (71)) and are
thus the natural representation for spin one (3 compo-
nents). Like for the Dirac equation, the operator does
not commute with the Hamiltonian in Equation (10), but
S
=+
J
LS does, as it can be easily verified (=

Lrp).

J. CUGNON 44
Actually,
(26)
with

(27)
The operator S2, which is equal to 2, owing to
Eq
rator

one has

,H=L

,H icSα
0.
0




H
H

2
uation (69), obviously commutes with the Hamil-
tonian.
The ope
0
0
i
=i






pH pS
p
H (28)
commutes with the Hamiltonian, as well as with the
momentum operator
p
. Therefore the eigenvectors of H
can be taken as eigeectors of nv
p
and of the helicity
operator
.h= 
 
p
S
p
p (29)
Actually, this is case for the eigenvectors of H built on
th
.4. The Velocity “Operator”
et us consider the velocity “operator” defined by the
e vectors of Equation (23). They are eigenvectors of
the helicity operator with eigenvalues h = 1, −1, 1, −1,
respectively. Here again, the negative eigenvalue solu-
tions duplicate the positive ones.
2
L
relation

d,.
d
i
=H
t
rr (30)
For the Hamiltonian defined in the wave Equation (10),
one has
0
d,
0 d=c t
 H
rα
H (31)
where

is given by Equation (27), which tells that

ofis the vcity operator in units of c. The eigenvalues
the operators i
α, i = 1,2,3, are equal to 0 (with no di-
vergenceless enfunctions) and ±1. The eigenfunctions
are the same as in Equations (21, 22) (or as in Equation
(23) for the ±1 energy eigenvalues, as can be checked
directly, despite the fact that
elo
ige

and H do not commute.
This strange property is alsolated to the fact that the
i
α operators do not commute and therefore one cannot
ild an eigenstate of 1
α which is an eigenstate of 2 α
at the same time. In oth words, a precise measurem
of the x-component of the velocity is incompatible with a
precise measurement of the y-component. A similar di-
fficulty exists for the Dirac equation. Incidentally, we
mention that, like for the Dirac equation, Equation (31) is
equivalent to
re
bu
er ent
3
dd,
d= t

r
j
r (32)
where
j
is the probability current defined by Equation
on-existence of
th
e
.5. Interp r e tation of the Wa ve Func t i on
is often argued that if the photon can be viewed as “an
relationship first for the wave Equa-
tio
(12). We will not elaborate any further on this point,
which is beyond the scope of this note.
These difficulties are linked with the n
e position operator for the photon. There is a large lit-
erature on this point (see [3] for a review). Heuristically,
the non-existence of the position operator derives simply
from the fact that the multiplication by r of a diver-
genceless function does not yield a divergnceless func-
tion. Therefore the position operator has no eigenvalue.
2
It
elementary excitation of the quantized electromagnetic
field” and as a particle at the same time, its wave func-
tion should be related to the (average) electric and mag-
netic fields that it carries, i.e. with physical electro-
magnetic fields.
We discuss this
n (4). We will closely follow here the arguments of
Reference [12]. The three-component Equation (4) may
be written
ic
t






 (33)
Assuming
=+iψ
E
B (34)
where
E
unct i ons2, one gets after and
B
are rea l f
tioepsubstitun and saration of real and imaginary parts,
1,
1,
ct
ct






E
B
B
E


(35)
which are nothing but the Maxwell equations for free
transverse fields. According to the authors of Reference
[12], the Maxwell equations provide a “correct relativis-
tic, quantum theory of the light quantum”. It is thus
tempting to interpret, as did the authors of this work,
E
and
B
as the electric and magnetic fields. This int
pretan raises a certain number of problems. First,
Equations (35) derive from the wave Equation (33) only
er-
tio
2This object, where E and Bare the electric and magnetic fields, is
known under the name of the Riemann-Silberstein wave function. Let
us remind however, that this object has been introduced by these
authors [13,14], long before the advent of quantum mechanics.

