.
The paper in question examines the possibility of formalizing
the Mach-Zehnder interferometer experiment
for one particle (the authors prefers the term quantum), using this as an
indicator of a possible system of logic that could account for the
corpuscular interpretation of quantum theory.
It should the mentioned that the symbol of entailment was not properly
embedded in the pdf file I received (appearing as ``plus/minus"), while
the symbol for the necessity modal operator was not printed at all;
these omissions did not impair reading, but care must be exercised
in printing a final copy.
Section one is devoted to a description of the Mach-Zehnder
interferometer,
while the second discusses briefly distinct point of view regarding
quantum
physics, indicating the author's opinion of why the ``natural" logic of
quantum mechanics should be non-classical, a theme touched again at the
end of section 3, which presents the short-comings of an interpretation
based on the orto-modular lattices that arise from the closed sub-spaces
of
a Hilbert space. The last paragraph of page 3, the author says that the
interpretation of the logical connective of disjunction in this type
of logic is distinct from the classical notion of disjunction.
This is not quite true, since the orthogonal sum of sub-spaces is
precisely
the join (or least upper bound) of the given sub-spaces, just as the union
is the least upper bound of two subsets of a set. Section 3 is mostly
devoted to showing, using a spin 1/2 particle (or is it quantum ?)
example,
originating with Hughes, that distributivity is violated in quantum logic
(precisely what is one referring to by this last expression ?).
In section 4, the author argues, using a simple formalization of the
Mach-Zehnder interferometer experiment for one photon that
if the the usual or-elimination rule of classical logic is valid, then
the premiss that the photon is either in A or in B (corresponding to
distinct paths in the experiment setup) has to be false.
It also argued that the attempted formalization is not inside quantum
logic,
since one of the sentences in the premisses involves probability, while
each
sentence of non-distributive quantum logic should be associated to a
subspace
of the Hilbert space of the system being examined. Again, this not quite
precise, since the square integrable functions on any probability space is
a perfectly good Hilbert space and, in fact, it can be shown that
spaces of this type are isometric (not only isomorphic) to any given
Hilbert space. The last sentence of section 4
``How should be proceed to offer..."
should read
``How should we proceed to offer..."
In section 5, the author presents his case for introducing modalities into
the picture. Since necessity does distribute over joins,
one is able to block the undesirable conclusion of the formalization
of the Mach-Zehnder interferometer experiment, while keeping the validity
of all premisses that are should be valid in a strictly corpuscular
interpretation of quantum physics.
Although the proposal blocks the undesirable argument,
there is need of considerably more evidence that it produces a system
capable of constituting a foundation of quantum physics for the
corpuscular interpretation. The situation here is akin to Russel's
paradox, showing that Frege's idea of a set was untenable. But it took
the Fraenkel's seminal idea of introducing the axiom if substitution
to remove the limitations imposed on what remained of Set Theory.
It seems clear that there is still a fair amount of work to be done before
one can talk of a ``modal quantum logic" (in this respect, perhaps the
first
sentence of footnote 4 in page 7 should, after careful thought, be
omitted.) :
--- What are its axioms and rules (the author himself
raises questions about the role of conjunction, the EPR paradox
and Bell's theorem) ?
--- What are the models of the ensuing system ? The author mentions
``stochastic corpuscular view, obeying a modal logic"; how would
one make this concrete, obtaining basic results such as
soundness and (perhaps) completeness ?
Understood as the proposal of a direction of research, and rewritten
to make that clearer,
the present paper merits publication, since it might stimulate
interesting philosophical discussion and mathematical work.
As a final comment, it would be interesting to see if (some form of)
linear logic, with its distinct conjunctions, disjunctions
(join and par) and implications, eventually with modalities, might
not add profitably to the foundations of quantum physics, particularly
to the strictly corpuscular interpretation of this discipline.