Schrödinger Representation

Jae-Hyun YANG

Akira Arai, Hiroaki Kawano

We consider a class of representations {(Hv, Fv )}v of the Heisenberg commutation relation (HvFv - FvHv = -i) which are deformations of the Schrodinger representation, where the index parameter V is in a class of real­valued C00-functions on R. The Schrodinger representation is given by the case V = l. We classify the representations in terms of properties of V and show that they are divided into three subclasses. All elements of one of these subclasses are unitarily equivalent to the Schrodinger representation. But the others contain no elements unitarily equivalent to the Schrodinger represen­tation. In particular a subclass consists of representations in each of which Hv has no self-adjoint extension. Moreover, we obtain exact solutions to the Heisenberg equation and the Schrodinger equation associated with Hv. We show that, even in the case where Hv has no self-adjoint extension, exact solutions local in time can be constructed.

Robert Oeckl

We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case, this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.

John S. Nodvik