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Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Geometric dualities and model theory
B. Zilber
University of Oxford
March 11, 2015
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
1 Dualities in logic and geometry
2 The Heisenberg formalism
3 The structure on the space of states
4 A pseudo-finite calculus on the space of states
5 Further work
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Tarskian duality
Theory T ←→ Class of models M(T )
For a κ-categorical T
Theory T ←→ Model MT (of cardinality κ)
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Tarskian duality
Theory T ←→ Class of models M(T )
For a κ-categorical T
Theory T ←→ Model MT (of cardinality κ)
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Geometric dualitiesAffine commutative C-algebra
R = C[X1, . . . ,Xn]/I
Commutative unital C∗-algebra
A
Affine k -scheme
R = k [X1, . . . ,Xn]/I
k -scheme of finite type
S
Complex algebraic variety
VR
Compact topological space
VA
The geometry of k -definablepoints, curves etc of an algebraicvariety VR
The geometry of k -definablepoints, curves etc of a “Zariskigeometry” VS
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Claim A
These are syntax – semantics dualities.
The dualities can be recast in the form of Tarskian dualities.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Claim A
These are syntax – semantics dualities.
The dualities can be recast in the form of Tarskian dualities.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
From the scheme – language – theory to the structure
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Claim A
These are syntax – semantics dualities.
The dualities can be recast in the form of Tarskian dualities.In general the syntax may come with a topology (as inC∗-algebras).Recall also the syntax of continuous model theory.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Claim A
These are syntax – semantics dualities.
The dualities can be recast in the form of Tarskian dualities.In general the syntax may come with a topology (as inC∗-algebras).
Recall also the syntax of continuous model theory.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Claim A
These are syntax – semantics dualities.
The dualities can be recast in the form of Tarskian dualities.In general the syntax may come with a topology (as inC∗-algebras).Recall also the syntax of continuous model theory.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
What is geometric semantics?
Non-example. Models of the theory of arithmetic do notprovide a semantics of geometric type.
Suggestions. The structures on the right hand side of dualitiesmust be stable (in a generalised sense).
There also may be a topology on the syntax and thence atopology on structures associated in a natural way with the thaton syntax.
This leads again to the notion of Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
What is geometric semantics?
Non-example. Models of the theory of arithmetic do notprovide a semantics of geometric type.
Suggestions. The structures on the right hand side of dualitiesmust be stable (in a generalised sense).
There also may be a topology on the syntax and thence atopology on structures associated in a natural way with the thaton syntax.
This leads again to the notion of Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
What is geometric semantics?
Non-example. Models of the theory of arithmetic do notprovide a semantics of geometric type.
Suggestions. The structures on the right hand side of dualitiesmust be stable (in a generalised sense).
There also may be a topology on the syntax and thence atopology on structures associated in a natural way with the thaton syntax.
This leads again to the notion of Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
What is geometric semantics?
Non-example. Models of the theory of arithmetic do notprovide a semantics of geometric type.
Suggestions. The structures on the right hand side of dualitiesmust be stable (in a generalised sense).
There also may be a topology on the syntax and thence atopology on structures associated in a natural way with the thaton syntax.
This leads again to the notion of Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
What is geometric semantics?
Non-example. Models of the theory of arithmetic do notprovide a semantics of geometric type.
Suggestions. The structures on the right hand side of dualitiesmust be stable (in a generalised sense).
There also may be a topology on the syntax and thence atopology on structures associated in a natural way with the thaton syntax.
This leads again to the notion of Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
L-Zariski geometries.
V is said to be a Noetherian L-Zariski if it satisfies
Closed subsets of V n are exactly those which areL-positive-quantifier-free definable.(There may be points which are not closed!)The projection of a closed set is constructible (positivequantifier-elimination).A good dimension notion on closed subsets is given. Theaddition formula and the fibre conditions hold.For every extension L(C) of the language by constants,any V′ � V as an L(C)-structure satisfies the above.
Theorem. Noetherian L-Zariski geometries are of finite Morleyrank.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
L-Zariski geometries.
V is said to be a Noetherian L-Zariski if it satisfies
Closed subsets of V n are exactly those which areL-positive-quantifier-free definable.(There may be points which are not closed!)The projection of a closed set is constructible (positivequantifier-elimination).A good dimension notion on closed subsets is given. Theaddition formula and the fibre conditions hold.For every extension L(C) of the language by constants,any V′ � V as an L(C)-structure satisfies the above.
Theorem. Noetherian L-Zariski geometries are of finite Morleyrank.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Affine k -schemes.
