dorsal/arxiv
View SchemaExtension of Quantum Mechanics to Individual Systems
| Authors | James Ax, Simon Kochen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9905077 |
| URL | https://arxiv.org/abs/quant-ph/9905077 |
Abstract
The Copenhagen Interpretation describes individual systems, using the same Hilbert space formalism as does the statistical ensemble interpretation (SQM). This leads to the well-known paradoxes surrounding the Measurement Problem. We extend this common mathematical structure to encompass certain natural bundles with connections over the Hilbert sphere S. This permits a consistent extension of the statistical interpretation to interacting individual systems, thereby resolving these paradoxes. Suppose V is a physical system in interaction with another system W. The state vector of V+W has a set of polar decompositions with a vector q of complex coefficients. These are parameterized by the right toroid T of amplitudes q, and comprise a singular toroidal bundle over S, which comprises the enlarged state space of V+W. We prove that each T has a unique natural convex partition yielding the correct SQM probabilities. In the extended theory V and W synchronously assume pure spectral states according to which member of the partition contains q. The apparent indeterminism of SQM is thus attributable to the effectively random distribution of initial phases.
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"abstract": "The Copenhagen Interpretation describes individual systems, using the same\nHilbert space formalism as does the statistical ensemble interpretation (SQM).\nThis leads to the well-known paradoxes surrounding the Measurement Problem. We\nextend this common mathematical structure to encompass certain natural bundles\nwith connections over the Hilbert sphere S. This permits a consistent extension\nof the statistical interpretation to interacting individual systems, thereby\nresolving these paradoxes.\n Suppose V is a physical system in interaction with another system W. The\nstate vector of V+W has a set of polar decompositions with a vector q of\ncomplex coefficients. These are parameterized by the right toroid T of\namplitudes q, and comprise a singular toroidal bundle over S, which comprises\nthe enlarged state space of V+W. We prove that each T has a unique natural\nconvex partition yielding the correct SQM probabilities. In the extended theory\nV and W synchronously assume pure spectral states according to which member of\nthe partition contains q. The apparent indeterminism of SQM is thus\nattributable to the effectively random distribution of initial phases.",
"arxiv_id": "quant-ph/9905077",
"authors": [
"James Ax",
"Simon Kochen"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "Extension of Quantum Mechanics to Individual Systems",
"url": "https://arxiv.org/abs/quant-ph/9905077"
},
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