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Register a new userRiccati equations as a scale-relativistic gateway to quantum mechanics
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Applying the resolution-scale relativity principle to develop a mechanics of non-differentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Ito process driven by the solutions of a Riccati equation. We verify that the corresponding Fokker-Planck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrodinger equation for the same problem. Inspired by the treatment of the one-dimensional case, we identify a generalization to time dependent problems in any number of dimensions. The Ito process is then driven by a function which is identified as establishing the link between non-differentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantum-like states as characterizing the entire system rather than the motion of its individual constituents.
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This article motivates and presents the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. It stems from the scale relativity proposal to extend the principle of relativity to resolution-scale transformations, which leads to considering non-differentiable dynamical paths. We first define a complex scale-covariant time-differential operator and show that mechanics of non-differentiable paths is implemented in the same way as classical mechanics but with the replacement of the time derivative and velocity with the time-differential operator and associated complex velocity. With this, the generalized form of Newtons fundamental relation of dynamics is shown to take the form of a Langevin equation in the case of stationary motion characterized by a null average classical velocity. The numerical integration of the Langevin equation in the case of a harmonic oscillator taken as an example reveals the same statistics as the stationary solutions of the Schrodinger equation for the same problem. This motivates the rest of the paper, which shows Schrodingers equation to be a reformulation of Newtons fundamental relation of dynamics as generalized to non-differentiable geometries and leads to an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the scale relativistic approach and, at the same time, it allows to envision macroscopic chaotic systems observed at resolution time-scales exceeding their horizon of predictability as candidates in which to search for quantum-like dynamics and structures.
This article is the second in a series of two presenting the Scale Relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. Here, we show Schroedingers equation to be a reformulation of Newtons fundamental relation of dynamics as generalized to non-differentiable geometries in the first paper cite{paper1}. It motivates an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the Scale Relativistic approach and, at the same time, it allows to identify macroscopic chaotic systems considered at time scales exceeding their horizon of predictability as candidates in which to search for quantum-like structuring or behavior.
When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian background is presented as an intrinsic fundamental classical presumption. However, the Hamiltonian formulation of classical analytical mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather than a philosophical framework. If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a presentation in which little of the purported difference between quantum and classical mechanics remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal description is not a condition for but a consequence of objectivity. It requires no postulates. This is achieved by evading spatial notions and assuming nothing but time translation invariance.
Can a simple microscopic model of space and time demonstrate Special Relativity as the macroscopic (aggregate) behavior of an ensemble ? The question will be investigated in three parts. First, it is shown that the Lorentz transformation formally stems from the First Relativity Postulate (FRP) {it alone} if space-time quantization is a fundamental law of physics which must be included as part of the Postulate. An important corollary, however, is that when measuring devices which carry the basic units of lengths and time (e.g. a clock ticking every time quantum) are `moving uniformly, they appear to be measuring with larger units. Secondly, such an apparent increase in the sizes of the quanta can be attributed to extra fluctuations associated with motion, which are precisely described in terms of a thermally agitated harmonic oscillator by using a temperature parameter. This provides a stringent constraint on the microscopic properties of flat space-time: it is an array of quantized oscillators. Thirdly, since the foregoing development would suggest that the space-time array of an accelerated frame cannot be in thermal equilibrium, (i.e. it will have a distribution of temperatures), the approach is applied to the case of acceleration by the field of {it any} point object, which corresponds to a temperature `spike in the array. It is shown that the outward transport of energy by phonon conduction implies an inverse-square law of force at low speeds, and the full Schwarzschild metric at high speeds. A prediction of the new theory is that when two inertial observers move too fast relative to each other, or when fields are too strong, anharmonic corrections will modify effects like time dilation, and will lead to asymmetries which implies that the FRP may not be sustainable in this extreme limit.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schrodingers and Heisenbergs formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of parity and displacement in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
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