Sheaf Logic, Quantum Set Theory and The Interpretation of Quantum Mechanicsweb.math.unifi.it/users/navarro/Web_Page_JohnBenavides/J.B._files... · Sheaf Logic, Quantum Set Theory

Deutsch-Everett interpretation Sheaves of Structures, Sheaf Logic and Quantum Set Theory.

Sheaf Logic, Quantum Set Theory and TheInterpretation of Quantum Mechanics

J. Benavides

October 26, 2012


The interpretation problemToday, more than one hundred years after Max Planckformulated the quantum hypothesis, we still do not have asettled agreement about what quantum reality is or if there issomething 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 beapproximated as a connected four dimensional time orientedLorentzian manifold where Einstein’s equation is valid.

On the other hand, whether quantum theory tell us somethingabout the structure of the world we inhabit or is just asophisticated formalism that allows to quantify the behaviour ofthe small scale world, is still a very controversial argument.










The classical quantum formalism

Postulate 1A quantum system is described by a unit vector |ψ(t)〉 (the statevector) in a complex Hilbert space H and an operator A knownas the Hamiltonian.

Postulate 2In absence of any external influence the state vector changessmoothly in time according to the time dependent Schrödingerequation

i}d |ψ(t)〉

dt= A|ψ(t)〉





i}d |ψ(t)〉

dt= A|ψ(t)〉





i}d |ψ(t)〉

dt= A|ψ(t)〉


Postulate 3The observables of the system are represented mathematicallyby self-adjoint operators acting on H.

Postulate 4 (Schema)

If an observable B is represented by a self-adjoint operator Bwith eigenvalues b1, ...,bm and respective eigenvectors|b1〉, ..., |bn〉, and the state vector |ψ(t)〉 is expressed in thebasis formed by the eigenvalues of B as:

|ψ(t)〉 = α1|b1〉+ ...+ αm|bm〉.

Then a measurement of B at time t will give as a result one ofthe eigenvalues bi with probability |αi |2 respectively, and noneother result that is not one of the eigenvalues is obtained.


Postulate 3The observables of the system are represented mathematicallyby self-adjoint operators acting on H.

Postulate 4 (Schema)

If an observable B is represented by a self-adjoint operator Bwith eigenvalues b1, ...,bm and respective eigenvectors|b1〉, ..., |bn〉, and the state vector |ψ(t)〉 is expressed in thebasis formed by the eigenvalues of B as:

|ψ(t)〉 = α1|b1〉+ ...+ αm|bm〉.

Then a measurement of B at time t will give as a result one ofthe eigenvalues bi with probability |αi |2 respectively, and noneother result that is not one of the eigenvalues is obtained.


Measurement ProblemThe quantum formalism is not able to distinguish the actualresult of measurement from all the possible results.

Hugh Everett, Many-Worlds(1957)Quantum Mechanics isconsistent with the idea that allpossible results of themeasurement actually happen,being the single universe wherewe perceive one single outcomepart of a larger structure ofmany-worlds where the differentoutcomes happen.


Measurement ProblemThe quantum formalism is not able to distinguish the actualresult of measurement from all the possible results.

Hugh Everett, Many-Worlds(1957)Quantum Mechanics isconsistent with the idea that allpossible results of themeasurement actually happen,being the single universe wherewe perceive one single outcomepart of a larger structure ofmany-worlds where the differentoutcomes happen.


David Deutsch, QuantumMultiverse 1984-2011-...Since 1984 David Deutsch, thefather of quantum computation,has improved Everett ideas inparallel with his work onquantum information.Unfortunately, even if quantumcomputation is maybe the mostremarkable result in theoreticalphysics in the last 37 years,Deutsch’s ideas about themany-worlds or multiversalinterpretation have not beenfully appreciated.


The Deutsch-Everett multiversal interpretation

The multiverse is a set with a measure, whose elements aremaximal sets of observables with definite values thatcorrespond to different universes or different histories.

In this context the terms |αi |2 associated to the expression

|ψ(t)〉 = α1|b1〉+ ...+ αm|bm〉

represent the values of the measure of the sets of universeswhere the observable B assume the value bi respectively.

Different sets of compatible observables determine differentexpressions of the state vector |ψ(t)〉, these different forms toexpress the state vector correspond to different foliations of themultiverse in the same sense that a region of spacetime can befoliated by spacelike surfaces in different ways.



