dorsal/arxiv
View SchemaA group of invariance transformations for nonrelativistic quantum mechanics
| Authors | B. Galvan |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0002081 |
| URL | https://arxiv.org/abs/quant-ph/0002081 |
Abstract
This paper defines, on the Galilean space-time, the group of asymptotically Euclidean transformations (AET), which are equivalent to Euclidean transformations at space-time infinity, and proposes a formulation of nonrelativistic quantum mechanics which is invariant under such transformations. This formulation is based on the asymptotic quantum measure, which is shown to be invariant under AET's. This invariance exposes an important connection between AET's and Feynman path integrals, and reveals the nonmetric character of the asymptotic quantum measure. The latter feature becomes even clearer when the theory is formulated in terms of the coordinate-free formalism of asymptotically Euclidean manifold, which do not have a metric structure. This mathematical formalism suggests the following physical interpretation: (i) Particles evolution is represented by trajectories on an asymptotically Euclidean manifold; (ii) The metric and the law of motion are not defined a priori as fundamental entities, but they are properties of a particular class of reference frames; (iii) The universe is considered as a probability space in which the asymptotic quantum measure plays the role of a probability measure. Points (ii) and (iii) are used to build the asymptotic measurement theory, which is shown to be consistent with traditional quantum measurement theory. The most remarkable feature of this measurement theory is the possibility of having a nonchaotic distribution of the initial conditions (NCDIC), an extremely counterintuitive but not paradoxical phenomenon which allows to interpret typical quantum phenomena, such as particle diffraction and tunnel effect, while still providing a description of their motion in terms of classical trajectories.
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"abstract": "This paper defines, on the Galilean space-time, the group of asymptotically\nEuclidean transformations (AET), which are equivalent to Euclidean\ntransformations at space-time infinity, and proposes a formulation of\nnonrelativistic quantum mechanics which is invariant under such\ntransformations. This formulation is based on the asymptotic quantum measure,\nwhich is shown to be invariant under AET\u0027s. This invariance exposes an\nimportant connection between AET\u0027s and Feynman path integrals, and reveals the\nnonmetric character of the asymptotic quantum measure. The latter feature\nbecomes even clearer when the theory is formulated in terms of the\ncoordinate-free formalism of asymptotically Euclidean manifold, which do not\nhave a metric structure. This mathematical formalism suggests the following\nphysical interpretation: (i) Particles evolution is represented by trajectories\non an asymptotically Euclidean manifold; (ii) The metric and the law of motion\nare not defined a priori as fundamental entities, but they are properties of a\nparticular class of reference frames; (iii) The universe is considered as a\nprobability space in which the asymptotic quantum measure plays the role of a\nprobability measure. Points (ii) and (iii) are used to build the asymptotic\nmeasurement theory, which is shown to be consistent with traditional quantum\nmeasurement theory. The most remarkable feature of this measurement theory is\nthe possibility of having a nonchaotic distribution of the initial conditions\n(NCDIC), an extremely counterintuitive but not paradoxical phenomenon which\nallows to interpret typical quantum phenomena, such as particle diffraction and\ntunnel effect, while still providing a description of their motion in terms of\nclassical trajectories.",
"arxiv_id": "quant-ph/0002081",
"authors": [
"B. Galvan"
],
"categories": [
"quant-ph",
"gr-qc"
],
"title": "A group of invariance transformations for nonrelativistic quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0002081"
},
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