The spectral theory of commutative C<sup>∗</sup>-algebras: The constructive spectrum

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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“…8 In simple non-relativistic systems, the state is specified at any given moment of time. Relativistic systems (particularly quantum gravity!)…”

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confidence: 99%

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

Döring¹,

Isham

2010

New Structures for Physics

104

197

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“…This allows the results to be applied in non-standard contexts. For instance, one can translate a theorem about one C*-algebra to a theorem about a continuous field of C*-algebras [3][4][5][6]. In turn, the results about commutative C*-algebras may be obtained directly using Riesz spaces [14].…”

Section: Theorem 1 [Stone-yosida] Let R Be An Archimedean Riesz Spacementioning

confidence: 99%

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

Spitters

2010

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In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that Archimedean almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters. Keywords Formal topology · Axiom of choice · Riesz space · Constructive analysisThe Stone-Yosida representation theorem [25,28] for Riesz spaces shows how to embed every Riesz space [20,29] into the Riesz space of continuous functions on its spectrum.Definition 1 A Riesz space is a vector space with compatible lattice operationsi.e. f ∧ g + f ∨ g = f + g and if f 0 and a 0, then af 0. A (strong) unit 1 in a Riesz space is an element such that for all f there exists a natural number n such that | f | n · 1. A Riesz space is Archimedean if for all n, n|x| y implies that x = 0.A Riesz space with a strong unit is Archimedean.Definition 2 A representation of a Riesz space R with unit is a linear map σ : R → Ê which respects the lattice operations and the unit. We denote by the space of representations equipped with the weakest topology which makes all the maps e r (σ ) := σ (r) continuous.

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“…This logical view is analogous to the Lindenbaum construction that allows theories in propositional geometric logic to be considered as presentations of locales by generators and relations, which in particular was exploited when proving in an arbitrary Grothendieck topos that the category of commutative unital C*-algebras is equivalent to the category of compact completely regular locales [3,4,5], thus generalising the classical Gelfand duality, which (in the presence of the Axiom of Choice) establishes an equivalence between (the dual of) the category of commutative unital C*-algebras and the category of compact Hausdorff spaces.…”

Section: Introductionmentioning

confidence: 99%

“…However, the logical view was one of the influences behind the introduction of the quantale Max A, in the sense that a presentation by generators and relations may be obtained as a generalisation of that of the locale Max A considered in the case of a commutative C*-algebra A [3,4,5]. It has also been used extensively in computer science when studying concurrent systems [1,25,26,27,28,30].…”

Section: Introductionmentioning

confidence: 99%

A Noncommutative Theory of Penrose Tilings

Mulvey

Resende²

2005

Int J Theor Phys

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Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.

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The spectral theory of commutative C^∗-algebras: The constructive spectrum

Cited by 27 publications

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

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

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

A Noncommutative Theory of Penrose Tilings

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The spectral theory of commutative C∗-algebras: The constructive spectrum

Cited by 27 publications

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

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

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

A Noncommutative Theory of Penrose Tilings

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The spectral theory of commutative C^∗-algebras: The constructive spectrum