Theoretical, Computational, and Statistical Physics

This book illustrates the interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group. The Lorentz group plays, in relativistic space-time, a role analogue to the rotations in Euclidean space. The hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. Hyperbolic geometry is presented via special relativity to benefit from physical intuition. Secondly, this book introduces basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral, and calculus. This introduction allows study in linear stochastic differential equations on groups of matrices. In this way the spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and the relativistic diffusion are constructed. Thirdly, quotients of the hyperbolic space under a discrete group of isometries are introduced. In this framework some elements of hyperbolic dynamics are presented, as the ergodicity of the geodesic and horocyclic flows. This book culminates with an analysis of the chaotic behaviour of the geodesic flow, performed using stochastic analysis methods. This main result is known as Sinai's central limit theorem.

Mathematical physics provides physical theories with their logical basis and the tools for drawing conclusions from hypotheses. Introduction to Mathematical Physics explains to the reader why and how mathematics is needed in the description of physical events in space. For undergraduates in physics, it is a classroom-tested textbook on vector analysis, linear operators, Fourier series and integrals, differential equations, special functions and functions of a complex variable. Strongly correlated with core undergraduate courses on classical and quantum mechanics and electromagnetism, it helps the student master these necessary mathematical skills. The book covers advanced topics of interest to graduate students on relativistic square-root spaces and nonlinear systems. It contains many tables of mathematical formulas and references to useful materials on the Internet. Short tutorials on basic mathematical topics are included to help readers refresh their mathematical knowledge. An appendix on Mathematica encourages the reader to use computer-aided algebra to solve problems in mathematical physics.

This book is based on lectures given to graduate students in physics at the University of Wisconsin-Madison. Group theory has been around for many years and the only thing new in this book is my approach to the subject, in particular the attempt to emphasize its beauty. The inspiration was this quote from Hermann Weyl: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other I usually chose the beautiful”. The book starts with the definition of basic concepts such as group, vector space, algebra, Lie group, Lie algebra, simple and semisimple groups, compact and non-compact groups. Next SO(3) and SU(2) are introduced as examples of elementary Lie groups and their relation to Physics and angular momentum. All irreducible representations, addition of angular momentum, Clebsch–Gordan coefficients and the Wigner–Eckart theorem are presented. Next the so(n) algebras and spinors are discussed using Clifford numbers. Conjugate, orthogonal and symplectic representations of spinors are described and the Clebsch–Gordan series for spinors is given, followed by a description of the center and outer automorphisms of Spin(n). The book next presents a diversion on Hurwitz’s theorem, quaternions and octonions, which leads into a discussion of the exceptional group G ₂. The discussion of orthogonal groups is concluded with a presentation of their Casimir operators. As a lead in to unitary groups the book discusses classical groups and obtain the dimensions of orthogonal, unitary and symplectic groups in one fell swoop by treating them as unitary groups over the reals, the complex and the quaternions, respectively. The symmetric group S is introduced and used to discuss irreducible representations of SU(n). The next three chapters involve the Cartan basis, the Cartan classification of semisimple algebras and Dynkin diagrams. The last three chapters are of particular importance to physicists as they describe the space-time groups known as the Lorentz, Poincaré and Liouville groups and the energy levels of the hydrogen atom in n space dimensions. At the end of each chapter brief biographical notes are given on the scientist(s) mentioned in that chapter for the first time.