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Noncommutative Mathematics for Quantum Systems

Uwe Franz,Adam Skalski

Author : Uwe Franz,Adam Skalski
Publisher : Cambridge University Press
Page : 200 pages
File Size : 49,6 Mb
Release : 2016-01-07
Category : Mathematics
ISBN : 9781316674048

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Noncommutative Mathematics for Quantum Systems by Uwe Franz,Adam Skalski Pdf

Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.

Noncommutative Structures in Mathematics and Physics

S. Duplij,Julius Wess

Author : S. Duplij,Julius Wess
Publisher : Springer Science & Business Media
Page : 472 pages
File Size : 46,7 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401008365

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Noncommutative Structures in Mathematics and Physics by S. Duplij,Julius Wess Pdf

A presentation of outstanding achievements and ideas, of both eastern and western scientists, both mathematicians and physicists. Their presentations of recent work on quantum field theory, supergravity, M-theory, black holes and quantum gravity, together with research into noncommutative geometry, Hopf algebras, representation theory, categories and quantum groups, take the reader to the forefront of the latest developments. Other topics covered include supergravity and branes, supersymmetric quantum mechanics and superparticles, (super) black holes, superalgebra representations, and SUSY GUT phenomenology. Essential reading for workers in the modern methods of theoretical and mathematical physics.

Noncommutative Geometry, Quantum Fields and Motives

Alain Connes,Matilde Marcolli

Author : Alain Connes,Matilde Marcolli
Publisher : American Mathematical Soc.
Page : 785 pages
File Size : 47,9 Mb
Release : 2019-03-13
Category : Electronic
ISBN : 9781470450458

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Noncommutative Geometry, Quantum Fields and Motives by Alain Connes,Matilde Marcolli Pdf

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Mathematical Results in Quantum Mechanics

Radu Purice

Author : Radu Purice
Publisher : World Scientific
Page : 312 pages
File Size : 45,6 Mb
Release : 2008
Category : Mathematics
ISBN : 9789812832382

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Mathematical Results in Quantum Mechanics by Radu Purice Pdf

The 10th Quantum Mathematics International Conference (Qmath10) gave an opportunity to bring together specialists interested in that part of mathematical physics which is in close connection with various aspects of quantum theory. It was also meant to introduce young scientists and new tendencies in the field.This collection of carefully selected papers aims to reflect recent techniques and results on SchrAdinger operators with magnetic fields, random SchrAdinger operators, condensed matter and open systems, pseudo-differential operators and semiclassical analysis, quantum field theory and relativistic quantum mechanics, quantum information, and much more. The book serves as a concise and well-documented tool for the more experimented scientists, as well as a research guide for postgraduate students.

Quantum Stochastic Processes and Noncommutative Geometry

Kalyan B. Sinha,Debashish Goswami

Author : Kalyan B. Sinha,Debashish Goswami
Publisher : Cambridge University Press
Page : 301 pages
File Size : 42,8 Mb
Release : 2007-01-25
Category : Mathematics
ISBN : 9781139461696

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Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.

Noncommutative Dynamics and E-Semigroups

William Arveson

Author : William Arveson
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 45,6 Mb
Release : 2012-11-06
Category : Mathematics
ISBN : 9780387215242

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Noncommutative Dynamics and E-Semigroups by William Arveson Pdf

This book introduces the notion of an E-semigroup, a generalization of the known concept of E_O-semigroup. These objects are families of endomorphisms of a von Neumann algebra satisfying certain natural algebraic and continuity conditions. Its thorough approach is ideal for graduate students and research mathematicians.

Quantum Stochastic Processes and Noncommutative Geometry

Kalyan B. Sinha,Debashish Goswami

Author : Kalyan B. Sinha,Debashish Goswami
Publisher : Unknown
Page : 290 pages
File Size : 44,6 Mb
Release : 2007
Category : Mathematics
ISBN : 0511269978

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Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega

Author : Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega
Publisher : Springer
Page : 341 pages
File Size : 54,5 Mb
Release : 2017-10-26
Category : Science
ISBN : 9783319654270

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Quantization, Geometry and Noncommutative Structures in Mathematics and Physics by Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega Pdf

