Empirical algebraic geometry

In a previous paper we introduced the notion of an orthogonal category and generalized the notion of a sheaf of sets on a complete Boolean algebraB to that of a sheaf on the complete Boolean algebraB with values in an orthogonal categoryA. By properly replacing the complete Boolean algebraB by a manualM of Boolean locales, we get a notion of a sheaf onM with values inA, which can be regarded as a quantum generalization of a sheaf onB. TakingA to be the category of sheaves of Abelian groups or that of schemes à la Grothendieck, we will discuss some fundamental aspects of the quantum generalizations of sheaves and schemes.     

Oscillatory Modules

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow.     

Quantum Event Structures from the Perspective of Grothendieck Topoi

We develop a categorical scheme of interpretation of quantum event structures from the viewpoint of Grothendieck topoi. The construction is based on the existence of an adjunctive correspondence between Boolean presheaves of event algebras and Quantum event algebras, which we construct explicitly. We show that the established adjunction can be transformed to a categorical equivalence if the base category of Boolean event algebras, defining variation, is endowed with a suitable Grothendieck topology of covering systems. The scheme leads to a sheaf theoretical representation of Quantum structure in terms of variation taking place over epimorphic families of Boolean reference frames.     

Boolean coverings of quantum observable structure: a setting for an abstract differential geometric mechanism

We develop the idea of employing localization systems of Boolean coverings, associated with measurement situations, in order to comprehend structures of quantum observables. In this manner, Boolean domain observables constitute structure sheaves of coordinatization coefficients in the attempt to probe the quantum world. Interpretational aspects of the proposed scheme are discussed with respect to a functorial formulation of information exchange, as well as, quantum logical considerations. Finally, the sheaf theoretical construction suggests an operationally intuitive method to develop differential geometric concepts in the quantum regime.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

A remark on a new category of supermanifolds

In a previous paper a new category of supermanifolds, called -supermanifolds, was introduced. The objects of that category are pairs (M, ), with M a topological space and a suitably defined sheaf of ₂ -graded commutative B_L - algebras, B_L being a Grassmann algebra with L generators. In this note we complete the analysis of that category by showing that is isomorphic with the sheaf of - maps M → B_L.     

Empirical quantum mechanics

Machida and Namiki developed a many-Hilbert-spaces formalism for dealing with the interaction between a quantum object and a measuring apparatus. Their mathematically rugged formalism was polished first by Araki from an operator-algebraic standpoint and then by Ozawa for Boolean quantum mechanics, which approaches a quantum system with a compatible family of continuous superselection rules from a notable and perspicacious viewpoint. On the other hand, Foulis and Randall set up a formal theory for the empirical foundation of all sciences, at the hub of which lies the notion of a manual of operations. They deem an operation as the set of possible outcomes and put down a manual of operations at a family of partially overlapping operations. Their notion of a manual of operations was incorporated into a category-theoretic standpoint into that of a manual of Boolean locales by Nishimura, who looked upon an operation as the complete Boolean algebra of observable events. Considering a family of Hilbert spaces not over a single Boolean locale but over a manual of Boolean locales as a whole, Ozawa's Boolean quantum mechanics is elevated into empirical quantum mechanics, which is, roughly speaking, the study of quantum systems with incompatible families of continuous superselection rules. To this end, we are obliged to develop empirical Hilbert space theory. In particular, empirical versions of the square root lemma for bounded positive operators, the spectral theorem for (possibly unbounded) self-adjoint operators, and Stone's theorem for one-parameter unitary groups are established.     

Indeterministic Objects in the Category of Effect Algebras and the Passage to the Semiclassical Limit

We consider a category of effect algebras and formulate an abstract-hidden variables problem for an object of this category. A notion of indeterministic object is introduced as of an object which induces a Kochen–Specker-type contradiction. A sufficient condition for an object to be indeterministic is derived. An abstract algebraic point of view on a no-hidden variables example constructed by Mermin is given. The notion of a passage to the semiclassical limit is analyzed and refined.     

The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.     

On Quantum Event Structures Part I: The Categorical Scheme

In this paper a mathematical scheme for the analysis of quantum event structures is being proposed based on category theoretical methods. It is shown that there exists an adjunctive correspondence between Boolean presheaves of event algebras and quantum event algebras. The adjunction permits a characterization of quantum event structures as Boolean manifolds of event structures.     

Noncommutative space from dynamical noncommutative geometries

We present noncommutative topology as a basis for noncommutative geometry phrased completely in terms of partially ordered sets with operations. In this note we introduce a noncommutative space-time starting from a dynamical system of noncommutative topologies based on the notion of temporal points. At every moment a commutative topological space is constructed and it is shown to approximate the noncommutative space in sheaf theoretical terms; this so called moment space should be the space where observed phenomena should be described, the commutative shadow of the noncommutative space is to be thought of as the usual space-time.     

