Second-quantized Darboux coordinates simplifying creation/annihilation operators

Identify and construct a change of second-quantized Darboux coordinates on the manifold of pure states that simplifies the creation and annihilation operators given in equation (5.35) and coincides, at the first-quantized level, with the holomorphic coordinate change in equation (4.67).

Background

In the geometrized formulation of QFT developed in the thesis, the authors define real and holomorphic second-quantized (s.q.) coordinates and express creation and annihilation operators in these coordinates. The expressions in real s.q. coordinates (equation 5.35) are algebraically involved.

They ask whether an appropriate change of Darboux s.q. coordinates exists that both simplifies these operator expressions and remains compatible with the holomorphic change at the first-quantized level (equation 4.67), leaving this as an open question.