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Contexts, Bi-Heyting Algebras and a New Logic for Quantum Systems Oberseminar Theoretische Informatik FAU Erlangen 4

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Contexts, Bi-Heyting Algebras and a New Logic for Quantum Systems Oberseminar Theoretische Informatik FAU Erlangen 4 Contexts, Bi-Heyting Algebras and a New Logic for Quantum Systems Oberseminar Theoretische Informatik FAU Erlangen 4. November 2014 Andreas D¨oring Theoretische Physik I, FAU Erlangen [email protected] Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 1 / 49 \Never express yourself more clearly than you are able to think." Niels Bohr Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 2 / 49 References This talk is based on AD,\Topos-based Logic for Quantum Systems and Bi-Heyting Algebras", to appear in Logic & Algebra in Quantum Computing, Lecture Notes in Logic, Association for Symbolic Logic in conjunction with Cambridge University Press (2012); [arXiv:1202.2750] Some good references on standard quantum logic are: G. Birkhoff, J. von Neumann, \The Logic of Quantum Mechanics", Annals of Mathematics 37, No. 4, 823{843 (1936). Varadarajan, Geometry of Quantum Theory, second ed., Springer (1985). M. Dalla Chiara, R. Giuntini, \Quantum Logics", in Handbook of Philosophical Logic, Kluwer (2002); [arXiv:quant-ph/0101028v2] Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 3 / 49 Standard quantum logic Standard quantum logic Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 4 / 49 Standard quantum logic Standard quantum logic Some basics of standard quantum logic: Birkhoff and von Neumann suggested to use the complete orthomodular lattice P(H) of projections on a separable, complex Hilbert space H as representatives of propositions. (Fine point: modular vs. orthomodular lattices.) The link between (pre-mathematical) propositions \A " ∆” and projections is provided by the spectral theorem. Pure quantum states j i (roughly) are models of this propositional logic, but only specific projections are assigned true or false: if P^ j i = j i, then the proposition represented by P^ is true in the state j i; if P^ j i = 0, then the proposition is false in the state j i. In general, only a probability between 0 and 1 for finding the outcome of a measurement of A to lie in a Borel set ∆ can be given. After measurement, the quantum state has changed to the eigenstate corresponding to the outcome. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 5 / 49 Standard quantum logic Some (problematic) features of quantum logic Propositions form a non-distributive orthomodular lattice. Quantum states do not assign true or false to all propositions \A " ∆”; in general, one can only obtain probabilities. Lack of two-valued models; instrumentalism. A disjunction \A " ∆ or B " Γ” can be true in a state j i despite the fact that neither \A " ∆” nor \B " Γ” are true in the state. This is due to superposition. There a many meets in P(H) that are physically meaningless. This is due to the failure to take contextuality into account. The implication problem: there is no material implication, and the implicative rule does not hold. This is due to non-distributivity. There is no algebra of subsets representing propositions, but an algebra of subspaces. Most problematic features result from this (insufficient?) geometric underpinning of quantum logic. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 6 / 49 A state space model for quantum systems A state space model for quantum systems Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 7 / 49 A state space model for quantum systems The topos approach The topos approach to the formulation of physical theories was initiated by Chris Isham '97 and Isham/Butterfield '98{'02, addresses certain structural and conceptual issues in the foundations of physics, uses topos theory to give new geometric and logical ideas for physics, aims to provide a new way of formulating physical theories in general, and quantum theory in particular { not based on Hilbert spaces, not `just another interpretation', is motivated by questions on the way to quantum gravity (QG) and quantum cosmology (QC), most work so far is on standard, non-relativistic quantum theory { natural starting point, testing ground. Other researchers include: Landsman, Heunen, Spitters, Nakayama, Vickers, Fauser, Flori, ... Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 8 / 49 A state space model for quantum systems Where are we now? The topos approach is very much work in progress, so think of Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 9 / 49 A state space model for quantum systems Classical physics In classical physics, a physical quantity A is described by a (measurable) real-valued function fA on the state space (phase space) P: In a given state s 2 P, a physical quantity A has the value fA(s) 2 R. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 10 / 49 A state space model for quantum systems Classical physics (2) Propositions such as \A " ∆”, i.e., \the physical quantity A has a value in the Borel set ∆ ⊆ R", are represented by subsets of the state space: The Borel subsets of P form a σ-complete Boolean algebra. Pure states s 2 P are models of this propositional logic. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 11 / 49 A state space model for quantum systems The Kochen-Specker theorem and instrumentalism But can quantum theory be cast in such a form? Obstacle: Theorem (Kochen-Specker 1967): If dim(H) ≥ 3, then there is no state space model of quantum theory (under mild and natural conditions). In logical terms, there is no way to assign true or false to all propositions \A " ∆" at the same time. Usual answer: interpret propositions in an instrumentalist way, and use probabilities, because repeatedly measuring the same physical quantity A in the same state j i can give different outcomes { states do not assign truth values to all propositions. This is not what we want to do { we want to provide a quantum state space in spite of the KS theorem. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 12 / 49 A state space model for quantum systems C ∗-algebras and von Neumann algebras As a reminder: Definition A C ∗-algebra A is a complex Banach algebra with an involution (−)∗ : A!A such that 8a 2 A : ja∗aj = jaj2: A W ∗-algebra is a C ∗-algebra that is the dual of a Banach space M. Every C ∗-algebra can be faithfully represented as a norm-closed subalgebra of B(H), the algebra of bounded linear operators on some Hilbert space H. Every W ∗-algebra can be represented as a weakly closed subalgebra of some B(H). Weakly closed subalgebras of B(H) are called von Neumann algebras. We will often use `hats' (as in A^; B^; :::) when denoting elements of operator algebras. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 13 / 49 A state space model for quantum systems The basic idea With a quantum system (described by a noncommutative von Neumann algebra N , e.g. N = B(H)), we associate a topos and define a spectral object Σ in the topos. The spectral presheaf Σ is a generalised set, playing the role of a state space for the quantum system (notwithstanding the Kochen-Specker theorem!). Structure of state space determines logical structure of a theory: Classical physics: Borel subsets of state space S represent propositions \A " ∆”, form Boolean algebra. Quantum physics: clopen subobjects of the state space Σ represent propositions \A " ∆”, form bi-Heyting algebra. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 14 / 49 A state space model for quantum systems The context category Let S be a quantum system, described by a noncommutative von Neumann algebra N ⊆ B(H). Let V be a non-trivial commutative von Neumann subalgebra of N which has the same unit element as N . This gives and is given by a set of commuting self-adjoint operators in N . We call V a context. Each context V is a partial classical perspective on the quantum system. The main idea: take all classical perspectives/contexts together to obtain a complete picture of the quantum system. Concretely, we consider the set V(N ) of all contexts, partially ordered under inclusion. This poset is called the context category. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 15 / 49 A state space model for quantum systems Gelfand spectra Each commutative von Neumann algebra V has a Gelfand spectrum ΣV , consisting of the algebra morphisms λ : V −! C: These maps are also pure states of V . The Gelfand topology is the ∗ (relative) weak -topology on ΣV . With respect to this topology, ΣV is an extremely disconnected compact Hausdorff space. For a commutative von Neumann algebra V , the projection lattice P(V ) is a complete Boolean algebra. Let λ 2 ΣV , and let P^ 2 P(V ), representing some proposition about an A^ 2 Vsa. Note that λ(P^) = λ(P^ 2) = λ(P^)2 2 f0; 1g ' ffalse; trueg; so the elements of ΣV are the models of the classical logic described by P(V ). Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 16 / 49 A state space model for quantum systems The spectral presheaf If V 0 ⊂ V is a von Neumann subalgebra, then there is a canonical restriction map rV ;V 0 :ΣV −! ΣV 0 λ 7−! λjV 0 : This map is continuous, closed, open and surjective. Definition Let N be a von Neumann algebra, and let V(N ) be its context category. The spectral presheaf Σ of N over V(N ) is defined (a) on objects: for all V 2 V(N ), let ΣV := ΣV , 0 (b) on arrows: for all inclusions iV 0V : V ,! V , let Σ(iV 0V ) = rV ;V 0 . Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 17 / 49 The bi-Heyting algebra of clopen subobjects The bi-Heyting algebra of clopen subobjects Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 18 / 49 The bi-Heyting algebra of clopen subobjects Bi-Heyting algebras { history Rauszer: bi-Heyting algebras in superintuitionistic logic ('73{'77) Lawvere: co-Heyting and bi-Heyting algebras in category and topos theory, in particular in connection with continuum physics ('86, '91) Reyes/Makkai ('95) and Reyes/Zolfaghari ('96): bi-Heyting algebras and modal logic Bezhanishvili et al.
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