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Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic, Quantum Set Theory and The Interpretation of Quantum Mechanics J. Benavides Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The interpretation problem Today, more than one hundred years after Max Planck formulated the quantum hypothesis, we still do not have a settled agreement about what quantum reality is or if there is something as a quantum reality at all. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The interpretation problem Today, more than one hundred years after Max Planck formulated the quantum hypothesis, we still do not have a settled agreement about what quantum reality is or if there is something as a quantum reality at all. General relativity provides a clear description of physical reality, this theory tell us that we inhabit a universe that can be approximated as a connected four dimensional time oriented Lorentzian manifold where Einstein’s equation is valid. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The interpretation problem Today, more than one hundred years after Max Planck formulated the quantum hypothesis, we still do not have a settled agreement about what quantum reality is or if there is something as a quantum reality at all. General relativity provides a clear description of physical reality, this theory tell us that we inhabit a universe that can be approximated as a connected four dimensional time oriented Lorentzian manifold where Einstein’s equation is valid. On the other hand, whether quantum theory tell us something about the structure of the world we inhabit or is just a sophisticated formalism that allows to quantify the behaviour of the small scale world, is still a very controversial argument. Deutsch-Everett interpretation The classical quantum formalism Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The classical quantum formalism Postulate 1 A quantum system is described by a unit vector |ψ(t)i (the state vector) in a complex Hilbert space H and an operator Ǎ known as the Hamiltonian. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The classical quantum formalism Postulate 1 A quantum system is described by a unit vector |ψ(t)i (the state vector) in a complex Hilbert space H and an operator Ǎ known as the Hamiltonian. Postulate 2 In absence of any external influence the state vector changes smoothly in time according to the time dependent Schrödinger equation d|ψ(t)i i} = Ǎ|ψ(t)i dt Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Postulate 3 The observables of the system are represented mathematically by self-adjoint operators acting on H. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Postulate 3 The observables of the system are represented mathematically by self-adjoint operators acting on H. Postulate 4 (Schema) If an observable B is represented by a self-adjoint operator B̌ with eigenvalues b1 , ..., bm and respective eigenvectors |b1 i, ..., |bn i, and the state vector |ψ(t)i is expressed in the basis formed by the eigenvalues of B̌ as: |ψ(t)i = α1 |b1 i + ... + αm |bm i. Then a measurement of B at time t will give as a result one of the eigenvalues bi with probability |αi |2 respectively, and none other result that is not one of the eigenvalues is obtained. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results. Hugh Everett, Many-Worlds (1957) Quantum Mechanics is consistent with the idea that all possible results of the measurement actually happen, being the single universe where we perceive one single outcome part of a larger structure of many-worlds where the different outcomes happen. Deutsch-Everett interpretation David Deutsch, Quantum Multiverse 1984-2011-... Since 1984 David Deutsch, the father of quantum computation, has improved Everett ideas in parallel with his work on quantum information. Unfortunately, even if quantum computation is maybe the most remarkable result in theoretical physics in the last 37 years, Deutsch’s ideas about the many-worlds or multiversal interpretation have not been fully appreciated. Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation The multiverse is a set with a measure, whose elements are maximal sets of observables with definite values that correspond to different universes or different histories. