Supermathematics and its Applications in Statistical Physics: Grassmann Variables and the Method of Supersymmetry: 920 (Lecture Notes in Physics, 920) - Softcover

Wegner, Franz

Softcover

ISBN 10: 3662491680 ISBN 13: 9783662491683

Publisher: Springer, 2016

Synopsis

This text presents the mathematical concepts of Grassmann variables and the method of supersymmetry to a broad audience of physicists interested in applying these tools to disordered and critical systems, as well as related topics in statistical physics. Based on many courses and seminars held by the author, one of the pioneers in this field, the reader is given a systematic and tutorial introduction to the subject matter.

The algebra and analysis of Grassmann variables is presented in part I. The mathematics of these variables is applied to a random matrix model, path integrals for fermions, dimer models and the Ising model in two dimensions. Supermathematics - the use of commuting and anticommuting variables on an equal footing - is the subject of part II. The properties of supervectors and supermatrices, which contain both commuting and Grassmann components, are treated in great detail, including the derivation of integral theorems. In part III, supersymmetric physical models are considered. While supersymmetry was first introduced in elementary particle physics as exact symmetry between bosons and fermions, the formal introduction of anticommuting spacetime components, can be extended to problems of statistical physics, and, since it connects states with equal energies, has also found its way into quantum mechanics.

Several models are considered in the applications, after which the representation of the random matrix model by the nonlinear sigma-model, the determination of the density of states and the level correlation are derived. Eventually, the mobility edge behavior is discussed and a short account of the ten symmetry classes of disorder, two-dimensional disordered models, and superbosonization is given.

About this title

About the Author

Franz Wegner obtained his PhD from the Technical University in Munich in 1968 and, as postdoctoral researcher, has worked amongst others with Leo Kadanoff at Brown University in Providence, Rhode Island (USA). 1974 he was appointed professor of theoretical physics at Heidelberg University where he became a professor emeritus in 2006. Prof. Wegner is recipient of numerous prizes and awards, among them the Walter-Schottky prize (1976), the Max-Planck medal of the German Physical Society (1986), and most recently of the Lars-Onsager prize of the American Physical Society (2015).

From the Back Cover

This text presents the mathematical concepts of Grassmann variables and the method of supersymmetry to a broad audience of physicists interested in applying these tools to disordered and critical systems, as well as related topics in statistical physics. Based on many courses and seminars held by the author, one of the pioneers in this field, the reader is given a systematic and tutorial introduction to the subject matter.

The algebra and analysis of Grassmann variables is presented in part I. The mathematics of these variables is applied to a random matrix model, path integrals for fermions, dimer models and the Ising model in two dimensions. Supermathematics - the use of commuting and anticommuting variables on an equal footing - is the subject of part II. The properties of supervectors and supermatrices, which contain both commuting and Grassmann components, are treated in great detail, including the derivation of integral theorems. In part III, supersymmetric physical models are considered. While supersymmetry was first introduced in elementary particle physics as exact symmetry between bosons and fermions, the formal introduction of anticommuting spacetime components, can be extended to problems of statistical physics, and, since it connects states with equal energies, has also found its way into quantum mechanics.

Several models are considered in the applications, after which the representation of the random matrix model by the nonlinear sigma-model, the determination of the density of states and the level correlation are derived. Eventually, the mobility edge behavior is discussed and a short account of the ten symmetry classes of disorder, two-dimensional disordered models, and superbosonization is given.

Edition details

• Publisher: Springer
• Publication date: 2016
• Language: English
• ISBN 10: 3662491680
• ISBN 13: 9783662491683
• Binding: Paperback
• Edition number: 1
• Number of pages: 391