  We establish a precise isomorphism between the Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger 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 K\"ahler polarization.