This monograph presents various ongoing approaches to the vast topic of quantization, which is the process of forming a quantum mechanical system starting from a classical one, and discusses their numerous fruitful interactions with mathematics.The opening chapter introduces the various forms of quantization and their interactions with each other and with mathematics.A first approach to quantization, called deformation quantization, consists of viewing the Planck constant as a small parameter. This approach provides a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables. When symmetries come into play, deformation quantization needs to be merged with group actions, which is presented in chapter 2, by Simone Gutt.The noncommutativity arising from quantization is the main concern of noncommutative geometry. Allowing for the presence of symmetries requires working with principal fiber bundles in a non-commutative setup, where Hopf algebras appear naturally. This is the topic of chapter 3, by Christian Kassel. Nichols algebras, a special type of Hopf algebras, are the subject of chapter 4, by Nicolás Andruskiewitsch. The purely algebraic approaches given in the previous chapters do not take the geometry of space-time into account. For this purpose a special treatment using a more geometric point of view is required. An approach to field quantization on curved space-time, with applications to cosmology, is presented in chapter 5 in an account of the lectures of Abhay Ashtekar that brings a complementary point of view to non-commutativity.An alternative quantization procedure is known under the name of string theory. In chapter 6 its supersymmetric version is presented. Superstrings have drawn the attention of many mathematicians, due to its various fruitful interactions with algebraic geometry, some of which are described here. The remaining chapters discuss further topics, as the Batalin-Vilkovisky formalism and direct products of spectral triples.This volume addresses both physicists and mathematicians and serves as an introduction to ongoing research in very active areas of mathematics and physics at the border line between geometry, topology, algebra and quantum field theory.

Mathematical Results In Quantum Mechanics - Proceedings Of The Qmath10 Conference

Radu Purice,Ingrid Beltita,Gheorghe Nenciu

Author : Radu Purice,Ingrid Beltita,Gheorghe Nenciu
Publisher : World Scientific
Page : 312 pages
File Size : 55,7 Mb
Release : 2008-08-11
Category : Science
ISBN : 9789814470544

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Mathematical Results In Quantum Mechanics - Proceedings Of The Qmath10 Conference by Radu Purice,Ingrid Beltita,Gheorghe Nenciu Pdf

The 10th Quantum Mathematics International Conference (Qmath10) gave an opportunity to bring together specialists interested in that part of mathematical physics which is in close connection with various aspects of quantum theory. It was also meant to introduce young scientists and new tendencies in the field.This collection of carefully selected papers aims to reflect recent techniques and results on Schrödinger operators with magnetic fields, random Schrödinger operators, condensed matter and open systems, pseudo-differential operators and semiclassical analysis, quantum field theory and relativistic quantum mechanics, quantum information, and much more. The book serves as a concise and well-documented tool for the more experimented scientists, as well as a research guide for postgraduate students.

Quantum Groups and Noncommutative Spaces

Matilde Marcolli,Deepak Parashar

Author : Matilde Marcolli,Deepak Parashar
Publisher : Springer Science & Business Media
Page : 247 pages
File Size : 42,6 Mb
Release : 2010-11-02
Category : Mathematics
ISBN : 9783834898319

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Quantum Groups and Noncommutative Spaces by Matilde Marcolli,Deepak Parashar Pdf

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

Quantum Theory, Deformation and Integrability

R. Carroll

Author : R. Carroll
Publisher : Elsevier
Page : 420 pages
File Size : 40,9 Mb
Release : 2000-11-09
Category : Science
ISBN : 0080540082

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Quantum Theory, Deformation and Integrability by R. Carroll Pdf

About four years ago a prominent string theorist was quoted as saying that it might be possible to understand quantum mechanics by the year 2000. Sometimes new mathematical developments make such understanding appear possible and even close, but on the other hand, increasing lack of experimental verification make it seem to be further distant. In any event one seems to arrive at new revolutions in physics and mathematics every year. This book hopes to convey some of the excitment of this period, but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical. There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc. but philosophy will not be emphasized; generally we want to enjoy the fruits of computation based on the operator formulation of QM and quantum field theory. In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone. Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings and Hirota formulas, and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior. In particular, connections to Seiberg-Witten theory (arising in N = 2 supersymmetric (susy) Yang-Mills (YM) theory) are discussed and we would still like to understand more deeply what is going on. Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. from both a physical and mathematical point of view. In Chapter 6 we continue the deformation quantization then by exhibiting material based on and related to noncommutative geometry and gauge theory.

