143 research outputs found
The Logic of Quantum Mechanics Derived from Classical General Relativity
For the first time it is shown that the logic of quantum mechanics can be
derived from Classical Physics. An orthomodular lattice of propositions,
characteristic of quantum logic, is constructed for manifolds in Einstein's
theory of general relativity. A particle is modelled by a topologically
non-trivial 4-manifold with closed timelike curves - a 4-geon, rather than as
an evolving 3-manifold. It is then possible for both the state preparation and
measurement apparatus to constrain the results of experiments. It is shown that
propositions about the results of measurements can satisfy a non-distributive
logic rather than the Boolean logic of classical systems. Reasonable
assumptions about the role of the measurement apparatus leads to an
orthomodular lattice of propositions characteristic of quantum logic.Comment: 16 pages Late
Linearity and Compound Physical Systems: The Case of Two Separated Spin 1/2 Entities
We illustrate some problems that are related to the existence of an
underlying linear structure at the level of the property lattice associated
with a physical system, for the particular case of two explicitly separated
spin 1/2 objects that are considered, and mathematically described, as one
compound system. It is shown that the separated product of the property
lattices corresponding with the two spin 1/2 objects does not have an
underlying linear structure, although the property lattices associated with the
subobjects in isolation manifestly do. This is related at a fundamental level
to the fact that separated products do not behave well with respect to the
covering law (and orthomodularity) of elementary lattice theory. In addition,
we discuss the orthogonality relation associated with the separated product in
general and consider the related problem of the behavior of the corresponding
Sasaki projections as partial state space mappingsComment: 25 page
Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality
We extend the Gelfand-Naimark duality of commutative C*-algebras, "A
COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A
C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a
C*-algebra is isomorphic to the convolution algebra of continuous regular Borel
measures on the topological equivalence relation given by the above mentioned
quotient. In commutative case this reduces to Gelfand-Naimark theorem.
Applications: 1) A simultaneous extension, to arbitrary Hilbert space
operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2)
A functional calculus for arbitrary operators. 3) Affirmative solution of
Invariant Subspace Problem. 4) Extension of Pontryagin duality to nonabelian
groups, and inevitably to groups whose underlying topological space is
noncommutative.Comment: 10 page
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