An Intuitionistic Model of Single Electron Interference

Corbett, J. V.; Durt, Thomas

doi:10.1007/s11225-010-9246-6

Cited by 5 publications

(14 citation statements)

References 8 publications

(15 reference statements)

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“…Nevertheless the experimental evidence [31,36,25] clearly shows single electrons building up an interference pattern. This building up process is described deterministically in the qr-number model [8].…”

Section: 74mentioning

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%

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confidence: 99%

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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“…Other work which emphasizes the close relationship between physics and mathematics is concerned with quantum theory representations of mathematical systems. This work includes papers on quantum set theory [10,11,12], quantum theory representations of real numbers [13,14,15,16,17,18,19], and the use of category theory in physics [20,21].…”

Section: Introductionmentioning

confidence: 99%

Reference frame fields based on quantum theory representations of real and complex numbers

Benioff¹

2009

Contemporary Mathematics

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A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k ≥ 2. Arithmetic and transformation properties of these states are given, both for basis states representing rational numbers and linear superpositions of these states. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of q k string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical representations as kit string states in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. In particular, states of qukit strings are elements of Hilbert spaces, which are vector spaces over the complex field. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Some approaches to integrating this work with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of states of qukit strings.

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An Intuitionistic Model of Single Electron Interference

Cited by 5 publications

References 8 publications

A Topos Theory Foundation for Quantum Mechanics

A Topos Theory Foundation for Quantum Mechanics

Entanglement and the quantum spatial continuum

Reference frame fields based on quantum theory representations of real and complex numbers

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