dorsal/arxiv
View SchemaHow to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms
| Authors | Giacomo Mauro D'Ariano |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0603011 |
| URL | https://arxiv.org/abs/quant-ph/0603011 |
Abstract
In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.
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"abstract": "In the present paper I show how it is possible to derive the Hilbert space\nformulation of Quantum Mechanics from a comprehensive definition of \"physical\nexperiment\" and assuming \"experimental accessibility and simplicity\" as\nspecified by five simple Postulates. This accomplishes the program presented in\nform of conjectures in the previous paper quant-ph/0506034. Pivotal roles are\nplayed by the \"local observability principle\", which reconciles the holism of\nnonlocality with the reductionism of local observation, and by the postulated\nexistence of \"informationally complete observables\" and of a \"symmetric\nfaithful state\". This last notion allows one to introduce an operational\ndefinition for the real version of the \"adjoint\"--i. e. the transposition--from\nwhich one can derive a real Hilbert-space structure via either the\nMackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in\ndetail only the Gelfand-Naimark-Segal construction, which leads to a real\nHilbert space structure analogous to that of (classes of generally unbounded)\nselfadjoint operators in Quantum Mechanics. For finite dimensions, general\ndimensionality theorems that can be derived from a local observability\nprinciple, allow us to represent the elements of the real Hilbert space as\noperators over an underlying complex Hilbert space (see, however, a still open\nproblem at the end of the paper). The route for the present operational\naxiomatization was suggested by novel ideas originated from Quantum Tomography.",
"arxiv_id": "quant-ph/0603011",
"authors": [
"Giacomo Mauro D\u0027Ariano"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
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"title": "How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms",
"url": "https://arxiv.org/abs/quant-ph/0603011"
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