🚀 go-pugleaf

Review - for the record

Dirac ket derivation of quantum connection phase signal.  Including
irrelevant beam splitter.

|a,b> = {|a,e,+>|b+> + |a,o,->|b->}/sqrt2

|a,e,+> -> e^iphi|a,e,+> -> e^iphi[|1><1|a,e,+> + |2><2|a,e,+>]

|a,o,-> -> |a,o,+> -> |1><1|a,o,+> + |2><2|a,o,+>

|a,b> -> |a,b>'

= {[e^iphi(|1><1|a,e,+> + |2><2|a,e,+>)]|b+>

+ [|1><1|a,o,+> + |2><2|a,o,+>]|b->}/sqrt2

<3|a,b>' = {[e^iphi(<3|1><1|a,e,+> + <3|2><2|a,e,+>)]|b+>

+ [<3|1><1|a,o,+> + <3|2><2|a,o,+>]|b->}/sqrt2

<b'+'|<3|a,b>' = {[e^iphi(<3|1><1|a,e,+> + <3|2><2|a,e,+>)]<b'+'|b+>

+ [<3|1><1|a,o,+> + <3|2><2|a,o,+>]<b'+'|b->}/sqrt2

<b'+'|b+> = cos(theta)

<b'+'|b-> = sin(theta)

<b'+'|<3|a,b>' = {[e^iphi(<3|1><1|a,e,+> + <3|2><2|a,e,+>)]cos(theta)

 + [<3|1><1|a,o,+> + <3|2><2|a,o,+>]sin(theta)}/sqrt2

|1><1| + |2><2| = 1

This shows the beam splitter is irrelevant if it is unitary, which I
assume.

<b'+'|<3|a,b>' = {e^iphi<3|a,e,+>cos(theta) + <3|a,o,+>sin(theta)}/sqrt2

|<b'+'|<3|a,b>'|^2 = |

{e^iphi<3|a,e,+>cos(theta)+<3|a,o,+>sin(theta)}/sqrt2|^2

<3|a,e,+> = |A|e^iphi(e)

<3|a,o,+> = |A|e^iphi(o)

Note |a,e,+> and |a,e,-> span the e-subspace and |a,o,+> and |a,o,-> span
the o-subspace.  Both e and o paths are dumped into |3> totally in first
approximation.  This is part of the physical geometry of guiding collimated
beams into a counter of extended area and volume.

|<b'+'|<3|a,b>'|^2 = (|A|^2/2){1 + sin(2theta)cos(phi+phi(e)-phi(o))}

Similarly,

|<b'-'|<3|a,b>'|^2 = (|A|^2/2){1 - sin(2theta)cos(phi+phi(e)-phi(o))}

Therefore, |A| = 1 and the local probabilities sum to 1 on both sides.  On
the other hand if we interpret them as probability densities. Let

PHI = phi+phi(e)-phi(o)

Integral{|A|^2 p(PHI)d(PHI)} = 1

So again |A| = 1

There is no reason to assume that it is a law of nature that under all
circumstances

Integral{cos(PHI)p(PHI)d(PHI)} = 0

where p(PHI) is the classical probability distribution of phase differences
PHI corresponding to different distinguishable non-interfering "spots" on
the transmitter detector photo-sensitive surface where the transmitter
photon a can be irreversibly and non-unitarily absorbed to make a record.

Indeed, the evolution is unitary between measurements, and non-unitary in
the collapse into |3> on the transmotter side which induces "fringes" at
a distance on the receiver side - no fringes on transmitter side. So
it is effectively unitary in that the sum of local probabilities add to
unity on each side (transmitter and receiver).  The only way to not get
a quantum phase connection polarization signal in standard quantum
mechanical formalism is to introduce an entirely ad-hoc non-quantum idea
that Integral{cos(PHI)p(PHI)d(PHI)} = 0, but even then there is still
a sin(2theta) nonlocal dependence in the local receiver root mean square
fluctuations of the receiver counter photo-currents analogous to the Hanbury
Brown-Twiss correlated intensity fluctuations. The quantum connection phase
signal is also analogous to the nonlocal phase shift of the Bohm Aharonov
effect.  What we have here is simply the classic Heisenberg microscope
double slit experiment in which the photon pair is an inseparable super-
photon. The transmitter part goes through the "two slits" (i.e. ordinary
abd extraordinary paths of transmitter crystal with half-wave plate in one
path - but the receiver part arrives at the (discrete) "screen" and
shows the complementary energy and probability conserving "fringes"
controlled from the transmitter even from the future state of transmitter
in a delayed choice mode. The probability at the transmitter is also
conserved, always 1 with no paradoxical fringes. Everything is globally
consistent and physical. All the abstract proofs to the ocntrary are not
worth the paper or computer space they occupy!