J. CUGNON45
if the fields
E
and
B
are taken real. This is incom-
patible with th princie of the invariance of the wave
function under a global phase shift, if the electromag-
netic fields
e pl
E
and
B
have a physical reality, because
they are changed by this phase shift. Second, the two
circular polarization plane waves solutions require two
different identifications of the components of their wave
function ψ (one with
E
and one with
B
as the real
part of ψ Finally, the wve function ψi  ). a
E
B does
not havee right properties under spac if the
real components are considered as physical electroma-
gnetic fields. Indeed, electric fields change sign, whereas
magnetic fields do not, and the Hamiltonian in Equation
(4) commutes with the parity operator.
These difficulties do not appear in ou
the irsion,
r fulatio
nve
ormn. If
the 3-vectors 1  and 2  are identified with the elec-
tric and magnetic fieldsspectively, the Equation (14)
above are also identical to the Maxwell equations for free
transverse fields. There is however an important differ-
ence in the two approaches. The functions 1  and 2
, re

need not to be real. If they need to be related physi
fields, it is sufficient to consider the latter as the real (or
imaginary) part of 1  and 2 , as it is customary in the
harmonic representn of ssical electromagnetism.
Applying a global phase shift to the wave function (9)
merely corresponds to introducing a constant phase shift
in the physical fields (at least for plane waves) and the
physical reality attached to these fields is preserved.
Furthermore, the plane wave solutions exhibit automati-
cally the two helicities in our formulation, without the
identification of the components with electric and mag-
netic fields. Finally, our wave function (9) has the right
properties under space inversion, as described in Section
4.
T lts of
of t
to
resu this
yhe electro
cal
at
tio
io
s
n th
cla
hesideratione main note
an
enat the dualit-
m
e cons are th
d show that if the wave function is to be related with
physical electric and magnetic fields, the Raymer and
Schmidt formulation (Equations (33,34)) does not seem
to be consistent with quantum mechanics, contrarily to
our formulation.
Let us also m
agnetic equations, namely that the equations are in-
variant under the substitution 
E
B, 
B
E is ful-
filled by both our wave eqanaymer-
Schmidt one. The mere inspection of Equation (10) shows
that they are invariant under the substitution 12 
uation de R th

,
2
. As for the wave Equation (33), the s
m Equation (35), but is obtained by consi-
dering ψ and iψ as solutions, if the real and imagi-
nary pa of th wave functions are identified with
physical electric and magnetic fields, respectively.
Finally, let us discuss a little bit the normalizati
duality i
evident fro
rts ese
on of
the wave function, since it interferes with the identifica-
tion of the wave function with electromagnetic fields. Let
us start with the probability density, which is given by
Equation (11) in our formalism, provided the wave func-
tion is properly normalized as


2


2
3
12
d1.=r (36)
The direct identification of to 1 
E
(and of 2

to
B
) is not possible since these quantis do not ha
theme dimension. If 1
tie ve
sa

and 2  are to be related
with physical electromagnic fieldat least the identi-
fication mentioned above should be corrected by a con-
stant factor. Indeed, the energy of the photon in state ψ
(Equation (9)) can be identified to the energy of t
corresponding electromagnetic field. One should then
require:
ets,
he


22
3
12
d,
8π
c=H
r (37)
where the rhs is the quantum average of the Hamiltonian.
This is a conserved quantity for a free photon. It can also
be written as
.H= iψψ
t

  (38)
Therefore it may be more appropriate to make the
identification
12
.
8π8π
cc
,
H
H