The multiverse is a set with a measure, whose elements aremaximal sets of observables with definite values thatcorrespond to different universes or different histories.In this context the terms |αi |2 associated to the expression

|ψ(t)〉 = α1|b1〉+ ...+ αm|bm〉





The multiverse is a set with a measure, whose elements aremaximal sets of observables with definite values thatcorrespond to different universes or different histories.In this context the terms |αi |2 associated to the expression

|ψ(t)〉 = α1|b1〉+ ...+ αm|bm〉





Each universe in any foliation is associated to a classicalsystem, which corresponds to the classical physical worldwhere we see the measuring apparatus taking one uniquevalue.

The universes interact via interferencephenomena, but such interactions aresuppressed at the classical leveldescribed by classical physics.At this classical level, the lack ofinterference allows to process classicalscale information in an autonomousway, for this reason our classicaltheories (e.g. General Relativity) cangive an accurate description at thislevel without appealing to the deepermultiversal structure.




The universes interact via interferencephenomena, but such interactions aresuppressed at the classical leveldescribed by classical physics.

At this classical level, the lack ofinterference allows to process classicalscale information in an autonomousway, for this reason our classicaltheories (e.g. General Relativity) cangive an accurate description at thislevel without appealing to the deepermultiversal structure.




The universes interact via interferencephenomena, but such interactions aresuppressed at the classical leveldescribed by classical physics.At this classical level, the lack ofinterference allows to process classicalscale information in an autonomousway, for this reason our classicaltheories (e.g. General Relativity) cangive an accurate description at thislevel without appealing to the deepermultiversal structure.


Evidence of the Multiverse I-Interference Phenomena

• Initial state= |0〉

• State after interacting withthe beam splitter=

1√2|0〉+

i√2|1〉

• State after interacting withthe full silvered mirrors=

i√2|0〉+

1√2|1〉

• Final state=i |0〉



• Initial state= |0〉• State after interacting with

the beam splitter=

1√2|0〉+

i√2|1〉


i√2|0〉+

1√2|1〉





the beam splitter=

1√2|0〉+

i√2|1〉


i√2|0〉+

1√2|1〉





the beam splitter=

1√2|0〉+

i√2|1〉


i√2|0〉+

1√2|1〉



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 quantumcomputer one afternoon, to find the factors on a classicalcomputer can take the whole history of the universe.



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 quantumcomputer one afternoon, to find the factors on a classicalcomputer can take the whole history of the universe.



David Deutsch on Shor’s algorithm“To those who still cling to a single-universe world-view, I issuethis challenge: explain how Shor’s algorithm works... WhenShor’s algorithm has factorized a number, using 10500 or sotimes the computational resources that can be seen to bepresent, where was the number factorized ? There are onlyabout 1080 atoms in the entirely universe, an utterly minusculenumber compared with 10500. So if the visible universe werethe extent of physical reality would not even remotely containthe resources required to factorize such large number. Who didfactorize it, then? How, and where, was the computationperformed”


Only an experimental test derived from a sound mathematicalformulation of the multiverse will be conclusive to settle thisinterpretation. This formulation does not exist yet.

Two hints towards a mathematical formulation of themultiverse.• The Sheaf Logic formulation of Cohen’s forcing use

something that looks very much like a multiverse ofmathematical universes that can be collapsed to a singleclassical mathematical universe.

• In 1975 Takeuti, using an alternative formulation ofCohen’s method (Boolean Valued Models), found amathematical universe where some self-adjoint operatorsof a Hilbert space correspond to the objects that representthe real numbers in the logic of the model.












Sheaves of Structures

Sheaves of structures (Motivation)

Figure: Galilean Spacetime

Galilean Spacetime• A topological space X =temporal line

• For each x ∈ X there exists astructure Ax constituted by:-A universe Ex formed by snapshotsof extended objects in time.- Functions f x

1 , fx2 , ... and relations

Rx1 ,R

x2 , ..., that give the

instantaneous properties of extendedobjects.The different worlds Ex attach in anextended universe E , in such a waythat the different functions andrelations attach in a continuous way.

• Ax = (Ex ,Rx1 ,R

x2 ..., f

x1 , f

x2 ...)