Representations of empirical set theories

Any manual of Boolean locales in the strong sense, namely a subcategory of the opposite category of the category of complete Boolean algebras and complete Boolean homomorphisms satisfying not only conditions (3.1)–(3.10) of our previous paper [International Journal of Theoretical Physics,32, 1293 (1993b)], but also conditions (4.1)–(4.4) of that paper, is shown to be representable as the second-class orthomodular manual of Boolean locales on an orthomodular poset In this sense the study on manuals of Boolean locales in the strong sense is tantamount to the study on a special class of orthomodular posets, though our viewpoint is radically different from the conventional one in the traditional approach to orthomodular posets. Then the notion of a manual of Hilbert spaces or exactly what is called a manual of Hilbert locales is introduced, over which a variant of the celebrated Gelfand-Naimark-Segal theorem for a manual of Boolean locales in the strong sense is established.     

Remark on a paper of Mņczyński

In his paper, Boolean Properties of Observables in Axiomatic Quantum Mechanics,M$\text{n}\text{?}$czyński formulates the notion of a Boolean representation of the magnitudes of a physical theory, and proves the theorem (4.1) that a family P of magnitudes has a Boolean representation if and only if there is a homomorphism from $\text{L}$into a Boolean algebra, where $\text{L}$ is the orthomodular poset associated with P. Exploiting Finch's representation theorem for orthomodular posets, M$\text{n}\text{?}$czyński is able to show that this condition is satisfied if and only if the Boolean subalgebras of $\text{L}$ have a nondegenerate direct limit. The direct limit is then a maximal Boolean representation of P. Homomorphic relations were introduced by Kochen and Specker as a weakening of the concept of an imbedding between partial algebras. The purpose of this remark is to show that the direct limit of the Boolean subalgebras of $\text{L}$ has a natural characterization in terms of the notion of a homomorphic relation between $\text{L}$ and the direct product of its Boolean subalgebras.     

Infinitesimal Deformation Quantization of Complex Analytic Spaces

The quantization problem for analytic algebras and for complex analytic spaces is discussed. The construction of Hochschild cohomology is modified for this category. It is proved that this cohomology is always a coherent analytic sheaf in each degree.     

Category-Theoretic Interpretative Framework of the Complementarity Principle in Quantum Mechanics

This study aims to provide an analysis of the complementarity principle in quantum theory through the establishment of partial structural congruence relations between the quantum and Boolean kinds of event structure. Specifically, on the basis of the existence of a categorical adjunction between the category of quantum event algebras and the category of presheaves of variable Boolean event algebras, we establish a twofold complementarity scheme consisting of a generalized/global and a restricted/local conceptual dimension, where the latter conception is subordinate to and constrained by the former. In this respect, complementarity is not only understood as a relation between mutually exclusive experimental arrangements or contexts of comeasurable observables, as envisaged by the original conception, but it is primarily comprehended as a reciprocal relation concerning information transfer between two hierarchically different structural kinds of event structure that can be brought into partial congruence by means of the established adjunction. It is further argued that the proposed category-theoretic framework of complementarity naturally advances a contextual realist conceptual stance towards our deeper understanding of the microphysical nature of reality.

     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(φ,A) of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(φ,A), where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,φ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B, s₀,φ), where B is a Boolean algebra and s₀ is a state on B.     

Empirical sets

The pillar concept of Foulis and Randall's school is surely that of a manual of operations. They chose to regard an operation as the set of possible outcomes, thereby taking a manual of operations to be a family of partially overlapping operations. Our previous work is a development of their ideas in two points. First, each operation is represented not by the set of possible outcomes, but by the complete Boolean algebra of observable events. Second, since each complete Boolean algebraB possesses the Scott-Solovay modelV ^(B) of classical set theory as its higher-order companion, the Scott-Solovay universes of all the operations in the manual lump together into a family of Boolean set theories interconnected by geometric morphisms, which we suggestively designated an empirical set theory. The principal concern of this paper is to show how to get a cross-operational set concept by choosing an internal set withinV ^(B) for each operationB in the manual and bundling them up. The resulting structure is denominated an empirical set. We show that the category of empirical sets is complete, is cocomplete, has a subobject classifier for well-rounded subobjects, and has exponentials only for degraded objects.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

Tropical mathematics and the financial catastrophe of the 17th century. Thermoeconomics of Russia in the early 20th century

In the paper, an example is presented concerning relationships (which cannot be neglected) between mathematics and other sciences. In particular, the relationship between the tropical mathematics and the humanitarian-economic catastrophe of 17th century (related to slavery of Africans) is considered. The notion of critical state of economy of the 19th century is introduced by using the refined Fisher equation. A correspondence principle for thermodynamics of fluids and economics of the 19th century is presented.