Is Light Heavy?
A. Cooney, J. Dudley, T. Fleming, P. Flynn, S. Keating (Sch).
March 2, 2001
Abstract
It is always valid to ask if we are justied in assuming a zero rest mass for
the photon. We follow Proca's treatment of a massive vector eld with external
source and derive the Symmetric Stress Tensor, its Conservation Laws, and the
Equations of Motion. We then use these to predict the eect a heavy photon
would have on the Earth's Magnetic Field, following a method proposed by
Schrödinger. Using information on the Earth's Magnetic eld, we put an upper
limit on the mass of the photon of ~4.10 −48 g.
. . . being heavy, I will bear the light . . . proclaimed the melancholy hero in
Shakespeare's Romeo and Juliet. Is light heavy? What evidence is there to support
the assumption of Classical Electrodynamics that the photon has a zero rest mass?
These are valid questions and we shall consider them here. First, we will assume
that the photon has indeed a non-zero rest mass and derive its properties. Then,
in section 2, following a method proposed by Schrödinger we will predict the eect
a massive photon would have on the Earth's magnetic eld and use recent data to
place an upper limit on the photon's rest mass.
1. The Proca Lagrangian
In 1930 Proca considered a Lagrangian density for a massive vector eld, in interaction with some external source J υ .
To calculate any observable eects of a
heavy photon we will need to consider such a Lagrangian density. For completeness,
we derive the Symmetric Energy-Momentum Stress Tensor (with its time-time and
space-time components in terms of
E
and
B)
and the dierential conservation laws
it obeys, as well as the equations of motion which we will need for the next section.
(a) The Stress Tensor
The Proca Lagrangian density for a massive vector eld in interaction with an
external source is given as
Lp =
with
m = mγ c/h̄,
m2
1
−1
Fµυ F µυ +
Aµ Aµ − J µ Aµ
16π
8π
c
the Compton wave number of the photon, and
1
mγ
the eld mass.
Now, dene the Energy-Momentum Stress Tensor as
T
µυ
"
#
∂Lp
=
∂ υ Aσ − g µυ Lp
∂ (∂µ Aσ )
so,
1 µυ σβ
m2 µυ
1
−1 µσ υ
F ∂ Aσ +
g F ∂σ Aβ −
g Aα Aα + g µυ Jµ Aµ
4π
8π
8π
c
T µυ =
=
1 µσ
1 µυ σβ
m2 µυ
1
1 βυ µσ
g F Fσβ −
F ∂σ Aυ +
g F ∂σ Aβ −
g Aα Aα + g µυ Jµ Aµ
4π
4π
8π
8π
c
1 βυ µσ
1
4π µ
=
g F Fσβ −
F µσ ∂σ Aυ + Aυ ∂σ F µσ + Aυ
J − m 2 Aµ
4π
4π
c
+
1 µυ σβ
m2 µυ
1
g F ∂σ Aβ −
g Aα Aα + g µυ Jµ Aµ
8π
8π
c
by equations of motion,
1 βυ µσ
1 υ 4π µ
1
g F Fσβ −
A
J − m 2 Aµ −
∂σ (F µσ Aυ )
=
4π
4π
c
4π
+
m2 µυ
1
1 µυ σβ
g F ∂σ Aβ −
g Aα Aα + g µυ Jµ Aµ
8π
8π
c
So, we may construct the Symmetric Energy Momentum Stress Tensor by dening
Θµυ = T µυ − TDµυ
where
1 βυ µσ
1 υ 4π µ
=
g F Fσβ −
A
J − m 2 Aµ
4π
4π
c
1
m2 µυ
1
g Aα Aα + g µυ Jµ Aµ
+ g µυ F σβ ∂σ Aβ −
8π
8π
c
1 υ 4π µ
1 µυ σβ
1 βυ µσ
−
g F Fσβ −
A
J − m 2 Aµ +
g F ∂σ Aβ
4π
4π
c
8π
)
m2 µυ
1 µυ
1
α
µ
µσ υ
− g Aα A + g Jµ A − 2. (∂σ (F A ))
8π
c
4π
2 TDµυ
So,
µυ
2Θ
1 βυ υσ
1 µβ υσ
2 µυ σβ
1 υ 4π µ
=
g F Fσβ +
g F Fσβ +
g F ∂σ Aβ −
A
J − m 2 Aµ
4π
4π
8π
4π
c
4π υ
1
1
2m2 µυ
1
J − m 2 A υ + J υ Aµ + J µ A υ −
g Aα Aα
− Aµ
4π
c
c
c
8π
1 µυ
m2
1
1 µι
συ
σβ
g Fισ F +
g Fσβ F +
Aµ Aυ − g µυ Aα Aα
=
4π
16π
4π
2
1
1
+ g µυ Fισ F ισ + m2 Aµ Aυ − g µυ Aι Aι
4
2
So, nally, we have that
µυ
4πΘ
µι
= g Fισ F
συ
2
(b) Equations of Motion
For the Proca Field we have the Euler-Lagrange equations of motion as
!