Sheaves of Structures (Definition)

DefinitionGiven a fix type of structures τ = (R1, ..., f1, ..., c1, ...) a sheaf ofτ -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 -structureAx = (Ex ,Rx

1 ,Rx2 ..., f

x1 , ..., c

x1 , ...), where Ex = p−1(x) (the fiber

that could be empty) is the universe of the τ -structure Ax , andthe following conditions are satisfied:i. RA =

⋃x Rx is open in

⋃x En

x seeing as subspace of En,where R is an n-ary relation symbol.ii. fA =

⋃x fx :

⋃x Em

x →⋃

x Ex is a continuous function, wheref is an m-parameter function symbol.iii. h : X → E such that h(x) = cx , where c is a constantsymbol, is continuous.




DefinitionGiven a fix type of structures τ = (R1, ..., f1, ..., c1, ...) a sheaf ofτ -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 -structureAx = (Ex ,Rx

1 ,Rx2 ..., f

x1 , ..., c



⋃x Rx is open in

⋃x En


⋃x fx :

⋃x Em

x →⋃





DefinitionGiven a fix type of structures τ = (R1, ..., f1, ..., c1, ...) a sheaf ofτ -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 -structureAx = (Ex ,Rx

1 ,Rx2 ..., f

x1 , ..., c


that could be empty) is the universe of the τ -structure Ax , andthe following conditions are satisfied:

i. RA =⋃

x Rx is open in⋃

x Enx seeing as subspace of En,

where R is an n-ary relation symbol.ii. fA =

⋃x fx :

⋃x Em

x →⋃





DefinitionGiven a fix type of structures τ = (R1, ..., f1, ..., c1, ...) a sheaf ofτ -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 -structureAx = (Ex ,Rx

1 ,Rx2 ..., f

x1 , ..., c



⋃x Rx is open in

⋃x En


⋃x fx :

⋃x Em

x →⋃




Presheaves of Structures

DefinitionA presheaf of structures of type τ over X is an assignation Γ,such that to each open set U ⊆ X is assigned a τ -structureΓ(U) = (Γ(U),RΓ(U)

1 , ..., f Γ(U)1 , ..., cΓ(U)

1 , ..); and for U,V ⊆ Xopen such that V ⊆ U, it is assigned an homomorphism ΓUVwhich satisfies ΓUU = IdΓ(U) and ΓVW ◦ ΓUV = ΓUW wheneverW ⊆ V ⊆ U.

The sheaf of germs GΓA associated to the presheaf of sectionsΓA of a sheaf A is naturally isomorphic to the original sheaf. Onthe other hand, given a presheaf Γ, the presheaf ΓGΓ associatedto the sheaf of germs GΓ turns out to be isomorphic to theoriginal presheaf if the presheaf is exact i.e. ...



Presheaves of Structures

DefinitionA presheaf of structures of type τ over X is an assignation Γ,such that to each open set U ⊆ X is assigned a τ -structureΓ(U) = (Γ(U),RΓ(U)

1 , ..., f Γ(U)1 , ..., cΓ(U)

1 , ..); and for U,V ⊆ Xopen such that V ⊆ U, it is assigned an homomorphism ΓUVwhich satisfies ΓUU = IdΓ(U) and ΓVW ◦ ΓUV = ΓUW wheneverW ⊆ V ⊆ U.

The sheaf of germs GΓA associated to the presheaf of sectionsΓA of a sheaf A is naturally isomorphic to the original sheaf. Onthe other hand, given a presheaf Γ, the presheaf ΓGΓ associatedto the sheaf of germs GΓ turns out to be isomorphic to theoriginal presheaf if the presheaf is exact i.e. ...



DefinitionA 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 somerelations RΓ(Ui )

i ,RΓ(Uj )

j which are sent by the homomorphismsΓUi ,Ui∩Uj , ΓUj ,Ui∩Uj to a same relation for all i , j , there exists anunique relation RΓ(U) which is sent by the homomorphism ΓUUi

to the relation RΓ(Ui )i for all i .

In categorical language a presheaf of structures of type τ is acontravariant functor from Op(X ) to Strτ , where the latter is theis the category of structures and morphisms of type τ .



DefinitionA 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 somerelations RΓ(Ui )

i ,RΓ(Uj )

j which are sent by the homomorphismsΓUi ,Ui∩Uj , ΓUj ,Ui∩Uj to a same relation for all i , j , there exists anunique relation RΓ(U) which is sent by the homomorphism ΓUUi

to the relation RΓ(Ui )i for all i .

In categorical language a presheaf of structures of type τ is acontravariant functor from Op(X ) to Strτ , where the latter is theis the category of structures and morphisms of type τ .



