58,748 research outputs found
Orthomodular-Valued Models for Quantum Set Theory
In 1981, Takeuti introduced quantum set theory by constructing a model of set
theory based on quantum logic represented by the lattice of closed linear
subspaces of a Hilbert space in a manner analogous to Boolean-valued models of
set theory, and showed that appropriate counterparts of the axioms of
Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model.
In this paper, we aim at unifying Takeuti's model with Boolean-valued models by
constructing models based on general complete orthomodular lattices, and
generalizing the transfer principle in Boolean-valued models, which asserts
that every theorem in ZFC set theory holds in the models, to a general form
holding in every orthomodular-valued model. One of the central problems in this
program is the well-known arbitrariness in choosing a binary operation for
implication. To clarify what properties are required to obtain the generalized
transfer principle, we introduce a class of binary operations extending the
implication on Boolean logic, called generalized implications, including even
non-polynomially definable operations. We study the properties of those
operations in detail and show that all of them admit the generalized transfer
principle. Moreover, we determine all the polynomially definable operations for
which the generalized transfer principle holds. This result allows us to
abandon the Sasaki arrow originally assumed for Takeuti's model and leads to a
much more flexible approach to quantum set theory.Comment: 25 pages, v2: to appear in Rev. Symb. Logic, v3: corrected typo
Łukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems
A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms
Conformally parametrized surfaces associated with CP^(N-1) sigma models
Two-dimensional conformally parametrized surfaces immersed in the su(N)
algebra are investigated. The focus is on surfaces parametrized by solutions of
the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the
CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for
immersion is determined and an explicit formula for a moving frame on a surface
is constructed. This allows us to determine the structural equations and
geometrical properties of surfaces in R^(N^2-1). The fundamental forms,
Gaussian and mean curvatures, Willmore functional and topological charge of
surfaces are given explicitly in terms of any holomorphic solution of the CP^2
model. The approach is illustrated through several examples, including surfaces
immersed in low-dimensional su(N) algebras.Comment: 32 page
M-brane Models from Non-Abelian Gerbes
We make the observation that M-brane models defined in terms of 3-algebras
can be interpreted as higher gauge theories involving Lie 2-groups. Such gauge
theories arise in particular in the description of non-abelian gerbes. This
observation allows us to put M2- and M5-brane models on equal footing, at least
as far as the gauge structure is concerned. Furthermore, it provides a useful
framework for various generalizations; in particular, it leads to a fully
supersymmetric generalization of a previously proposed set of tensor multiplet
equations.Comment: 17 pages, v2: new SUSY models, version published in JHE
Generalized Gibbs Ensembles in Discrete Quantum Gravity
Maximum entropy principle and Souriau's symplectic generalization of Gibbs states have provided crucial insights leading to extensions of standard equilibrium statistical mechanics and thermodynamics. In this brief contribution, we show how such extensions are instrumental in the setting of discrete quantum gravity, towards providing a covariant statistical framework for the emergence of continuum spacetime. We discuss the significant role played by information-theoretic characterizations of equilibrium. We present the Gibbs state description of the geometry of a tetrahedron and its quantization, thereby providing a statistical description of the characterizing quanta of space in quantum gravity. We use field coherent states for a generalized Gibbs state to write an effective statistical field theory that perturbatively generates 2-complexes, which are discrete spacetime histories in several quantum gravity approaches
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