Quantum-to-classical transition and the emergence of quantum Darwinism with measurements distributed in time -- a path integral approach

We present a new formulation for the emergence of classical dynamics in a quantum world by considering a path integral approach that also incorporates continuous measurements. Our program is conceptually different from the decoherence program as well as the quantum-to-classical transition framework with coarse-grained measurements. The path integral formulation provides the joint statistics of a sequence of measurements with each Feynman path picking up an additional random phase due to measurements. The magnitude of this phase is proportional to the measurement strength, and we give conditions under which the dominant contribution to the probability amplitude comes from the trajectories in the vicinity of the classical paths. The proliferation of this information accross the environment, an essential feature of quantum Darwinism, takes place via scattering of plane-wave probes by the system. Extending to repeated measurements, we show that in the continuous limit, each system trajectory picks up an additional phase due to work done by the momentum kicks from the probes - origin of the back-action force. We provide conditions for which the measurement provides sufficient ''which-path" information and keeps the wave packet sufficiently localized. This allows for description of quantum-to-classical transition at the level of individual trajectories in contrast to the statistical ensemble interpretation provided by density matrices in the decoherence program.