Spatial Localization in Quantum Theory Based on qr-numbers

Corbett, J. V.; Durt, Thomas

doi:10.1007/s10701-010-9424-4

Cited by 4 publications

(9 citation statements)

References 16 publications

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“…The qr-numbers approach maintains Einstein locality in the quantum space of the entangled two particle system because the single particles are always close to each other in their qr-number space and hence can always interact [7]. It gives the usual quantum mechanical results for the experiment.…”

Section: 74mentioning

confidence: 99%

“…The paper is based on much collaborative work [5,7,8] with Thomas Durt whose important contributions are acknowledged. I also wish to thank Ross Street and members of the Centre of Australian Category Theory for their encouragement and Huw Price and members of the Foundations of QFT group for interesting discussions.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…The set of operators form a * -algebra A and the state space E S (A ) of the system is the space of normalized linear functionals on A . The state space has the weak topology generated by the real -valued functions We have argued elsewhere that many of the conceptual enigmas of quantum theory, including the measurement problem [5], the double slit [8] and the non-locality of entangled systems [7], result from trying to fit the microscopic phenomena into a conceptual framework in which standard real numbers are taken to be the only possible quantitative values for the attributes of quantum systems. We claim that it is as unreasonable to assume that the only numerical values of physical quantities are classical real numbers as it is to assume that the only geometry of space, or space-time, is Euclidean.…”

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confidence: 99%

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A Topos Theory Foundation for Quantum Mechanics

Corbett¹

2012

Electron. Proc. Theor. Comput. Sci.

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The theory of quantum mechanics is examined using non-standard real numbers, called quantum real numbers (qr-numbers), that are constructed from standard Hilbert space entities. Our goal is to resolve some of the paradoxical features of the standard theory by providing the physical attributes of quantum systems with numerical values that are Dedekind real numbers in the topos of sheaves on the state space of the quantum system. The measured standard real number values of a physical attribute are then obtained as constant qr-number approximations to variable qr-numbers. Considered as attributes, the spatial locations of massive quantum particles form non-classical spatial continua in which a single particle can have a quantum trajectory which passes through two classically separated slits and the two particles in the Bohm-Bell experiment stay close to each other in quantum space so that Einstein locality is retained.1 The quantum real number interpretation of quantum mechanics.The quantum real number (qr-number) interpretation of quantum mechanics is based on the claim that the change from classical physics to quantum physics is achieved by changing in the type of numerical values that the physical variables can possess. It assumes that the properties of microscopic entities differ from those of classical because their numerical values are not standard real numbers but are Dedekind real numbers in a spatial topos built upon the quantum state space of the microscopic entities. The Dedekind reals differ from standard real is that each holds true to a non-trivial extent given by an open subsets of the quantum state space. These are its conditions, they replace the individual states of standard quantum theory. The internal logic is intuitionistic. The "direct connection between observation properties and properties possessed by the independently existing object" [10] is cut, an indirect connection is made through the experimental measurement processes. The interpretation builds on the fact that any experimental measurement has a limited level of accuracy. Within this level of accuracy an attribute's qr-number value is approximated by a standard real number.Its mathematical structure is built from elements of standard quantum mechanics: we start from von Neumann's assumption [37] that to any quantum system we can associate a Hilbert space H . H is the carrier space of a unitary representation of a Lie group G, the symmetry group of the quantum system. Then, as in the standard interpretation, the physical attributes (measurable qualities) of the quantum system are represented by essentially self-adjoint operators that act on a dense subset D ⊂ H . The set of operators form a * -algebra A and the state space E S (A ) of the system is the space of normalized linear functionals on A . The state space has the weak topology generated by the real -valued functions a Q : E S (A ) → R given by a Q (ρ) = Tr(ρ.Â) : ∀ρ ∈ E S (A ) and labeled by the operatorsÂ ∈ A . A typical open set is a finite intersection of open sets N (ρ 0 ;...

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Section: 74mentioning

confidence: 99%

Section: Acknowledgmentsmentioning

confidence: 99%

mentioning

confidence: 99%

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A Topos Theory Foundation for Quantum Mechanics

Corbett¹

2012

Electron. Proc. Theor. Comput. Sci.

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“…Bose National Centre for Basic Science, Kolkata, 6-10 January 2011, I wish to thank Dipankar and Archan for their warm hospitality during and after the symposium. The talk is based on much collaborative work [4,6,5] with Thomas Durt whose important contributions are acknowledged.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…We have argued that many of the conceptual enigmas of quantum theory: such as the two slit experiment [5], the measurement problem [4] and the non-locality of entangled systems [6], result from trying to fit the microscopic phenomena into a conceptual framework in which standard real numbers are taken to be the only possible quantitative values for the attributes of quantum systems. Our solution is use non-standard real numbers.…”

Section: Introductionmentioning

confidence: 99%

Entanglement and the quantum spatial continuum

Corbett¹

2011

AIP Conference Proceedings

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The non-locality of entangled systems provides more evidence that the spatial continuum of quantum particles is not classical. We assume that physical quantities take Dedekind real numbers in a topos for their numerical values. This means that the quantum spatial continuum is isomorphic to R D (E S (M )) 3 , where R D (E S (M )) the sheaf of Dedekind real numbers in the topos Shv(E S (M ) of sheaves on the state space of the quantum system. In such a continuum, a single particle can have a quantum trajectory which passes through two classically separated slits and two particles in an entangled condition stay close to each other in their quantum space and hence Einstein locality is retained.

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Quantum dynamics is infinitesimal qr-number dynamics

Corbett

2019

Preprint

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The Heisenberg equations of motion for a quantum particle of mass m are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass m describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qrnumber equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass m expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.

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Spatial Localization in Quantum Theory Based on qr-numbers

Cited by 4 publications

References 16 publications

A Topos Theory Foundation for Quantum Mechanics

A Topos Theory Foundation for Quantum Mechanics

Entanglement and the quantum spatial continuum

Quantum dynamics is infinitesimal qr-number dynamics

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