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Publication# The research program of Stochastic Deformation (with a view toward Geometric Mechanics)

2012

Report or working paper

Report or working paper

Abstract

This is an overview of a program of stochastic deformation of the mathematical tools of classical mechanics, in the Lagrangian and Hamiltonian approaches. It can also be regarded as a stochastic version of Geometric Mechanics.The main idea is to construct well defined probability measures strongly inspired by Feynman Path integral method in Quantum Mechanics. In contrast with other approaches, this deformation preserves the invariance under time reversal of the underlying classical (conservative) dynamical systems.

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Christian Ferrari, Christian Gruber

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2010The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbeinlike formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a nontrivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics.