arXiv subjects

Wladyslaw Marcinek

Publications and source records attributed to Wladyslaw Marcinek.

16 recordsLinked to original sources

arXiv2002

Quantum-Classical Interactions and Galois Type Extensions

An algebraic model for the relation between a certain classical particle system and the quantum environment is proposed. The quantum environment is described by the category of possible quantum states. The initial particle system is represented by an associative algebra in the category of states. The key new observation is that particle interactions with the quantum environment can be described in terms of Hopf-Galois theory. This opens up a possibility to use quantum groups in our model of particle interactions.

math.QA↗

arXiv2002

Quantum-Classical Correspondence and Galois Extensions

A proposal of an algebraic model for the relation between a quantum environment and certain classical particle system is given. The quantum environment is described by a category of possible quantum states, the initial particle system is represented by an associative algebra in the category of states. The result of particle interactions with quantum environment is described as Galois extension. A physical applications to highly organized structures of matter are discussed.

math.QA↗

arXiv2002

Regular obstructed categories and TQFT

A proposal of the concept of $n$-regular obstructed categories is given. The corresponding regularity conditions for mappings, morphisms and related structures in categories are considered. An n-regular TQFT is introduced. It is shown the connection of time reversibility with the regularity.

math.QA↗

arXiv2001

Regular graded algebras and obstructed categories with duality

Regular and higher regular graded algebras (in simplest case satisfying Von Neumann regularity $Θ_{1}Θ_{2}Θ_{1}=Θ_{1}$ instead of anticommutativity) are introduced and their properties are studied. They are described in terms of obstructed categories with nonclosed invertible and noninvertible morphisms for which generalized (obstructed) functors and natural transformations are given. Regular algebras and bialgebras are considered with examples. Corresponding regularizations of the cross product and Wick algebras are made.

math.QA↗

arXiv2001

n-Categories Admissible by n-graph

The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.

math.CT↗

arXiv2001

Particles, Generalized Statistics and Categories

Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.

math.QA↗

arXiv2000

Noninvertibility, semisupermanifolds and categories regularization

The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions. The concepts of regular n-cycles, obstruction and the regularization procedure are introduced and investigated. It is shown that the regularization of a category with nonivertible morphisms and obstruction form a 2-category. The generalization of functors, Yang-Baxter equation, (co-) algebras, (co-) modules and some related structures to the regular case is given.

math-ph↗

arXiv2000

Higher regularity properties of mappings and morphisms

We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and ``semicommutative'' cases is measured by non-zero obstruction proportional to the difference of some self-mappings (obstructors) $e^{(n)}$ from the identity. This allows us to generalize the notion of functor and to ``regularize'' braidings and related structures in monoidal categories. A ``noninvertible'' analog of the Yang-Baxter equation is proposed.

math-ph↗

arXiv2000

Some Comments on Categories, Particles and Interactions

An algebraic formalism for the study of interacting particle systems is developed. Particle processes are described in terms of the category theory. The problem for the unique description of these processes is discussed. Categories relevant for this subject are described. The concept of generalized transmutations of interacting particle systems is considered. The connection with a system with some generalized statistics is explained.

math.QA↗

arXiv1999

On Generalized Statistics and Interactions

The concept of exchange braid statistics is generalized. The cross statistics is studied as a result of interaction. An algebraic model of a system of particles equipped with such statistics is described. The corresponding Fock space representation is also given.

math.QA↗

arXiv1999

Remarks on Quantum Statistics II

Some problems related to an algebraic approach to quantum statistics are discussed. Generalized quantum statistics is described as a result of interactions. The Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. An example of physical effect in system with generalized statistics is also given.

math.QA↗

arXiv1998

On composite systems and quaqntum statistics

An algebraic formalism for the study of a system of charged particles interacting with an external quantum field is developed. The notion of monoidal categories with duality is used for the description of composite systems and corresponding quantum statistics. The Fock space representation is also discussed.

math.QA↗

arXiv1998

Particles and Quantum Symmetries

The concept of quantum commutativity with respect to an action or coaction of a given Hopf algebra is used for the algebraic description of a system of particles and their interaction with certain quantum field. Graded commutativity and particles in singular magnetic field are considered as a particular example of the general formalism. The corresponding statistics and generalized Pauli exclusion principle are also mentioned.

math.QA↗

arXiv1997

Particles in Singular Magnetic Field

An algebraic formalism for description of quantum states of charged particle with spin moving in two-dimensional space under influence of singular magnetic field is developed in terms of graded algebras. The fundamental assumption is that the particle is transformed into a composite system which consists quasiparticles, quasiholes and magnetic fluxes. Such system is endowed with generalized statistics determined by a grading group and a commutation factor on it. Composite systems corresponding to the quantum Hall effect and the electronic magnetotransport anomaly are described. The Fock space representation are also given.

q-alg↗

arXiv1997

Topology and Quantization

A simple algebraic model for charged particle moving in two dimensional space under influence of singular magnetic field is given. The fundamental assumption for the model is that every charged particle coupled to the magnetic field is transformed into a system of quasiparticles. It is also assumed for simplicity that there is in average N fluxes per particle. If the number N of fluxes is even, then the system can be identified as composite fermions. If N is odd, then we obtain composite bosons. Quantization is described as gradation by certain abelian group G. Statistics is determined by a commutation factor on the grading group G. It is schown that for N = 2 there is a state for which the magnetic field is completely compensatad by the system of quasiparticles and for N = 3 the Landau levells are fractionally filled like in the fractional quantum Hall effect.

hep-th↗

arXiv1995

On Commutation Relations for Quons

The model of generalized quons is described in a purely algebraic way. Commutation relations and corresponding consistency conditions for our generalized quons system are studied in terms of quantum Weyl algebras. Fock space representation and corresponding scalar product is also given.

q-alg↗