Complex Structures and Quantization

The Schrodinger 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.