🚀 go-pugleaf

Some gifted high school students could follow the elementary
math and logic of the following:

I here re-examine the question: Does standard quantum
mechanics, in the Feynman history interpretation, permit
faster-than-light and backwards-in-time (i.e. strong delayed
choice) communication in which the Einstein-Podolsky-Rosen-
Bohm nonlocal quantum connection is the locally decodable
channel?

That is, there is no need to compare data from both ends.
The polarization of the receiver light will depend on the
nonlocal parameter @ controlled from the transmitter.

If the answer is affirmative, does this mean a conflict
between the Feynman version of quantum mechanics and the
Dirac "bra-ket" formalism which prevents such communication
by positing the invariance of "bra-kets" under time
evolution between measurements?

                                     5
                                     |
                                     |
                                     |
       <--------@------->            |
1-----|         S        |-3---------\&---------6
      |---------|--------|           |
2-----|                  |-4---|-----/
      P                  p     H
receive                  transmit

Experiment I

S = photon pair source (Ca atomic cascade as in Aspect
experiment J = 0 -> 1 -> 0)
P = receiver calcite
p = transmitter calcite
@ = relative angle between calcites at times (T, t)
of detection of receiver photon and twin transmitter photon
in same individual pair, respectively.  That is,
@ = @(T) - @(t)
so that @ is a controllable nonlocal parameter (unhidden
variable)
& is relative reflection-transmission phase shift at the
half-silvered beam recombiner in the transmiotter
interferometer
H is a half-wave plate that rotates the plane of
polarization of the transmitter photon by 90 degrees

Given the intial entangled photon pair state

Pair = [|+>|+> + |->|->]/rt2

standard quantum mechanics implies the following nonlocal
Feynman amplitudes

A(1,3) = A(2,4) = cos@/rt2

A(1,4) = -A(2,3) = sin@/rt2

For example, in the standard Aspect experiment, without
the additional apparatus beyond 3 and 4, the light at each
end is unpolarized independent of @ so that there is no
quantum connection communication in that particular total
experimental arrangement.  Thus, using the rule to square
amplitudes before adding for distinguishable alternatives.
The local probability for a "click" at detector 1 is

p(1) = |A(1,3)|^2 + |A(1,4)|^2 = 1/2

Let us do the same calculation for the new apparatus in
Experiment 1 above.

A(1,5) = A(1,3)e^i& + A(1,4)

A(1,6) = A(1,3) + A(1,4)e^i&

Notice that the half-wave plate is essential for the above
equations since it makes the two paths 3 and 4
indistinguishable. That is, the half-wave plate H "quantum
erases" the polarization distinguishability of the
transmitter calcite p.

Therefore,

p(1) = |A(1,5)|^2 + |A(1,6)|^2

     = (1 + sin2@ cos&)/2

similarly,

p(2) = (1 - sin2@ cos&)/2

Therefore, the polarization of the receiver light is

p(1) - p(2) = sin2@cos&

the sin2@ reminds one of the Josephson effect.

The above computation is correct quantum mechanics. However,
it is incomplete. As Aephraim Steinberg has pointed out,
conservation of energy and time reversal symmetry for a
passive (i.e., no laser amplification) beam recombiner force

cos& = 0

in order that destructive interference occur so that energy
is conserved in the time reversal.  Penrose has discussed
some paradoxical features of this argument in THE EMPEROR'S
NEW MIND.

Making the recombiner a laser amplifier might pull & away
from pi/2. We may then come up against the "no photon clone"
issue?  One may also wonder whether fluctuations in & may
lead to an observable effect? That is, even though cos& = 0,
the cosine of the root mean square fluctuation of & will not
vanish and this may make it possible to see a communication
effect by a fractal analysis on the time series of the
receiver data?

There is still another possibility presented in Experiment
II below.

                                                      7
                                                    /
                                                  /
                                     /---------_|_?
                                     |          |
                                     |          |
                                     |          |
       <--------@------->      &     |          |
1-----|         S        |-3---------\C---------/
      |---------|--------|           |
2-----|                  |-4---|-----/
      P                  p     H
receive                  transmit

Experiment II

If a detector ? can be made with an output to 7 that cannot
distinguish which path, then

p(1) = |A(1,5) + A(1,6)|^2

     = (1 + sin2@)/2

p(2) = (1 - sin2@)/2

so that the locally observable polarization at the receiver
is precisely the Josephson form sin2@.