The resolvent algebra of the canonical commutation relations

Abstract: 

The canonical commutation relations (CCR's) are at the foundation of quantum mechanics and quantum field theory. To properly analyse the CCR's, which involve unbounded operators, we encode them in a C$^*$-algebra. This C*-algebra however, is not unique. The Weyl algebra has been the standard C$^*$-algebra of the CCR's used by mathematical physicists since the 1960s, but it does not admit many natural dynamics and does not contain some objects of physical interest, such as bounded functions of the Hamiltonian. In this seminar I will discuss the resolvent algebra which is the C$^*$-algebra generated by the resolvents of the canonical field operators. We will see its key properties, that it admits many families of dynamics, and that it contains natural physical objects which the Weyl algebra does not contain.