EB
 (39)
The same considerations apply to the Raymer and
Sc
arguments against the interpreta-
tio
hmidt formulation.
There are however
n of the components of the wave function as con-
nected to the physical electric and magnetic fields at-
tached to the photon. First of all, any physical quantity
attached to the (free) photon should be associated to a
Hermitian operator. But the only operators available in
the quantum mechanics of a photon are linked with the
translational and spin degrees of freedom and have no
relation with electromagnetic properties. Strictly speak-
ing, the electromagnetic field quantum mechanical op-
erators are defined in the Fock space of field theory and
have no effect in the Hilbert space of a single photon
(like there is no operator linked to the electric field of the
electron in the quantum mechanics of a single electron).
Actually, the average value of the electric field or the
magnetic field is vanishing. In face of these considera-
tions, one may wonder whether the consistency between
the photon wave equation and the Maxwell equations
should not be interpreted differently, considering the
components of the wave function as merely behaving as
classical electromagnetic fields. They fulfill the same

J. CUGNON 46
n, some of the criticisms
to
Maxwell equations, transform in the same way under
Lorentz transformations (see below) and their equations
are invariant under dual transformations. In other words,
they may simply be objects with the same mathematical
properties as the classical electromagnetic fields, but
devoid of physical electromagnetic properties. They, of
course, keep their physical meaning concerning the
quantum probability density.
Within this new interpretatio
the wave Equation (4) disappear. Let us consider a
specific solution ψ. According to the previous interpre-
tation, attaching se physical meaning to the real and
imaginary part, the solution i ψe
om

where

is a real
constant, is not equivalent to ince the components
are changed. On the contrary, in the new interpretation,
the real and imaginary parts of the wave function have
no physical meaning, but just behave as classical elec-
tromagnetic fields, both ψ and i ψe
ψ, s

are acceptable
solutions of Equation (33).hey hoer correspond to
the same quantum mechanical reality (probability density
and current).
Twev
3. Invariance under Transformations of the
he wave Equation (10) acting in the space of diver-




Hamiltonian
T
genceless 1  and 2  has a special property: there are
infinite numrs of Hiltonians equivalent to the one of
Equation (10), i.e. having the same solutions. Indeed it
suffices to add any term generating the divergence of
1
 and/or 2 . For instance the substitution
123
ap apap

be am
0000
0000
0
cc
c
c

 










p
Hp
pH
pH
H
(40)
where a is an arbitrary constant, generates an equivalent
Hamiltonian. The transformed Hamiltonian does not look
Hermitian, but since the line containing the p-operators
gives a vanishing contribution in the space of divergen-
celess 1  and 2 , this Hamiltonian is automatically
Hermit It has actly the same matrix elements as the
original Hamiltonian in the space of divergenceless3 1
ian. ex

and 2 . In fact, one can add a similar set of operators
any l of any of the four submatrices to generate equi-
valent Hamiltonians. Of course, it is much easier to drop
all terms generating a divergence of 1  or 2 . Never-
theless, this property will be used to diuss t Lorentz
invariance in the next Section. Generalizing the argu-
ment, the Hamiltonians
63
to
ine
sc he
66
3
11 14
,
kiki kik
k=i=k= i=
i
H
'=H+apU+bpU

  (41)
where the Uik are the 6 × 6 matrices defined in Equation
variance
ariance of the wave equa-
(66), are equivalent to the original Hamiltonian, i.e. they
have the same eigenvalues and the same (divergenceless)
eigenfunctions.
. Lorentz In4
.1. Introductory Remark4
he question of the Lorentz invT
tion (10) may appear as a trivial issue, since the compo-
nents 1

and 2

behave as classical electromagnetic
fields. Some pois however need a clarification. First,
Equation (10) is restricted to divergenceless functions
and the condition for vanishing divergence has no obvi-
ous Lorentz covariance property. Second, 1  and 2
nt

are associated in a 6-vector and not in a tenslike or
F


,
e. Iwhich is the usual basis to discuss Lorentz invarianct
would then be desirable to prove explicitly the Lorentz
invariance of the wave equation. Here below, we restrict
ourselves to show explicitly the Lorentz invariance for a
Lorentz boost and for space inversion, following closely
the method ordinarily used for the Dirac equation.
.2. Lorentz Invariance for a Boost 4
e start with the remark that the wave EquWation (10) can
be cast, after multiplication on the left by the a non-
singular 0

matrix, in the following Dirac form (for a
massless particle)
0
μ
μ
γψ=, (42)
with

