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.

Presents step-by-step constructions of various non-commutative and fuzzy spaces Offers an accessible explanation of various quantum properties of non-commutative scalar phi-four theory and non-commutative gauge theory Includes a detailed discussion of matrix models, multitrace, Monte Carlo and other methods used in studying fuzzy/non-commutative spaces and their matrix field theories Includes supplementary material: sn.pub/extras

Autorentext
Badis Ydri, currently a Professor of theoretical particle physics, teaching at the Institute of Physics, Badji Mokhtar Annaba University, Algeria, received his PhD (2001) from Syracuse University, NY, USA, and his Habilitation (2011) from Annaba University, Algeria. He is also a research associate at the Dublin Institute for Advanced Studies, Ireland, and a regular ICTP associate at the Abdus Salam Center for Theoretical Physics, Trieste, Italy. His post-doctoral experience comprises a Marie Curie fellowship (2006-2008) at Humboldt University Berlin, Germany, and a Hamilton fellowship (2001-2005) at the Dublin Institute for Advanced Studies, Ireland. Research directions: Renormalization group equation and Monte Carlo method for matrix and non-commutative field theories. Emergent physics (gauge, matter, gravity, time, etc.) and emergent geometry from matrix models. Cosmological Yang-Mills models. AdS/CFT and Yang-Mills matrix models. Other interests include stringtheory, causal dynamical triangulation, Horava-Lifshitz gravity, and supersymmetric gauge theory in four dimensions.His hobbies are reading philosophic works and history of science.

Inhalt
Preface.- Introductory Remarks.- The Non-Commutative Moyal-Weyl Spaces Rd.- The Fuzzy Sphere.- Quantum Non-Commutative Phi-Four.- The Multitrace Approach.- Non-Commutative Gauge Theory.- Appendix A - The Landau States.- Appendix B - The Traces TrtAtB and TrtAtBtCtD.- Index.