Supermathematics and its Applications in Statistical Physics

Grassmann Variables and the Method of Supersymmetry

  • Franz Wegner

Part of the Lecture Notes in Physics book series (LNP, volume 920)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Grassmann Variables and Applications

    1. Front Matter
      Pages 1-1
    2. Franz Wegner
      Pages 3-5
    3. Franz Wegner
      Pages 7-12
    4. Franz Wegner
      Pages 13-27
    5. Franz Wegner
      Pages 29-36
    6. Franz Wegner
      Pages 45-46
    7. Franz Wegner
      Pages 47-66
    8. Franz Wegner
      Pages 67-73
    9. Franz Wegner
      Pages 75-100
  3. Supermathematics

    1. Front Matter
      Pages 101-101
    2. Franz Wegner
      Pages 103-111
    3. Franz Wegner
      Pages 113-116
    4. Franz Wegner
      Pages 117-121
    5. Franz Wegner
      Pages 139-154
    6. Franz Wegner
      Pages 171-179
  4. Supersymmetry in Statistical Physics

    1. Front Matter
      Pages 181-182
    2. Franz Wegner
      Pages 183-191
    3. Franz Wegner
      Pages 203-225
    4. Franz Wegner
      Pages 227-259
    5. Franz Wegner
      Pages 261-301
    6. Franz Wegner
      Pages 303-334
    7. Franz Wegner
      Pages 335-339
  5. Back Matter
    Pages 341-374

About this book


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.


Disordered systems Supermatrices and superdeterminants Supersymmetric and superreal matrices Supersymmetric quantum mechanics Random matrix theory Nonlinear sigma model Schäfer-Wegner parametrization

Authors and affiliations

  • Franz Wegner
    • 1
  1. 1.Institut für Theoretische PhysikUniversität HeidelbergHeidelbergGermany

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2016
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-3-662-49168-3
  • Online ISBN 978-3-662-49170-6
  • Series Print ISSN 0075-8450
  • Series Online ISSN 1616-6361
  • Buy this book on publisher's site
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