(43)
Note that these gamma matrices are 6 × 6 matrices.
Th
0.
0
0i
i
i
H
=H




γ 
e 0

matrix is arbritrary, except that its square should
be equal to the identity matrix. It is tempting to take
010

.
01
=


γ (44)
The i

matrices are then given by

(45)
Using these matrices and momentum operators, the
wave Equation (42) can be written as:
0H

.
0
i
i
i
=H




3Note however that one has to be careful when applying the Hamil-
tonian on the bras; then one has to use the Hermitian conjugate of the
6 × 6 matrix in Equation (65).

J. CUGNON47
01
02
10,
1
pc
cp




 

pH
pH

 (46)
which, with

0
p
=i,t is equivalen
(10). The gamma matrices introduced here do not follow
ommutation re
t to Equation
the same anticlations as the Dirac matrices.
The reason is that the square of the operator in Equation
(42) is not equal to p2, but to an operator which reduces
to p2 for divergenceless functions. Presumably, the
gamma matrices (43) are not unique and Equation (42) is,
like the Dirac equation, independant of the representation.
We did not investigate this point.
We shall not attempt to derive the Lorentz invariance
in general. Following the method described in Reference
[15], we will verify this property for one particular
transformation, namely a boost along the z axis. Ac-
cording to this reference, it is sufficient to show that
there exists a matrix S, relating the wave functions in the
two different frames by


S, which is such that if

is a solution of Equation (42),

 is solution of the
wave equation written in tormed frame:
0.
μ
u
γψ'=

 (47)
This equation may be rewritten as
he transf
0,
μν
γaSψ=
μν
where
(48)
ν
μ
a
e usu
is the Lorentz transform
ing thal method, it is thus su
ist
However, according to the discu
Section, one must admit in th
Eq
where the dots stand for such terms, an
giving vanishing contributions in the w
ation matrix. Follow-
fficient to show that
there exs a matrix S satisfying
.
μν ν
μ
γa= γS (49)
ssion of the preceding
e right hand side of
uation (48), additional terms, which give a vanishing
contribution for divergenceless 1  and 2 , and the
previous condition becomes
μν ν
μν ν
γa=γS (50) +
d any other terms
ave equation.
We merely show that the following matrix

3
3
1
1
=
H
H





 








S



(51)
satisfies these requirements. In this equation, –β is the
velocity of the primed frame with respect to the un-
primed one. The corresponding Lorentz transformation
matrix in Equation (50) is given by
00
0100
0010
=
00










a (52)
We leave the detail of the calculat
We collect the results for the operator of the lhs of
Equation (48) from Equations (75-78):
ion in Appendix C.



64 165266 3
31 132233
apSpUp UpUp
Up UpUp
 
 


 

 
 
3
33035 1342
66062161 2
1
1
UpUp Up
UpUpUp


  
 
(53)
where the 6 × 6 matrices Uab have all vanishing el
except for the one at the crossing of line a and column b,
which is equal to one. It is then very easy to see that the
ements
second and third terms of the rhs of the last equation
gives vanishing contributions when applied to diver-
genceless 1

and 2

. It can also easily be seen that
the terms proportional to (γ − 1) are simply the z-com-
ponents of the two Equations (14), giving thus also a
vanishing ctributio One then recovers the Equations
(46) or the Equations (14), except that the third and sixth
equations are multiplied by (γ − 1).
There is no secret beyond the matrix (51). It is nothing
but the matrix expressing the transformation of classical
transverse electromagnetic fields form one frame to the
ot
on n.
her:
,S

 

 

