dorsal/arxiv
View SchemaOn the representation of contextual probabilistic dynamics in the complex Hilbert space: linear and nonlinear evolutions, Schroedinger dynamics
| Authors | Andrei Yu. Khrennikov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409128 |
| URL | https://arxiv.org/abs/quant-ph/0409128 |
| DOI | 10.1393/ncb/i2004-10142-6 |
Abstract
We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a ``prespace''. The basic condition for representing of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert space projection of the ``prespace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).
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"abstract": "We constructed the representation of contextual probabilistic dynamics in the\ncomplex Hilbert space. Thus dynamics of the wave function can be considered as\nHilbert space projections of realistic dynamics in a ``prespace\u0027\u0027. The basic\ncondition for representing of the prespace-dynamics is the law of statistical\nconservation of energy -- conservation of probabilities. Construction of the\ndynamical representation is an important step in the development of contextual\nstatistical viewpoint to quantum processes. But the contextual statistical\nmodel is essentially more general than the quantum one. Therefore in general\nthe Hilbert space projection of the ``prespace\u0027\u0027 dynamics can be nonlinear and\neven irreversible (but it is always unitary). There were found conditions of\nlinearity and reversibility of the Hilbert space dynamical projection. We also\nfound conditions for the conventional Schroedinger dynamics (including\ntime-dependent Hamiltonians). We remark that in general even the Schroedinger\ndynamics is based just on the statistical conservation of energy; for\nindividual systems the law of conservation of energy can be violated (at least\nin our theoretical model).",
"arxiv_id": "quant-ph/0409128",
"authors": [
"Andrei Yu. Khrennikov"
],
"categories": [
"quant-ph"
],
"doi": "10.1393/ncb/i2004-10142-6",
"title": "On the representation of contextual probabilistic dynamics in the complex Hilbert space: linear and nonlinear evolutions, Schroedinger dynamics",
"url": "https://arxiv.org/abs/quant-ph/0409128"
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