On 06-07-2021 14:10, Bruce Kellett wrote:
On Tue, Jul 6, 2021 at 5:21 PM smitra <[email protected]> wrote:
On 05-07-2021 12:18, Bruce Kellett wrote:
On Mon, Jul 5, 2021 at 7:39 PM smitra <[email protected]> wrote:
arXiv:quant-ph/0412003v2
This is then what I said previously, what you denied, i.e. that
you are
only considering part of the system which is defined by the
reduced
density matrix. The complete system of buckyball plus photons
will
show interference, even if the wavelength is small enough to
resolve the
slits provided you perform the right sort of measurement on the
balls and photons.
That is false.
This is easy to see. Denote the buckyball state of a buckball moving
through the left slit by |L> and moving through the right slit by
|R>.
Suppose that a photon is emitted by the by the buckyballs such that
the
ball moving through the left slit emits a photon in a state |PL>
that
will be orthogonal to the state |PR> of the photon emitted by the
ball
moving through the right slit . The state of the system after the
ball
passes the slits is then:
|psi> = 1/sqrt(2) [|L>|PL> + |R>|PR>]
This state then evolves under unitary time evolution, we can write
the
state just before the ball hits the screen as:
|psi_s> = 1/sqrt(2) [|L_s>|PL_s> + |R_s>|PR_s>]
There is then no interference patter on the screen for the
buckyballs
because |PL_s> and |PR_s> are orthogonal, the unitary time evolution
preserves the orthogonality of the initial states. The probability
to
observe a buckyball on position x on the screen is:
P(x) = ||<x|psi_s>||^2 = 1/2 [|<x|L_s>|^2 + |<x|R_s>|^2] +
Re[<x|L_s>
<x|R_s>* <PR_s|PL_s>]
And the last interference term is zero because <PR_s|PL_s> = 0
But if we also observe the photon on another screen and keep the
joint
count for buckyballs landing on spot x on the buckyball screen and
for
photons landing on spot y on the photon screen as a function of x
and y,
then we do have an interference pattern as a function of x for fixed
y.
If we de note by U the unitary time evolution for the photons until
they
hit their screen, and put |PL_t> =U|PL_s> and |PR_t> = U|PR_s>, then
the
probability distribution is:
P(x,y) = |<x,y|U|psi_s>|^2 = 1/2 [|<x|L_s>|^2|<y|PL_t>|^2 +
|<x|R_s>|^2|<y|PR_t>|^2] +Re[<x|L_s> <x|R_s>* <y|PL_t><y|PR_t>*]
The interference term Re[<x|L_s> <x|R_s>* <y|PL_t><y|PR_t>*] does
not
vanish as it involves evaluating the components of the buckyball
and
photon states in the position basis and so there is no inner product
involved anymore. For fixed y the quantity <y|PL_t><y|PR_t>* will
have
some value that will be nonzero in general, so if we keep y fixed
then
there will be an interference term.
So, we can conclude that invoking escaping IR photons does not male
any
sense in this discussion because all it does is it scrambles the
interference pattern to make it invisible in a way that allows it to
be
recovered in principle using measurements on those IR photons. You
can,
of course, erase the interference patter by measuring the observable
for
the photons that has |PR> and |PL> as its eigenstates. But even in
that
case the information will still be there in the state of all the
atoms
of the measurement apparatus for the photons. But if you don't
perform
any measurement then the information will simply continue to exists
in
the escaping photons.
So, in general we can conclude by generalizing this to any large
number
of particles that even with what we consider to be permanent
records,
you don't get rid of the theoretical possibility of interference
between
the sectors where those records are different. So, the existence of
parallel worlds cannot be made fully 100% irrelevant if QM is
rigorously
correct, and we cannot therefore argue that QM is exactly equivalent
to
an alternative theory that leaves out parallel worlds. Even though
the
difference may be almost 100% insignificant FAPP, it's not exactly
100%
even in the macroscopic realm.
It is a shame that your fancy analysis is contradicted by the actual
experimental results. I leave it as an exercise for you to determine
where your mistake is. But I suggest that you actually reads the paper
quant-ph/0412003.
The analysis is 100% rigorous, based on standard QM. The results of the
paper quant-ph/0412003 are irrelevant as they don't measure the joint
interference pattern. In that case the interference pattern of the balls
will vanish due to the photon states being orthogonal, as I've shown
above.
The argument against the existence of parallel worlds by invoking
decoherence that makes superposition hard to detect for complex
systems
is thus analogous to the defense of creationists when they invoke a
God
of ever smaller gaps of things that have not yet been fully
explained.
My dear, you really have lost the plot, haven't you Saibal?
I'm sticking to QM, your position depends on some unproven effect that
would make pure states evolve into mixed states.
Saibal
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