Collimation processes in quantum mechanics interpreted in quantum real numbers

Corbett, J. V.; Durt, Thomas

doi:10.1016/j.shpsb.2008.07.006

Cited by 9 publications

(49 citation statements)

References 30 publications

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“…The associated instrumentalist 18 The integers, and associated rationals, have a "natural" interpretation from a physical perspective since we can all count. On the other hand, the Cauchy-sequence and/or the Dedekind-cut definitions of the reals are distinctly un-intuitive from a physical perspective.…”

Section: Why Are Physical Quantities Assumed To Be Real-valued?mentioning

confidence: 99%

“…One argument (due to Penrose) is based on the observation that any attempt to localise a "thing" is bound to fail beyond a certain point because of the quantum production of pairs of particles from the energy/momentum uncertainty caused by the spatial localisation. Another argument concerns the artificiality 18 of the use of real numbers as coordinates with which to identify a space-time point. There is also Einstein's famous "hole argument" in general relativity which asserts that the notion of a space-time point (in a manifold) has no physical meaning in a theory that is invariant under the group of space-time diffeomorphisms.…”

Section: Why Are Physical Quantities Assumed To Be Real-valued?mentioning

confidence: 99%

“…There are no further obvious connections between this approach and the topos approach, but it would be interesting to compare both constructions with respect to the underlying geometric structures: both approaches formulate an intuitionistic form of quantum logic using Heyting algebras, and every complete Heyting algebra is a locale and hence a generalised topological space. Finally there is the work by Corbett et al [3,18] on "quantum" real numbers, or "qr-numbers". Of course, as always these days, Google will speedily reveal all that we have omitted.…”

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confidence: 99%

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“What is a Thing?”: Topos Theory in the Foundations of Physics

Döring¹,

Isham

2010

New Structures for Physics

104

197

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The goal of this article is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question "What is a thing?".Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this article, a key goal is to represent any physical quantity A with an arrowȂ φ : Σ φ → R φ where Σ φ and R φ are two special objects (the "state object" and "quantity-value object") in the appropriate topos, τ φ .We discuss two different types of language that can be attached to a system, S. The first, PL (S), is a propositional language; the second, L(S), is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of PL(S) we expand and develop some of the earlier work 1 on topos theory and quantum physics. A key step is a process we term "daseinisation" by which a projection operator is mapped to a sub-object of the spectral presheaf Σ-the topos quantum analogue of a classical state space. The topos concerned is Sets V(H)op : the category of contravariant set-valued functors on the category (partially ordered set) V(H) of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space H.There are two types of daseinisation, called "outer" and "inner": they involve approximating a projection operator by projectors that are, respectively, larger and smaller in the lattice of projectors on H.We then introduce the more sophisticated language L(S) and use it to study "truth objects" and "pseudo-states" in the topos. These objects play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics.One of the main mathematical achievements is finding a topos representation for self-adjoint operators. This involves showing that, for any bounded, self-adjoint operatorÂ, there is a corresponding arrowδ o (Â) : Σ → R where R is the quantity-value object for this theory. The construction ofδ o (Â) is an extension of the daseinisation of projection operators.The object R can serve as the quantity-value object if only outer daseinisation of self-adjoint operators is used in the construction of arrowsδ o (Â) : Σ → R . If both inner and outer daseinisation are used, then a related presheaf R ↔ is the appropriate choice. Moreover, in order to enhance the applicability of the quantity-value object, one can consider a topos analogue of the Grothendieck extension of a monoid to a group, applied to R (resp. R ↔ ). The resulting object, k(R ) (resp. k(R ↔ )), is an abelian group-object in τ φ .Finally we...

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Section: Why Are Physical Quantities Assumed To Be Real-valued?mentioning

confidence: 99%

Section: Why Are Physical Quantities Assumed To Be Real-valued?mentioning

confidence: 99%

mentioning

confidence: 99%

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“What is a Thing?”: Topos Theory in the Foundations of Physics

Döring¹,

Isham

2010

New Structures for Physics

104

197

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“…In previous works we showed how in principle certain paradoxes or counter-intuitive properties of the quantum theory can be explained provided we enlarge our conception of number [9,11]. Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of non-standard real numbers that we call qr-numbers.…”

Section: Introductionmentioning

confidence: 99%

“…This is not part of the usual paradigm of quantum theory but it is obtained using mathematical entities from the standard Hilbert space framework and by adopting it new light is shed on the measurement problem [11] and other counterintuitive aspects of the standard interpretation (see also [17] for a similar approach in the framework of relational interpretations of quantum mechanics).…”

Section: Introductionmentioning

confidence: 99%

Spatial Localization in Quantum Theory Based on qr-numbers

Corbett

Durt²

2010

Found Phys

Self Cite

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We show how trajectories can be reintroduced in quantum mechanics provided that its spatial continuum is modelled by a variable real number (qr-number) continuum. Such a continuum can be constructed using only standard Hilbert space entities. In this approach, the geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare trajectories in this new description of space-time with the corresponding Bohmian picture.

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An Intuitionistic Model of Single Electron Interference

Corbett

Durt²

2010

Stud Logica

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The double slit experiment for a massive scalar particle is described using intuitionistic logic with quantum real numbers as the numerical values of the particle's position and momentum. The model assigns physical reality to single quantum particles. Its truth values are given open subsets of state space interpreted as the ontological conditions of a particle. Each condition determines quantum real number values for all the particle's attributes. Questions, unanswerable in the standard theories, concerning the behaviour of single particles in the experiment are answered.

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Collimation processes in quantum mechanics interpreted in quantum real numbers

Cited by 9 publications

References 30 publications

“What is a Thing?”: Topos Theory in the Foundations of Physics

“What is a Thing?”: Topos Theory in the Foundations of Physics

Spatial Localization in Quantum Theory Based on qr-numbers

An Intuitionistic Model of Single Electron Interference

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