Download Introduction Vacuum effects due to Dirac Sea When do the

Can Light Signals Travel Faster-than-C in Vacuum?
Presented at CCFP 2006, Nizhny Novgorod
Prof. Heidi Fearn • Department of Physics, California State University Fullerton, Fullerton, CA 92834
Time Dilation
Introduction
Relativistic causality implies that the front velocity of the pulse
cannot travel faster than the speed of light in a vacuum. The front
velocity of the pulse is that part of the wave packet that has the
highest UV frequency components. This corresponds to a discontinuity (non-analytic cut) in the pulse. This is the speed at which information flows.
Light Clock Experiment
L0
2. Barton, Phys. Letts. B237, 559 (1990) Mar.
s
C
∆t
c∆t = 2
Square
c = V
According to [1] above, ∆c/c
“obviously is far beyond any experimental reach”.
⇒ ∆t = γ ∆t0
Vacuum effects due to Dirac Sea
• Toll, Phys. Rev. 104, 1760 (1956)- dispersion relations connect
the Re. and Im. parts of the diagonal elements of the scattering
matrix.
• Dispersion relations used in QFT first by Gell- Mann, Thirring
and Goldberger , Phys. Rev. 95, 1612 (1954).
• Is there a minimum length scale involved which the wavelength
of light is not allowed to fall below? How many atoms constitutes
the minimum number before you can apply the idea of a refractive
index? Is something defined for the vacuum?
• Are we misusing the dispersion relations and idea of refractive
index for vacuum?
½
• Adler- strong magnetic field effects? Faster-than-c signals?
2
, γ=
Clarify: Basics of QED
Quantization of ϕ for the KG field
s[ϕ] = ∫d 4 x L(x) =
l
lo
1
d 4 x [∂µϕ(x)∂µϕ(x) – m 2ϕ 2(x)]
∫
2
The quantized Hamiltonian is; H =
1
2
∫d x (π
4
2
+ | ∇ϕ| 2 + m 2ϕ 2)
V
d1
d2
v ∆t1
∆t1 to get from left to right
∆t2 to get from back to left
V
v ∆t2
proper length of clock = lo (as measured in sʹ)
proper time for one round trip = ∆t0 = 2l0 /c (as measured in sʹ)
d1 = l + v ∆t1 = c ∆t1 ⇒ l = ∆t1 (c – v)
d2 = l – v ∆t2 = c ∆t2 ⇒ l = ∆t2 (c + v)
∆t = ∆t1 + ∆t2 =
l
l
2lc
2l/c
+
= 2
=
(c – v) (c + v) (c – v2)
(1 – v2/c2)
time dilation ∆t = γ ∆t0 = γ 2L0 /c
2L0 /c
2l/c
2 2 ½
⇒ ∆t = ∆t1 + ∆t2 =
=
→
l
=
l
0 (1 – v /c )
(1 – v2/c2) (1 – v2/c2)
Another more conclusive
Light Clock experiment
Remember, the change in c
predicted by the Scharnhorst
effects is dependent on the
distance between the parallel
mirrors. If the distance between
the mirrors is different in one
frame to another in relative
motion-velocity of light becomes
frame dependent.
What value of c do we use in
the Lorentz transformations?
How do we agree on time
dilation and Lorentz contraction.
Stenner et al., Nature 425, 695 (2003).
Barton & Scharnhorst, J. Phys. Math Gen. 26,
2037 (1993).
r
r
r r
[ϕ(x,t), π( y,t)] = iδ (x – y)
Scharnhorst, “The velocities of light in modified
QED vacua” http://arxiv.org/hep-th/9810221
Dittrich & Giess, Phys. Rev. D58, 025004
(1998), also their book “Probing the quantum
vacuum”, Springer Press.
Loudon, R.“Quantum theory of Light” Oxford
University Press.
Acknowledgments
I would like to thank the organizers for inviting
me to CCFP 2006.
Air Force Office of Scientific Research
ϕ = i[H, ϕ] = π (x)
Russian Foundation for Basic Research
π = i[H, π] ⇒ ϕ + m 2ϕ = 0
•
European Office of Aerospace Research
and Development
Page 108, Quantum field theory by Itzykson and Zuber.