 
E
E
B
B (54)
see [16]. However, we consider here wave functions and
thus we have to take care of some
requirements. First, Equation (10) acts on divergenceless
quantum mechanical
functions. Therefore, one has to verify that this trans-
formation preserves the vanishing divergence of the trans-
formed components 1

and 2 . Once again, we limit
the demonstration to the Lorentz boost described by Eq-
uation (52). In this case, one has, with








111
123
,,

1




2222
123
,: 






11 2131
123
   

 

 
 

 
11 211 31
123 3
1221
12 21
131
33
11 2131
123
12 222
213
)(
t
t
t
t
 
 
 


 
 




2
(55)
In the second line, the explicit form of S (Equation

J. CUGNON 48
(51)) has been used. The last parenthesis vanishes, by
virtue of Equation (35) or Equation (14). This shows that
is divergenceless if is divergenceless. The simi-
roperty for ised on exactly the same way.
at this noan that any divergen
tion in a f ismatically divergenceless in
other framhvation of Equation (55), ex-
licit use of thveation (10) (or the Maxwell
tions), link
1

lar p
Note th
func
any
p
equa
1

obtain
t me
auto
e deri
Equ
and
2

does
rame
e. In t
e wa
ing
celess
1
2

, has been made.
Let us notice that the transformation


S, with S
given by Equation (51) is not a unitary transformation
and therefore does not conserve the norm of the wave
function. This, of course, is consistent with the fact that,
for a classical electromagnetic field (
E
,
B
), the quantity

22
33
E
+B is an invariant, whereas its integral (over
coordinate r in one case and coordinate

r in the
other case) is not invariant under a Lorentz transfor-
mation (along the z-direction). In other words, if one has
normalized the wave function as in Equation (36) in a
givsformed en frame, the wave function tranthrough

 S (with S given by Equation (51)) in another
frame does not automatically fulfill Equation (36) in this
new frame.
4.3. Loren Invariance for Space Inversion
It is interesting to consider the Lorentz transformation
corresponding to space inversion. Again, we have to find
a matrix
tz
P
S which satisfies4
,a




 
PP
SS (56)
where ν
μ
a is
1000
0100




00 10
=


a
00 01

It is easy to see that
(57)
P
S is the 6 × 6 matrix
10
01
=



P
S


etame result as for the Dirac equation (ex-
cept that in the latter case
(58)
We g the s
P
S
ince o
c equat
electrom
is a 4 × 4 matrix). Here
also, there is no surprise, sur Equation (42) has the
same structure as the Diraion.
tion is behaving like anagnetic wave, space in-
version leaves unchanged and flips the sign of
Since the wave func-
.
5.
We have re-analyzed the prop
for a six-component wave function of a photon. This
w
era
f f t
ve equation and wave function pro-
We have shown that the diffi-
ropagation of
W
ium
co, 4-7 January 1994.
the Wave Function of the Pho-
olf Ed., Progress in Optics XXXVI, Elsevier, Amster-
irula, “Exponential Localization of Pho-
tons,” Physical Review Letters, Vol. 80, No. 24, 1998, pp.
1
 2

Conclusions
erties of a wave equation
ave equation and this wave function have been already
proposed in the past by sevl authors. The purpose of
this work was a careful analysis of the quantum and
invariance properties of the formalism. We have shown
that the properties of the latter are more consistent with
the principles oquantum mechanics than those ohe
three-component wa
osed in Reference [12]. p
culty for the latter choice comes from the interpretation
of the wave function as an observable electromagnetic
field. If this interpretation is abandoned, the incon-
sistency of the formalism of Reference [12] with quan-
tum mechanics disappears, except for the problems
which are linked with the position operator, that survive
in our formulation as well. Let us however mention that
the three-component wave equation does not admit plane
wave solutions

