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

TL;DR

This work addresses how the cocycle condition from deriving the Quantum Hamilton-Jacobi Equation (QHJE) via the Equivalence Postulate yields a unique Schwarzian derivative and links its infinitesimal variation to the exponentiation of the stress tensor transformation in conformal field theory (CFT), with a view toward higher dimensions. It demonstrates that energy quantization follows from the existence of the QHJE under duality transformations implied by the EP, by showing the QHJE is equivalent to the Schwarzian form ${\{w,q\\}} = -\frac{4m}{\hbar^2}{\cal W}(q)$ for $w=\psi^D/\psi$ and requiring $L^2(\mathbb{R})$ solutions. A two-particle model is developed to illustrate how the quantum potential arising from the EP drives nontrivial relative-motion dynamics, yielding a decomposed Schrödinger–Jacobi structure with a negative definite quantum potential and potential boundary-condition constraints on expansion coefficients that affect interactions. Collectively, the results connect non-probabilistic QHJE structure and Schwarzian calculus to stress-tensor transformation properties in CFT, suggesting a higher-dimensional Schwarzian role and a concrete mechanism for energy quantization.

Abstract

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.