Sheaf Logic (Motivation)

• A sheaf of structures is a space extended over the basespace X of the sheaf as Galilean spacetime extends over time.The elements of this space are not the points of E but thesections of the sheaf conceived as extended objects. Thesingle values of these sections represent just point-wisedescriptions of the extended object.• As the objects of a sheaf of structures are the sections of thesheaf, the logic which governs them should define when aproperty for an extended object holds in a point of its domain ofdefinition.

Contextual Truth ParadigmIf a property for an extended object holds in some point of its

domain then it has to hold in a neighbourhood of that point.



Sheaf Logic (Motivation)

• A sheaf of structures is a space extended over the basespace X of the sheaf as Galilean spacetime extends over time.The elements of this space are not the points of E but thesections of the sheaf conceived as extended objects. Thesingle values of these sections represent just point-wisedescriptions of the extended object.• As the objects of a sheaf of structures are the sections of thesheaf, the logic which governs them should define when aproperty for an extended object holds in a point of its domain ofdefinition.

Contextual Truth ParadigmIf a property for an extended object holds in some point of its

domain then it has to hold in a neighbourhood of that point.


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)] thenA y ψ[σ1(y), ..., σn(y)].• A x ∀vϕ(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)].


Sheaf Logic

Sheaf Logic, Local Semantics

Given an open subset U ⊆ X and sections defined over U, we say that aproposition about these sections holds in U if it holds at each point in U or inother 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 thatU = V ∪W, A V ϕ[σ1, ..., σn] and A W ψ[σ1, ..., σn].• A U ∃vϕ(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 ∀vϕ(v , σ1, ..., σn)⇔ for any open set W ⊆ U and µ defined on W,A W ϕ(µ, σ1, ..., σn).


Sheaf Logic

Sheaf Logic, Local Semantics

Given an open subset U ⊆ X and sections defined over U, we say that aproposition about these sections holds in U if it holds at each point in U or inother 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 thatU = V ∪W, A V ϕ[σ1, ..., σn] and A W ψ[σ1, ..., σn].• A U ∃vϕ(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 ∀vϕ(v , σ1, ..., σn)⇔ for any open set W ⊆ U and µ defined on W,A W ϕ(µ, σ1, ..., σn).


Sheaf Logic

• The logic just defined can be seen as a multivalued logic withtruth values that variate over the Heyting algebra of the opensets of the base space X .

• Let σ1, ..., σn be sections of a sheaf A defined over an openset 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 valuationas a topological valuation on formulas:

TU : ϕ 7→ [[ϕ(σ1, ..., σn)]]U .

• The definition of the logic allows to define the value of thelogic operators in terms of the operations of the algebra of opensets.


Sheaf Logic

• The logic just defined can be seen as a multivalued logic withtruth values that variate over the Heyting algebra of the opensets of the base space X .• Let σ1, ..., σn be sections of a sheaf A defined over an openset 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)


TU : ϕ 7→ [[ϕ(σ1, ..., σn)]]U .



Sheaf Logic

• The logic just defined can be seen as a multivalued logic withtruth values that variate over the Heyting algebra of the opensets of the base space X .• Let σ1, ..., σn be sections of a sheaf A defined over an openset 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)


TU : ϕ 7→ [[ϕ(σ1, ..., σn)]]U .



Sheaf Logic

The importance of this Sheaf-Logic approach is that rendersevident how sheaves of a Grothendieck topos yields a naturalmodel theory of variable structures, providing geometricfoundations for intuitionistic logic based in a local conception ofthe notion of truth. Furthermore based in this approach it ispossible to introduce a notion of generic filter over the domainof the variation of the sheaf of structures, that allows to prove acorresponding generic model theorem that arise as a naturalgeneralization of the forcing theorem in set theory and the Łostheorem on ultraproducts. This construction allows to unify thedifferent forcing approaches to Cohen’s forcing and the basictheorems of classical model theory as completeness,compactness, omitting types, Fraïssé limits etc., since all theseresults can be reduced to the construction of generic modelsover suitable sheaves.


Quantum Set Theory

Variable Sets


Quantum Set Theory

Variable Sets

Using the comprehension axiom in classical set theory, given aproposition ϕ(x) and a set A, we can construct a set B suchthat x ∈ B if and only if x ∈ A and ϕ(x) is “truth” for x , or inother words,

B = {x ∈ A : ϕ(x)}.

Consider the following proposition:ϕ(x) ≡ “x is an even number greater or equal than 4 and x canbe 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 Goldbachconjecture is valid.


Quantum Set Theory

Variable Sets


B = {x ∈ A : ϕ(x)}.

