College of Liberal Arts & Sciences

# Math Physics Seminar

**Mr. Marc Herrmann; Department of Physics & Astronomy; University of Iowa**

**“C*-algebras of the canonical commutation relations.”**

**Abstract:** In quantum mechanics, it is possible to construct an operator algebra and a vacuum vector that can be used to describe the behavior of a general representation of the canonical commutation relations. However, the selection of a vacuum vector becomes problematic when transitioning to quantum field theory.

To deal with this fact, we can start by defining a class of abstract C*-algebras corresponding to the canonical commutation relations. Specifying a positive linear functional on these algebras is then ufficient to construct a field theory.

This formulation can be used to compare the properties of algebras constructed from free fields that have been restricted to different hyperplanes, and discuss the requirements these algebras need satisfy in order to construct a physically relevant field theory.