∂Lp
∂Lp
−
=0
∂ (∂µ Aυ )
∂Aυ
∂µ
Thus
∂µ
= ∂µ
∂Lp
∂ (∂µ Aυ )
!
= ∂µ
∂
−1
Fισ F ισ
∂ (∂µ Aυ ) 16π
!
!
−1
∂
(∂ι Aσ − ∂σ Aι ) (∂ ι Aσ − ∂ σ Aι )
∂ (∂µ Aυ ) 16π
!
−1
∂
=
∂µ
(2 (∂ι Aσ ) (∂ ι Aσ ) − 2 (∂σ Aι ) (∂ σ Aι ))
16π
∂ (∂µ Aυ )
=
−1
∂µ (2∂ µ Aυ + 2∂ µ Aυ − 2∂ υ Aµ − 2∂ υ Aµ )
16π
=
−1
∂µ F µυ
4π
Similarly,
−1
m2
1
Fµυ F µυ +
A µ Aµ − J µ A µ
16π
8π
c
∂Lp
∂
=
∂Aυ
∂Aυ
m2 µυ
1
g Aµ Aυ − g µυ Jµ Aυ
8π
c
∂
=
∂Aυ
=
!
!
m2 µυ
1
g Aµ − g µυ Jµ
4π
c
So,
−1
m2 µυ
1
∂µ F µυ −
g Aµ + g µυ Jµ = 0
4π
4π
c
So, the Proca equations of motion for the massive eld are given as,
∂µ F µυ + m2 Aυ =
3
4π υ
J
c
(c) Conservation Laws
Look at the dierential conservation laws for the massive vector eld,
∂µ Θµυ =
=
=
1
1
1
∂µ g µι Fισ F συ + g µυ Fισ F ισ + m2 Aµ Aυ − g µυ Aι Aι
4π
4
2
1
1
1
∂ ι (Fισ ) F συ + Fισ ∂ ι (F συ ) + Fισ ∂ υ (F ισ ) + m2 ∂µ (Aµ Aυ ) − ∂ υ (Aι Aι )
4π
2
2
1
4π
4π
1
1
Jσ − m2 Aσ F συ + Fισ ∂ ι (F συ ) + Fισ ∂ υ (F ισ ) + m2 ∂µ (Aµ Aυ ) − ∂ υ (Aι Aι )
c
2
2
by the equations of motion. Therefore,
1
1
1
4π ∂µ Θµυ − Jσ F συ = Fισ ∂ ι (F συ )+ Fισ ∂ υ (F ισ )+m2 ∂µ (Aµ Aυ )− ∂ υ (Aι Aι )−m2 Aσ F συ
c
2
2
Examine the term
1
F µσ ∂µ Fσν + F µσ ∂ν Fµσ
2
1
= Fϕσ [(∂ µ F ϕσ + ∂ ϕ F σν ) + ∂ ϕ F σν ]
2
now,
µν = 0
∂µ Fg
therefore
∂ν (3εµνϕσ Fϕσ ) = 0
which implies
6(2∂ν Fϕµ + ∂ϕ Fµν + ∂µ Fνϕ ) = 0
so
1
Fϕσ [(∂ µ F ϕσ + ∂ ϕ F σν ) + ∂ ϕ F σν ]
2
1
= Fϕσ [∂ σ F ϕν + ∂ ϕ F σν ] = 0
2
which gives nally:
1
1
4a(∂µ Θµν Jσ F σν ) = m2 (∂µ (Aµ Aν ) − ∂ ν (Aϕ Aϕ ) − Aσ F σν )
c
2
and in particular if we employ the Lorentz gauge we have that:
1
∂µ Θµν = Jϕ F ϕν
c
as for the massless electromagnetic eld.
4
(d) Components
Here we give the time-time components of the Proca symmetric stress momentum
energy tensor
1
1
1
8πΘ00 = 2F0i F i0 + F0i F 0i + F0i F i0 + Fij F ij + 2m2 (A0 A0 − Ai Ai )
2
2
2
1
= 2E i E i − E i E i + (−εijk B k )(−εijl B l ) + m2 (A0 A0 − Ai Ai )
2
~2 + B
~ 2 + m2 (A0 A0 − Ai Ai )
= E
Here, we exhibit the space-time components of the Proca tensor as,
8πΘi0 = g i0 F0l F l0 + g (ii) Fil F l0 + m2 Ai A0 − 21 g l0 Ai Ai
= g (ii) Fij F j0 + m2 (Ai A0 )
= εijk E i B k + m2 Ai A0
~ × B)
~ i + m 2 Ai A 0
= (E
2. On a Geomagnetic Limit for the Photon Rest Mass
In 1943 Schrödinger proposed a method of estimating the mass of a photon. This,
he claimed, would have a measurable eect on the Earth's magnetic eld, and using
sparse and innaccurate data puts forward a conservative estimate. We improve his
method using the machinery we developed in the previous section, and propose a
signicantly smaller number for the mass of the photon with the aid of more accurate
data from recent satellite and surface observations.
