The cocycle of the quantum HJ equation and the stress tensor of CFT

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.

On the Ricci flow and emergent quantum mechanics

“…The Ricci flow has provided many far-reaching insights into long-standing problems in topology and geometry [1,2]. Recent works [8,9,10,11,12,13,14,15] have shed light on applications of conformal symmetry and the Ricci flow to foundational issues in quantum mechanics. In this contribution we have established a 1-to-1 correspondence between conformally flat metrics on configuration space, and quantum mechanics on that same space.…”

“…After setting ϕ = f we appear to have a contradiction, since we have two different flow equations in (9) for just one field f . That there is in fact no contradiction can be seen as follows.…”

“…That there is in fact no contradiction can be seen as follows. In (9) we have two independent flow equations for the two independent fields f and ϕ. Equating the latter two fields implies that the two flow equations must reduce to just one. This can be achieved by substituting one of the two flow eqns.…”

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“…In the WKB approximation to quantum mechanics [3], one substitutes e iS/ for the wave function ψ in the time-independent Schrödinger equation. To zeroth order in , the result equals the Hamilton-Jacobi equation of classical mechanics (for an alternative presentation of this subject see [4] and refs. therein).…”