Review of the Topos Approach to Quantum Theory (original) (raw)
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Review of the topos approach to quantum theory 1
Canadian Journal of Physics, 2013
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate, in general, physical theories. However, the motivation for using it to express quantum theory, lies in the desire to solve certain interpretational problems inherent in the standard formulation of the theory. In fact, the topos approach to quantum theory overcomes the instrumentalist–Copenhagen interpretation, rendering the theory more realist. The caveat is that one ends up with a multivalued–intuitionistic logic, rather than a Boolean logic.
Topos theory and 'neo-realist' quantum theory
In "Quantum Field Theory, Competitive Models", eds. B. Fauser, J. Tolksdorf, E. Zeidler, 25--47, Birkhäuser (2009)
Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a `mathematical universe' with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
Journal of Mathematical Physics 49, 053516 (2008)
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.
Topos Theoretic Quantum Realism
Topos Quantum Theory (TQT) is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics. In this paper, we study the extent to which TQT can really be characterised as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, we show that the notion of the quantum state is problematic for those who view TQT as a realist reformulation of quantum theory.
Some possible roles for topos theory in quantum theory and quantum gravity
Foundations of Physics, 2000
We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Section 1, we introduce these problems. In Section 2, we introduce topos theory, especially the idea of a topos of presheaves. In Section 3, we discuss several possible applications of topos theory to the problems in Section 1. In Section 4, we draw some conclusions.
'What is a Thing?': Topos Theory in the Foundations of Physics
In "New Structures for Physics", ed. Bob Coecke, Springer Lecture Notes in Physics 813, 753--940, Springer, Heidelberg (2011)
The goal of this paper 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 paper, a key goal is to represent any physical quantity AAA with an arrow breveAphi:SiphimapRphi\breve{A}_\phi:\Si_\phi\map\R_\phibreveAphi:SiphimapRphi where Siphi\Si_\phiSiphi and Rphi\R_\phiRphi are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, tauphi\tau_\phitauphi. We discuss two different types of language that can be attached to a system,$S$. The first, PLS\PL{S}PLS, is a propositional language; the second, LS\L{S}LS, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of PLS\PL{S}PLS we expand and develop some of the earlier work (By CJI and collaborators.) 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 Sig\SigSig--the topos quantum analogue of a classical state space. The topos concerned is SetH\SetH{}SetH: the category of contravariant set-valued functors on the category (partially ordered set) V\V{}V of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space Hi\HiHi.
Journal of Mathematical Physics 49, 053517 (2008)
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 brevedeltao(A):SigmapSR\breve{\delta}^o(A):\Sig\map\SRbrevedeltao(A):SigmapSR, where SR\SRSR is the quantity-value object for this theory. The construction of brevedeltao(A)\breve{\delta}^o(A)brevedeltao(A) is an extension of the daseinisation of projection operators that was discussed in paper II. The object SR\SRSR is a monoid-object only in the topos, tauphi\tau_\phitauphi, of the theory, and to enhance the applicability of the formalism, we apply to SR\SRSR a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, kSR\kSRkSR, is an abelian group-object in tauphi\tau_\phitauphi. We also discuss another candidate, PRmathR\PR{\mathR}PRmathR, 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.
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
Journal of Mathematical Physics 49, 053515 (2008)
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.
Toposes and categories in quantum theory and gravity
The topos theory is discussed from the physical point of view. Basic ideas of topos are presented and explained. The connection with algebras of classical and quantum observables, alternative concepts of space-time, theory of relativity and quantum gravity, the generalized histories approach to a quantum theory of the whole universe are reviewed. Using developed by the authors formalism of n-regular obstructed categories the concept of a topos is properly generalized.
A Second Course in Topos Quantum Theory
2018
Logic of Propositions in Topos Quantum Theory -- Developments on Self Adjoint Operators in Topos Quantum Theory -- Physical quantities interpreted as modal operators -- Group action in Topos Quantum Theory take two -- Quantization in Topos Quantum Theory -- Groethendieck Topoi -- Locales -- Spacetime -- Topos and Logic -- Internalizing Objects in Topos Theory -- What information can be recovered from the abelian subalgebras of a von-Neumann algebra -- Extending the spectral presheaf to non-abelian unital C*-algebras.