From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: FUTURE PHYSICS I
Organization: Whole Earth 'Lectronic Link
Date: Thu, 13 Aug 1992 06:53:49 GMT


Superluminal Debate II

Sarfatti's proof of Solovay's Theorem.

I do it in a special simple case relevant to the physics. This is not
the way Solovay would do it - but it illustrates Solovay's point in a
physically clear way. We are not trying to be mathematicians here.

A and B are two commuting operators. A will measure the polarization of
photon a relative to the actual orientation of the doubly-refracting
crystal that locally interacts with photon a. The two eigenvalues of
operator A will be e for the extraordinary path and o for the ordinary
path. Similarly, B measures the polarization of the other twin photon
relative to the actual orientation of the other crystal. The eigenvalues
of B are E and O.

The total experimental arrangement is a pair source between the two
crystals. Photons a and b go off in opposite directions. Paths e and o
each go to separate counters as do paths E and O. This is the topology
of the actual experiments done by Clauser, Aspect et-al.

This experiment has one nonlocal parameter x which is the relative
mismatch between the two crystals. There is a subtle point here in that
the times of detection do not have to be simultaneous. Indeed,
simultaneity is not a frame-invariant idea in special relativity. Choose
a direction in space. Measure the orientation of crystal A relative to
that direction at the moment that photon a locally interacts with A.
Call that angle x(a). Similarly for crystal B. Then x = x(a) - x(b).
Thus, x is a nonlocal parameter specifying the boundary conditions that
define the vector space on which the operators operate.

Consider the actual entangled photon pair state |a,b> produced by the
atomic cascade in the source used by Clauser et-al. In terms of the
actual orientations of the crystals at the generally different moments
that their respective photons locally interact with them, the pair state
may be written

|a,b> = [|e>|E> + |o>|O>]cosx/sqrt2 + [|e>|O> - |o>|E>]sinx/sqrt2  (1)

Note that we assume "causality"

[A,B] = 0        (2)

at the operator level so that it is certainly true in the expectation

<a,b|[A,B]|a,b> = 0  (3)

The operators are defined by

A|e> = |e>    (4a,b,c,d,e,f)

A|o> = -|o>

B|E> = |E>

B|O> = -|O>

<e|e> = <o|o> = <E|E> = <O|O> =1

<e|o> = <E|O> = 0

So that A and B look like 2x2 Pauli z matrices in their eigenvector
bases corresponding to the actual orientations of the crystals.

For example, look at the local probability p(e) to measure photon a with
eigenvalue e.
p(e) = <a,b| |e><e| |a,b>

=  {[<e|<E| + <o|<O|]cosx/sqrt2 + [<e|<O| - <o|<E|]sinx/sqrt2}

|e><e|{[|e>|E> + |o>|O>]cosx/sqrt2 + [|e>|O> - |o>|E>]sinx/sqrt2}

= {<E|cosx/sqrt2 + <O|sinx/sqrt2}{|E>cosx/sqrt2 + |O>sinx/sqrt2}

= [cosx/sqrt2]^2 + [sinx/sqrt2]^2  = 1/2   (5)

similarly for p(o), p(E) and p(O). The light at both ends is locally
unpolarized.

There is no dependence of the local eigenvalue probability distributions
on the nonlocal parameter x. Hence for this entangled state there is no
superluminal communication. However, this is far from being the end of
the story!

The way to get superluminal communication is to disentangle the pair
state so that it is no longer given by equation (1). The physical
transformation from an entangled to a disentangled state (caused by the
transmitter) induces a dependence of the local eigenvalue probability
distribution at the receiver on a set of parameters (not eigenvalues!)
controlled from the transmitter.

This disentanglement is caused by placing a variable phase plate in the
E path. The E path of the transmitter photon is reflected by a mirror as
an input to a beam recombiner. The O path passes through a half-wave
plate at 45 degrees to the O plane of polarization. This rotates the
polarization to coincide with that of the E path. The rotated O path is
the second input to the beam recombiner. The half-wave plate and the
beam recombiner together form a "nonlocal quantum eraser" that induces a
coherent polarization of the receiver photon.

The mixing of the E and O transmitter paths together with the rotation
of E to O polarization in the O path allow interference terms to be
detected locally at the receiver provided that the receiver crystal is
misaligned from the transmitter crystal. The difference in count rates
at the receiver, in the ideal case of no absorption and 100% efficient
photon counters, depends upon sin2x. It also depends upon the cosine of
the reflection phase shift b of the transmitter recombiner and upon the
cosine of the sum of the transmitter phase plate shift phi and the
transmitter mirror phase shift m. Elementary application of Feynman's
two rules (1 add amplitudes before squaring for indistinguishable
histories and 2 square amplitudes before adding for distinguishable
histories) shows the the receiver photon polarization eigenvalue
probability distribution obeys the simple equation

p(o) - p(e) = sin (2x) cos(phi + m) cos(b)       (6)

The total experimental arrangement has one nonlocal parameter x and
three local transmitter parameters phi, m and b. As a practical matter
keep x fixed at 45 degrees and used the standard external reflection
shifts m = b = 180 degrees so that we have a simple cos(phi) dependence
giving phase-modulated FTL/BIT messages by a sequence of cutting the
quantum connections for successive photon pairs.

However, if we monitor the x dependence at a distance we can distinguish
a left-handed rotation of the far away transmitter crystal from a right-
handed rotation. Thus, communication by breaking the quantum connection
also is a mechanism for parity violation that may explain the weak force
of beta radioactivity in a new way. Indeed the Higgs mechanism involves
subluminal quantized vibrations in a tachyonic superfluid (spontaneous
symmetry breaking of vacuum)

In summary, superluminal (FTL) and "delayed choice" backwards-in-time
(BIT) communication is the result of quantum action-at-a-distance (QAAD)
induced by breaking the quantum connection! It is a kind of transient
effect vaguely analogous to the induced EMF of Faraday's law in
Maxwell's classical electromagnetic field theory. Bits of information on
the parameters describing the settings of macroscopic parts of the
transmitter disappear from the local eigenvalue probability distribution
(because of "memory" of the connection) and reappear in the local
eigenvalue probability distribution at the receiver. The light at the
receiver will be polarized at a distance in both space and time by the
transmitter.

One can think of the transmitter parameter "phase bits" as being
"teleported" by what Professor Costa de Beauregard calls a "Feynman zig
zag". The bits travel backwards in time from the future detection of the
transmitter photon to the past photon pair emission event in the atom.
The bits then travel forwards in time from the pair emission event to
the future detection of the twin receiver photon. Therefore, direct
space-time separation between the detections of both photons in the same
pair is irrelevant to this communication between those same two events!

This picture demands a "pilot wave" type of theory in the sense of
Bohm's highly nonlocal "quantum potential". The quantum waves are
physically real and distinct from the particles. The advanced quantum
waves carry positive energy backwards in time in the opposite space
direction (i.e., complex conjugate wave). These advanced waves modulate
the retarded waves that carry positive energy forwards in time.
Therefore,each wave loop has exactly zero energy transport in the same
direction of time. The actual energy transport is done by the particle.