Time Evolution in Quantum Systems and Stochastics

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The time evolution problem for non-self-adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral via the use of certain underlying stochastic differential equations, which naturally emerge when computing the path integral, leads to a universal expression for the associated measure regardless of the form of the differential operators. The discrete non-linear hierarchy (DNLS) is then considered and the corresponding hierarchy of solvable, in principle, SDEs is extracted. The first couple members of the hierarchy correspond to the discrete stochastic transport and heat equations. The discrete stochastic Burgers equation is also obtained through the analogue of the Cole–Hopf transformation. The continuum limit is also discussed.
Original languageEnglish
Title of host publicationQuantum Theory and Symmetries
Number of pages10
ISBN (Electronic)9783030557775
ISBN (Print)9783030557768
Publication statusPublished - 27 Mar 2021

Publication series

NameCRM Series in Mathematical Physics


  • math-ph
  • math.MP
  • nlin.SI


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