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A SUBLUMINOUS SCHR ¨ ODINGER EQUATION

Philip Rosenau, Z. Schuss

Published Jul 11, 2010• 0 citations
OpenAlex

Abstract

AbstractThe standard derivation of Schro¨dinger’s equation from a Lorentz-invariant Feynman path integral consists in taking first the limit ofinfinite speed of light and then the limit of short time slice. In this or-der of limits the light cone of the path integral disappears, giving riseto an instantaneous spread of the wave function to the entire space.We ascribe the failure of Schro¨dinger’s equation to retain the lightcone of the path integral to the very nature of the limiting process: itis a regular expansion of a singular approximation problem, becausethe boundary conditions of the path integral on the light cone are lostin this limit. We propose a distinguished limit, which produces anintermediate model between non-relativistic and relativistic quantummechanics: it produces Schro¨dinger’s equation and preserves the zeroboundary conditions on and outside the original light cone of the pathintegral. These boundary conditions relieve the Schro¨dinger equationof several annoying, seemingly unrelated unphysical artifacts, includ-ing non-analytic wave functions, spontaneous appearance of disconti-nuities, non-existence of moments when the initial wave function hasa jump discontinuity (e.g., a collapsed wave function after a measure-ment), the EPR paradox, and so on. The practical implications of thepresent formulation are yet to be seen.

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AbstractThe standard derivation of Schro¨dinger’s equation from a Lorentz-invariant Feynman path integral consists in taking first the limit ofinfinite speed of light and then the limit of short time slice. In this or-der of limits the light cone of the path integral disappears, giving riseto an instantaneous spread of the wave function to the entire space.We ascribe the failure of Schro¨dinger’s equation to retain the lightcone of the path integral to the very nature of the limiting process: itis a regular expansion of a singular approximation problem, becausethe boundary conditions of the path integral on the light cone are lostin this limit. We propose a distinguished limit, which produces anintermediate model between non-relativistic and relativistic quantummechanics: it produces Schro¨dinger’s equation and preserves the zeroboundary conditions on and outside the original light cone of the pathintegral. These boundary conditions relieve the Schro¨dinger equationof several annoying, seemingly unrelated unphysical artifacts, includ-ing non-analytic wave functions, spontaneous appearance of disconti-nuities, non-existence of moments when the initial wave function hasa jump discontinuity (e.g., a collapsed wave function after a measure-ment), the EPR paradox, and so on. The practical implications of thepresent formulation are yet to be seen.
The standard derivation of Schroedinger'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 Schroedinger'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 boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit, which produces an intermediate model between non-relativistic and relativistic quantum mechanics: it produces Schroedinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. These boundary conditions relieve the Schroedinger equation 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), the EPR paradox, and so on. The practical implications of the present formulation are yet to be seen.
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
In his extremely important paper [1] E. Schro¨dinger proposed a wave equation which is the basic equation of the “undulatorical” mechanics and can be considered as a substitution to the Hamilton-Jacobi partial differential equations (HJ) of the ordinary (classical) mechanics. The wave equation was established on condition that the Lagrange function contains no terms linear in velocities. Schro¨dinger writes (footnote on p. 514 l.c.): “In relativistic mechanics and in the case of a magnetic field the expression of the (HJ) is more complicated. In the case of a single electron this equation means the constancy of the four-dimensional gradient diminished by a given vector (the four-dimensional potential). The wave mechanical translation of this Ansatz meets some difficulties.”
Extreme kinematic limits of space-time symmetries reveal alternative forms of quantum behaviour. In the Carrollian limit the speed of light is taken to zero ([Formula: see text]): causal light cones collapse onto the time axis, spatial points become disconnected, and the usual notion of particle motion disappears. The corresponding quantum equation is the Carroll–Schrödinger equation—the structural mirror image of the ordinary Schrödinger equation, being first order in space and second order in time. We explore the mathematical connections between these two equations in one space and one time dimension. Using operator methods, we find conditions on external potentials that allow both equations to share solutions, and show that any Carrollian problem can be mapped to an equivalent Schrödinger problem via an explicit coordinate transformation. Probability densities and currents are related by removing the interaction potential and swapping space and time coordinates. An extreme relativistic boost combined with a classical approximation yields the Carrollian dispersion relation and conditions for coinciding classical trajectories. By formulating the dynamics on a Hilbert space of time rather than space, we prove that evolution in the spatial coordinate preserves probability. Closed-form solutions illustrate each result and together offer a practical framework for translating between the two descriptions.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.
The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.