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Physical Review

A Calculus of Finite Precision

Benjamin Liebowitz

Published Dec 1, 1944• 5 citations
Publisher

Abstract

Sec. I. This paper inverts certain quantum mechanical ideas by constructing a "probability of presence" from first principles. If ${x}_{i}={f}_{i}(s,{a}_{1},{a}_{2})$ represents a 2-parameter family of curves in 3-dimensional space, the density of the functions ${f}_{i}$ with respect to the parameters ${a}_{1}{a}_{2}$ is defined by means of an expression for the normal cross section of a sheaf of curves. This density plays the role of a probability. Its derivative along a curve of the family is independent of the parameters ${a}_{1}{a}_{2}$. This circumstance permits a special density to be constructed, called the "characteristic" or "smoothest" density, which is independent of the parameters and which can be found by solving a linear second-order partial differential equation. The extension to $n$-dimensional space and ($n\ensuremath{-}1$) parameters is given. Sec. II. The concept of orbital density is the dynamical equivalent of the density of Sec. I. Dividing the orbital density by the velocity of the representative point gives the quantity we call kinematical probability $P$, which satisfies the fundamental equation (2.8), and which has the same force as any other probability of presence. Equation (2.8) is used to derive from the Hamilton-Jacobi equation a "wave" equation for $\ensuremath{\surd}P$ for a system of $n$ planets moving around a sun under the condition that the space average of the mutual interactions of the planets is a minimum. This provides a new approach to planetary dynamics and yields an equation that is very much like Schroedinger's equation for the stationary states of $n$ particles, and has the same basic interpretation. Sec. III. A non-conservative system of $n$ particles is assumed to be in a state such that the action function can be written $W=\ensuremath{-}Et+R({x}_{i},t)$ in which $R$ still contains the time explicitly. Starting with the Hamilton-Jacobi equation, using Eq. (2.8), and imposing the condition that the space average of $\frac{\ensuremath{\partial}R}{\ensuremath{\partial}t}$ is a minimum, an equation [viz., (3.10)] is derived which differs from Schroedinger's equation for stationary states only in certain respects that cannot be regarded as fundamental. Despite this basic equivalence of Schroedinger's equation and our equation (3.10), conflicts in interpretation and formalism arise which are briefly discussed. Sec. IV. Complex $U$' s are introduced into Eq. (3.10) in a simple way which fails to bridge the gap between our formalism and that of quantum mechanics. The kinematical probability defined by Eq. (3.10) is shown to have the "smoothest" property. To account physically for the stationary states an electromagnetic wave system is proposed which has the property that a particle moving in it may be in "phase-mechanical" resonance with it. Macroscopic examples of this kind of resonance are given. This concept represents an inversion of physical ideas of quantum mechanics corresponding to the mathematical inversion with which we began.