The same holds for 2nd Quantization and for
particles of zero mass
d 3k 1
–i(k.x)
+
i(k.x)
{a(k)e
+
a
(k)e
}
3
(2π) 2ωk
a(k) = ∫ d 3x e ik.x [ωkϕ(x) + iπ (x)]
Bjorken & Drell, “Relativistic quantum fields”,
McGraw Hill, NY (1965).
•
r
r
r
r
[ϕ(x,t), ϕ( y,t)] = [π (x, t), π ( y,t)] = 0
ϕ(x) = ∫
Itzykson & Zuber, “Quantum field theory”,
McGraw Hill, NY (1965).
Conference Sponsers:
With the equal time commutators defined as;
sʹ
Wang, et al., Nature 406, 277 (2000).
Hatfield, B. “Quantum field theory of point
particles and strings”
We define the conjugate momentum as, π (x) = ∂L = ∂tϕ = ϕ•
∂(∂tϕ)
Lorentz Contraction
S
r
Metric + ---, x → (ct, x)
Classical action s[ϕ] is given by,
1
(1 – v2/c2)½
Brillouin, “Wave propagation and Group Velocity”,
Academic Press, New York (1960).
Ben-Menahem, Phys. Letts. B250, 133 (1990).
2
2 L0 / c
=
(1 – v2/c2)½
Literature cited
Milloni & Svozil, Phys. Letts. B248, 437 (1990).
• Example of the scalar field quantization follows for simplicity purposes
here-
∆to = 2 L0 /c
Proper time
as measured
on the moving
clock
∆c
1.6 × 10–60
∆c
–36
and
for
L
=
1µm
we
get
∼
≃
1.6
×
10
c
L4
c
• Topic of current research--
2
2 L0
∆t =
(c2 – v2)½
c
α
11π
∆n≃ –
and c⊥ =
≈ (1 – ∆n)c > c
4
2 4 2
n
(mL) 2 3 5
When do the Dispersion relations apply?
• High energy QED may have problems.. May not depend on SR. Radiation
reaction could be re-addressed for example… possibility of removing
Landau poles and all inconsistencies like run-aways and pre-acceleration.
∆t (c – v ) = 4 L0
2
2
• The non-linearity’s, jointly with mirror induced changes in the
ZP Maxwell field, (between parallel mirrors) can cause c to
change or the vacuum to amplify. Not sure which ?
( ))
2
c2∆t2 = 4 L02 + v2∆t2
Mirror
• The electron-positron (Dirac) field profoundly alters the properties
of the vacuum in QED relative to those of the classical vacuum:
It induces non-linearity’s in Maxwell’s eqns. and a consequent
scattering of light by light.
(
L02 +
v ∆t
2
• QED as we know it (at low to semi high energies) requires SR and causality!
The commutation relations depend on it.
• The method of quantization of the fundamental Lagrangian in QED
requires commutators of the form which do not allow for signals
(interaction between particles/information of any kind) to pass between
two space-like separated points.
Path length of moving pulse measured in s = AB + BC
Casimir Vacuum
Photon
Assume you have a Scharnhorst
type experiment in progress
between the mirrors. The velocity
in different frames (one with a
Scharnhorst and one without)
will measure different velocities
perpendicular to the mirrors and
hence register different times.
L0
sʹ frame of moving light pulse clock
s is stationary lab frame.
3. Scharnhorst, Annals. of Phys. 7, 700 (1998).
2
v
A
v ∆t
2
1. Scharnhorst, Phys. Letts. B236, 354 (1990) Feb.
V⊥ > c
B
sʹ
Scharnhorst Effect
L
Serious problems with violation of SR…
π (x) = ϕ
•
[a(k), a+ (k')] = (2π)3 2ωkδ 3 (k – k')
• E and B-fields and quantized and then vector potentials are quantized
into harmonic oscillator raising and lowering operators, a and a+ terms,
with similar commutators as above, FT as needed.
For further information
Please contact [email protected].
More information on this and related
projects can be obtained at
http://physics.fullerton.edu/~heidi
An older e-print relevant here can be found at,
http://arxiv.org/quant-ph/0310059. The paper
will be published in the Laser Physics journal(3)
v17, (2007).