What is"Dynamic Operational Quantum
Logic", or "Intuitionistic Quantum Logic", or "New-Style Operational Quantum
Logic", or "Quantum Logicality"?
A lot of names used for essentially
the same thing. In short, the "failure" of old-style quantum logic,
either operational or not (we exclude here Foulis-Randall-style empirical
statistics which did survive) in the following two senses,
The rejection of "orthomodular logic",
that is, Sasaki hook based logic, by the traditional logic community, because
of being essentially "deductively useless".
are essentially due to a failure
to recognize a fundamentally dynamic ingredient in the kinematics of quantum
theory itself. Formally, the core of construction relies on two pillars:
The stagnation of "old-style operational
quantum logic" a la Jauch-Piron including a lack of progress for some twenty
years, what made it vanish from the dictionary of theoretical physics.
A principle (that can be proved)
called "Causal Duality". Causal duality in itself provides an interpretation
for orthomodularity, formally due to the equivalence of the latter with
Sasaki adjunction. Orthomodularity as such encodes the fundamental
dynamic nature of quantum kinematics.
(a more subtle less sloganistic
outline will be put here soon)
The (trivial) observation that lattice
of closed subspaces of a Hilbert space can be equivalently represented
as a set that goes equipped with an "operational resolution". The
non-triviality of the operational resolution (that is, not being an identity)
is what distinguishes quantum from classical. The construction operationally
coincides with the injective hull construction for meet-semilattices (yielding
a complete Heyting algebra).
B. Coecke: 'Quantum Logic
in Intuitionistic Perspective' (arXiv: math.LO/0011208), Studia Logica
B. Coecke, and S. Smets: 'The
Sasaki-Hook is not a Static Implicative Connective but Induces a Backward
(in Time) Dynamic One that Assigns Causes' (quant-ph/0111076).
I. Stubbe ``What
can we certainly say about what may possibly be? The operational resolution,
quantales of induced state/property transitions, and causal duality as
crux of the biscuit" Notes of talk at Comlab, Oxford. SURVEY
+ REFS (2001).
Survey of the "old-style results"
and the subject in general, including Foulis-Randall-style empirical statistics
which is still an active field:
D.J. Foulis: 'A
Half Century of Quantum Logic What Have we Learned?', Talk (1995).
B. Coecke, D.J. Moore and A. Wilce: 'Operational
Quantum Logic: An Overview' (arXiv: quant-ph/0008019) In: Current Research
in Operational Quantum Logic, B.Coecke, D.J. Moore and A. Wilce (Eds.),
Kluwer Academic Publishers, 2000.