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arXiv:gr-qc/0610142 · Journal of Physics A Mathematical and Theoretical

Is quantum mechanics based on an invariance principle?

L. Brenig

Published Apr 11, 2007• 10 citations
arXivHTMLPDF

Abstract

Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through an invariance principle under transformations that preserve the Heisenberg position–momentum inequality. These transformations are induced by isotropic space dilations. This invariance imposes a change in the laws of classical mechanics that exactly corresponds to the transition-to-quantum mechanics. The Schrödinger equation appears jointly with a second nonlinear equation describing non-unitary processes. Unitary and non-unitary evolutions are exclusive and appear sequentially in time. The non-unitary equation admits solutions that seem to correspond to the collapse of the wavefunction.

DOI
10.1088/1751-8113/40/17/012

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Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through an invariance principle under transformations that preserve the Heisenberg position–momentum inequality. These transformations are induced by isotropic space dilations. This invariance imposes a change in the laws of classical mechanics that exactly corresponds to the transition-to-quantum mechanics. The Schrödinger equation appears jointly with a second nonlinear equation describing non-unitary processes. Unitary and non-unitary evolutions are exclusive and appear sequentially in time. The non-unitary equation admits solutions that seem to correspond to the collapse of the wavefunction.
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction.
A unique postulate is shown to underly the whole quantum mechanics theory: the invariance of the Heisenberg uncertainty inequality under a group of special nonlinear gauge transformations (NLGT). With this postulate, the quantum mechanics of a free particle is derived from classical mechanics, including the statements of the postulates of quantum mechanics, except for the wave-function-collapse postulate. An explanatory mechanism for the latter postulate is derived by performing an analytical continuation of the NLGTs. This extension results in a Schrödinger-bridge process, intertwined under the NLGT with the standard unitary quantum evolution, and revealing non-quantum (or beyond-quantum) phenomena. Mechanisms of that latter kind, like the ones associated to the quantum measurement process, occur in a new space-like dimension and hence are non causal in nature, in opposition to a time evolution. The present exercice focusses on the free particle in order to highlight the features of the performed derivation in the simplest possible way. Work is in progress to extend the performed derivation beyond that simple case.
Abstract For a free non-relativistic particule, the postulates of quantum mechanics are derived from classical mechanics by imposing a single one. It states that the Heisenberg inequality is fundamental and invariant under a group of special nonlinear gauge transformations (NLGT) with real group parameter. Imposing this postulate to classical mechanics with random initial positions generates a twin dynamical structure: one is the standard unitary quantum evolution. The other is not unitary. The two dynamics tightly intertwine under a NLGT. Moreover, the new dynamics and the unitary evolution are directly mapped one into the other by a NLGT with imaginary parameter. This mapping reveals that the novel dynamics is a Schrödinger bridge process (SBP), a classical random process that naturally involves the Born rule. Any solution admitted by a SBP requires both an initial and a final condition and its evolution parameter is structurally analogous to a space-like dimension, different of the usual time and revealing a process that is non causal in nature. Remarkably, this evolution parameter and the usual unitary time form a two-dimensional Minkowski space. The SBP provides a possible description of the collapse of the wave-function during a measurement. A functional Lie-algebra framework is used and is an essential mathematical tool for the presented theory.
Abstract A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group. The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles. To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are considered
The free Schrodinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrodinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.