DOI
10.1103/physrev.66.343

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Sec. I. This paper inverts certain quantum mechanical ideas by constructing a "probability of presence" from first principles. If xi=fi(s,a1,a2){x}_{i}={f}_{i}(s,{a}_{1},{a}_{2})xi​=fi​(s,a1​,a2​) represents a 2-parameter family of curves in 3-dimensional space, the density of the functions fi{f}_{i}fi​ with respect to the parameters a1a2{a}_{1}{a}_{2}a1​a2​ is defined by means of an expression for the normal cross section of a sheaf of curves. This density plays the role of a probability. Its derivative along a curve of the family is independent of the parameters a1a2{a}_{1}{a}_{2}a1​a2​. This circumstance permits a special density to be constructed, called the "characteristic" or "smoothest" density, which is independent of the parameters and which can be found by solving a linear second-order partial differential equation. The extension to nnn-dimensional space and (n\ensuremath−1n\ensuremath{-}1n\ensuremath−1) parameters is given. Sec. II. The concept of orbital density is the dynamical equivalent of the density of Sec. I. Dividing the orbital density by the velocity of the representative point gives the quantity we call kinematical probability PPP, which satisfies the fundamental equation (2.8), and which has the same force as any other probability of presence. Equation (2.8) is used to derive from the Hamilton-Jacobi equation a "wave" equation for \ensuremath√P\ensuremath{\surd}P\ensuremath√P for a system of nnn planets moving around a sun under the condition that the space average of the mutual interactions of the planets is a minimum. This provides a new approach to planetary dynamics and yields an equation that is very much like Schroedinger's equation for the stationary states of nnn particles, and has the same basic interpretation. Sec. III. A non-conservative system of nnn particles is assumed to be in a state such that the action function can be written W=\ensuremath−Et+R(xi,t)W=\ensuremath{-}Et+R({x}_{i},t)W=\ensuremath−Et+R(xi​,t) in which RRR still contains the time explicitly. Starting with the Hamilton-Jacobi equation, using Eq. (2.8), and imposing the condition that the space average of \ensuremath∂R\ensuremath∂t\frac{\ensuremath{\partial}R}{\ensuremath{\partial}t}\ensuremath∂t\ensuremath∂R​ is a minimum, an equation [viz., (3.10)] is derived which differs from Schroedinger's equation for stationary states only in certain respects that cannot be regarded as fundamental. Despite this basic equivalence of Schroedinger's equation and our equation (3.10), conflicts in interpretation and formalism arise which are briefly discussed. Sec. IV. Complex UUU' s are introduced into Eq. (3.10) in a simple way which fails to bridge the gap between our formalism and that of quantum mechanics. The kinematical probability defined by Eq. (3.10) is shown to have the "smoothest" property. To account physically for the stationary states an electromagnetic wave system is proposed which has the property that a particle moving in it may be in "phase-mechanical" resonance with it. Macroscopic examples of this kind of resonance are given. This concept represents an inversion of physical ideas of quantum mechanics corresponding to the mathematical inversion with which we began.
Implementation of the equivalence principle, stating that all physical systems can be connected by a coordinate transformation (trivializing map) to the free system with vanishing energy, uniquely leads to the quantum analogue of the Hamilton–Jacobi (HJ) equation. A property of the formalism is manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of the recently observed relation with second–order differential equations. This makes clear the intrinsic ψD –ψ duality between linearly independent solutions of the Schrödinger equation. Being a third–order non–linear differential equation, the Quantum Stationary HJ Equation (QSHJE) gives rise to initial conditions which cannot be seen in the Schrödinger equation. Solutions of the QSHJE provide a trajectory representation of quantum mechanics, depending on the Planck length. Unlike Bohm’s theory, there is a non– trivial action even for bound states and no wave guide is present. A fundamental aspect of the formulation is that no use of any axiomatic interpretation of the wave function is made. The QSHJE is defined only if the ratio ψD /ψ is a local homeomorphism of the extended real line into itself. In this respect the quantization of the energy is strictly related to the S duality of the Schwarzian derivative. As a result, the equivalence principle, according to which all the potentials are equivalent under coordinate transformations, by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization. Finally, we derive the quantum HJ equation also in the higher dimensional case with time dependent potentials and in the relativistic case. Contents
A novel theory is outlined for describing the dynamics of relativistic electrons and positrons.By introducing the Lorentz-invariant universal time as a fifth independent variable, the Hamilton-Jacobi formalism of classical mechanics is extended from three to four spatial dimensions.This approach allows one to incorporate gravitation and spin interactions in the extended fivedimensional Lagrangian in a covariant form.The universal time has the function of a hidden Bell parameter.By employing the method of variation with respect to the four coordinates of the particle and the components of the electromagnetic field, the path equation and the electromagnetic field produced by the charge and the spin of the moving particle are derived.In addition the covariant equations for the dynamics of the components of the spin tensor are obtained.These equations can be transformed to the familiar BMT equation in the case of homogeneous electromagnetic fields.The quantization of the five-dimensional Hamilton-Jacobi equation yields a five-dimensional spinor wave equation, which degenerates to the Dirac equation in the stationary case if we neglect gravitation.The quantity which corresponds to the probability density of standard quantum mechanics is the four-dimensional mass density which has a real physical meaning.By means of the Green method the wave equation is transformed into an integral equation enabling a covariant relativistic path integral formulation.Using this approach a very accurate approximation for the four-dimensional propagator is derived.The proposed formalism makes Dirac's hole theory obsolete and can readily be extended to many particles.
Abstract In stark contrast to classical Newtonian mechanics, the standard interpretation of quantum mechanics associates the Schrödinger wavefunction, not in a determ- inistic way with definite predictions about a specific, individual experimental result, but only in a probabilistic manner, with |ψ(x, t )|2 acting as a probability density, in this case, for position measurements. Before analyzing this approach in more detail, we will find it profitable to briefly review some of the basic concepts of discrete and
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,\Psi(\mathbf r,t)=R(\mathbf r,t)e^{iS(\mathbf r,t)/\hbar},Ψ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂R2∂t+∇⋅[R2∇S−qAm]=0,\frac{\partial R^2}{\partial t}+\nabla\cdot\left[R^2\frac{\nabla S-q\mathbf A}{m}\right]=0,∂t∂R2​+∇⋅[R2m∇S−qA​]=0,∂S∂t+(∇S−qA)22m+qϕ+U+Q=0,\frac{\partial S}{\partial t}+\frac{(\nabla S-q\mathbf A)^2}{2m}+q\phi+U+Q=0,∂t∂S​+2m(∇S−qA)2​+qϕ+U+Q=0,whereQ=−ℏ22m∇2RR.Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}.Q=−2mℏ2​R∇2R​. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,\boldsymbol{\Pi}=\nabla S-q\mathbf A,Π=∇S−qA,and the associated local propagation velocity isv=Πm.\mathbf v=\frac{\boldsymbol{\Pi}}{m}.v=mΠ​. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term QQQ provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdvdt=q(E+v×B)−∇U−∇Q,m\frac{d\mathbf v}{dt}=q(\mathbf E+\mathbf v\times\mathbf B)-\nabla U-\nabla Q,mdtdv​=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.