Finite And Profinite Quantum Systems
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Finite And Profinite Quantum Systems
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Author : Apostolos Vourdas
language : en
Publisher: Springer
Release Date : 2017-07-17
Finite And Profinite Quantum Systems written by Apostolos Vourdas and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-07-17 with Science categories.
This monograph provides an introduction to finite quantum systems, a field at the interface between quantum information and number theory, with applications in quantum computation and condensed matter physics. The first major part of this monograph studies the so-called `qubits' and `qudits', systems with periodic finite lattice as position space. It also discusses the so-called mutually unbiased bases, which have applications in quantum information and quantum cryptography. Quantum logic and its applications to quantum gates is also studied. The second part studies finite quantum systems, where the position takes values in a Galois field. This combines quantum mechanics with Galois theory. The third part extends the discussion to quantum systems with variables in profinite groups, considering the limit where the dimension of the system becomes very large. It uses the concepts of inverse and direct limit and studies quantum mechanics on p-adic numbers. Applications of the formalism include quantum optics and quantum computing, two-dimensional electron systems in magnetic fields and the magnetic translation group, the quantum Hall effect, other areas in condensed matter physics, and Fast Fourier Transforms. The monograph combines ideas from quantum mechanics with discrete mathematics, algebra, and number theory. It is suitable for graduate students and researchers in quantum physics, mathematics and computer science.
Finite Mathematics As The Foundation Of Classical Mathematics And Quantum Theory
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Author : Felix Lev
language : en
Publisher: Springer Nature
Release Date : 2020-11-03
Finite Mathematics As The Foundation Of Classical Mathematics And Quantum Theory written by Felix Lev and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-03 with Science categories.
This book delves into finite mathematics and its application in physics, particularly quantum theory. It is shown that quantum theory based on finite mathematics is more general than standard quantum theory, whilst finite mathematics is itself more general than standard mathematics.As a consequence, the mathematics describing nature at the most fundamental level involves only a finite number of numbers while the notions of limit, infinite/infinitesimal and continuity are needed only in calculations that describe nature approximately. It is also shown that the concepts of particle and antiparticle are likewise approximate notions, valid only in special situations, and that the electric charge and baryon- and lepton quantum numbers can be only approximately conserved.
Noncommutative Geometry Quantum Fields And Motives
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Author : Alain Connes
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-03-13
Noncommutative Geometry Quantum Fields And Motives written by Alain Connes and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-13 with Mathematics categories.
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.
Causality The P Adic Theory
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Author : Vladimir Anashin
language : en
Publisher: Springer Nature
Release Date : 2025-04-24
Causality The P Adic Theory written by Vladimir Anashin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-24 with Mathematics categories.
This book delves into the mathematical theory of causal functions over discrete time, offering a fresh perspective on causality beyond its philosophical roots. By exploring the intricate world of p-adic 1-Lipschitz functions, this volume bridges the gap between abstract mathematical concepts and their practical applications in fields such as automata theory, combinatorics, and applied computer science. Readers will uncover a wealth of insights as the book investigates key topics including the nature of causal functions, the role of discrete time in causality, and the application of non-Archimedean metrics. With contributions from eminent scholars, this work invites readers to ponder critical questions: How do we define causality in mathematical terms? What are the implications of using p-adic analysis in understanding complex systems especially quantum ones? The author's unique approach makes this book an essential read for anyone interested in the intersection of mathematics and real-world applications. Ideal for researchers and practitioners with a background in mathematics, computer science, or physics, this book is a valuable resource for those seeking to deepen their understanding of causal functions. Whether you're a scholar exploring theoretical perspectives or a professional looking to apply these concepts practically, this volume offers a comprehensive guide to navigating the complexities of causality. Part of an ongoing series on advanced mathematical theories, it is an indispensable addition to any academic library.
Mathematical Reviews
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Author :
language : en
Publisher:
Release Date : 2004
Mathematical Reviews written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
An Approach To The Foundations Of Quantum Mechanics Using The Concept Of A Filter As Primitive
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Author : Hans Kummer
language : en
Publisher: Kingston, Ont. : Queen's University
Release Date : 1984
An Approach To The Foundations Of Quantum Mechanics Using The Concept Of A Filter As Primitive written by Hans Kummer and has been published by Kingston, Ont. : Queen's University this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Filters (Mathematics). categories.
Compact Matrix Quantum Groups And Their Combinatorics
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Author : Amaury Freslon
language : en
Publisher: Cambridge University Press
Release Date : 2023-07-27
Compact Matrix Quantum Groups And Their Combinatorics written by Amaury Freslon and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-27 with Mathematics categories.
An organised step-by-step introduction to the theory of compact quantum groups, starting with examples coming from quantum physics, which stems from the basic undergraduate mathematics curriculum. Introducing more abstract concepts along the way when needed, the reader is led from the fundamentals of the theory to recent research results. The emphasis is put on the combinatorics underlying compact quantum groups, which is very elementary to describe but leads to profound results. This book includes many exercises to help students work through new concepts and ideas and consolidate their understanding. The theory itself is illustrated by an array of examples, some related to other fields of Mathematics such as free probability theory or graph theory. The book is intended for graduate students, motivated undergraduate students and researchers.
The Block Theory Of Finite Group Algebras
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Author : Markus Linckelmann
language : en
Publisher: Cambridge University Press
Release Date : 2018
The Block Theory Of Finite Group Algebras written by Markus Linckelmann and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Blocks categories.
This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.
Representations Of Finite Groups Of Lie Type
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Author : François Digne
language : en
Publisher: Cambridge University Press
Release Date : 2020-03-05
Representations Of Finite Groups Of Lie Type written by François Digne and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-05 with Mathematics categories.
An up-to-date and self-contained introduction based on a graduate course taught at the University of Paris.
Encyclopaedia Of Mathematics Supplement Iii
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Author : Michiel Hazewinkel
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-11-23
Encyclopaedia Of Mathematics Supplement Iii written by Michiel Hazewinkel and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-11-23 with Mathematics categories.
This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.