Complex Structures and Quantization

Abstract

The Schrödinger representation $\Gamma _S$ of $H_{2d+1}$ uses a specific choice of extra structure on classical phase space: a decomposition of its coordinates into positions $q_j$ and momenta $p_j$. For the unitarily equivalent Bargmann–Fock representation, a different sort of extra structure is needed, a decomposition of coordinates on phase space into complex coordinates $z_j$ and their complex conjugates $\overline{z}_j$. Such a decomposition is called a “complex structure" J and will correspond after quantization to a choice that distinguishes annihilation and creation operators.