From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Newsgroups: sci.physics
Subject: Correction Aephraim's Subtle Error -FTL Alive!
Date: 18 Aug 92 01:03:36 GMT
Organization: Whole Earth 'Lectronic Link


Sarfatti's correction of error in Aephraim's energy-conservation
argument that the phase shift of a beam recombiner is 90 degrees.
Aerphraim's subtle conceptual error was to assume that the same plate
acting in two different modes (i.e., as a beam splitter with one input,
and as a beam recombiner with two inputs) will have the same net phase
shift between the outputs.

#1 How do you reconcile your micro-model of the reflection phase shift
of the plate i.e. 2kd + pi -> kd + pi/2 (average) with transmission kd
with time reversal symmetry etc. if the forward path d1 is different
from return path d2  (by local optical inhomogeneity in return path
back to front) so that  k(d1+d2)/2 + pi/2 -kd1 = k(d2-d1)/2 + pi/2 = b.
Thus cos(b) not equal to zero. There would have to be some kind of
breakdown of Fresnel equations, some kind of screening of the optical
inhomogeneity so that b is forced to be pi/2? However, there is an
alternative.


#2 The net reflection phase shift b of plate used as a beam splitter
need not be the same as the net reflection phase shift B of the same
plate used a beam recombiner since the total experimental arrangement
is different in the two cases. Only the assumption of locality, which
contradicts quantum mechanics, would require such an identification. It
is B that is relevant to the superluminal communication problem. Thus,
given: 1 amplitude into I1, 0 amplitude into I2, t amplitude out O1, r
amplitude out O2. This defines experiment I in which the plate acts
like a beam splitter of one incident beam. Now consider a different
experiment (i.e., II) on the same plate. The input amplitude into O1 is
t* and the input amplitude into O2 is r*. There is no apriori reason to
assume that the same plate will respond to two inputs into both output
ports (i.e. phase shift B) with the same net phase shift b that it
responds to one input into one of the two input ports.  We can still
conserve energy! There is no need to perturb the vacuum. The time-
reversed amplitude into O1 is t*. Similarly, r* is into O2. Since we
are breaking time-reversal symmetry, the output amplitude from I1 is
t*T + r*R where T is not same as t and R is not same as r as it would
be if time-reversal symmetry were true. Similarly, the output from I2
is t*R + r*T. Conservation of energy is
1 = r*r + t*t = (t*T + r*R)(tT* + rR*) + (t*R + r*T)(tR* + rT*)
=|t|^2|T|^2+|r|^2|R|^2+r*tRT*+rt*R*T+|t|^2|R|^2+|r|^2|T|^2+r*tR*T+r*tR*
T
But |r|^2 + |t|^2 = 1 and |R|^2 + |T|^2 = 1, Therefore,
1 = |t|^2|T|^2+|r|^2|R|^2+|t|^2|R|^2+|r|^2|T|^2
and
0 =r*tRT*+rt*R*T+r*tR*T+r*tR*T
r = |r|e^iphi(r) etc.
b = phi(r) - phi(t) is net reflection phase shift in beam splitter mode
B = phi(R) - phi(B) is net reflection phase shift in beam recombiner
mode
Therefore,
cos(b+B) + cos(b-B) = 0
or
cosb cosB = 0
From time reversal symmetry in the beam splitter mode b = pi/2 so that
B is free to be anything without violating conservation of energy in
the beam recombiner mode.

We can assume that |T| = |t| and |r| = |R|. Explicitly, in the beam
recombiner mode, with two non-zero inputs and two non-zero outputs. The
input intensities are |t|^2 into O1 and |r|^2 into O2. The output
intensities are:

(t*T + r*R)(tT* + rR*) = |t|^2|T|^2+ |r|^2|R|^2 + r*tRT*+ rt*R*T
= |t|^4 + |r|^4 + 2|r|^2|t|^2 cos (b-B)
= |t|^4 + |r|^4 + 2|r|^2|t|^2 cos (pi/2-B)
= |t|^4 + |r|^4 + 2|r|^2|t|^2 sinB     for I1,
the output intensity from I2 is
(t*R + r*T)(tR* + rT*) = |t|^2|R|^2 + |r|^2|T|^2 + r*tR*T + r*tR*T
= |t|^2|r|^2 + |r|^2|t|^2 + 2|r|^2|t|^2 cos (b+B)
= |t|^2|r|^2 + |r|^2|t|^2 + 2|r|^2|t|^2 cos (pi/2+B)
= 2|r|^2|t|^2 - 2|r|^2|t|^2 sinB
Therefore, I modify my recent remarks. We can get superluminal
communication since cosB in the recombiner mode can be non zero. We do
not require breaking time reversal symmetry and we do not require
perturbing the vacuum, though that might also happen because the
question in #1 above in the beam splitter mode is not resolved.