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arXiv:1307.1697 · Journal of Physics Conference Series

Classical and quantum-mechanical axioms with the higher time derivative formalism

Timur Kamalov

Published Jun 10, 2013• 31 citations
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Abstract

A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibration) reference frames with the random initial conditions? One of the most generalized ways of descriptions (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher temporal derivatives of the coordinates in the Lagrange function. Such formalism describing physical objects in the infinite dimensions space does not contradict to the quantum mechanics and infinite dimensions Hilbert space.

DOI
10.1088/1742-6596/442/1/012051

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A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibration) reference frames with the random initial conditions? One of the most generalized ways of descriptions (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher temporal derivatives of the coordinates in the Lagrange function. Such formalism describing physical objects in the infinite dimensions space does not contradict to the quantum mechanics and infinite dimensions Hilbert space.
A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibrational) reference frames with the random initial conditions? One of the most general description (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher order temporal derivatives of the coordinates in the Lagrange function. Such formalism describes physical objects in the infinite dimensional space does not contradict quantum mechanics and infinite dimensional Hilbert space.
Abstract Refusal to use inertial reference frames in favor of non-inertial reference frames means significant changes in the axiomatics of both classical and quantum physics. Taking into account the influences of random small forces and fields makes the equations of classical physics, expanding, subject not only to deterministic laws, but also expands the equations of classical physics with additional variables that have an indeterministic, probabilistic behavior. Extending the scope of consideration of physical systems to non-inertial frames of reference leads to the already well-known new field of study known as quantum correction of Newton’s laws of motion. In this case, not only stable deterministic trajectories are considered, but also unstable, random ones, which usually constitute the measurement error in experiments in cases of applying classical physics and the probability distribution of quantum mechanics. Such a quantum correction can extend the action of Newton’s laws to the microworld by abandoning the basic idea of classical physics that physical systems should be described by differential equations of no higher than second order in favor of supplementing them with higher derivative variables that describe unstable random trajectories.
The spatially-temporal structure which we are going to introduce is a distinctive feature in the description of mechanical objects. In classical mechanics the motion of point masses is described in a frame of reference which consists of a spatial Cartesian coordinate system and a clock. The distinguished class is formed by the so called inertial frames which are characterized by the property that a free point mass performs uniform rectilinear motion with respect to such a frame. It follows that the coordinates (ξ, τ) and (ξ′, τ′) of a point mass in any two inertial frames of reference are related by the Galilei transformation 3.1.1 ξ′=Rξ+x+vτ,τ′+τ+t, \xi ' = R\xi + x + v\tau , \tau ' + \tau + t, ξ′=Rξ+x+vτ,τ′+τ+t, where R is an orthogonal matrix (rotation) which describes the orientation of the new spatial axes with respect to the old ones, x is the vector of the spatial shift of the origin, ν is the relative velocity, and τ is the time shift showing the difference between the readings of the two clocks.
Abstract The main idea of I. Newton in Principia is a description of the laws of motion by a second-order differential equation. Classical Newtonian physics can describe stability trajectories in Inertial Reference Frames and some Non-Inertial ones. The variation of the action functional S of stability trajectories equals to zero. The observational error is including the influence of the random fields’ background to the particle. Can we use another description in the form of a high-order differential equation? The high-order derivatives can be used as additional variables accounting for the influence of random fields background. Trajectories due the influence of the random fields’ background can be called instability random trajectories. They can be described by high order derivatives. Then the stability classical trajectories to be complemented by additional instability random trajectories. Quantum objects are described by the trajectory with neighborhoods. Quantum Probability can describe quantum objects in random fields. The variation of the action functional S is defined by the Planck constant. For the common description of quantum theory and high-order theory, let us compare r-neighborhoods of quantum action functional with r-neighborhoods of the action functional of a high-order theory.
The non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid non-inertial frames, containing the standard rigidly linear accelereted and rotating ones, is given in the framework of parametrized Galilei theories. Then the multi-temporal quantization, in which the gauge variables, describing the non-inertial effects, are not quantized but considered as c-number generalized times, is applied to non relativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of center of mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid non-inertial frames are recovered as special cases.