Given a field k and an affine k -algebra R we introduce alanguage LR and an LR-Zariski structure VR.
The language LR of two-sorted structures:sort K for an algebraically closed field K containing k withnames for elements of k ;sort V fibred as the union of all irreducible (1-dimensional)representations of R;
LR has means to describe the action of the field K and theadditive structure on fibres of V ;
LR has a name a for each a ∈ R interpreted as a linearoperator on each fibre (each vector space) of V .
The structure VR has exactly one fibre for each isomorphismtype of irreducible representations.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Illustration: sort V and a section
.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
We define natural morphisms between the LR-structures (fordifferent R).
Duality Theorem. The category of affine k-algebras for allk ⊆ K = K alg is isomorphic to the category of respectiveZariski stuctures.Points (non necessarily closed) of VR correspond to irreducibleK -representations of R.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
A noncommutative duality Theorem
The above duality can be extended to non-commutativegeometry “at roots of unity”.
AV ←→ VA.
AV – co-ordinate algebra, VA – Zariski geometry.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
A non-commutative example “at root of unity”
Non-commutative 2-torus at q = e2πi mN has
co-ordinate ring A = Aq =⟨U,V : U∗ = U−1, V ∗ = V−1, UV = qVU
⟩
Points have structure of (an orthonormal basis)of a N-dim Hilbert space.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
A non-commutative example “at root of unity”
Non-commutative 2-torus at q = e2πi mN has
co-ordinate ring A = Aq =⟨U,V : U∗ = U−1, V ∗ = V−1, UV = qVU
⟩Points have structure of (an orthonormal basis)of a N-dim Hilbert space.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Affine commutative C-algebra R
Commutative C∗-algebra A
Affine k -scheme R
k -scheme of finite type S
C∗-algebra A at roots of unity
Weyl-Heisenberg algebra〈Q,P : QP − PQ = i~〉
The integers Z
Complex algebraic variety VR
Compact topological space VA
The k -definable structure on analgebraic variety VR
The k -definable structure on aZariski geometry VS
Zariski geometry VA
? shut up and calculate!
A.Connes, 2014: Topos
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Affine commutative C-algebra R
Commutative C∗-algebra A
Affine k -scheme R
k -scheme of finite type S
C∗-algebra A at roots of unity
Weyl-Heisenberg algebra〈Q,P : QP − PQ = i~〉
The integers Z
Complex algebraic variety VR
Compact topological space VA
The k -definable structure on analgebraic variety VR
The k -definable structure on aZariski geometry VS
Zariski geometry VA
?
shut up and calculate!
A.Connes, 2014: Topos
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Affine commutative C-algebra R
Commutative C∗-algebra A
Affine k -scheme R
k -scheme of finite type S
C∗-algebra A at roots of unity
Weyl-Heisenberg algebra〈Q,P : QP − PQ = i~〉
The integers Z
Complex algebraic variety VR
Compact topological space VA
The k -definable structure on analgebraic variety VR
The k -definable structure on aZariski geometry VS
Zariski geometry VA
? shut up and calculate!
A.Connes, 2014: Topos
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Affine commutative C-algebra R
Commutative C∗-algebra A
Affine k -scheme R
k -scheme of finite type S
C∗-algebra A at roots of unity
Weyl-Heisenberg algebra〈Q,P : QP − PQ = i~〉
The integers Z
Complex algebraic variety VR
Compact topological space VA
The k -definable structure on analgebraic variety VR
The k -definable structure on aZariski geometry VS
Zariski geometry VA
? shut up and calculate!
A.Connes, 2014: Topos
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
QP − PQ = i~
“The whole of quantum mechanics is in this canonicalcommutation relation”.
An analogy:
Y = X 2 + aX + b
is (the equation of) a parabola.
H =12(P2 + ω2Q2)
is (the Hamiltonian of) a quantumharmonic oscillator.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
QP − PQ = i~
“The whole of quantum mechanics is in this canonicalcommutation relation”.
An analogy:
Y = X 2 + aX + b
is (the equation of) a parabola.
H =12(P2 + ω2Q2)
is (the Hamiltonian of) a quantumharmonic oscillator.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
QP − PQ = i~
This does not allow the C∗-algebra (Banach algebra) setting.
On suggestion of Herman Weyl and following Stone – vonNeumann Theorem replace the Weyl-Heisenberg algebra bythe category of Weyl ∗-algebras
Aa,b =⟨
Ua,V b : UaV b = eiab~V bUa⟩,
a,b ∈ R, Ua = eiaQ, V b = eibP .
where it is also assumed that Ua and V b are unitary.
