Abstract
In physics, experiments ultimately inform us about what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space) can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold and provides a crucial first link for any theoretical model of quantum space-time at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.
Highlights
Preface for the Special Issue: “Planck-Scale Deformations of Relativistic Symmetries.” Our group has been working on a relativity deformation scheme within the Lie group/algebra framework
As can hardly be emphasized enough, every precise formulation of any physical concept is really only a model, or part of a model, of nature. All such concepts need to have their mathematical and physical content reevaluated as theories develop. Quantum mechanics as it is to date inherits, with little critical revision, many Newtonian conceptual notions, while we see that perhaps a lot more fundamental changes are called for, even down to the most basic one: that of physical space and position within it
We are supposed to learn from experiments what constitutes a good/correct theoretical/mathematical model of any physical concept, and physical space should not be an exception
Summary
Preface for the Special Issue: “Planck-Scale Deformations of Relativistic Symmetries.” Our group has been working on a relativity deformation scheme within the Lie group/algebra framework. It was easy to accept the mathematical formulation of the theory but a lot more difficult to adopt a fundamental change in our basic perspective It is not a surprise, that even the great physicists who created the theory kept trying to think and talk about it in terms of Newtonian concepts, many of which are really not compatible with quantum mechanics. All such concepts need to have their mathematical and physical content reevaluated as theories develop Quantum mechanics as it is to date inherits, with little critical revision, many Newtonian conceptual notions, while we see that perhaps a lot more fundamental changes are called for, even down to the most basic one: that of physical space and position within it. Relativity symmetry, the Galilean symmetry for the case of Newtonian mechanics, is the crucial link It is as fundamental as the assumption of the structure of the physical space itself.
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