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Described here is Feynman's path integral approach to quantum
mechanics and quantum field theory from a functional integral point
of view. Therein lies the main focus of Euclidean field theory. The
notion of Gaussian measure and the construction of the Wiener
measure are covered. As well, the notion of classical mechanics and
the Schrödinger picture of quantum mechanics are recalled. There,
the equivalence to the path integral formalism is shown by deriving
the quantum mechanical propagator from it. Additionally, an
introduction to elements of constructive quantum field theory is
provided for readers.Â
This book provides an introduction to deformation quantization and
its relation to quantum field theory, with a focus on the
constructions of Kontsevich and Cattaneo & Felder. This subject
originated from an attempt to understand the mathematical structure
when passing from a commutative classical algebra of observables to
a non-commutative quantum algebra of observables. Developing
deformation quantization as a semi-classical limit of the
expectation value for a certain observable with respect to a
special sigma model, the book carefully describes the relationship
between the involved algebraic and field-theoretic methods. The
connection to quantum field theory leads to the study of important
new field theories and to insights in other parts of mathematics
such as symplectic and Poisson geometry, and integrable systems.
Based on lectures given by the author at the University of Zurich,
the book will be of interest to graduate students in mathematics or
theoretical physics. Readers will be able to begin the first
chapter after a basic course in Analysis, Linear Algebra and
Topology, and references are provided for more advanced
prerequisites.
This book provides a first introduction to the methods of
probability theory by using the modern and rigorous techniques of
measure theory and functional analysis. It is geared for
undergraduate students, mainly in mathematics and physics majors,
but also for students from other subject areas such as economics,
finance and engineering. It is an invaluable source, either for a
parallel use to a related lecture or for its own purpose of
learning it.The first part of the book gives a basic introduction
to probability theory. It explains the notions of random events and
random variables, probability measures, expectation values,
distributions, characteristic functions, independence of random
variables, as well as different types of convergence and limit
theorems. The first part contains two chapters. The first chapter
presents combinatorial aspects of probability theory, and the
second chapter delves into the actual introduction to probability
theory, which contains the modern probability language. The second
part is devoted to some more sophisticated methods such as
conditional expectations, martingales and Markov chains. These
notions will be fairly accessible after reading the first part. --
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