We may assume that ~2π ∈ Q and so, when a,b ∈ Q the algebra
Aa,b is at root of unity. We call such algebras rational Weylalgebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
QP − PQ = i~
This does not allow the C∗-algebra (Banach algebra) setting.On suggestion of Herman Weyl and following Stone – vonNeumann Theorem replace the Weyl-Heisenberg algebra bythe category of Weyl ∗-algebras
Aa,b =⟨
Ua,V b : UaV b = eiab~V bUa⟩,
a,b ∈ R, Ua = eiaQ, V b = eibP .
where it is also assumed that Ua and V b are unitary.
We may assume that ~2π ∈ Q and so, when a,b ∈ Q the algebra
Aa,b is at root of unity. We call such algebras rational Weylalgebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
QP − PQ = i~
This does not allow the C∗-algebra (Banach algebra) setting.On suggestion of Herman Weyl and following Stone – vonNeumann Theorem replace the Weyl-Heisenberg algebra bythe category of Weyl ∗-algebras
Aa,b =⟨
Ua,V b : UaV b = eiab~V bUa⟩,
a,b ∈ R, Ua = eiaQ, V b = eibP .
where it is also assumed that Ua and V b are unitary.
We may assume that ~2π ∈ Q and so, when a,b ∈ Q the algebra
Aa,b is at root of unity. We call such algebras rational Weylalgebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Sheaf of Zariski geometries over the category ofrational Weyl algebras
The category Afin has objects Aa,b, rational Weyl algebras, andmorphisms = embeddings.
This corresponds to the surjective morphism in the dualcategory Vfin of Zariski geometries
VAa,b → VAc,d .
The duality functorA 7→ VA
can be interpreted as defining a sheaf of Zariski geometriesover the category of rational Weyl algebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Sheaf of Zariski geometries over the category ofrational Weyl algebras
The category Afin has objects Aa,b, rational Weyl algebras, andmorphisms = embeddings.This corresponds to the surjective morphism in the dualcategory Vfin of Zariski geometries
VAa,b → VAc,d .
The duality functorA 7→ VA
can be interpreted as defining a sheaf of Zariski geometriesover the category of rational Weyl algebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Sheaf of Zariski geometries over the category ofrational Weyl algebras
The category Afin has objects Aa,b, rational Weyl algebras, andmorphisms = embeddings.This corresponds to the surjective morphism in the dualcategory Vfin of Zariski geometries
VAa,b → VAc,d .
The duality functorA 7→ VA
can be interpreted as defining a sheaf of Zariski geometriesover the category of rational Weyl algebras.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Completions of Afin and Vfin.
The completion of Afin is A, the category of all Weyl algebras inthe Lie-groups topology (not the Banach algebra topology).
Completing Vfin is the main difficulty of the project.
We use structural approximation, which in basic cases isequivalent to the ultraproduct construction of continuous modeltheory.
Exercise. Use the ultraproduct of continuous model theory toconstruct the universal cover of the algebraic torus C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Completions of Afin and Vfin.
The completion of Afin is A, the category of all Weyl algebras inthe Lie-groups topology (not the Banach algebra topology).Completing Vfin is the main difficulty of the project.
We use structural approximation, which in basic cases isequivalent to the ultraproduct construction of continuous modeltheory.
Exercise. Use the ultraproduct of continuous model theory toconstruct the universal cover of the algebraic torus C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Completions of Afin and Vfin.
The completion of Afin is A, the category of all Weyl algebras inthe Lie-groups topology (not the Banach algebra topology).Completing Vfin is the main difficulty of the project.
We use structural approximation,
which in basic cases isequivalent to the ultraproduct construction of continuous modeltheory.
Exercise. Use the ultraproduct of continuous model theory toconstruct the universal cover of the algebraic torus C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Completions of Afin and Vfin.
The completion of Afin is A, the category of all Weyl algebras inthe Lie-groups topology (not the Banach algebra topology).Completing Vfin is the main difficulty of the project.
We use structural approximation, which in basic cases isequivalent to the ultraproduct construction of continuous modeltheory.
Exercise. Use the ultraproduct of continuous model theory toconstruct the universal cover of the algebraic torus C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Completions of Afin and Vfin.
The completion of Afin is A, the category of all Weyl algebras inthe Lie-groups topology (not the Banach algebra topology).Completing Vfin is the main difficulty of the project.
We use structural approximation, which in basic cases isequivalent to the ultraproduct construction of continuous modeltheory.
Exercise. Use the ultraproduct of continuous model theory toconstruct the universal cover of the algebraic torus C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
The space of states.