.iωt
ψ= ekr a with a 3-vector a for
real ω5. That is the reason why the introduction of
polarization is not natural in the wave Equation (4) or
(33).
We have also demonstrated explicitly the Lorentz in-
variance of the six-component wave equation for Lorentz
boosts and space inversion.
Let us finally mention that the results are rather trans-
parent for free photon solutions in vacuum. As under-
lined in [3], the real interest of the formalism lies in the
treatment of the p photons in media.
6. Acknowledgments
e are very thankful to Dr. Jean-René Cudell for in-
teresting discussions and a careful reading of the manu-
script.
7. References
[1] V. V. Dvoeglazov, “2(2S + 1)-Component Model and Its
Connection with Other Field Theories,” XVII Sympos
on Nuclear Physics, Mexi
2] I. Bialynicki-Birula, “On [
ton,” Acta Physica Polonica A, Vol. 86, 1994, pp. 97-
116.
[3] I. Bialynicki-Birula, “Photon Wave Function,” In: E.
W
dam, 1996.
4] I. Bialynicki-B[
4Since the wave Equation (42) is homogeneous and linear in the
gamma matrices, one can require in general that , where
μν ν
μ
γa=γSA
A
is a non-singular matrix: then implies . For
the invariance under a Lorentz boost, discussed earlier, the matrix A
has been implicitly taken equal to the unit matrix.
0
μ
μ
γψ=A0
μ
μ
γψ=5This pro
p
erty is mathematically consistent with the fact that plane
wave solutions of the Maxwell equations correspond to electric and
magnetic components with a phase shift of 90 degrees.

J. CUGNON 49
5247-5250. doi :10. 1103 /Ph ysRev Lett. 80 .52 47
[5] J. E. Sipe, “Photon Wave Functions,” Physical Review A,
Vol. 52, No. 3, 1995, pp. 1875-1883.
do i:10. 1103 /P hysRev Lett. 80 .52 47
[6] D. H. Kobe, “A Relativistic Schrödinger-Like Equation
for a Photon and Its Second Quantization,” Foundations
of Physics, Vol. 29, No. 8, 1999, pp. 1203-1231.
doi:10.1023/A:1018855630724
[7] M. Hawton, “Lorentz-Invariant Photon Number Density,”
Physical Review A, Vol. 78, No. 1, 2008, Article ID 012111.
doi:10.1103/PhysRevA.78.012111
[8] H. Feshbach and F. Villars, “Elementary Relativistic
. 1, 1958, pp. 24-
Wave Mechanics of Spin 0 and Spin 1/2 Particles,” Re-
views of Modern Physics, Vol. 30, No
45. doi:10.1103/RevModPhys.30.24
[9] Z.-Y. Wang, C.-D. Xiong and O. Keller, “The First-
Quantized Theory of Photons,” Chinese Physics Letters,
Vol. 24, No. 2, 2007, pp. 418-420.
doi:10.1088/0256-307X/24/2/032
ished Research Notes
in, 2009.
[10] Z.-Y. Wang, C.-D. Xiong and O. Keller, “Photon Wave
Mechanics,” Chinese Physics Letters, Vol. 24, No. 2,
2007, arXiv:quant-ph/0511181v5.
[11] S. Esposito, E. Recami, A. Van der Merwe and R.
Battiston, “Ettore Majorana: Unpubl
on Theoretical Physics,” Springer, Berl
doi:10.1007/978-1-4020-9114-8
[12] M. G. Raymer and B. J. Smith, “The Maxwell Wave
Function of the Photon,” SPIE Conference Optics and
rtiellen
ematischen Physik Nach
rundgleichungen in
Photonics, San Diego, August 2005.
[13] H. Weber, “Notes on Differential Equations by Riemann,”
In: Riemann and G. F. Berhard, Eds., Die Pa
Differential-Gleichungen der Math
Riemann’s Vorlesungen, Friedrich Vieweg und Sohn, Bruns-
wick, 1901, p. 348.
[14] L. Silberstein, “Elektromagnetische G
Bivectorieller Behandlung,” Annalen der Physik, Vol. 22,
1907, p. 579. doi:10.1002/andp.19073270313
[15] J. J. Sakurai, “Advanced Quantum Mechanics,” Addison-
Wesley Publication Co., London, 1967.
[16] J. D. Jackson, “Classical Electrodynamics,” John Wiley,
New York, 1975, pp. 552-554.

