Sharp quantum logics

Chiara, M. Dalla; Giuntini, Roberto; Greechie, Richard J.

doi:10.1007/978-94-017-0526-4_8

Cited by 8 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One might argue that (i) the use of (triples of) real numbers to model space is based on empirically-based reflections about the nature of "distances" between objects; and (ii) the use of real numbers to model time reflects our experience that "instants of time" appear to be totally ordered, and that intervals of time are always divisible. 20 However, what does it really mean to say that two particles are separated by a distance of, for example, √ 2cms? From an empirical perspective, it would be impossible to make a measurement that could unequivocally reveal precisely that value from among the continuum of real numbers that lie around it.…”

Section: Why Are Space and Time Modelled With Real Numbers?mentioning

confidence: 99%

See 1 more Smart Citation

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

Döring¹,

Isham

2010

New Structures for Physics

104

197

View full text Add to dashboard Cite

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...

show abstract

Section: Why Are Space and Time Modelled With Real Numbers?mentioning

confidence: 99%

“…Developing this line 33 For an excellent survey of quantum logic see [20]. This includes a discussion of a first-order axiomatisation of quantum logic, and with an associated sequent calculus.…”

Section: The Basics Of the Language L(s)mentioning

confidence: 99%

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

Döring¹,

Isham

2010

New Structures for Physics

104

197

View full text Add to dashboard Cite

show abstract

“…18 For an excellent survey of quantum logic see [14]. This includes a discussion of a first-order axiomatisation of quantum logic, and with an associated sequent calculus.…”

Section: The Failure To Represent Pl(s) In Standard Quantum Theorymentioning

confidence: 99%

A topos foundation for theories of physics: I. Formal languages for physics

Döring

Isham

2008

Journal of Mathematical Physics

112

189

View full text Add to dashboard Cite

This paper is the first in a series whose goal is to develop 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.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 arises when the topos is the category of sets. Other types of theory employ a different topos.In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S).Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.

show abstract

“…It is worth noticing that the orthodox quantum logic (Birkhoff and von Neumann [34]) used projectors as primitive elemants (propositions). The great problem of quantum logic is that it is so general that it is not useful as model of quantum mechanics.…”

Section: Relations With Logicmentioning

confidence: 99%

A probabilistic approach to quantum mechanics based on ‘tomograms’

Caponigro

Mancini

Man’ko³

2006

Fortschritte der Physik

View full text Add to dashboard Cite

Abstract. It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.

show abstract

Sharp quantum logics

Cited by 8 publications

References 0 publications

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

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

A topos foundation for theories of physics: I. Formal languages for physics

A probabilistic approach to quantum mechanics based on ‘tomograms’

Contact Info

Product

Resources

About