Consider the following proposition:ϕ(x) ≡ “x is an even number greater or equal than 4 and x canbe written as the sum of two prime numbers”.

"Now and here" the following is valid in our temporal sheaf




Quantum Set Theory

Variable Sets


B = {x ∈ A : ϕ(x)}.

Consider the following proposition:ϕ(x) ≡ “x is an even number greater or equal than 4 and x canbe written as the sum of two prime numbers”."Now and here" the following is valid in our temporal sheaf




Quantum Set Theory

Variable Sets

Instead of conceiving sets as absolute entities, we canconceive them as variable structures which variate over ourLibrary of the states of knowledge.

It is natural then to conceive the set of nodes where our statesof Knowledge variates as nodes in a partial order or points in atopological space, that can represent, for instance, the causalstructure of spacetime.

Our “states of Knowledge” will be then structures that representthe sets as we see them in our nodes. Therefore from eachnode we will see arise a cumulative Hierarchy of variable sets,which structure will be conditioned by the perception of thevariable structures in the other nodes that relate to it. Or moreprecisely.


Quantum Set Theory

Variable Sets





Quantum Set Theory

Variable Sets





Quantum Set Theory

Definition (The Hierarchy of Variable Sets)

Let X be an arbitrary topological space. Given U ∈ Op(X ) we defineinductively:

V0(U) =∅

Vα+1(U) ={f : Op(U)→⋃

W⊆U

P(Vα(W )) : 1. If W ⊆ U then f (W ) ⊆ Vα(W ),

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) =⋃α<λ

Vα(U) if λ is a limit ordinal,

V (U) =⋃α∈On

Vα(U).

The valuation V over the open sets constitute an exact presheaf ofstructures, the respective sheaf of germs VX constitute the cumulativehierarchy of variable sets.


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 forV ∈ 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) asclassical sets.

Theorem

For any topological space XVX ZF

V ↪→ V (U)

To each classical set a we can associate a constant set

a(U) : Op(U)→⋃

W⊆U

V (W ) a(U)(W ) = {b(U) �Op(W ): b ∈ a}.

a 7→ a(U) defines an embedding of V in V (U) for any open set U ⊆ X .


Quantum Set Theory


the ∈ relation

f ∈U g (i.e. U f ∈ g)⇔ f ∈ g(U),


Theorem


V ↪→ V (U)


a(U) : Op(U)→⋃

W⊆U

V (W ) a(U)(W ) = {b(U) �Op(W ): b ∈ a}.



Quantum Set Theory


the ∈ relation

f ∈U g (i.e. U f ∈ g)⇔ f ∈ g(U),


Theorem


V ↪→ V (U)


a(U) : Op(U)→⋃

W⊆U

V (W ) a(U)(W ) = {b(U) �Op(W ): b ∈ a}.



Quantum Set Theory


the ∈ relation

f ∈U g (i.e. U f ∈ g)⇔ f ∈ g(U),


Theorem


V ↪→ V (U)


a(U) : Op(U)→⋃

W⊆U

V (W ) a(U)(W ) = {b(U) �Op(W ): b ∈ a}.



Quantum Set Theory

Using this embedding, it can be proved that:

N(U) = N(U), Z(U) = Z(U), Q(U) = Q(U)

for any open set U ⊆ X . (e.g to prove that N(U) = N(U) we define thesuccessor function

Suc(f ) : Op(U)→⋃

W∈Op(U)

V (W )

W 7→ {f �Op(W )} ∪ f (W );

and then we prove that N(U) is the minimum inductive set in the sheaf-logicsense. )

These tools provide a mechanism to construct new mathematical universesover arbitrary topological spaces. If we find a topological space able tocapture the essence of quantum logic this will provide a mathematicalquantum universe that will probably improves our understanding of quantummechanics.


Quantum Set Theory

Using this embedding, it can be proved that:

N(U) = N(U), Z(U) = Z(U), Q(U) = Q(U)

for any open set U ⊆ X . (e.g to prove that N(U) = N(U) we define thesuccessor function

Suc(f ) : Op(U)→⋃

W∈Op(U)

V (W )

W 7→ {f �Op(W )} ∪ f (W );

and then we prove that N(U) is the minimum inductive set in the sheaf-logicsense. )

These tools provide a mechanism to construct new mathematical universesover arbitrary topological spaces. If we find a topological space able tocapture the essence of quantum logic this will provide a mathematicalquantum universe that will probably improves our understanding of quantummechanics.


