## and (2007)

### BibTeX

@MISC{Döring07and,

author = {A. Döring and C. J. Isham},

title = {and},

year = {2007}

}

### OpenURL

### Abstract

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. 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. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow ˘ δo (A) : Σ → IR ≽ , where IR ≽ is the quantity-value object for this theory. The construction of ˘ δo ( Â) is an extension of the daseinisation of projection operators that was discussed in paper II., of The object IR ≽ is a monoid-object only in the topos, τφ = SetsV(H)op the theory, and to enhance the applicability of the formalism, we apply to IR ≽ a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, k(IR ≽), is an abelian group-object in τφ. We also discuss another candidate, IR ↔ , for the quantity-value object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.

### Citations

71 | 2002): A topos perspective on the Kochen-Specker theorem: IV. Interval Valuations - Butterfield, Isham |

60 |
Sheaves in geometry and logic: A first introduction to topos theory
- MacLane, Moerdijk
- 1994
(Show Context)
Citation Context ...itten as δo ( ( Â)V ′ Σ(iV ′ V )(λ) ) ≥ δo ( Â)V (λ) (3.4) for all λ ∈ Σ V . It is a standard result that the real-number object, IR, in a presheaf topos Sets Cop is the constant functor from C to IR =-=[1]-=-. It follows that the family of Gel’fand transforms, δ o ( Â)V , V ∈ Ob(V(H)), of the daseinised operators δ o ( Â)V , V ∈ Ob(V(H)), cannot define an arrow from Σ to IR, as this would require an equal... |

17 | Kochen-Specker theorem for von Neumann algebras - Döring - 2005 |

12 |
The Selfadjoint Operators of a von Neumann Algebra form a Conditionally Complete Lattice
- Olson
- 1971
(Show Context)
Citation Context ...uous. 4 I.e., there are a, b ∈ IR such that Êλ = ˆ0 for all λ ≤ a and Êλ = ˆ1 for all λ ≥ b. IR 5The spectral order. A key element for our work is the so-called spectral order that was introduced in =-=[17]-=-. 5 It is defined as follows. Let Â and ˆ B be (bounded) self-adjoint operators with spectral families { ÊA λ }λ∈IR and { ÊB λ }λ∈IR, respectively. Then define: Â ≼s ˆ B if and only if ÊB λ ≼ ÊA λ for... |

11 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory
- Döring, Isham
- 2008
(Show Context)
Citation Context ...propositional language PL(S) are all brought ‘inside’ the local language L(S). The second paper in the series dealt with the topos representation of the propositional language PL(S) in quantum theory =-=[3]-=-. In a sense this paper is a side-line to our main programme which is concerned with the local language L(S). In fact, logically speaking, we could have omitted our study of the representations of PL(... |

6 |
A topos foundation for theories of physics: IV. Categories of systems. arXiv:quant-ph/0703.066
- Döring, Isham
- 2008
(Show Context)
Citation Context ...under the transformations |ψ〉 ↦→ Û |ψ〉, Â ↦→ ÛÂÛ −1 . 5.2.3 The Û-twisted Presheaf Let us return once more to the definition (5.10) of the functor ℓÛ : V(H) → V(H). As we shall see in the next paper, =-=[4]-=-, any such functor induces a ‘geometric morphism’ from Sets V(H)op to Sets V(H)op . The exact definition is not needed here: it suffices to ∗Û remark that part of this geometric morphism is an arrow ℓ... |

5 | A sheaf theoretic approach to measure theory
- Jackson
- 2006
(Show Context)
Citation Context ...he Kochen-Specker theorem, we do not expect to be able to assign (constant) real numbers as values of physical quantities, at least not globally. Instead, we draw on some recent results of M. Jackson =-=[16]-=-, obtained as part of his extensive study of measure theory on a topos of presheaves. Here, we use a single construction in Jackson’s thesis: the presheaf of ‘order-preserving functions’ over a partia... |

4 | A Topos Perspective on State-Vector Reduction
- Isham
- 2006
(Show Context)
Citation Context ...urce of examples in texts on topos theory. It would be intriguing to experiment with constructing model theories of physics using one of these simple topoi. One possible use of M-sets is discussed in =-=[11]-=- in the context of reduction of the state vector, but there will surely be others. Acknowledgements This research was supported by grant RFP1-06-04 from The Foundational Questions Institute (fqxi.org)... |

3 |
On a canonical lattice structure on the effect algebra of a von Neumann algebra. arXiv:math-ph/0410.018v2
- Groote
- 2004
(Show Context)
Citation Context ... theorem, define self-adjoint operators in V . This leads to the definition of the two daseinisations of an arbitrary self-adjoint operator: 5The spectral order was later reinvented by de Groote, see =-=[5]-=-. 6 The ‘usual’ ordering is Â ≼ ˆ B if 〈ψ| Â |ψ〉 ≤ 〈ψ| ˆ B |ψ〉 for all vectors |ψ〉 ∈ H. 7 i The reason (2.13) and (2.14) have a different form is that λ ↦→ δ ( ˆ λ ↦→ δo ( ˆ Eλ)V is not. On the other ... |

3 |
An Introduction to Philosophical Logic
- Grayling
- 1990
(Show Context)
Citation Context ...s) for there is nothing external to which a proposition can ‘correspond’. Instead, what is needed is more like a coherence theory of truth in which a whole body of propositions is considered together =-=[18]-=-. This is a fascinating subject, but further discussion must be deferred to later work. 4 The Presheaf k(IR ≽ ) 4.1 Some Background Information 4.1.1 Preliminary Remarks We have shown how each self-ad... |

2 |
Observables as functions: Antonymous functions
- Döring
- 2005
(Show Context)
Citation Context ...inisation, and certain functions on the dual ideals/filters in the projection lattice P(H) of B(H). We only give a very brief sketch here: details can be found in de Groote’s work [6, 7], the article =-=[9]-=-, and a forthcoming paper [10]. This subsection serves as a preparation for the physical interpretation of the arrows ˘ δ(A) : Σ → IR ↔ . Spectral elements and ultrafilters. Let V ∈ Ob(V(H)), and let ... |

1 |
Daseinisation and functions on dual ideals
- Döring
- 2007
(Show Context)
Citation Context ...ons on the dual ideals/filters in the projection lattice P(H) of B(H). We only give a very brief sketch here: details can be found in de Groote’s work [6, 7], the article [9], and a forthcoming paper =-=[10]-=-. This subsection serves as a preparation for the physical interpretation of the arrows ˘ δ(A) : Σ → IR ↔ . Spectral elements and ultrafilters. Let V ∈ Ob(V(H)), and let λ ∈ Σ V be a spectral element ... |