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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)