Quantum Fields Algebras Processes

Are we living in a golden age? It is now more than half a century that Einstein and Heisenberg have given us the theories of relativity and of quantum mechanics, but the great challenge of 20th century science remains unre solved: to assemble these building blocks into a fundamental theory of matter. And yet, for anyone watching the interplay of mathematics and theoretical physics to-day, developing symbiotically through the stimulus of a lively, even essential interdisciplinary dia logue, this is a time of fascination and great satisfaction. It is also a time of gratitude to those who had the courage to in sist that "a rudimentary knowledge of the Latin and Greek alpha bets" was not enough, and tore down the barriers between the disciplines. On the basis of this groundwork there is now so much progress, and, notably, such strengthening of the dia].ogue with phenomenology that - reaching out for The Great Break through - this may indeed turn out to be the golden age.

The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles. The presented techniques emerge from interdisciplinary fields: mathematics, computer science and theoretical physics; the articles are written by mathematicians and physicists with the goal that both groups can learn from the other field, including most recent developments. Besides that, the collection of articles also serves as an up-to-date handbook of available algorithms/software that are commonly used or might be useful in the fields of mathematics, physics or other sciences.

Causal analysis in terms of white noise; Introduction to stochastic differential calculus; A generalized stochastic calculus in homogenization; Interaction picture for stochastic differential equations; Path integrals, stationary phase approximations and complex histories; Stochastic dynamics and the semiclassical limit of quantum mechanics; Asymptotic expansion of fresnel integrals relative to a non-singular quadratic form; Scaling limits of generalized random processes; Renormalization group analysis of some higly bifurcated families; Anticommutative integration and fermi fields; Homogenous self-dual cones and jordan algebras; Generators of one-parameter groups of *-automorphisms on UHF-algebras; Automorphisms of certain simple C*-algebras; Non-commutative group duality and the kubo-martin-schwinger condition; A uniqueness theorem for central extensions of discrete products of cyclic groups; Introduction to w*-categories; Net cohomology and its application to field theory; Construction of specifications; On the global markov property; Uniqueness and global markov property for euclidean fields and lattice systems; Martingale convergence and the exponencial interaction in R; On dia-and paramagnetic properties of yang-mills potentials; A new look at generalized, non-linear o-models and yang-mills theory; 1/N expansions and the O(N) nonlinear o-model in two dimensions; On the Z2 lattice higgs system; Fluctuation of the interface of the two-dimensional ising model; The stability problem in o4 scalar field theories.

Do quantum field theory without Feynman diagrams! Use the combinatorics behind cumulants, correlations, Green's functions and quantum fields.

This ambitious and original book sets out to introduce to mathematicians (even including graduate students ) the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in use. This in turn promotes the interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, though mathematicians are the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras.

Operational Quantum Theory II is a distinguished work on quantum theory at an advanced algebraic level. The classically oriented hierarchy with objects such as particles as the primary focus, and interactions of the objects as the secondary focus is reversed with the operational interactions as basic quantum structures. Quantum theory, specifically relativistic quantum field theory is developed the theory of Lie group and Lie algebra operations acting on both finite and infinite dimensional vector spaces. This book deals with the operational concepts of relativistic space time, the Lorentz and Poincaré group operations and their unitary representations, particularly the elementary articles. Also discussed are eigenvalues and invariants for non-compact operations in general as well as the harmonic analysis of noncompact nonabelian Lie groups and their homogeneous spaces. In addition to the operational formulation of the standard model of particle interactions, an attempt is made to understand the particle spectrum with the masses and coupling constants as the invariants and normalizations of a tangent representation structure of a an homogeneous space time model. Operational Quantum Theory II aims to understand more deeply on an operational basis what one is working with in relativistic quantum field theory, but also suggests new solutions to previously unsolved problems.