We construct a Vfin-projective limit object VA, corresponding tothe algebra
⋃A.
By construction
A ∈ Afin ⇒ VA � VA.
This object is closely related to the Hilbert space of quantummechanics. We call the object the space of states.
Remark. The same construction for the category ofcommutative algebras 〈Ua,U−a〉, a ∈ Q, (equivalent to thecategory of etalé covers of C×) produces the universal cover(C,+) of C× and the morphism exp : C→ C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
The space of states.
We construct a Vfin-projective limit object VA, corresponding tothe algebra
⋃A.
By construction
A ∈ Afin ⇒ VA � VA.
This object is closely related to the Hilbert space of quantummechanics. We call the object the space of states.
Remark. The same construction for the category ofcommutative algebras 〈Ua,U−a〉, a ∈ Q, (equivalent to thecategory of etalé covers of C×) produces the universal cover(C,+) of C× and the morphism exp : C→ C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
The space of states.
We construct a Vfin-projective limit object VA, corresponding tothe algebra
⋃A.
By construction
A ∈ Afin ⇒ VA � VA.
This object is closely related to the Hilbert space of quantummechanics. We call the object the space of states.
Remark. The same construction for the category ofcommutative algebras 〈Ua,U−a〉, a ∈ Q, (equivalent to thecategory of etalé covers of C×) produces the universal cover(C,+) of C× and the morphism exp : C→ C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
The space of states.
We construct a Vfin-projective limit object VA, corresponding tothe algebra
⋃A.
By construction
A ∈ Afin ⇒ VA � VA.
This object is closely related to the Hilbert space of quantummechanics. We call the object the space of states.
Remark. The same construction for the category ofcommutative algebras 〈Ua,U−a〉, a ∈ Q, (equivalent to thecategory of etalé covers of C×) produces the universal cover(C,+) of C× and the morphism exp : C→ C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
The space of states.
We construct a Vfin-projective limit object VA, corresponding tothe algebra
⋃A.
By construction
A ∈ Afin ⇒ VA � VA.
This object is closely related to the Hilbert space of quantummechanics. We call the object the space of states.
Remark. The same construction for the category ofcommutative algebras 〈Ua,U−a〉, a ∈ Q, (equivalent to thecategory of etalé covers of C×) produces the universal cover(C,+) of C× and the morphism exp : C→ C×.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
How noncommutative VA deforms into VA.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Operators P, Q on VA.
We define in each member of the ultraproduct
Q :=Ua − U−a
2ia, P :=
V b − V−b
2ibin accordance with
Ua = eiaQ, V b = eibP.
Then in the limit, we can calculate in VA, for any state e
(QP− PQ)e = i~e.
So, in the space of states:
QP− PQ = i~I.
In other words, there is a Lie algebra 〈Q,P〉 acting on VA (theHeisenberg algebra).
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Operators P, Q on VA.
We define in each member of the ultraproduct
Q :=Ua − U−a
2ia, P :=
V b − V−b
2ibin accordance with
Ua = eiaQ, V b = eibP.
Then in the limit, we can calculate in VA, for any state e
(QP− PQ)e = i~e.
So, in the space of states:
QP− PQ = i~I.
In other words, there is a Lie algebra 〈Q,P〉 acting on VA (theHeisenberg algebra).
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Time evolution operators on the space of states
Theorem. Automorphisms of Afin give rise to certain operatorson VA. These are definable in the sense of continuous modeltheory.
Such operators K (= Kparticle) are typically the “time evolutionoperators for a given particle”.
The one-parameter subgroups {K t : t ∈ R} describe the timeevolution of the particle corresponding to K .
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Time evolution operators on the space of states
Theorem. Automorphisms of Afin give rise to certain operatorson VA. These are definable in the sense of continuous modeltheory.
Such operators K (= Kparticle) are typically the “time evolutionoperators for a given particle”.
The one-parameter subgroups {K t : t ∈ R} describe the timeevolution of the particle corresponding to K .
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Restriction to a commutative algebra case
Theorem. A “time evolution operator” on (C,+) (as the cover ofC×) has the form
z 7→ κz, some κ ∈ R>0.
This corresponds to ’raising to power’ κ on C×.
Recall: the theory of raising to real power is superstable,provided Schanuel’s conjecture is true.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Restriction to a commutative algebra case
Theorem. A “time evolution operator” on (C,+) (as the cover ofC×) has the form
z 7→ κz, some κ ∈ R>0.
This corresponds to ’raising to power’ κ on C×.
