English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Semiclassical path integral expression for a quantum system coupled to aharmonic bath is derived based on the stationary phase condition. It isdiscovered that the system path is non-Markovian. Most strikingly, the systempath not only couples to its past (as in the Langevin equation), but also toits future, i.e. the equation of motion for the system is anintegro-differential equation that involves all times. Numerical tests areperformed to confirm that the future-involved term is indeed necessary. Becauseof the future-non-Markovian nature of the equation, the numerical solutioncannot be obtained by iterative methods. Instead, root search algorithms mustbe employed.

Non-Markovian trajectories involving future in the semi-classical path integral expression