Cohomological Quantum Mechanics And Calculability Of Observables

TL;DR

This paper explores cohomological quantum mechanics on the punctured plane, revealing a consistency condition for the prepotential, identifying a hidden scale, and providing topological interpretations of invariants.

Contribution

It introduces a consistency integral equation for the prepotential, analyzes the role of a hidden scale, and connects topological invariants to the punctured plane within cohomological quantum mechanics.

Findings

Prepotential V obeys a specific integral equation.

Existence of a hidden scale from infrared regularization.

Topological invariants relate to winding number and homotopy.

Abstract

We reconsider quantum mechanical systems based on the classical action being the period of a one form over a cycle and elucidate three main points. First we show that the prepotenial V is no longer completely arbitrary but obeys a consistency integral equation. That is the one form dV defines the same period as the classical action. We then apply this to the case of the punctured plane for which the prepotential is of the form $V= \alpha \theta + \Phi ( \theta )$. The function $ \Phi $ is any but a periodic function of the polar angle. For the topological information to be preserved, we further require that $ \Phi $ be even. Second we point out the existence of a hidden scale which comes from the regularization of the infrared behaviour of the solutions. This will then be used to eliminate certain invariants preselected on dimensional counting grounds. Then provided we discard nonperiodic solutions as being non physical we compute the expectation values of the BRST- exact observables with the general form of the prepotential using only the orthonormality of the solutions (periodic). Third we give topological interpretations of the invariants in terms of the topological invariants wich live naturally on the punctured plane as the winding number and the fundamental group of homotopy,but this requires a prior twisting of the homotopy structure.