Proca Equations and the Photon Imaginary Mass

Proca Equations and the Photon Imaginary Mass

Fran De Aquino
Maranhao State University, Physics Department, S.Luis/MA, Brazil.
Copyright © 2011 by Fran De Aquino. All Rights Reserved

It has been recently proposed that the photon has imaginary mass and null real mass. Proca equations are
the unique simplest relativistic generalization of Maxwell equations. They are the theoretical expressions
of possible nonzero photon rest mass. The fact that the photon has imaginary mass introduces relevant
modifications in Proca equations which point to a deviation from the Coulomb’s inverse square law.

Key words: Quantum electrodynamics, Specific calculations, Photons
PACS: 12.20.-m, 12.20.Ds, 14.70.Bh.

For quite a long time it has been
known that the effects of a nonzero photon
rest mass can be incorporated into
electromagnetism through the Proca
equations [1-2]. It is also known that
particles with imaginary mass can be
described by a real Proca field with a
negative mass square [3-5]. They could be
generated in storage rings, jovian
magnetosphere, and supernova remnants.
The existence of imaginary mass associated
to the neutrino is already well-known. It has
been reported by different groups of
experimentalists that the mass square of the
neutrino is negative [6]. Although the
imaginary mass is not a measurable amount,
its square is [7]. Recently, it was shown that
an imaginary mass exist associated to the
electron and the photon too [8]. The photon
imaginary mass is given by ( ) ( )12
3
2 ichfm =γ
This means that the photon has null real
mass and an imaginary mass, , expressed
by the previous equation.
γm
Proca equations may be found in many
textbooks [9-11]. They provide a complete
and self-consistent description of
electromagnetic phenomena [12]. In the
presence of sources ρ and jr , these
equations may be written as (in SI units)
( )
( )
( )
( )5
4
30
2
2
000
2
0
A
t
E
jB
t
B
E
B
E
rrrr
rr
r
r
γ
γ
μεμμ
φμε
ρ
−∂
∂+=×∇
∂
∂−=×∇
=⋅∇
−=⋅∇

where hcmγγμ = , with the real variables
γμ and . However, according to Eq. (1)
is an imaginary mass. Then,
γm
γm γμ must be
also an imaginary variable. Thus, is a
negative real number similarly to .
Consequently, we can write that
2γμ
2γm
( )6
3
42
3
4 2
2
2
22
2
rk
cm =⎟⎠
⎞⎜⎝
⎛== λ
πμ γγ h
whence we recognize λπ2=rk as the real
part of the propagation vector k
r
;
( )722 irir kkikkkk +=+== r
Substitution of Eq. (6) into Proca
equations, gives
( )
( )
( )
( )11
3
4
10
90
8
3
4
2
000
2
0
Ak
t
E
jB
t
B
E
B
kE
r
r
rrrr
rr
r
r
−∂
∂+=×∇
∂
∂−=×∇
=⋅∇
−=⋅∇
εμμ
φε
ρ

In four-dimensional space these
equations can be rewritten as
( )12
3
41
0
2
2
2
2
2 μμ μ jAk
tc
r
r−=⎟⎟⎠
⎞
⎜⎜⎝
⎛ −∂
∂−∇
where and μA μj
r
are the 4-vector of
potential ( )ciA φ, and the current density ( )ρicj ,r , respectively. In free space the above
equation reduces to

( )130
3
41 2
2
2
2
2 =⎟⎟⎠
⎞
⎜⎜⎝
⎛ −∂
∂−∇ μAk
tc
r

2
which is essentially the Klein-Gordon
equation for the photon.
Therefore, the presence of a photon in
a static electric field modifies the wave
equation for all potentials (including the
Coulomb potential) in the form
( )14
3
41
0
2
2
2
2
2
ε
ρφ −=⎟⎟⎠
⎞
⎜⎜⎝
⎛ −∂
∂−∇ rk
tc
For a point charge, we obtain
( ) ( ) ( )15
4
1 3
2
0
rk
e
r
q
r
r−= πεφ
and the electric field