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation The multiverse is a set with a measure, whose elements are maximal sets of observables with definite values that correspond to different universes or different histories. In this context the terms |αi |2 associated to the expression |ψ(t)i = α1 |b1 i + ... + αm |bm i represent the values of the measure of the sets of universes where the observable B̌ assume the value bi respectively. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation The multiverse is a set with a measure, whose elements are maximal sets of observables with definite values that correspond to different universes or different histories. In this context the terms |αi |2 associated to the expression |ψ(t)i = α1 |b1 i + ... + αm |bm i represent the values of the measure of the sets of universes where the observable B̌ assume the value bi respectively. Different sets of compatible observables determine different expressions of the state vector |ψ(t)i, these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value. The universes interact via interference phenomena, but such interactions are suppressed at the classical level described by classical physics. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. The Deutsch-Everett multiversal interpretation Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value. The universes interact via interference phenomena, but such interactions are suppressed at the classical level described by classical physics. At this classical level, the lack of interference allows to process classical scale information in an autonomous way, for this reason our classical theories (e.g. General Relativity) can give an accurate description at this level without appealing to the deeper multiversal structure. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse I-Interference Phenomena • Initial state= |0i Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse I-Interference Phenomena • Initial state= |0i • State after interacting with the beam splitter= i 1 √ |0i + √ |1i 2 2 Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse I-Interference Phenomena • Initial state= |0i • State after interacting with the beam splitter= i 1 √ |0i + √ |1i 2 2 • State after interacting with the full silvered mirrors= i 1 √ |0i + √ |1i 2 2 Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse I-Interference Phenomena • Initial state= |0i • State after interacting with the beam splitter= i 1 √ |0i + √ |1i 2 2 • State after interacting with the full silvered mirrors= i 1 √ |0i + √ |1i 2 2 • Final state=i|0i Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse II-Quantum Computation In 1994 Peter Shor found an algorithm that: • Can only be run on a quantum computer • Can find quickly the prime factors of a very large number. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse II-Quantum Computation In 1994 Peter Shor found an algorithm that: • Can only be run on a quantum computer • Can find quickly the prime factors of a very large number. To find the factors of a large number will take the quantum computer one afternoon, to find the factors on a classical computer can take the whole history of the universe. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Evidence of the Multiverse II-Quantum Computation David Deutsch on Shor’s algorithm “To those who still cling to a single-universe world-view, I issue this challenge: explain how Shor’s algorithm works... When Shor’s algorithm has factorized a number, using 10500 or so times the computational resources that can be seen to be present, where was the number factorized ? There are only about 1080 atoms in the entirely universe, an utterly minuscule number compared with 10500 . So if the visible universe were the extent of physical reality would not even remotely contain the resources required to factorize such large number. Who did factorize it, then? How, and where, was the computation performed” Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Only an experimental test derived from a sound mathematical formulation of the multiverse will be conclusive to settle this interpretation. This formulation does not exist yet. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Only an experimental test derived from a sound mathematical formulation of the multiverse will be conclusive to settle this interpretation. This formulation does not exist yet. Two hints towards a mathematical formulation of the multiverse. • The Sheaf Logic formulation of Cohen’s forcing use something that looks very much like a multiverse of mathematical universes that can be collapsed to a single classical mathematical universe. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Only an experimental test derived from a sound mathematical formulation of the multiverse will be conclusive to settle this interpretation. This formulation does not exist yet. Two hints towards a mathematical formulation of the multiverse. • The Sheaf Logic formulation of Cohen’s forcing use something that looks very much like a multiverse of mathematical universes that can be collapsed to a single classical mathematical universe. • In 1975 Takeuti, using an alternative formulation of Cohen’s method (Boolean Valued Models), found a mathematical universe where some self-adjoint operators of a Hilbert space correspond to the objects that represent the real numbers in the logic of the model. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaves of structures (Motivation) Galilean Spacetime • A topological space X =temporal line Figure: Galilean Spacetime • For each x ∈ X there exists a structure Ax constituted by: -A universe Ex formed by snapshots of extended objects in time. - Functions f1x , f2x , ... and relations R1x , R2x , ..., that give the instantaneous properties of extended objects. The different worlds Ex attach in an extended universe E, in such a way that the different functions and relations attach in a continuous way. • Ax = (Ex , R1x , R2x ..., f1x , f2x ...) Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R1 , ..., f1 , ..., c1 , ...) a sheaf of τ -structures A over a topological space X is given by: Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R1 , ..., f1 , ..., c1 , ...) a sheaf of τ -structures A over a topological space X is given by: a-) A sheaf (E, p) over X ( i.e a local homeomorphism). Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R1 , ..., f1 , ..., c1 , ...) a sheaf of τ -structures A over a topological space X is given by: a-) A sheaf (E, p) over X ( i.e a local homeomorphism). b-) For each x ∈ X , a τ -structure Ax = (Ex , R1x , R2x ..., f1x , ..., c1x , ...), where Ex = p−1 (x) (the fiber that could be empty) is the universe of the τ -structure Ax , and the following conditions are satisfied: Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R1 , ..., f1 , ..., c1 , ...) a sheaf of τ -structures A over a topological space X is given by: a-) A sheaf (E, p) over X ( i.e a local homeomorphism). b-) For each x ∈ X , a τ -structure Ax = (Ex , R1x , R2x ..., f1x , ..., c1x , ...), where Ex = p−1 (x) (the fiber that could be empty) is the universe of the τ -structure Ax , and the following S conditions are S satisfied: A i. R = x Rx is open in x Exn seeing as subspace of E n , where RSis an n-ary S mrelation S symbol. A ii. f = x fx : x Ex → x Ex is a continuous function, where f is an m-parameter function symbol. iii. h : X → E such that h(x) = cx , where c is a constant symbol, is continuous. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Presheaves of Structures Definition A presheaf of structures of type τ over X is an assignation Γ, such that to each open set U ⊆ X is assigned a τ -structure Γ(U) Γ(U) Γ(U) Γ(U) = (Γ(U), R1 , ..., f1 , ..., c1 , ..); and for U, V ⊆ X open such that V ⊆ U, it is assigned an homomorphism ΓUV which satisfies ΓUU = IdΓ(U) and ΓVW ◦ ΓUV = ΓUW whenever W ⊆ V ⊆ U. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Presheaves of Structures Definition A presheaf of structures of type τ over X is an assignation Γ, such that to each open set U ⊆ X is assigned a τ -structure Γ(U) Γ(U) Γ(U) Γ(U) = (Γ(U), R1 , ..., f1 , ..., c1 , ..); and for U, V ⊆ X open such that V ⊆ U, it is assigned an homomorphism ΓUV which satisfies ΓUU = IdΓ(U) and ΓVW ◦ ΓUV = ΓUW whenever W ⊆ V ⊆ U. The sheaf of germs GΓA associated to the presheaf of sections ΓA of a sheaf A is naturally isomorphic to the original sheaf. On the other hand, given a presheaf Γ, the presheaf ΓGΓ associated to the sheaf of germs GΓ turns out to be isomorphic to the original presheaf if the presheaf is exact i.e. ... Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Definition S A presheaf of structures Γ is said to be exact, if given U = i Ui and σi ∈ Γ(Ui ), such that ΓUi ,Ui ∩Uj (σi ) = ΓUj ,Ui ∩Uj (σj ) for all i, j; there exists an unique σ ∈ Γ(U) such that ΓUUi (σ) = σi for all i. And the same holds for the relations; i.e if we have some Γ(U ) Γ(U ) relations Ri i , Rj j which are sent by the homomorphisms ΓUi ,Ui ∩Uj , ΓUj ,Ui ∩Uj to a same relation for all i, j, there exists an unique relation R Γ(U) which is sent by the homomorphism ΓUUi Γ(U ) to the relation Ri i for all i. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Definition S A presheaf of structures Γ is said to be exact, if given U = i Ui and σi ∈ Γ(Ui ), such that ΓUi ,Ui ∩Uj (σi ) = ΓUj ,Ui ∩Uj (σj ) for all i, j; there exists an unique σ ∈ Γ(U) such that ΓUUi (σ) = σi for all i. And the same holds for the relations; i.e if we have some Γ(U ) Γ(U ) relations Ri i , Rj j which are sent by the homomorphisms ΓUi ,Ui ∩Uj , ΓUj ,Ui ∩Uj to a same relation for all i, j, there exists an unique relation R Γ(U) which is sent by the homomorphism ΓUUi Γ(U ) to the relation Ri i for all i. In categorical language a presheaf of structures of type τ is a contravariant functor from Op(X ) to Strτ , where the latter is the is the category of structures and morphisms of type τ . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaf Logic (Motivation) • A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. • As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaves of Structures Sheaf Logic (Motivation) • A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. • As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition. Contextual Truth Paradigm If a property for an extended object holds in some point of its domain then it has to hold in a neighbourhood of that point. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic Sheaf Logic, Point-Wise Semantics Sheaf Logic (Definition) • A x ¬ϕ[σ1 , ..., σn ] ⇔ exists U neighbourhood of x such that for all y ∈ U, A 1y ϕ[σ1 (y ), ..., σn (y )]. • A x (ϕ → ψ)[σ1 , ..., σn ] ⇔ Exists U neighbourhood of x such that for all y ∈ U if A y ϕ[σ1 (y ), ..., σn (y )] then A y ψ[σ1 (y ), ..., σn (y )]. • A x ∀v ϕ(v , σ1 , ..., σn ) ⇔ exists U neighbourhood of x such that for all y ∈ U and all σ defined in y , A y ϕ[σ(y ), σ1 (y ), ..., σn (y )]. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U ⊆ X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ1 , ..., σn ] ⇔ ∀x ∈ U, A x ϕ[σ1 , ..., σn ] Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U ⊆ X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ1 , ..., σn ] ⇔ ∀x ∈ U, A x ϕ[σ1 , ..., σn ] Lemma (Kripke-Joyal Semantics) • A U (ϕ ∨ ψ)[σ1 , ..., σn ] ⇔ there exist open sets V , W such that U = V ∪ W , A V ϕ[σ1 , ..., σn ] and A W ψ[σ1 , ..., σn ]. • A U ∃v ϕ(v , σ1 , ..., σn ) ⇔ there exists {Ui }i an open cover of U and µi sections defined on Ui such that A Ui ϕ[µi , σ1 , ..., σn ] for all i. • A U ∀v ϕ(v , σ1 , ..., σn ) ⇔ for any open set W ⊆ U and µ defined on W , A W ϕ(µ, σ1 , ..., σn ). Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic • The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic • The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X . • Let σ1 , ..., σn be sections of a sheaf A defined over an open set U, we define the “truth value” of a proposition ϕ in U as: [[ϕ(σ1 , ..., σn )]]U := {x ∈ U : A x ϕ[σ1 , ..., σn ]} (1) [[ϕ(σ1 , ..., σn )]]U is an open set, thus we can define a valuation as a topological valuation on formulas: TU : ϕ 7→ [[ϕ(σ1 , ..., σn )]]U . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic • The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X . • Let σ1 , ..., σn be sections of a sheaf A defined over an open set U, we define the “truth value” of a proposition ϕ in U as: [[ϕ(σ1 , ..., σn )]]U := {x ∈ U : A x ϕ[σ1 , ..., σn ]} (1) [[ϕ(σ1 , ..., σn )]]U is an open set, thus we can define a valuation as a topological valuation on formulas: TU : ϕ 7→ [[ϕ(σ1 , ..., σn )]]U . • The definition of the logic allows to define the value of the logic operators in terms of the operations of the algebra of open sets. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Sheaf Logic The importance of this Sheaf-Logic approach is that renders evident how sheaves of a Grothendieck topos yields a natural model theory of variable structures, providing geometric foundations for intuitionistic logic based in a local conception of the notion of truth. Furthermore based in this approach it is possible to introduce a notion of generic filter over the domain of the variation of the sheaf of structures, that allows to prove a corresponding generic model theorem that arise as a natural generalization of the forcing theorem in set theory and the Łoś theorem on ultraproducts. This construction allows to unify the different forcing approaches to Cohen’s forcing and the basic theorems of classical model theory as completeness, compactness, omitting types, Fraïssé limits etc., since all these results can be reduced to the construction of generic models over suitable sheaves. Deutsch-Everett interpretation Quantum Set Theory Variable Sets Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x ∈ B if and only if x ∈ A and ϕ(x) is “truth” for x, or in other words, B = {x ∈ A : ϕ(x)}. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x ∈ B if and only if x ∈ A and ϕ(x) is “truth” for x, or in other words, B = {x ∈ A : ϕ(x)}. Consider the following proposition: ϕ(x) ≡ “x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers”. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x ∈ B if and only if x ∈ A and ϕ(x) is “truth” for x, or in other words, B = {x ∈ A : ϕ(x)}. Consider the following proposition: ϕ(x) ≡ “x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers”. "Now and here" the following is valid in our temporal sheaf Now and Here ¬({x ∈ N : ϕ(x)} = {x ∈ N : x ≥ 4 ∧ (x is even )}), because now and here we do not know if the Goldbach conjecture is valid. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime. Our “states of Knowledge” will be then structures that represent the sets as we see them in our nodes. Therefore from each node we will see arise a cumulative Hierarchy of variable sets, which structure will be conditioned by the perception of the variable structures in the other nodes that relate to it. Or more precisely. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Definition (The Hierarchy of Variable Sets) Let X be an arbitrary topological space. Given U ∈ Op(X ) we define inductively: V0 (U) =∅ Vα+1 (U) ={f : Op(U) → [ P(Vα (W )) : 1. If W ⊆ U then f (W ) ⊆ Vα (W ), W ⊆U 2. If V ⊆ W ⊆ U, then for all g ∈ f (W ), g Op(V ) ∈ f (V ), 3. Given {Ui }i an open cover of U and gi ∈ f (Ui ) such that gi op(Ui ∩Uj ) = gj op(Ui ∩Uj ) for any i, j, there exists g ∈ f (U) such that g op(Ui ) = gi for all i} [ Vλ (U) = Vα (U) if λ is a limit ordinal, α<λ V (U) = [ Vα (U). α∈On The valuation V over the open sets constitute an exact presheaf of structures, the respective sheaf of germs VX constitute the cumulative hierarchy of variable sets. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory For each U ∈ Op(X ) the set V (U) is a set of functions defined over Op(U) which values for W ∈ Op(U) are functions over Op(W ) which values for V ∈ Op(W ) are functions over Op(V ) and so on. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory For each U ∈ Op(X ) the set V (U) is a set of functions defined over Op(U) which values for W ∈ Op(U) are functions over Op(W ) which values for V ∈ Op(W ) are functions over Op(V ) and so on. the ∈ relation f ∈U g (i.e. U f ∈ g) ⇔ f ∈ g(U), i.e. that respect to the context U, f belongs to g if and only if f ∈ g(U) as classical sets. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory For each U ∈ Op(X ) the set V (U) is a set of functions defined over Op(U) which values for W ∈ Op(U) are functions over Op(W ) which values for V ∈ Op(W ) are functions over Op(V ) and so on. the ∈ relation f ∈U g (i.e. U f ∈ g) ⇔ f ∈ g(U), i.e. that respect to the context U, f belongs to g if and only if f ∈ g(U) as classical sets. Theorem For any topological space X VX ZF Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory For each U ∈ Op(X ) the set V (U) is a set of functions defined over Op(U) which values for W ∈ Op(U) are functions over Op(W ) which values for V ∈ Op(W ) are functions over Op(V ) and so on. the ∈ relation f ∈U g (i.e. U f ∈ g) ⇔ f ∈ g(U), i.e. that respect to the context U, f belongs to g if and only if f ∈ g(U) as classical sets. Theorem For any topological space X VX ZF V ,→ V (U) To each classical set a we can associate a constant set [ b b a(U) : Op(U) → V (W ) b a(U)(W ) = {b(U) Op(W ) : b ∈ a}. W ⊆U a 7→ b a(U) defines an embedding of V in V (U) for any open set U ⊆ X . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Using this embedding, it can be proved that: b N(U) = N(U), b Z(U) = Z(U), b Q(U) = Q(U) b for any open set U ⊆ X . (e.g to prove that N(U) = N(U) we define the successor function Suc(f ) : Op(U) → [ V (W ) W ∈Op(U) W 7→ {f Op(W ) } ∪ f (W ); b and then we prove that N(U) is the minimum inductive set in the sheaf-logic sense. ) Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Using this embedding, it can be proved that: b N(U) = N(U), b Z(U) = Z(U), b Q(U) = Q(U) b for any open set U ⊆ X . (e.g to prove that N(U) = N(U) we define the successor function Suc(f ) : Op(U) → [ V (W ) W ∈Op(U) W 7→ {f Op(W ) } ∪ f (W ); b and then we prove that N(U) is the minimum inductive set in the sheaf-logic sense. ) These tools provide a mechanism to construct new mathematical universes over arbitrary topological spaces. If we find a topological space able to capture the essence of quantum logic this will provide a mathematical quantum universe that will probably improves our understanding of quantum mechanics. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ ∈ U admits a spectral decomposition in U, i.e a family of projections {P̌r }r ∈R ⊆ U such that Z Ǎ = rd P̌r Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ ∈ U admits a spectral decomposition in U, i.e a family of projections {P̌r }r ∈R ⊆ U such that Z Ǎ = rd P̌r We perceive the quantum system through an abelian Von Neumann frame of observables in analogous way as we perceive classical spacetime through an inertial frame which determines a particular foliation of a spacetime region. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets The base space=The space of Histories The Gelfand spectrum SA of U is the space of positive linear functions σ : U → C of norm 1 such that σ(AB) = σ(A)σ(B) for all A, B ∈ U. These are the histories because when restricted to the self-adjoint operators of U they become valuations. A valuation is a function λ from the set of self-adjoint operators Bsa (U) on U to the real numbers, λ : Bsa (H) → R, which satisfies: 1.λ(Ǎ) belongs to the spectrum of Ǎ, for all Ǎ ∈ Bsa (U) 2.λ(B̌) = f (λ(Ǎ)) whenever B̌ = f (Ǎ) with f : R → R a continuous function. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ ∈ U such that Ǎ = N X an P̌n n=1 is the spectral representation of Ǎ in U. Fix m such that 1 ≤ m ≤ N; and λ ∈ SU such that λ(Ǎ) = am , then λ(P̌m ) = 1. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ ∈ U such that Ǎ = N X an P̌n n=1 is the spectral representation of Ǎ in U. Fix m such that 1 ≤ m ≤ N; and λ ∈ SU such that λ(Ǎ) = am , then λ(P̌m ) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(P̌m ) = 1, the physical observable A assumes the value am . Therefore, given a projection P̌ ∈ P(U) the set P = {λ ∈ SU : λ(P̌) = 1} (2) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ ∈ U such that Ǎ = N X an P̌n n=1 is the spectral representation of Ǎ in U. Fix m such that 1 ≤ m ≤ N; and λ ∈ SU such that λ(Ǎ) = am , then λ(P̌m ) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(P̌m ) = 1, the physical observable A assumes the value am . Therefore, given a projection P̌ ∈ P(U) the set P = {λ ∈ SU : λ(P̌) = 1} (2) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history. {P}P̌∈P(U ) defines a topology in SU . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory A single foliation Perspective of The Quantum Multiverse The Cumulative hierarchy of variable sets VSU constructed over the topological space hSU , {P}P̌∈P(U ) i where U is an abelian Von Neumann Algebra is what we will call The Cumulative Hierarchy of Quantum Variable Sets. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory A single foliation Perspective of The Quantum Multiverse The Cumulative hierarchy of variable sets VSU constructed over the topological space hSU , {P}P̌∈P(U ) i where U is an abelian Von Neumann Algebra is what we will call The Cumulative Hierarchy of Quantum Variable Sets. The objects of this model will be the sections of the sheaf VX , which result to be extended objects that variate over the space of histories or universes X = SU , in a few words multiversal objects. Thanks to the strong internal interference that it is continuously undergoing, a typical electron is an irreducibly multiversal object, and not a collection of parallel-universe or parallel-histories objects. (D. Deutsch) Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Quantum Continuum Theorem Given a self adjoint operator Ǎ ∈ U, let {P̌r }r ∈R be the spectral family of operators associated to Ǎ. The operator Ǎ defines a real number in V SU given by: [ V (Q) UǍ (P) : Op(P) → Q⊆P b b(Q) ∈ Q(Q)(Q) Q 7→ {q : ∃r ∈ Q, r < q, Q * Prc } LǍ (P) : Op(P) → [ V (Q) Q⊆P b b(Q) ∈ Q(Q)(Q) Q 7→ {q : Q ⊆ Pqc } Deutsch-Everett interpretation Quantum Set Theory Sketch of the Proof Given a self adjoint operator Ǎ ∈ U , let {P̌r }r ∈R be the spectral family of operators associated to Ǎ. The family {P̌r }r ∈R is a family of operators contained in P(U) which satisfy: 1 2 3 4 P̌s ∧ P̌r = P̌min{r ,s} , V r ∈R P̌r = 0̌, W r ∈R P̌r = Ǐ, V q≤r P̌r = P̌q for every q ∈ R the above spectral family defines a family of open subsets, {Pr }r ∈R which satisfy analogous properties. Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Sketch of the Proof Given a self adjoint operator Ǎ ∈ U , let {P̌r }r ∈R be the spectral family of operators associated to Ǎ. The family {P̌r }r ∈R is a family of operators contained in P(U) which satisfy: 1 2 3 4 P̌s ∧ P̌r = P̌min{r ,s} , V r ∈R P̌r = 0̌, W r ∈R P̌r = Ǐ, V q≤r P̌r = P̌q for every q ∈ R the above spectral family defines a family of open subsets, {Pr }r ∈R which satisfy analogous properties. Using this properties we prove, for instance, that in any open set P, LǍ satisfies b b P q ∈ Q(P)(q ∈ LǍ → ∃r ∈ Q(P)((r ∈ LǍ ∧q < r )). Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Sketch of the Proof Given a self adjoint operator Ǎ ∈ U , let {P̌r }r ∈R be the spectral family of operators associated to Ǎ. The family {P̌r }r ∈R is a family of operators contained in P(U) which satisfy: 1 2 3 4 P̌s ∧ P̌r = P̌min{r ,s} , V r ∈R P̌r = 0̌, W r ∈R P̌r = Ǐ, V q≤r P̌r = P̌q for every q ∈ R the above spectral family defines a family of open subsets, {Pr }r ∈R which satisfy analogous properties. Using this properties we prove, for instance, that in any open set P, LǍ satisfies b b P q ∈ Q(P)(q ∈ LǍ → ∃r ∈ Q(P)((r ∈ LǍ ∧q < r )). Let Q ⊆ P be an open set and b(Q) ∈ Q(Q)(Q) be such that q b(Q) ∈ LǍ (Q). q Since Q ⊆ Pqc , {Q ∩ Prc }r ∈Q,r >q is an open cover of Q; indeed [ r ∈Q,r >q c (Q ∩ Pr ) = Q ∩ ( c [ Pr ) = Q ∩ ( r ∈Q,r >q \ c Pr ) = r ∈Q,r >q c = Q ∩ Pq = Q. We have then Q ∩ Prc ⊂ Prc , which implies b r (Q ∩ Prc ) ∈ LǍ (Q ∩ Prc ). Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Theorem If U, L : Op(P) → S Q⊆P V (Q) define a real number, consider [ b(P) ∈ U}, Pr = {P ∈ Op(X ) : ∀q > r , P q for r ∈ Q, then for s ∈ R the projections associated to the open set \ Pq , Ps = q∈Q,s<q define a spectral family; therefore, a self-adjoint operator. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Theorem If U, L : Op(P) → S Q⊆P V (Q) define a real number, consider [ b(P) ∈ U}, Pr = {P ∈ Op(X ) : ∀q > r , P q for r ∈ Q, then for s ∈ R the projections associated to the open set \ Pq , Ps = q∈Q,s<q define a spectral family; therefore, a self-adjoint operator. Sketch of the proof S • r ∈Q Pr = X : Let σ ∈ X and P an open neighbourhood of σ. From the “non empty” property of a Dedekind cut there exists an open set Qi , and b i )(Qi ) such that σ ∈ Qi ⊆ P and Q qbi (Q) ∈ U. Then by the qbi (Qi ) ∈ Q(Q i b “unbounded” property of upper cuts for all q > qi we S have Qi q (Qi ) ∈ U. Therefore σ ∈ Pqi , since σ was arbitrary we have r ∈Q Pr = X . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Each quantum state |hi ∈ H defines a measure µ|hi over SU given by: µ|hi : Op(SU ) → R P 7→ ||P̌|hi||2 . Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Each quantum state |hi ∈ H defines a measure µ|hi over SU given by: µ|hi : Op(SU ) → R P 7→ ||P̌|hi||2 . Open sets are of the form P = [[ϕ]]X = {λ :λ ϕ}, where ϕ is a proposition about the quantum system. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Each quantum state |hi ∈ H defines a measure µ|hi over SU given by: µ|hi : Op(SU ) → R P 7→ ||P̌|hi||2 . Open sets are of the form P = [[ϕ]]X = {λ :λ ϕ}, where ϕ is a proposition about the quantum system. For instance, given c, r ∈ R, Ǎ a self-adjoint operator and {P̌r }r ∈R the respective spectral family, b S = Pd \ Pc [[b c ≤ Ǎ ≤ d]] U and b S ) = ||(P̌d − P̌c )|hi||2 . µh ([[b c ≤ Ǎ ≤ d]] U This last expression coincides with the quantum probabilistic prediction that the observable Ǎ assumes a value between c and d when the system is in the state |hi. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Each quantum state |hi ∈ H defines a measure µ|hi over SU given by: µ|hi : Op(SU ) → R P 7→ ||P̌|hi||2 . Open sets are of the form P = [[ϕ]]X = {λ :λ ϕ}, where ϕ is a proposition about the quantum system. For instance, given c, r ∈ R, Ǎ a self-adjoint operator and {P̌r }r ∈R the respective spectral family, b S = Pd \ Pc [[b c ≤ Ǎ ≤ d]] U and b S ) = ||(P̌d − P̌c )|hi||2 . µh ([[b c ≤ Ǎ ≤ d]] U This last expression coincides with the quantum probabilistic prediction that the observable Ǎ assumes a value between c and d when the system is in the state |hi. But in this context this prediction is the measure of b S = {λ ∈ SU :λ b b [[b c ≤ Ǎ ≤ d]] c ≤ Ǎ ≤ d}, U i.e. the measure of the space of histories where the proposition is verified, Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Genericity: Collapsing to a Classical World Definition A filter of open sets F in a topological space X is a generic filter of VX if: • For all ϕ, ∃Q ∈ F s.t. Q ϕ or Q ¬ϕ. • If in P ∈ F , P ∃v ϕ(v ) then there exist Q ∈ F and σ defined over Q, s.t. q ϕ(σ). Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Genericity: Collapsing to a Classical World Definition A filter of open sets F in a topological space X is a generic filter of VX if: • For all ϕ, ∃Q ∈ F s.t. Q ϕ or Q ¬ϕ. • If in P ∈ F , P ∃v ϕ(v ) then there exist Q ∈ F and σ defined over Q, s.t. q ϕ(σ). Generic Model Theorem Let F be a generic filter of VX , then VX [F] = lim V (P) = →P∈F [ ˙ P∈F V (P)/∼F is a classical two valued model s.t. VX [F] |= ϕ([σ1 ], ..., [σn ]) ⇔ there exists P ∈ F such that P ϕG (σ1 , ..., σn ) ⇔ {λ ∈ X :λ ϕG (σ1 , ..., σn )} ∈ F, where ϕG is the Gödel translation of the formula ϕ. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Collapse to a Classic Unique History World In VSU for each history λ ∈ SU the set Fλ = {P ∈ Op(X ) : λ(P̌) = 1} is a generic filter. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Collapse to a Classic Unique History World In VSU for each history λ ∈ SU the set Fλ = {P ∈ Op(X ) : λ(P̌) = 1} is a generic filter. This allows to define rigorously what it means that a self-adjoint operator converges to a real number in each history λ. VSU SU Ǎ ∈ R(SU ) Fλ VSU [Fλ ] |= a ∈ R −−−−→ Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Collapse to a Classic Unique History World In VSU for each history λ ∈ SU the set Fλ = {P ∈ Op(X ) : λ(P̌) = 1} is a generic filter. This allows to define rigorously what it means that a self-adjoint operator converges to a real number in each history λ. VSU SU Ǎ ∈ R(SU ) Fλ VSU [Fλ ] |= a ∈ R −−−−→ In the same way this mechanism should provide a sound formulation of how the quantum dynamical variables converge to the classical ones. Consider for example the momentum operator, this is derived using that in classical mechanics the quantity whose conservation in a closed system follows from the homogeneity of space is the momentum, then using this “axiomatic” definition it is shown that the operator of spatial derivation, up to multiplication for a constant factor, satisfies this definition within the context of quantum theory. Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory. Quantum Set Theory Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned. (Deutsch, Ekert, Lupacchini, Machines, Logic and Quantum Physics)