Lectures SS 2026

Speaker: Michael Preeg, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by K.-H. Neeb

Abstract: In quantum field theory the CCR and CAR algebras play an important role. The are defined as the algebra generated by elements of complex Hilbert space $\mathcal{H}$ with respect to their corresponding relations either involving the symplectic form or the symmetric bilinear form defined by the inner product of $\mathcal{H}$. Both algebras have a natural way of acting on elements of the corresponding Fock space (symmetric or antisymmetric) defined via creation and annihilation operators. Since orthogonal and symplectic operators on $\mathcal{H}$ (understood as the underlying real Hilbert space) are preserving the corresponding CCR or CAR relations, one could ask whether the representations of the algebras are unitary equivalent before and after application of these operators.

This is answered by D. Shale and W. F. Stinespring. They are unitarily equivalent if and only if the orthogonal or symplectic operators are restricted in the sense that their antilinear part with respect to the complex strucure defined by the multiplication by $i \in \mathbb{C}$ is Hilbert-Schmidt. They are Banach-Lie groups, but this is not the appropriate topology. In this talk we will define the strongly restricted topology, explore them as topological groups and show the continuity of the projective unitary representation via the implementation on their corresponding Fock spaces.