Quantum Set Theory

Quantum Variable Sets

FoliationsLet U be an abelian Von Neumann subalgebra of the algebra ofoperators of the Hilbert space of a quantum system. Eachself-adjoint operator A ∈ U admits a spectral decomposition inU , i.e a family of projections {Pr}r∈R ⊆ U such that

A =

∫rdPr

We perceive the quantum system through an abelian VonNeumann frame of observables in analogous way as weperceive classical spacetime through an inertial frame whichdetermines a particular foliation of a spacetime region.


Quantum Set Theory


FoliationsLet U be an abelian Von Neumann subalgebra of the algebra ofoperators of the Hilbert space of a quantum system. Eachself-adjoint operator A ∈ U admits a spectral decomposition inU , i.e a family of projections {Pr}r∈R ⊆ U such that

A =

∫rdPr

We perceive the quantum system through an abelian VonNeumann frame of observables in analogous way as weperceive classical spacetime through an inertial frame whichdetermines a particular foliation of a spacetime region.


Quantum Set Theory


The base space=The space of HistoriesThe Gelfand spectrum SA of U is the space of positive linearfunctions σ : U → C of norm 1 such that σ(AB) = σ(A)σ(B) forall A,B ∈ U . These are the histories because when restricted tothe self-adjoint operators of U they become valuations. Avaluation is a function λ from the set of self-adjoint operatorsBsa(U) on U to the real numbers, λ : Bsa(H)→ R, whichsatisfies:

1.λ(A) belongs to the spectrum of A, for all A ∈ Bsa(U)

2.λ(B) = f (λ(A)) whenever B = f (A)

with f : R→ R a continuous function.


Quantum Set Theory


The Topology: Similar Histories InterfereConsider a self-adjoint operator A ∈ U such that

A =N∑

n=1

anPn

is the spectral representation of A in U . Fix m such that 1 ≤ m ≤ N; andλ ∈ SU such that λ(A) = am, then λ(Pm) = 1.

Thus if A represents a physicalobservable, we have that in all the histories λ such that λ(Pm) = 1, thephysical observable A assumes the value am. Therefore, given a projectionP ∈ P(U) the set

P = {λ ∈ SU : λ(P) = 1} (2)

is a context of histories which are similar in the sense that some physicalobservables assume the same values or the values satisfy the sameinequalities in each history.{P}P∈P(U) defines a topology in SU .


Quantum Set Theory



A =N∑

n=1

anPn

is the spectral representation of A in U . Fix m such that 1 ≤ m ≤ N; andλ ∈ SU such that λ(A) = am, then λ(Pm) = 1. Thus if A represents a physicalobservable, we have that in all the histories λ such that λ(Pm) = 1, thephysical observable A assumes the value am. Therefore, given a projectionP ∈ P(U) the set

P = {λ ∈ SU : λ(P) = 1} (2)

is a context of histories which are similar in the sense that some physicalobservables assume the same values or the values satisfy the sameinequalities in each history.

{P}P∈P(U) defines a topology in SU .


Quantum Set Theory



A =N∑

n=1

anPn

is the spectral representation of A in U . Fix m such that 1 ≤ m ≤ N; andλ ∈ SU such that λ(A) = am, then λ(Pm) = 1. Thus if A represents a physicalobservable, we have that in all the histories λ such that λ(Pm) = 1, thephysical observable A assumes the value am. Therefore, given a projectionP ∈ P(U) the set

P = {λ ∈ SU : λ(P) = 1} (2)

is a context of histories which are similar in the sense that some physicalobservables assume the same values or the values satisfy the sameinequalities in each history.{P}P∈P(U) defines a topology in SU .


Quantum Set Theory

A single foliation Perspective of The Quantum Multiverse

The Cumulative hierarchy of variable sets VSU constructed overthe topological space 〈SU , {P}P∈P(U)〉 where U is an abelianVon Neumann Algebra is what we will call The CumulativeHierarchy 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 spaceof histories or universes X = SU , in a few words multiversalobjects.

Thanks to the strong internal interference that it is continuouslyundergoing, a typical electron is an irreducibly multiversal

object, and not a collection of parallel-universe orparallel-histories objects. (D. Deutsch)


Quantum Set Theory

A single foliation Perspective of The Quantum Multiverse

The Cumulative hierarchy of variable sets VSU constructed overthe topological space 〈SU , {P}P∈P(U)〉 where U is an abelianVon Neumann Algebra is what we will call The CumulativeHierarchy 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 spaceof histories or universes X = SU , in a few words multiversalobjects.

