Complex Structures and Quantization

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.