🚀 go-pugleaf

OK boys and girls. I may have found an error in my "retraction proof" that
standard unitary reversible (forget dissipation) quantum mechanics does not
permit the quantum connection signal.  If anyone can find an error in the
following argument I would like to know.

Assume that the source of back-to-back photon pairs emits the standard
state used by Aspect in the Paris experiment.

|1,2> = [|1e+>|2e+> + |1o->|2o->]/rt2   (1)

Let's go over the notation.  It is assumed that at the relativistic space-
time "point" (approximation) event where and when "transmitter" photon 1
locally interacts with a doubly-refracting polarizing beam splitter
oriented in space such that its polarization basis of that is (e+,o-) ( "e"
= extraordinary, "o" = ordinary in sense of classical crystal optics).

Note that nonlocal quantum action-at-a-distance is already built into the
particular entangled structure of equation (1) because both connected
photons 1 and 2 are written in the same basis!  This is why standard
quantum mechanics violates the local classical relativistic Einsteinian
causality built into Bell's inequality.  This is why Professor Costa De
Beauregard (Institut Henri Poincare, Paris) uses the Feynman "zig zag"
picture in which advanced information from the detection of photon 1 is
transmitted backwards in time the the past pair emission to prepare photon
2 in the same (e+,o-) polarization basis that photon 1 is prepared in.
Thus, photon 1 is locally polarized by its future interaction with its beam
splitter and its twin photon 2 is nonlocally retroactively polarized and
subsequently locally analyzed when it scatters off its beam splitter
spatially oriented in basis (e'+.o'-) at angle @ relative to (e+.o-) about
the common line of flight in opposite directions from the source.

The two narrow collimated beams e and o (i.e. alternative paths of single
photon 1) are brought to a spot represented by |1&> in which & is the
relative phase difference.  There are several diffraction limited spots and
the completeness condition (i.e. summing over all spots where photon 1 can
be detected) is

S(&)|1&><1&| = 1     (2)

where S(&) means "take the sum over all &".

|1e+> -> e^i&|1e+>  (3a)

|1o-> -> |1o->      (3b)

The distorted pair state made by the "transmitter interferometer" is then

|1,2>' = [e^i&|1e+>|2e+> + |1o->|2o->]/rt2   (4)

Note the orthogonality

<1e+|1o-> = 0     (5)

I am omitting the half-wave plate in the 1o- path of previous designs since
we do not care about getting first order fringes at the transmitter.  Such
fringes would be impossible anyway for the correlated light because of the
the orthogonality

<2e+|2o-> = 0     (5)

of the twin photon 2 states in the nonlocal entangled structure of equation
(4).

The nonlocal joint probability amplitude to detect photon 1 at spot & and
photon 2 in path e'+ is the inner product (unitary time evolution operators
sandwiched between bras and kets not explicitly represented since they
would only severely complicate the formalism and detract from the physical
understanding)

<2e'+|<1&|1,2>' = [e^i&<1&|1e+><2e'+|2e+> + <1&|1o-><2e'+|2o->]/rt2  (6)

the nonlocal joint probability is the mod square of the amplitude

|<2e'+|<1&|1,2>'|^2 = |<1&|1e+><2e'+|2e+> + <1&|1o-><2e'+|2o->|^2/2  (7)

The local probability at the counter e'+ for "receiver" photon 2 is the sum
(S(&)) of this nonlocal joint probability over all places & where photon 1
can be absorbed.  This is

S(&)|<2e'+|<1&|1,2>'|^2 = S(&)|<1&|1e+><2e'+|2e+> + <1&|1o-><2e'+|2o->|^2/2

= {|<2e'+|2e+>|^2S(&)<1e+|&><&|1e+> +  |<2e'+|2o->|^2S(&)<1o-|&><&|1o->

+ <2e+|2e'+><2e'+|2o->S(&)e^-i&<1e+|1&><1&|1o->

+ <2e'+|2e+><2o-|2e'+>S(&)e^i&<1o-|1&><1&|1e+>}/2    (8a)

Let us simplfy this using the standard group theory formulas for the spin 1
photon

<2e'+|2e+> = cos@   (9a)

<2e'+|2o-> = sin@   (9b)

Therefore,

S(&)|<2e'+<1&|1,2>'|^2 = S(&)|<1&|1e+><2e'+|2e+> + <1&|1o-><2e'+|2o->|^2/2

= {(cos@)^2S(&)<1e+|&><&|1e+> +  (sin@)^2S(&)<1o-|&><&|1o->

+ cos@sin@S(&)e^-i&<1e+|1&><1&|1o->

+ cos@sin@S(&)e^i&<1o-|1&><1&|1e+>}/2    (8b)

The completeness equation (2) which describes conservation of local
probability for the detection of transmitter photon 1 implies

S(&)<1e+|&><&|1e+> = <1e+|1e+> = 1  (10a)

similarly

S(&)<1o-|&><&|1o-> = <1o-|1o-> = 1  (10b)

*But now we come to the important point. Look at complex conjugate pair of
interference terms which carries the quantum connection signal. We have a
"Fourier transform" (at unit "angular momentum" canonically conjugate to
phase &). For example,

S(&)e^-i&<1e+|1&><1&|1o-> = <1e+|{S(&)e^-i&|1&><1&|}|1o->       (11a)

If the modulating phase factor e^-i& were not there, then, the completeness
equation (2) would make the interference term vanish in accord with Henry
Stapp's expectations. But the phase factor is there and we cannot conclude
that

S(&)e^-i&|1&><1&| = 1   (11b)   (FALSE!)

simply because

S(&)|1&><1&| = 1   (2)

Therefore, even though the "bare" inner product vanishes, i.e.,

<1e+|1o-> = 0  (5)

We cannot logically conclude that the "dressed" inner product (i.e., matrix
element of operator S(&)e^-i&|1&><1&| sandwiched between these orthogonal
states) also vanishes.

The quantum connection signal s would then depend upon the nonlocal
controllable unhidden parameter @ (the angle between the two polarizing
beam splitters) and it would be the difference in the outputs of the two
(i.e. e'+ and o'-) receiver photon 2 counters given by

s = cos@sin@S(&){e^-i&<1e+|1&><1&|1o->+ e^i&<1o-|1&><1&|1e+>}   (12)