The Cocycle of the Quantum HJ Equation and the Stress Tensor of CFT

We consider two theorems formulated in the derivation of the Quantum Hamilton–Jacobi Equation from the EP. The first one concerns the proof that the cocycle condition uniquely defines the Schwarzian derivative. This is equivalent to show that the infinitesimal variation of the stress tensor “exponentiates” to the Schwarzian derivative. The cocycle condition naturally defines the higher dimensional version of the Schwarzian derivative suggesting a role in the transformation properties of the stress tensor in higher dimensional CFT. The other theorem shows that energy quantization is a direct consequence of the existence of the quantum Hamilton–Jacobi equation under duality transformations as implied by the EP. 1 HJ Equation and Coordinate Transformations Let us consider the Hamilton–Jacobi (HJ) equation for a one dimensional system. This is obtained by considering the canonical transformation (q, p) → (Q,P ) leading to a vanishing Hamiltonian