V5B7 - Advanced Topics in Analysis

Functional integrals involving commuting and anticommuting variables

 

Prof. M. Disertori

Lectures:

Tuesday    14(c.t)-16   Endenicher Allee 60  SemR 0.011

Friday        10(c.t)-12   Endenicher Allee 60 SemR 1.008

 

Lecture notes: see here

 

Description:

This course will introduce some tools to study integrals in N variables as N diverges and/or other parameters vanish. These integrals arise in several models both in theoretical physics and probability, among which random operators for quantum diffusion in disordered materials, spin systems in statistical mechanics and memory dependent random walks. They may involve, together with standar real and/or complex variables (which are commuting) also so-called Grassman variables (which are anticommuting). These variables are necessary to reformulate the starting integral as a new integral (duality) where estimates are easier. The resulting integral involves often a Gaussian measure perturbed by some local potential.

Tentative program:

  • Some motivation, dualities between integrals
  • Gaussian integrals in many variables (with real and complex parameters)
  • Introduction to Grassman variables
  • Some applications of integrals involving only Grassmann variables
  • Integrals involving both commuting and anti-commuting variables (supersymmetry)
  • Estimating the resulting integrals: saddle analysis in many variables, cluster expansions...  

 

 

 

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