Quantum Observables for Free Fermions With Boundary Conditions

Speaker:

Stephan Stolz

Zugehörigkeit:

University of Notre Dame

Datum:

Don, 23/05/2024 - 12:00 - 13:30

Location:

MPIM Lecture Hall [2]

Parent event:

Physical Math Seminar [3]

Costello and Gwilliam developed a systematic approach to perturbative quantization of field theories that describes classical and quantum observables as factorization algebras on the space-time manifold. In the simple case of the linear sigma model whose fields are smooth maps from the real line to an inner product space $V$ (aka the free boson), the algebra of quantum observables is the Weyl algebra of differential operators on $V$. For the free fermion, where $V$ is replaced by its odd analog, the algebra of quantum observables is the Clifford algebra generated by $V$.

In this talk, based on joint work with Bill Dwyer, I will discuss how to enhance the free fermion by suitable boundary conditions to obtain a factorization algebra of “bulk-boundary quantum observables”, which can be evaluated on open intervals as well as half-open intervals. The challenge is that there is a space of boundary conditions, each of which yields a bulk-boundary factorization algebra, but unlike for the free boson (where there are Dirichlet and Neumann boundary conditions), there is no distinguished boundary condition. Our main result is that a spin structure on $V$ yields a distinguished bulk-boundary factorization algebra of quantum observables.