Unitary dilations of commutation relations associated to alternating bilinear forms
We study second quantization of symplectic spaces \(V\) with given alternating bilinear form. An algebraic framework is developed using the associated central extension Lie algebras. We then impose the further condition that \(V\) carries a reflection operator satisfying an Osterwalder-Schrader type positivity. Our construction of representations is based on induction from representations of associated (non-degenerate) Heisenberg algebras obtained as unitary dilations of given \(V\). When Osterwalder-Schrader positivity is assumed, we show that \(V\) has a Heisenberg extension inducing representations of \(V\) if and only if \(V\) decomposes in a specific way such that the given symplectic form satisfies a certain a priori estimate relative to the subspaces of the decomposition. Our induction provides in particular representations of the free chiral quantum field on a higher genus Riemann surface.
second quantization of symplectic spaces
scientific article; zbMATH DE number 561298
scientific article