Noncommutative Geometry

Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts

Author : Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 52,9 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 3540203575

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Noncommutative Geometry by Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts Pdf

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements

Konstantin A Makarov,Eduard R Tsekanovskii

Author : Konstantin A Makarov,Eduard R Tsekanovskii
Publisher : World Scientific
Page : 300 pages
File Size : 53,8 Mb
Release : 2021-12-24
Category : Science
ISBN : 9789811241246

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Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements by Konstantin A Makarov,Eduard R Tsekanovskii Pdf

This book presents new developments in the open quantum systems theory with emphasis on applications to the (frequent) measurement theory.In the first part of the book, the uniqueness theorems for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions are discussed. The major theme involves an intrinsic characterization of the simplest symmetric operator solutions to the Heisenberg uncertainty relations, the problem posed by Jørgensen and Muhly, followed by the proof of the uniqueness theorems for the simplest solutions to the restricted Weyl commutation relations. The detailed study of unitary invariants of the corresponding dissipative and symmetric operators opens up a look at the classical Stone-von Neumann uniqueness theorem from a new angle and provides an extended version of the uniqueness result relating various realizations of a differentiation operator on the corresponding metric graphs.The second part of the book is devoted to mathematical problems of the quantum measurements under continuous monitoring. Among the topics discussed are the complementarity of the Quantum Zeno effect and Exponential Decay scenario in frequent quantum measurements, and a rigorous treatment, within continuous monitoring paradigm, of the celebrated 'double-slit experiment' where the renowned exclusive and interference measurement alternatives approach in quantum theory is presented in a way that is accessible for mathematicians. One of the striking applications of the generalized (1-stable) central limit theorem is the mathematical evidence of exponential decay of unstable states of the quantum pendulum under continuous monitoring.

Noncommutative Geometry and Number Theory

Caterina Consani,Matilde Marcolli

Author : Caterina Consani,Matilde Marcolli
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 49,8 Mb
Release : 2007-12-18
Category : Mathematics
ISBN : 9783834803528

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Noncommutative Geometry and Number Theory by Caterina Consani,Matilde Marcolli Pdf

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

Quantum Mechanics Using Computer Algebra

W.-H. Steeb,Yorick Hardy

Author : W.-H. Steeb,Yorick Hardy
Publisher : World Scientific
Page : 245 pages
File Size : 40,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814307178

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Quantum Mechanics Using Computer Algebra by W.-H. Steeb,Yorick Hardy Pdf

This book collects standard and advanced methods in quantum mechanics and implements them using SymbolicC++ and Maxima, two popular computer algebra packages. Throughout, the sample programs and their outputs are accompanied with explantory text of the underlying mathematics and physics explained in detail. Selected problems have also been implemented using two other popular packages --- Mathematica and Maple --- while some problems are implemented in C++. --

Noncommutative Mathematics For Quantum Systems

Noncommutative Mathematics for Quantum Systems by Uwe Franz,Adam Skalski Pdf

Noncommutative Structures in Mathematics and Physics by S. Duplij,Julius Wess Pdf

Noncommutative Geometry, Quantum Fields and Motives by Alain Connes,Matilde Marcolli Pdf

Mathematical Results in Quantum Mechanics by Radu Purice Pdf

Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

Noncommutative Dynamics and E-Semigroups by William Arveson Pdf

Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics by Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega Pdf

Mathematical Results In Quantum Mechanics - Proceedings Of The Qmath10 Conference by Radu Purice,Ingrid Beltita,Gheorghe Nenciu Pdf

Quantum Groups and Noncommutative Spaces by Matilde Marcolli,Deepak Parashar Pdf

Quantum Theory, Deformation and Integrability by R. Carroll Pdf

Noncommutative Geometry by Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts Pdf

Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements by Konstantin A Makarov,Eduard R Tsekanovskii Pdf

Noncommutative Geometry and Number Theory by Caterina Consani,Matilde Marcolli Pdf

Quantum Mechanics Using Computer Algebra by W.-H. Steeb,Yorick Hardy Pdf

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