Thanks to the strong internal interference that it is continuouslyundergoing, a typical electron is an irreducibly multiversal

object, and not a collection of parallel-universe orparallel-histories objects. (D. Deutsch)


Quantum Set Theory

Quantum Continuum

Theorem

Given a self adjoint operator A ∈ U , let {Pr}r∈R be the spectralfamily of operators associated to A. The operator A defines areal number in V SU given by:

UA(P) : Op(P)→⋃

Q⊆P

V (Q)

Q 7→ {q(Q) ∈ Q(Q)(Q) : ∃r ∈ Q, r < q,Q * Pcr }

LA(P) : Op(P)→⋃

Q⊆P

V (Q)

Q 7→ {q(Q) ∈ Q(Q)(Q) : Q ⊆ Pcq}


Quantum Set Theory

Sketch of the Proof

Given a self adjoint operator A ∈ U ,let {Pr}r∈R be the spectral family ofoperators associated to A. Thefamily {Pr}r∈R is a family ofoperators contained in P(U) whichsatisfy:

1 Ps ∧ Pr = Pmin{r,s},2

∧r∈R Pr = 0,

3∨

r∈R Pr = I,

4∧

q≤r Pr = Pq for every q ∈ Rthe above spectral family defines afamily of open subsets, {Pr}r∈Rwhich satisfy analogous properties.

Using this properties we prove, for instance,that in any open set P, LA satisfies

P q ∈ Q(P)(q ∈ LA → ∃r ∈ Q(P)((r ∈ LA∧q < r)).

Let Q ⊆ P be an open set andq(Q) ∈ Q(Q)(Q) be such that q(Q) ∈ LA(Q).Since Q ⊆ Pc

q , {Q ∩ Pcr }r∈Q,r>q is an open

cover of Q; indeed⋃r∈Q,r>q

(Q ∩ Pcr ) = Q ∩ (

⋃r∈Q,r>q

Pcr ) = Q ∩ (

⋂r∈Q,r>q

Pr )c =

= Q ∩ Pcq = Q.

We have then Q ∩ Pcr ⊂ Pc

r , which impliesr(Q ∩ Pc

r ) ∈ LA(Q ∩ Pcr ).


Quantum Set Theory

Sketch of the Proof



∧r∈R Pr = 0,

3∨

r∈R Pr = I,

4∧







(Q ∩ Pcr ) = Q ∩ (

⋃r∈Q,r>q

Pcr ) = Q ∩ (

⋂r∈Q,r>q

Pr )c =

= Q ∩ Pcq = Q.



r ) ∈ LA(Q ∩ Pcr ).


Quantum Set Theory

Sketch of the Proof



∧r∈R Pr = 0,

3∨

r∈R Pr = I,

4∧







(Q ∩ Pcr ) = Q ∩ (

⋃r∈Q,r>q

Pcr ) = Q ∩ (

⋂r∈Q,r>q

Pr )c =

= Q ∩ Pcq = Q.



r ) ∈ LA(Q ∩ Pcr ).


Quantum Set Theory

Theorem

If U, L : Op(P)→⋃

Q⊆P V (Q) define a real number, consider

Pr =⋃{P ∈ Op(X ) : ∀q > r , P q(P) ∈ U},

for r ∈ Q, then for s ∈ R the projections associated to the open set

Ps =⋂

q∈Q,s<q

Pq ,

define a spectral family; therefore, a self-adjoint operator.

Sketch of the proof

•⋃

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 , andqi (Qi ) ∈ Q(Qi )(Qi ) such that σ ∈ Qi ⊆ P and Qi qi (Q) ∈ U. Then by the“unbounded” property of upper cuts for all q > qi we have Qi q(Qi ) ∈ U.Therefore σ ∈ Pqi , since σ was arbitrary we have

⋃r∈Q Pr = X .


Quantum Set Theory

Theorem

If U, L : Op(P)→⋃

Q⊆P V (Q) define a real number, consider

Pr =⋃{P ∈ Op(X ) : ∀q > r , P q(P) ∈ U},

for r ∈ Q, then for s ∈ R the projections associated to the open set

Ps =⋂

q∈Q,s<q

Pq ,

define a spectral family; therefore, a self-adjoint operator.

