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arXiv:hep-th/0103027 · Physics Letters A

Renormalisation in quantum mechanics

H. Jirari, H. Kröger, X.Q. Luo, G. Melkonyan, K.J.M. Moriarty

Published Oct 1, 2002• 9 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(02)01269-0

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We use the functional Renormalisation Group (fRG) to describe the equilibrium state of stochastic processes governed by an overdamped Langevin equation in highly anharmonic potentials. Exploiting the connection between Langevin dynamics and supersymmetric quantum mechanics in imaginary time, we obtain renormalisation flow equations for the effective action, approximated in terms of the Local Potential Approximation and Wavefunction Renormalisation which we solve numerically. The obtained effective potential determines exactly the equilibrium position ⟨x⟩\langle x \rangle⟨x⟩, the variance ⟨(x−⟨x⟩)2⟩\langle \left(x- \langle x \rangle\right)^2\rangle⟨(x−⟨x⟩)2⟩, as well as all higher order cumulants. The equilibrium 2-point correlation function ⟨x(0)x(t)⟩\langle x(0)x(t) \rangle⟨x(0)x(t)⟩ and the relevant correlation time can also be obtained to percent accuracy when Wavefunction Renormalisation is utilized. The numerical computations determining the effective action are much faster than the direct simulation of the stochastic dynamics to which we compare our fRG results. A companion paper will examine the application of the fRG to this system when the initial state is not that of equilibrium.