Lectures on Matrix Field Theory

1. Front Matter

   Pages i-xii

   PDF

2. Introductory Remarks

   • Badis Ydri

   Pages 1-18

3. The Noncommutative Moyal-Weyl Spaces R ^d _θ

   • Badis Ydri

   Pages 19-71

4. The Fuzzy Sphere

   • Badis Ydri

   Pages 73-118

5. Quantum Noncommutative Phi-Four

   • Badis Ydri

   Pages 119-206

6. The Multitrace Approach

   • Badis Ydri

   Pages 207-275

7. Noncommutative Gauge Theory

   • Badis Ydri

   Pages 277-313

8. Back Matter

   Pages 315-352

   PDF

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.

• Noncommutative field theory
• Noncommutative geometry
• Noncommutative scalar phi-four
• Noncommutative gauge theory
• Matrix models
• Fuzzy spaces
• Fuzzy physics
• LNP Lecture Notes in Physics
• Noncommutative Moyal-Weyl Spaces

“The book collects almost all that has been achieved on the topic within the recent years, including all major results of many authors. As such, it is a nice reference work for graduate students and beginning researchers who want to pursue research in this area. Having all the results and different approaches collected in one place, together with the exhaustive list of references make this a valuable compendium to everyone working on noncommutative models of quantum field theory.” (Andrzej Sitarz, zbMATH 1371.81013, 2017)

• Institute of Physics, BM Annaba University, Annaba, Algeria

  Badis Ydri

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.

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