dorsal/arxiv
View SchemaQuantum mechanics as a theory of probability
| Authors | Itamar Pitowsky |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0510095 |
| URL | https://arxiv.org/abs/quant-ph/0510095 |
Abstract
We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". We apply the approach to the analysis of entanglement, Bell inequalities, and the quantum theory of macroscopic objects. We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem.
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"abstract": "We develop and defend the thesis that the Hilbert space formalism of quantum\nmechanics is a new theory of probability. The theory, like its classical\ncounterpart, consists of an algebra of events, and the probability measures\ndefined on it. The construction proceeds in the following steps: (a) Axioms for\nthe algebra of events are introduced following Birkhoff and von Neumann. All\naxioms, except the one that expresses the uncertainty principle, are shared\nwith the classical event space. The only models for the set of axioms are\nlattices of subspaces of inner product spaces over a field K. (b) Another axiom\ndue to Soler forces K to be the field of real, or complex numbers, or the\nquaternions. We suggest a probabilistic reading of Soler\u0027s axiom. (c) Gleason\u0027s\ntheorem fully characterizes the probability measures on the algebra of events,\nso that Born\u0027s rule is derived. (d) Gleason\u0027s theorem is equivalent to the\nexistence of a certain finite set of rays, with a particular orthogonality\ngraph (Wondergraph). Consequently, all aspects of quantum probability can be\nderived from rational probability assignments to finite \"quantum gambles\". We\napply the approach to the analysis of entanglement, Bell inequalities, and the\nquantum theory of macroscopic objects. We also discuss the relation of the\npresent approach to quantum logic, realism and truth, and the measurement\nproblem.",
"arxiv_id": "quant-ph/0510095",
"authors": [
"Itamar Pitowsky"
],
"categories": [
"quant-ph"
],
"title": "Quantum mechanics as a theory of probability",
"url": "https://arxiv.org/abs/quant-ph/0510095"
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