From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Add to FUTURE PHYSICS I
Organization: Whole Earth 'Lectronic Link
Date: Thu, 13 Aug 1992 08:39:55 GMT


Mathematical Appendix

Elementary derivation of Equation (6) for superluminal signal.

The actions of the phase plate, the mirror and the beam recombiner on
the transmitter photon kets is the linear transformation

|E> -> e^i(phi+m)2^-1/2|I> +  e^i(phi+m+b)2^-1/2 |II> (7a)

|O> -> e^ib 2^-1/2|I> + 2^-1/2 |II>                   (7b)

where |I> and |II> are the orthonormal inputs into the two transmitter
counters I and II. This transformation of the states of the single
transmitter photon is not unitary but it is physically what happens. It
does not have to be unitary because it is only a description of a part
of the whole. Who can deny that a phase plate makes a phase shift, that
a mirror has a reflection phase plate and that a half-wave plate
properly positioned rotates the incident plane of polarization by 90
degrees? What is the alternative to equations (7a&b) for this apparatus
if not equations (7a&b))? Equations (7a&b) is the what makes the device
work! I call this the spin disentanglement transformation. It changes
the spin part of the pure initial entangled pair state into an impure
disentangled pair state in spin space. There remains, however, a kind of
memory of the initial entanglement in position space. I mean that before
we trace over the measurements on the transmitter side, we still have an
entangled state when we include the position degrees of freedom. The
disentanglement is strictly in spin space beyond space-time which is
where this new kind of communication happens.

Thus, the initial pair state |a,b> of equation (1) becomes

|a,b> = [|e>|E> + |o>|O>]cosx 2^-1/2  + [|e>|O> - |o>|E>]sinx 2^-1/2

-> |a,b>' = [|e> {e^i(phi+m)2^-1/2|I> +  e^i(phi+m+b)2^-1/2 |II>}  +

|o>{e^ib 2^-1/2|I> + 2^-1/2 |II>}]cosx 2^-1/2 +

[|e>{e^ib 2^-1/2|I> + 2^-1/2 |II>}

- |o>{e^i(phi+m)2^-1/2|I> + e^i(phi+m+b)2^-1/2 |II>}]sinx 2^-1/2

= |e>|I>[e^i(phi+m)cosx + e^ib sinx]/2 + |e>|II>[e^i(phi+m+b)cosx + 1]/2

+ |o>|I>[e^ib cosx - e^i(phi+m)sinx]/2 + |o>|II>[cosx - e^i(phi+m+b)]/2

(8)

Therefore, at the receiver

p(e) = |[e^i(phi+m)cosx + e^ib sinx]/2|^2 + |[e^i(phi+m+b)cosx + 1]/2|^2

= [1 + sin2x cos(phi+m)cosb]/2  (9a)

p(o) = |[e^ib cosx - e^i(phi+m)sinx]/2|^2 + |[cosx - e^i(phi+m+b)]/2|^2

= [1 - sin2x cos(phi+m)cosb]/2  (9b)

So that the locally observable probabilities to measure the receiver
photon polarization eigenvalues is conserved and we get the connection
communication equation

p(e) - p(o) = sin2x cos(phi+m)cosb  (6)

This is the complete solution so we can see what happens at the
transmitter

p(I)=|[e^i(phi+m)cosx+e^ib sinx]/2|^2 +|[e^ib cosx-e^i(phi+m)sinx]/2|^2

= 1/2

Similarly p(II) = 1/2 so that probability is conserved on the
transmitter side and there is no trace of the coherent phase parameters
phi,m and b as we would expect for ordinary light on the same apparatus!
The coherent phase parameters have teleported to the other receiver
side.

It only remains to check that the total transformation |a,b> -> |a.b>'
in equation (8) in pair space is unitary. I leave that as a homework
problem!










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