From an e-mail I wrote to Jerry Finkelstein
on Jan 10, 2001; I refer to Kim and Shih (KS),
quant-ph/9905039 and my paper quant-ph/0010030:
I think the situation
in Popperīs experiment in KSīs version is finally becoming clear to me.
Because of the finite momentum spread \delta(p) of the initial
state KSīs virtual slit at distance d from the
crystal with size \delta(x) has a slit width s which fulfills:
s >= (\delta(p)/p) d = (h/(\delta(x) p)) d (1)
Now we ask: what is the angular spread \delta(\phi)
(the quantity measured by KS) induced by s
according to the uncertainty relation?
\delta(\phi)= h/sp = \delta(x)/d (2)
where I inserted eq (1).
Next question:
what is the angular spread at the virtual slit BEFORE the
photons position at the real slit is measured?
Because the photons necessarily come from all parts of the crystal
in KSīs setup from
elementary geometry it is clear that the spread is
\delta(\phi)= \delta(x)/d (3)
i.e. the spread from "virtual diffraction" was already present
before, and independent of, any localisation of the opposite photon.
Therefore: the photon obeys the uncertainty relation in the virtual slit
(your point) but there is no special "virtual diffraction" effect
due to the localisation of the opposite photon in a real slit
(Asherīs point).
KS overlooked that their virtual slit width is s (eq(1)) rather than
the real-slit width. If they assume a virtual slit width s instead of 0.16 mm
their experimental result seems to be in agreement with the uncertainty
relation (my previous email and Tonyīs point).
My paper is wrong because I uncritically believed their conclusion.
"CONCL" is a misguided attempt to understand their wrong conclusion.
Actually KSīs whole idea for the setup is flawed.
Collettīs point in in 87 Nature paper was:
the necessary finite size of the source precludes precise localisation.
KS found a set-up around this, but what they overlook is that this setup
introduces an angular spread at the
virtual slit that enforces the uncertainty relation already
in the initial state before virtual localisation.
So in final consequence it now seems to me that Popperīs fundamental
idea (virtual localisation of a state with \delta(p)<