de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based o n the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigo rously defined to be chaotic if the Lyapunov number, associated with the qu antum trajectories in de Broglie-Bohm phase space, is positive definite. Th is definition of quantum chaos which under classical conditions goes over t o the well-known definition of classical chaos in terms of positivity of Ly apunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positiv e Lyapunov numbers in a simple quantum system is given analytically, provin g the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated . It is shown that the rigorous deterministic quantum chaos provides an int rinsic mechanism towards irreversibility of the Schrodinger evolution of th e wave function, without invoking 'wave function collapse' or 'measurements '.