(a) Massive Vector Potential
The Proca equations of motion are:
∂ β Fβα + µ2 Aα =
4π
Jα
c
where
Fβα = ∂β Aα − ∂α Aβ
Imposing the Lorentz guage condition
∂ β Aβ = 0
this implies that
(∂ β ∂β + µ2 )Aα =
4π
Jα
c
For a steady-state distribution of current this becomes
~ = − 4π J~
(∇2 − µ2 )A
c
5
We require a Green's Function satisfying
(∇2 − µ2 )G(r) = −4πδ(r)
Consider r 6=0 :
(∇2 − µ2 )G(r) = 0
as G = G(r) we can write
1 d2
∇ G=
(rG)
r dr2
2
thus
As
φ
:
goes to as r →
d2
(rG) = µ(rG)
dr2
rG = K exp ±µr
∞
we take
rG = K exp −
µr
r
the (spherically symmetric) Yukawa potential. We require
1
K = − 4π
so that
:
(∇2 − µ2 )G(r) = −4πδ(r)
for all r and:
G(~x, x~, ) = −
1 exp −µ | ~x − x~0 |
4π
| ~x − x~0 |
~ = − 4π
A
c
~ x~0 )d3 x0
G(~x, x~0 )J(
thus
Z
1 Z exp −µ | ~x − x~0 | ~ ~0 3 0
=
J(x )d x
c
| ~x − x~0 |
If there is a magnetisation
~ = mf
M
~ (~x)
where
m
~
is a xed vector and
f (~x)
is a
localised scalar function then
J~ = c∇ × mf
~ (x)
noting that in a steady state distribution of current the Lorentz guage becomes the
Coulomb guage, so
~=0
∇·A
and we obtain
~ = −m
A
~ ×∇
Z
f (x)
exp −µ | ~x − x~0 | 3 0
dx
| ~x − x~0 |
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(b) Magnetic eld
f (x) = δ(x)
If the magnetic dipole is a point dipole at the origin then
and
−µr
e
~ = −m
A
~ ×∇
with
~ = ∇×A
~.
B
r
We have
∇
e−µr
e−µr
= − 3 (µr + 1)~r ≡ R~r
r
r
Thus
~ = ∇ × ((R~r) × m)
B
~
= (m
~ · ∇)(R~r) − m
~ (∇ · (R~r))
where we have used the identity
~ = G
~ · ∇ F~ − G
~ ∇ · F~ − F~ · ∇ G
~ + F~ ∇ · G
~
∇ × F~ × G
and the fact that
m
~
is a constant vector. Consider now
∂
(m
~ .∇)(φ~r)≡ma ∂xa (φxb~eb )
using Einstein's summation convention (a=1,2,3) with
basis vectors. Thus
∂φ
(m
~ .∇)(φ~r)≡ma ∂x
xb e~b + φδba e~b
a
~
≡ (m.∇φ)~
r+mφ
~
e~b
the usual orthonormal
µ2 r 2
e−µr
e−µr
(rb.m)
~ rb − 3 (µr + 1) m
~
= 3 3 1 + µr +
r
3
r
!
Now
∇. (φ~r) = φ (∇.~r) + (∇φ) .~r = 3φ + (∇φ) .~r =
So we obtain,
~ = [3 (rb.m)
B
~ rb − m]
~ 1 + µr +
µ2 r 2
3
e−µr
r
−µr
− 23 µ2 e r m
~
(c) Photon mass limit
J. D. Jackson tells us that
Hext
∼ 4 × 10−3
HAD
(upper limit) on the surface of the earth (r=R). Thus
1+
2
(µR)2
3
µR + 13 (µR)2
yielding,
7
= 4 × 10−3
e−µr 2 2 µr
r3
µR = 0.0806
This gives us a lower limit on
µ−1 of
µ−1 = 12.407
earth radii. Schrödinger, in his study, calls this the characteristic length. It is, of
course, the reciprocal Compton wavelength of a photon
µ=
c
m
h̄
Thus, an upper limit of the rest mass of the photon is
m ∼ 4 × 10−48
g.
[A review of past and current studies of possible long-distance, low-frequency deviations from Maxwell electrodynamics and Einstein gravity has appeared in [4],
nhb.]
References
[1] E. Schrödinger, Proc. Roy. Irish Acad., A 49, 135 (1943).
[2] A. S. Goldhaber and M. M. Nieto, Phys. Rev. 21, 567 (1968).
[3] J. D. Jackson, Classical Electrodynamics, Wiley 3e (1998).
[4] A. S. Goldhaber and M. M. Nieto, arXiv:0809.1003 [hep-ph].
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