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These lecture notes provide a systematic introduction to matrix
models of quantum field theories with non-commutative and fuzzy
geometries. The book initially focuses on the matrix formulation of
non-commutative and fuzzy spaces, followed by a description of the
non-perturbative treatment of the corresponding field theories. As
an example, the phase structure of non-commutative phi-four theory
is treated in great detail, with a separate chapter on the
multitrace approach. The last chapter offers a general introduction
to non-commutative gauge theories, while two appendices round out
the text. Primarily written as a self-study guide for postgraduate
students - with the aim of pedagogically introducing them to key
analytical and numerical tools, as well as useful physical models
in applications - these lecture notes will also benefit experienced
researchers by providing a reference guide to the fundamentals of
non-commutative field theory with an emphasis on matrix models and
fuzzy geometries.
This book is divided into two parts. In the first part we give an
elementary introduction to computational physics consisting of 21
simulations which originated from a formal course of lectures and
laboratory simulations delivered since 2010 to physics students at
Annaba University. The second part is much more advanced and deals
with the problem of how to set up working Monte Carlo simulations
of matrix field theories which involve finite dimensional matrix
regularizations of noncommutative and fuzzy field theories, fuzzy
spaces and matrix geometry. The study of matrix field theory in its
own right has also become very important to the proper
understanding of all noncommutative, fuzzy and matrix phenomena.
The second part, which consists of 9 simulations, was delivered
informally to doctoral students who were working on various
problems in matrix field theory. Sample codes as well as sample key
solutions are also provided for convenience and completeness.
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