TY - JOUR

T1 - Tempered wave functions

T2 - Schrödinger's equation within the light cone

AU - Rosenau, Philip

AU - Schuss, Zeev

N1 - Funding Information:
We thank Y. Gefen and A. Marchewka for useful discussions. The work of the first author was supported by the ISF grant No. 801/07 .

PY - 2011/1/31

Y1 - 2011/1/31

N2 - The standard derivation of Schrödinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of the propagation in time according to Schrödinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the time-dependent boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit of the time-sliced relativistic path integral, which produces Schrödinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. This produces an intermediate model between non-relativistic and relativistic mechanics of a single particle quantum particle. These boundary conditions relieve the solutions of Schrödinger's equation in the entire space of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), and so on. The practical implications of the present formulation are yet to be seen.

AB - The standard derivation of Schrödinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of the propagation in time according to Schrödinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the time-dependent boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit of the time-sliced relativistic path integral, which produces Schrödinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. This produces an intermediate model between non-relativistic and relativistic mechanics of a single particle quantum particle. These boundary conditions relieve the solutions of Schrödinger's equation in the entire space of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), and so on. The practical implications of the present formulation are yet to be seen.

UR - http://www.scopus.com/inward/record.url?scp=78651356160&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2010.12.067

DO - 10.1016/j.physleta.2010.12.067

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AN - SCOPUS:78651356160

SN - 0375-9601

VL - 375

SP - 891

EP - 897

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

IS - 5

ER -