( ) ( ) ( ) ( )16
3
2
1
4
3
2
2
0
rk
erk
r
q
rE
r
r
−
⎥⎦
⎤⎢⎣
⎡ += πε

Note that only in the absence of the photon
the expression of reduces to
the well-known expression:
( 0=rk ) ( )rE
( ) 204 rqrE πε= .
Thus, these results point to an exponential
deviation from Coulomb’s inverse square
law, which, as we know, is expressed by the
following equation (in SI units):
( )17
4
1
3
12
1221
0
2112
r
rqq
FF r
rrr
πε=−=
As seen in Eq. (16), the term
( )rkr
3
2
only becomes significant if
( )1810~ 4λ−>r
This means that the Coulomb’s law is a good
approximation when . However, if
, the expression of the force
departs from the prediction of Maxwell’s
equations.
λ410~ −<r
λ410~ −>r
The lowest-frequency photons of the
primordial radiation of 2.7K is about
[Hz810 13]. Therefore, the wavelength of
these photons is m1≈λ . Consider the
presence of these photons in a terrestrial
experiment designed to measure the force
between two electric charges separated by a
distance r . According to Eq. (18), the
deviation from the Coulomb’s law only
becomes relevant if . Then, if we
take
mr 410−>
mr 1.0= , the result is

( ) 73.0
3
4
3
2 =⎟⎠
⎞⎜⎝
⎛= λ
π r
rkr
and
( ) ( ) 83.0
3
2
1 3
2
=−⎥⎦
⎤⎢⎣
⎡ + rkerk rr

Therefore, a deviation of 17% in respect to
the value predicted by the Coulomb’s law.
Then, why the above deviation is not
experimentally observed? Theoretically
because of the presence of Schumann
radiation ( )mHzf 711 108.3,83.7 ×== λ
[14-15]. According to Eq. (18),
for , the deviation only
becomes significant if
m71 108.3 ×=λ
Kmr 8.310~ 1
4 => − λ
Since the values of r in usual experiments
are much smaller than the result is
that the deviation is negligible. In fact, this is
easy to verify. For example, if , we
get
Km8.3
mr 1.0=

( ) 8
7
1
109.1
108.3
1.0
3
4
3
4
3
2 −×=⎟⎠
⎞⎜⎝
⎛
×=⎟⎟⎠
⎞
⎜⎜⎝
⎛= πλ
π r
rkr

and
( ) ( ) 999999999.0
3
2
1 3
2
=−⎥⎦
⎤⎢⎣
⎡ + rkerk rr
Now, if we put the experiment inside an
aluminum box whose thickness of the walls
are equal to 21cm * the experiment will be
shielded for the Schumann radiation. By
putting inside the box a photons source of
m1≈λ , and making , then it will be
possible to observe the deviation previously
computed of 17% in respect to the value
predicted by the Coulomb’s law.
mr 1.0=

* The thickness δ necessary to shield the experiment
for Schumann radiation can be calculated by means of
the well-known expression [16]: fz πμσδ 2105 ==
where μ and σ are, respectively, the permeability
and the electric conductivity of the material; is the
frequency of the radiation to be shielded.
f

3

References

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[2] Proca A 1937 J. Phys. Radium Ser. VII 8 23–8

[3] Tomaschitz, R. (2001) Physica A 293, 247.

[4] Tomaschitz, R. (2004) Physica A 335, 577.

[5] Tomaschitz, R. (2005) Eur. Phys. J. D. 32, 241.

[6] S. Eidelman et al. (Particle Data Group), Phys. Lett. B
592, 1 (2004) and 2005 partial update for edition 2006
(URL: http://pdg.lbl.gov).

[7] Pecina-Cruz, J. N., (2006) arXiv: physics/0604003v2

[8] De Aquino, F. (2010) Mathematical Foundations of the
Relativistic Theory of Quantum Gravity, Pacific Journal
of Science and Technology, 11 (1), pp. 173-232.

[9] Greiner, W. (2000) Relativistic Quantum Mechanics,
Springer, Berlin, 3rd edition.

[10] Itzykson, C., and Zuber, J.B., (1980) Quantum Field
Theory, McGraw-Hill, N.Y.).

[11] Morse. P.M., Feshbach, H., (1953) Methods of
Theoretical Physics (McGraw-Hill, N.Y.).

[12] Byrne J C 1977 Astrophys. Space Sci. 46 115–32

[13] Audouze, J. (1980) An introduction to Nuclear
Astrophysics, D. Reidel Publishing Company-Holland,
p. 22.

[14] Schumann W. O. (1952). "Über die strahlungslosen
Eigenschwingungen einer leitenden Kugel, die von
einer Luftschicht und einer Ionosphärenhülle umgeben
ist". Zeitschrift und Naturfirschung 7a: 149–154

[15] Volland, H. (1995), Handbook of Atmospheric
Electrodynamics, CRC Press, vol.I, Chapter11.

[16] Quevedo, C. P. (1977) Eletromagnetismo, McGraw-
Hill, p.270.