Supermathematics And Its Applications In Statistical Physics: Grassmann Variables And The Method Of Supersymmetry (lecture Notes In Physics)
by Franz Wegner /
2016 / English / PDF
3.5 MB Download
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.
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.
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.
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.