Bücher der Reihe SpringerBriefs in Mathematical Physics

The SLE and Dyson's BM model may be considered as "children" of the Bessel process with parameter D, BES(D), and the SLE and Dyson's BM model as "grandchildren" of BM.

This book is a volume of the Springer Briefs in Mathematical Physics and serves as an introductory textbook on the theory of Macdonald polynomials. It is based on a series of online lectures given by the author at the Royal Institute of Technology (KTH), Stockholm, in February and March 2021. Macdonald polynomials are a class of symmetric orthogonal polynomials in many variables. They include important classes of special functions such as Schur functions and Hall¿Littlewood polynomials and play important roles in various fields of mathematics and mathematical physics. After an overview of Schur functions, the author introduces Macdonald polynomials (of type A, in the GLn version) as eigenfunctions of a q-difference operator, called the Macdonald¿Ruijsenaars operator, in the ring of symmetric polynomials. Starting from this definition, various remarkable properties of Macdonald polynomials are explained, such as orthogonality, evaluation formulas, and self-duality, with emphasis on the roles of commuting q-difference operators. The author also explains how Macdonald polynomials are formulated in the framework of affine Hecke algebras and q-Dunkl operators.

This book explains the mathematical structures of supersymmetric quantum field theory (SQFT) from the viewpoints of functional and infinite-dimensional analysis. The main mathematical objects are infinite-dimensional Dirac operators on the abstract Boson-Fermion Fock space. The target audience consists of graduate students and researchers who are interested in mathematical analysis of quantum fields, including supersymmetric ones, and infinite-dimensional analysis. The major topics are the clarification of general mathematical structures that some models in the SQFT have in common, and the mathematically rigorous analysis of them. The importance and the relevance of the subject are that in physics literature, supersymmetric quantum field models are only formally (heuristically) considered and hence may be ill-defined mathematically. From a mathematical point of view, however, they suggest new aspects related to infinite-dimensional geometry and analysis. Therefore, it is important to show the mathematical existence of such models first and then study them in detail. The book shows that the theory of the abstract Boson-Fermion Fock space serves this purpose. The analysis developed in the book also provides a good example of infinite-dimensional analysis from the functional analysis point of view, including a theory of infinite-dimensional Dirac operators and Laplacians.

The isomonodromic deformation equations such as the Painleve and Garnier systems are an important class of nonlinear differential equations in mathematics and mathematical physics. In choosing the suitable approximation problem, the linear differential equations give the Lax pair for some isomonodromic equations.

This book is an exposition of recent progress on the Donaldson¿Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi¿Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov¿Witten/Donaldson¿Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi¿Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar¿Vafa invariant, which was firstproposed by Gopakumar¿Vafa in 1998, but its precise mathematical definition has not been available until recently.This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.

This book focuses on the Symmetric Informationally Complete quantum measurements (SICs) in dimensions 2 and 3, along with one set of SICs in dimension 8.

Rich information-theoretic structure in out-of-equilibrium thermodynamics exists in both the classical and quantum regimes, leading to the fruitful interplay among statistical physics, quantum information theory, and mathematical theories such as matrix analysis and asymptotic probability theory.

This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories.

The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science.

This is the first book on elliptic quantum groups, i.e., quantum groups associated to elliptic solutions of the Yang-Baxter equation.

In this book, a statistical mechanical interpretation of AIT is introduced while explaining the basic notions and results of AIT to the reader who has an acquaintance with an elementary theory of computation. A simplification of the setting of AIT is the noiseless source coding in information theory.

This book provides self-contained proofs of the existence of ground states of several interaction models in quantum field theory. We show the existence of the ground state of the Pauli-Fierz mode, the Nelson model, and the spin-boson model, and several kinds of proofs of the existence of ground states are explicitly provided.

This Brief presents in a self-contained, non-technical and illustrative fashion the state-of-the-art results and techniques for the dynamics of extremal black holes. The book presents various recently discovered characteristic phenomena (such as the horizon instability) that have enhanced our understanding of the dynamics of extremal black holes.

This book gives a rigorous treatment of entanglement measures in the general context of quantum field theory. It covers a broad range of models and the use of fields allows us to properly take the localization of systems into account. The required mathematical techniques are introduced in a self-contained way.

This book is a short, but complete, introduction to the Loewner equation and the SLEs, which are a family of random fractal curves, as well as the relevant background in probability and complex analysis.

The Kadison-Singer property deals with the following question: given a Hilbert space H and an abelian unital C*-subalgebra A of B(H), does every pure state on A extend uniquely to a pure state on B(H)?

This book offers a guided tour through the mathematical habitat of noncommutative geometry a la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given.

This book explains what quantum states of light look like. Moreover, squeezed states of light are explained as "longitudinal" waves of light and Schroedinger's cat states as macroscopic superposition states.

These notes investigate the time evolution of quantum systems, and in particular the rigorous derivation of effective equations approximating the many-body Schroedinger dynamics in certain physically interesting regimes.

This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology;

The smooth entropy framework is discussed next and provides a natural means to lift many arguments from information theory to the quantum setting.Finally selected applications of the theory to statistics and cryptography are discussed. The book is aimed at graduate students in Physics and Information Theory.

The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher order differential equations.

This book is a survey of the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by Kontsevich. The main topics covered are the theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem.

This book explains the basic ideas of supergravity, collecting the relevant formulae in one place. Covers vielbein formulation of gravity; supergravities in four dimensions; superalgebras and supermultiplets; supergravities in higher dimensions and more.

This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities.

In this work the question whether noncommutative geometry allows for supersymmetric theories is addressed.

This book focuses on Erdélyi-Kober fractional calculus from a statistical perspective inspired by solar neutrino physics. Results of diffusion entropy analysis and standard deviation analysis of data from the Super-Kamiokande solar neutrino experiment lead to the development of anomalous diffusion and reaction in terms of fractional calculus. The new statistical perspective of Erdélyi-Kober fractional operators outlined in this book will have fundamental applications in the theory of anomalous reaction and diffusion processes dealt with in physics.A major mathematical objective of this book is specifically to examine a new de¿nition for fractional integrals in terms of the distributions of products and ratios of statistically independently distributed positive scalar random variables or in terms of Mellin convolutions of products and ratios in the case of real scalar variables. The idea will be generalized to cover multivariable cases as well as matrix variable cases. In the matrix variable case, M-convolutions of products and ratios will be used to extend the ideas. We then give a de¿nition for the case of real-valued scalar functions of several matrices.