Sketch of the proof

•⋃

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 , andqi (Qi ) ∈ Q(Qi )(Qi ) such that σ ∈ Qi ⊆ P and Qi qi (Q) ∈ U. Then by the“unbounded” property of upper cuts for all q > qi we have Qi q(Qi ) ∈ U.Therefore σ ∈ Pqi , since σ was arbitrary we have

⋃r∈Q Pr = X .


Quantum Set Theory

Each quantum state |h〉 ∈ H defines a measure µ|h〉 over SU given by:

µ|h〉 : Op(SU )→ R

P 7→ ||P|h〉||2.

Open sets are of the form P = [[ϕ]]X = {λ : λ ϕ}, where ϕ is a propositionabout the quantum system.For instance, given c, r ∈ R, A a self-adjoint operator and {Pr}r∈R therespective spectral family,

[[c ≤ A ≤ d ]]SU = Pd \ Pc

andµh([[c ≤ A ≤ d ]]SU ) = ||(Pd − Pc)|h〉||2.

This last expression coincides with the quantum probabilistic prediction thatthe observable A assumes a value between c and d when the system is inthe state |h〉.But in this context this prediction is the measure of

[[c ≤ A ≤ d ]]SU = {λ ∈ SU : λ c ≤ A ≤ d},

i.e. the measure of the space of histories where the proposition is verified,just as in the Deutsch-Everett interpretation.


Quantum Set Theory



P 7→ ||P|h〉||2.

Open sets are of the form P = [[ϕ]]X = {λ : λ ϕ}, where ϕ is a propositionabout the quantum system.

For instance, given c, r ∈ R, A a self-adjoint operator and {Pr}r∈R therespective spectral family,

[[c ≤ A ≤ d ]]SU = Pd \ Pc



[[c ≤ A ≤ d ]]SU = {λ ∈ SU : λ c ≤ A ≤ d},



Quantum Set Theory



P 7→ ||P|h〉||2.


[[c ≤ A ≤ d ]]SU = Pd \ Pc


This last expression coincides with the quantum probabilistic prediction thatthe observable A assumes a value between c and d when the system is inthe state |h〉.

But in this context this prediction is the measure of

[[c ≤ A ≤ d ]]SU = {λ ∈ SU : λ c ≤ A ≤ d},



Quantum Set Theory



P 7→ ||P|h〉||2.


[[c ≤ A ≤ d ]]SU = Pd \ Pc



[[c ≤ A ≤ d ]]SU = {λ ∈ SU : λ c ≤ A ≤ d},



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 ∃vϕ(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→P∈F

V (P) =⋃

P∈FV (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.


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 ∃vϕ(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→P∈F

V (P) =⋃

P∈FV (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.


Quantum Set Theory

Collapse to a Classic Unique History World

In VSU for each history λ ∈ SU the set

Fλ = {P ∈ Op(X ) : λ(P) = 1}

is a generic filter.

This allows to define rigorously what it means that a self-adjoint operatorconverges to a real number in each history λ.

VSU SU A ∈ R(SU ) Fλ−−−−→VSU [Fλ] |= a ∈ R

In the same way this mechanism should provide a sound formulation of howthe quantum dynamical variables converge to the classical ones. Consider forexample the momentum operator, this is derived using that in classicalmechanics the quantity whose conservation in a closed system follows fromthe homogeneity of space is the momentum, then using this “axiomatic”definition it is shown that the operator of spatial derivation, up to multiplicationfor a constant factor, satisfies this definition within the context of quantumtheory.


Quantum Set Theory



Fλ = {P ∈ Op(X ) : λ(P) = 1}

is a generic filter.This allows to define rigorously what it means that a self-adjoint operatorconverges to a real number in each history λ.




Quantum Set Theory



Fλ = {P ∈ Op(X ) : λ(P) = 1}

is a generic filter.This allows to define rigorously what it means that a self-adjoint operatorconverges to a real number in each history λ.




Quantum Set Theory

Though the truths of logic and pure mathematics are objective andindependent of any contingent facts or laws of nature, our knowledge of thesetruths depends entirely on our knowledge of the laws of physics. Recentprogress in the quantum theory of computation has provided practicalinstances of this, and forces us to abandon the classical view thatcomputation, and hence mathematical proof, are purely logical notionsindependent of that of computation as a physical process. Henceforward, aproof must be regarded not as an abstract object or process but as a physicalprocess, a species of computation, whose scope and reliability depend onour knowledge of the physics of the computer concerned.(Deutsch, Ekert, Lupacchini, Machines, Logic and Quantum Physics)

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