J. CUGNON 50


Appendix A: Properties of the Hi Matrices
The matrices
123
00 00 01010
001,0 00,1 0 0
010100 000








HHH
(59)
possess the following property:

i ijk
jk =

H (60)
Therefore, the scalar product of an arbitrary vector
with is a 3 × 3 matrix, whose matrix elements are
given by
a
H

ljk l
jk l
=ε

aH a (61)
If this matrix multiplies a column vector , the j-th
element of the resulting column vector is given by
b
  
.
jlkl kj
jlk
=εab= 

aHba b (62)
One can thus write

.=aHbab (63)
The square of is a 3 × 3 matrix, whose ele-
ments are given by

aH





2
2.
jm mk
jk m
l ljmll'jmjkjk
ll'
=
=aεa'ε=δ+a a




aHaH aH
a
(64)
The matrices i have also some other remarkable
properties. The product of two of them , for
is given by
H
ij
HH
ij

.
ijijik jl
kl kl
U

 HH (65)
i.e., for instance, H1H2 has all vanishing elements, but
the element (1,2), which is equal to one. Note that, in this
paper, the matrices Uij are defined by

.
ik jl
kl

 
ij
U (66)
even for i = j. The squares of the matrices are given
by
i
H
21
ii
=H (67)
where 1i is the unit 3 × 3 matrix in which the i-th diagonal
element has been replaced by 0. More generally,
414 24 34
,1, ,
nn nn
ii
iiii ii
 1,

 HHHHHH (68)
and
21.=

HH (69)
Properties (65) and (67)are summarized formally in


ijikmjmlij klikjl
kl m
=

   

HH (70)
Finally, the matrices Hi satisfy the following com-
mutation relations
,
ijijk k
= 



HH H. (71)
Thus, the matrices iHj, j = 1,2,3, are the adjoint repre-
sentation of the SU(3) algebra.
Appendix B: Probability Density and
Current
Starting with Equation (14), multiying the first line on
the left by *
1

, and the second line by and sum-
ming, one readily gets
*
2

 
**
112 2
**
122 .
tt
cc







1

 
(72)
Summing this equation and its complex conjugate
leads to


 

22 **
12 122 1
**
122
.
cc
t
cc



 



1
(73)
Using



 
, abba ab one readily
gets



22 **
12 1221
..cc
t



 (74)
Appendix C: Demonstration of
Equation (53)
Let us consider the left hand side of Equation (50) for ν =
0. Using Equations (52) and (51), one has

J. CUGNON 51













33
03
33
1
1
1
1
HH
=
HH
=



 
 


 

 
 






























S
0
33 66
1
1
1
1
(1)(1),UU


 












 (75)
where Uij is the 6 × 6 matrix defined in Equation (66),
whose elements are vanishing, except the one at the cross-
ing of line i and column j, which is equal to one.
For ν = 1, 2, one has successively

13
1
13
1
31 64
1
1
1
11
1
11
1
1
1
HH
HH
UU






 
 
 








































 
S

35 62
1,UU   (76)

J. CUGNON 52





















 
23
2
23
2
32 653461
1
1
1
11
1
11
1
11.
1
HH
HH
UUU U






 
 
  

 
 

 
 

 
 

 
 
 

 
 
 
 
 

 
 

 
 
 





 








S
(77)
Finally, one has, for ν = 3,

33
03
33
1
1
1
1
HH
HH
 


 


 
 

 
































S
3
33 66
1
1
.
1
1
UU
  














 (78)