E-raamat: Supermathematics and its Applications in Statistical Physics: Grassmann Variables and the Method of Supersymmetry

This text presents the mathematical concepts of Grassmannvariables and the method of supersymmetry to a broad audience of physicistsinterested in applying these tools to disordered and critical systems, as wellas related topics in statistical physics. Based on many courses and seminarsheld by the author, one of the pioneers in this field, the reader is given asystematic and tutorial introduction to the subject matter.The algebra and analysis of Grassmann variables ispresented in part I. The mathematics of these variables is applied to a randommatrix model, path integrals for fermions, dimer models and the Ising model intwo dimensions. Supermathematics - the use of commuting and anticommutingvariables on an equal footing - is the subject of part II. The properties ofsupervectors and supermatrices, which contain both commuting and Grassmanncomponents, are treated in great detail, including the derivation of integraltheorems. In part III, supersymmetric physical models

are considered. Whilesupersymmetry was first introduced in elementary particle physics as exactsymmetry between bosons and fermions, the formal introduction of anticommutingspacetime components, can be extended to problems of statistical physics, and,since it connects states with equal energies, has also found its way intoquantum mechanics.Several models are considered in the applications, afterwhich the representation of the random matrix model by the nonlinearsigma-model, the determination of the density of states and the levelcorrelation are derived. Eventually, the mobility edge behavior is discussedand a short account of the ten symmetry classes of disorder, two-dimensionaldisordered models, and superbosonization is given.

Part I Grassmann Variables and Applications.- Part II Supermathematics.- Part III Supersymmetry in Statistical Physics.- Summary and Additional Remarks.- References.- Solutions.- Index.