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Path integrals for boundaries in diffusion and quantum mechanics

Pontificia Universidad Católica de Chile. Instituto de Física, T. B. Lantaño

Published Jan 1, 2020• 0 citations
Publisher
DOI
10.7764/tesisuc/fis/48240

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Discontinuous initial wave functions, or wave functions with a discontinuous derivative and a bounded support, arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a unidirectional current at the boundary of the support. We use path integrals to define current and unidirectional current, and to provide a direct derivation of the expression for current from the path-integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short-time propagation of an initial wave function with compact support for cases of both a discontinuous derivative and a discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{3/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{1/2}).$ This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{1/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{\ensuremath{-}1/2}).$ This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(\ensuremath{\Delta}t).$ This gives a decay law.
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.
In this work, we advance the development of the Probability Bracket Notation (PBN), a formalism inspired by Dirac's notation in quantum mechanics, to provide a unified framework for probability modeling. We demonstrate that under a Special Wick Rotation (SWR), an imaginary-time map, the Schrödinger equation, the transition amplitude, and its associated path integral in Dirac notation transform into the master equation, the transition probability, and its Euclidean path integral in the PBN, from which we can reproduce the master equation, representing induced micro-diffusion processes. By extending this approach through a General Wick Rotation (GWR) and employing an anti-Hermitian wave-number operator, we perform parallel derivations of path integrals in both the Dirac and PBN frameworks. This leads to the formulation of the Euclidean Lagrangian for induced diffusions and the strong-damping harmonic oscillator (described by the Smoluchowski diffusion equation). Our findings highlight the versatility of the PBN in bridging quantum mechanics and stochastic processes, offering a coherent notation system for analyzing time evolution and path integrals across these domains.