p2

where T is the temperature of the vacuum and k B is the Boltzmann constant. Here the speed of light is
decreased.
For light (photons) propagating in an anisotropic vacuum given by an external constant uniform
magnetic field B:
c
∗
c
0
c
c
∗
⊥
0
2
⎛ 8 2 B 2
⎞
= ⎜1− α sin θ 1 ( c
4 ⎟< =
0
= ε
0
= µ
0
= 1)
⎝ 45 me
⎠
2
⎛ 14 2 B 2
⎞
= ⎜1− α sin θ 1
4 ⎟<
⎝ 45 me
⎠
(2.17),
where the speed of light is decreased in this vacuum for polarizations coplanar (||) with and perpendicular
(⊥) to the plane defined by B and the direction of propagation, and θ is the angle between B and the
direction of propagation. Latorre et al. (1995) calculated the polarization-average of (2.17) to give the
averaged (modified) speed of light in the B-field:
c
c
∗ 2
⎛ 22 2 B ⎞
⎜
4 ⎟
0
m
= 1− α < 1 ( = c0 = ε
0
= µ
0
= 1) (2.18).
⎝ 135
e ⎠
For light (photons) propagating in an anisotropic vacuum given by an external constant uniform
electric field E, the polarization-averaged modified speed of light is:
c
c
∗ 2
⎛ 22 2 E ⎞
⎜
4 ⎟
0
m
= 1− α < 1 ( = c0 = ε
0
= µ
0
= 1) (2.19).
⎝ 135
e ⎠
Here the speed of light is decreased.
Equations (2.16) – (2.19) are the result of vacuum modifications that populate the vacuum with
virtual or real particles that induce coherent (light-by-light) scattering, which reduces the speed of
massless particles. By examining the form of equations (2.13) and (2.15) – (2.19) Latorre et al. (1995)
discovered that the low energy modification of the speed of light is proportional to the ratio of the
modified vacuum energy density (as compared to the standard vacuum energy density, ρ vac = 0) over m e 4 ,
with a universal numerical coefficient and the corresponding coupling constants. And a general rule
became apparent from their analysis that is applicable to modified vacua for massive and massless
quantum field theories, for low energy:
c * > c 0 (vacuum refraction index < 1) when the modified vacuum has a lower energy density
c * < c 0 (vacuum refraction index > 1) when the modified vacuum has a higher energy density
c * = c 0 (vacuum refraction index = 1) when the vacuum is free (or un-modified) with ρ vac = 0
The first two rules explain the sign of the change of the speed of light. From this rule and the
mathematical commonality between the form of (2.13) and (2.15) – (2.19) Latorre et al. (1995) found a
single unifying expression to replace these equations:
c
c
∗
0
44 2 ρ
= 1 − α ( = c
4
0
= ε
0
= µ
0
= 1) (2.20),
135 me
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17