dorsal/arxiv
View SchemaReconstruction of quantum theory on the basis of the formula of total probability
| Authors | Andrei Khrennikov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0302194 |
| URL | https://arxiv.org/abs/quant-ph/0302194 |
| DOI | 10.1063/1.1874572 |
| Journal | Proceedings of Conference Foundations of Probability and Physics-3, American Institute of Physics, Ser. Conference Proceedings, 750, pp. 187-219, 2005 |
Abstract
The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: ``first context and then probability.'' We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference.
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"abstract": "The notion of context (complex of physical conditions) is basic in this\npaper. We show that the main structures of quantum theory (interference of\nprobabilities, Born\u0027s rule, complex probabilistic amplitudes, Hilbert state\nspace, representation of observables by operators) are present in a latent form\nin the classical Kolmogorov probability model. However, this model should be\nconsidered as a calculus of contextual probabilities. In our approach it is\nforbidden to consider abstract context independent probabilities: ``first\ncontext and then probability.\u0027\u0027 We start with the conventional formula of total\nprobability for contextual (conditional) probabilities and then we rewrite it\nby eliminating combinations of incompatible contexts from consideration. In\nthis way we obtain interference of probabilities without to appeal to the\nHilbert space formalism or wave mechanics. However, we did not just reconstruct\nthe probabilistic formalism of conventional quantum mechanics. Our contextual\nprobabilistic model is essentially more general and, besides the projection to\nthe complex Hilbert space, it has other projections. The most important new\nprediction is the possibility (at least theoretical) of appearance of\nhyperbolic interference.",
"arxiv_id": "quant-ph/0302194",
"authors": [
"Andrei Khrennikov"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1874572",
"journal_ref": "Proceedings of Conference Foundations of Probability and\n Physics-3, American Institute of Physics, Ser. Conference Proceedings, 750,\n pp. 187-219, 2005",
"title": "Reconstruction of quantum theory on the basis of the formula of total probability",
"url": "https://arxiv.org/abs/quant-ph/0302194"
},
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