Stochastic geometric model for a system of n fermions
Seminars
Seminar of the Sector #1, Department “Modern Mathematical Physics”
Date and Time: Wednesday, 11 October 2017, at 2:30 PM
Venue: Lecture Hall (2nd floor), Bogoliubov Laboratory of Theoretical Physics
Seminar topic: «Stochastic geometric model for a system of n fermions»
Speaker: Luigi M. Borasi (Institute for Applied Mathematics, University of Bonn, Germany)
Abstract:
We consider a “finite dimensional system of Fermions without spin” represented by an element of the exterior algebra of the n-dimensional complex space. We associate invariant vector fields on SO(2n + 1) to the Fermionic creation-annihilation operators. These vector fields implement the regular representation of the Lie algebra so(2n + 1). As such, they do not satisfy the canonical anti-commutation relations in general, however once they have been projected onto an appropriate subspace of L2 (SO(2n + 1)) these relations are satisfied. We choose a symmetric positive-definite quadratic form in the creationannihilation operators as Hamiltonian for our system of Fermions. The realization of Fermionic creation-annihilation operators in terms of (invariant) vector fields allows us to interpret the (Wick rotated) time evolution of the system of Fermions as a stochastic process generated by a hypoelliptic operator.