dorsal/arxiv
View SchemaA Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
| Authors | A. Doering, C. J. Isham |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0703062 |
| URL | https://arxiv.org/abs/quant-ph/0703062 |
| DOI | 10.1063/1.2883742 |
| Journal | J.Math.Phys.49:053516,2008 |
Abstract
This paper is the second 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 study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf--the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study `truth objects' in the topos. These are objects in the topos that 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. Truth objects therefore play a crucial role in our formalism.
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"abstract": "This paper is the second in a series whose goal is to develop a fundamentally\nnew way of constructing theories of physics. The motivation comes from a desire\nto address certain deep issues that arise when contemplating quantum theories\nof space and time. Our basic contention is that constructing a theory of\nphysics is equivalent to finding a representation in a topos of a certain\nformal language that is attached to the system. Classical physics arises when\nthe topos is the category of sets. Other types of theory employ a different\ntopos. In this paper, we study in depth the topos representation of the\npropositional language, PL(S), for the case of quantum theory. In doing so, we\nmake a direct link with, and clarify, the earlier work on applying topos theory\nto quantum physics. The key step is a process we term `daseinisation\u0027 by which\na projection operator is mapped to a sub-object of the spectral presheaf--the\ntopos quantum analogue of a classical state space. In the second part of the\npaper we change gear with the introduction of the more sophisticated local\nlanguage L(S). From this point forward, throughout the rest of the series of\npapers, our attention will be devoted almost entirely to this language. In the\npresent paper, we use L(S) to study `truth objects\u0027 in the topos. These are\nobjects in the topos that play the role of states: a necessary development as\nthe spectral presheaf has no global elements, and hence there are no\nmicrostates in the sense of classical physics. Truth objects therefore play a\ncrucial role in our formalism.",
"arxiv_id": "quant-ph/0703062",
"authors": [
"A. Doering",
"C. J. Isham"
],
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"doi": "10.1063/1.2883742",
"journal_ref": "J.Math.Phys.49:053516,2008",
"title": "A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory",
"url": "https://arxiv.org/abs/quant-ph/0703062"
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