Quantum to Classical Transition - A Path Integral Approach

We present a path-integral formulation of quantum-to-classical transition for systems undergoing continuous measurements. Quantum mechanical interferences are sensitive to measurements; the cost of which-path information is paid by loss of coherence and back-action disturbance. We consider scattering of the system and plane-wave probes by employing the S-Matrix formalism to represent outgoing momentum-scrambled eigenstates. Extending to repeated measurements, we show that in the continuous limit each system trajectory picks up an additional phase due to work done by momentum kicks from the probes. Hence, in the path integral formulation of sum-of-phases, each phase is randomized due to this back-action force. We show that the contribution to the joint system-probe amplitude at Ο(hⁿ) is proportional to the product of strength of quantum fluctuations (given by the n^th functional derivative of the Action) and 1/λⁿ . We provide conditions on the S-Matrix for which the measurement provides sufficient which-path information and localization, in spirit with phenomenological master equations which construct quantum-to-classical transition based on wave-packet localization.

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