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In 2004, Shahriar Afshar promoted widely in the yellow
press the bombastic claim that
Bohr’s principle of complementarity is false, based
on a mathematically inconsistent
interpretation of a simple lens experiment that has been
called ‘Afshar’s experiment‘ (Chown,
2004). Despite the fact that Afshar’s manuscript (Afshar,
2003) was submitted and rejected
by Physical Review Letters, Afshar did not get frustrated
and started the propaganda rapidly
in popular science magazines and newspapers like: El Cultural
(September 9, 2004),
The Philosophers’ Magazine (October 2, 2004), The Independent
(October 6, 2004), Galileu
Magazine (December 2004), and OE Magazine (January 2005).
Nevertheless, the most cited
source for discussing Afshar’s experiment and Afshar’s
interpretation was undoubtedly the
cover story in the July 24, 2004 edition of New Scientist
(Chown, 2004). Possibly there
wouldn’t be any fuss around Afshar’s experiment
if Afshar’s interpretation was not backed
up by the authority of Prof. John G Cramer, known as the
father of the Transactional
Interpretation of Quantum Mechanics (Cramer, 2004 and 2005).
Immediately after the appearance of the New Scientist
article, there appeared two main critiques by Unruh (2004)
and Motl (2004). While both of them wrongly agreed that
without wire grid, Afshar’s experiment does measure
the which way information, there were major differences
in the arguments explaining why Afshar was wrong. Motl did
some wrong calculations in order to argue that putting the
wire grid decreases the which way information due to diffraction
from the wires, while Unruh made substantial progress in
replacing Afshar’s setup with an equivalent one consisting
of two Mach-Zehnder interferometers, called later ‘Unruh’s
experiment’ (Georgiev, 2007a and 2007b). Unfortunately,
Unruh then made wrong calculations, and also wrongly claimed
that without obstacles there is which way information. It
seemed that as if the physicists concerned with Afshar’s
setup were blind to the fact that Quantum Mechanics is governed
by non-Aristotelian logic, and therefore, the extrapolation
of mixed setups to coherent setups is not permissible. Fortunately,
late in 2006, there appeared the first complete proof of
non-existent which way information (Georgiev, 2006), which
was independently backed up by two more preprints (Reitzner,
2007; and Qureshi, 2007). The main dispute between Unruh
and Georgiev took place on the pages of the journal Progress
Ii Physics, where exact mapping between Afshar’s and
Unruh’s setup was presented, and it was shown that
Unruh’s interpretation is mathematically inconsistent.
Due to possibility of misunderstanding of the main paper
(Georgiev, 2007a) disproving Unruh’s interpretation
due to minor slipped typos and ambiguities in the text,
we present here the proof of Unruh’s inconsistency,
using quantum operators.1 The present exposition is intended
to be a self-consistent, polished version of the proof of
non-existent which way information in Unruh’s and Afshar’s
setups, and is written in the formalism of quantum operators.
Let us now discuss in some detail the mathematical inconsistency
resulting from improper
usage of ‘quantum operators’ as done in Unruh
(2004 and 2007). Since the attempt is to
discuss intermediate events, we have to use for each history
a sequence of operators. In the
current paper as well as in previous work (Georgiev, 2007a
and 2007b), the correct notion of
photon path is described by a ‘sequence of operators’
forming a history or Feynman path.
Only in this scenario one may ask questions on what happens
in intermediary steps such as
interferometer arms 5 or 6, and may infer the existence
of quantum interference.
Each complete history Si , where i = {1, 2, ... , 8}, cannot
be thought of as a single event,
and is therefore, mathematically not equivalent to other
histories, even in case where the
final output at detectors is identical to the output of
alternative history. It is exactly the
existence of intermediate events that makes alternative
histories mathematically inequivalent.
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