Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes. This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields. It starts with a general introduction to the ideas and methods involving refection positive Hilbert spaces and the Osterwalder--Schrader transform. It then turns to Reflection Positivity in Lie group representations. Already the case of one-dimensional groups is extremely rich. For the real line it connects naturally with Lax--Phillips scattering theory and for the circle group it provides a new perspective on the Kubo--Martin--Schwinger (KMS) condition for states of operator algebras. For Lie groups Reflection Positivity connects unitary representations of a symmetric Lie group with unitary representations of its Cartan dual Lie group. A typical example is the duality between the Euclidean group E(n) and the Poincare group P(n) of special relativity. It discusses in particular the curved context of the duality between spheres and hyperbolic spaces. Further it presents some new integration techniques for representations of Lie algebras by unbounded operators which are needed for the passage to the dual group. Positive definite functions, kernels and distributions and used throughout as a central tool.

Presents a comprehensive and coherent account of the theory of quantum fields on a lattice.

This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area of ongoing research for both graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.

This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modem physics: statistical physics, quantum mechanics, and quantum fields. The unity of mathemati cal structure for problems of diverse origin in physics should be no surprise. For classical physics it is provided, for example, by a common mathematical formalism based on the wave equation and Laplace's equation. The unity transcends mathematical structure and encompasses basic phenomena as well. Thus particle physicists, nuclear physicists, and con densed matter physicists have considered similar scientific problems from complementary points of view. The mathematical structure presented here can be described in various terms: partial differential equations in an infinite number of independent variables, linear operators on infinite dimensional spaces, or probability theory and analysis over function spaces. This mathematical structure of quantization is a generalization of the theory of partial differential equa tions, very much as the latter generalizes the theory of ordinary differential equations. Our central theme is the quantization of a nonlinear partial differential equation and the physics of systems with an infinite number of degrees of freedom. Mathematicians, theoretical physicists, and specialists in mathematical physics are the three audiences to which the book is addressed. Each of the three parts is written with a different scientific perspective.

Describes fifteen years' work which has led to the construc- tion of solutions to non-linear relativistic local field e- quations in 2 and 3 space-time dimensions. Gives proof of the existence theorem in 2 dimensions and describes many properties of the solutions.

The interactions between concentration, isoperimetry and functional inequalities have led to many significant advances in functional analysis and probability theory. Important progress has also taken place in combinatorics, geometry, harmonic analysis and mathematical physics, with recent new applications in random matrices and information theory. This will appeal to graduate students and researchers interested in the interplay between analysis, probability, and geometry.

This volume presents a collection of courses introducing the reader to the recent progress with attention being paid to laying solid grounds and developing various basic tools. It presents new results on phase transitions for gradient lattice models.