Recall: the theory of raising to real power is superstable,provided Schanuel’s conjecture is true.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Restriction to a commutative algebra case
Theorem. A “time evolution operator” on (C,+) (as the cover ofC×) has the form
z 7→ κz, some κ ∈ R>0.
This corresponds to ’raising to power’ κ on C×.
Recall: the theory of raising to real power is superstable,provided Schanuel’s conjecture is true.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
The Hamiltonian:H =
12(P2 + Q2)
The time evolution operator :
K t = K tH := e−i H
~ t , t ∈ R.
This “induces” the automorphism of the category of algebras
Ua 7→ e−2πa2 sin t cos t
2 Ua sin tV a cos t
V a 7→ e2πa2 sin t cos t
2 U−a cos tV a sin t
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
The Hamiltonian:H =
12(P2 + Q2)
The time evolution operator :
K t = K tH := e−i H
~ t , t ∈ R.
This “induces” the automorphism of the category of algebras
Ua 7→ e−2πa2 sin t cos t
2 Ua sin tV a cos t
V a 7→ e2πa2 sin t cos t
2 U−a cos tV a sin t
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
The Hamiltonian:H =
12(P2 + Q2)
The time evolution operator :
K t = K tH := e−i H
~ t , t ∈ R.
This “induces” the automorphism of the category of algebras
Ua 7→ e−2πa2 sin t cos t
2 Ua sin tV a cos t
V a 7→ e2πa2 sin t cos t
2 U−a cos tV a sin t
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Scheme of calculations
rewrite the formula over VA in terms of Zariski-regularpseudo-finite sums and products over VA, A ∈ Afin;
calculate uniformly in VA (using e.g. the Gauss quadraticsums formula)apply approximation limit to the result and get the result interms of the standard reals.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
K t = K tH := e−i H
~ t , t ∈ R.
To calculate K t we approximate assuming sin t , cos t ∈ Q. Thistransfers us to the rational category Vfin and to the calculationsin (pseudo)finite-dimensional spaces.Then the matrix element on row x1 and column x2 ( kernel ofthe Feynman propagator) is calculated as
〈x1|K tx2〉 =√
12πi~ sin t
exp i(x2
1 + x22 ) cos t − 2x1x2
2~ sin t.
The trace of K t ,
Tr(K t) =
∫R〈x |K tx〉 = 1
sin t2.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
K t = K tH := e−i H
~ t , t ∈ R.
To calculate K t we approximate assuming sin t , cos t ∈ Q. Thistransfers us to the rational category Vfin and to the calculationsin (pseudo)finite-dimensional spaces.
Then the matrix element on row x1 and column x2 ( kernel ofthe Feynman propagator) is calculated as
〈x1|K tx2〉 =√
12πi~ sin t
exp i(x2
1 + x22 ) cos t − 2x1x2
2~ sin t.
The trace of K t ,
Tr(K t) =
∫R〈x |K tx〉 = 1
sin t2.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Example. Quantum harmonic oscillator.
K t = K tH := e−i H
~ t , t ∈ R.
To calculate K t we approximate assuming sin t , cos t ∈ Q. Thistransfers us to the rational category Vfin and to the calculationsin (pseudo)finite-dimensional spaces.Then the matrix element on row x1 and column x2 ( kernel ofthe Feynman propagator) is calculated as
〈x1|K tx2〉 =√
12πi~ sin t
exp i(x2
1 + x22 ) cos t − 2x1x2
2~ sin t.
The trace of K t ,
Tr(K t) =
∫R〈x |K tx〉 = 1
sin t2.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Tr(K t) =
∫R〈x |K tx〉 = 1
sin t2.
Note that in terms of conventional mathematical physics wehave calculated
Tr(K t) =∞∑
n=0
e−it(n+ 12 ),
a non-convergent infinite sum.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Tr(K t) =
∫R〈x |K tx〉 = 1
sin t2.
Note that in terms of conventional mathematical physics wehave calculated
Tr(K t) =∞∑
n=0
e−it(n+ 12 ),
a non-convergent infinite sum.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Scheme of calculations
rewrite the formula over VA in terms of Zariski-regularpseudo-finite sums and products over VA, A ∈ Afin;
calculate uniformly in VA
apply the approximation limit to the result and get the resultin terms of the standard reals.
B. Zilber University of Oxford
Geometric dualities and model theory
Dualities in logic and geometry The Heisenberg formalism The structure on the space of states A pseudo-finite calculus on the space of states Further work
Which formulas the scheme of calculations isapplicable to?
In a precise form this is equivalent to the Feynman path integralhypothesis (for quantum mechanics).
B. Zilber University of Oxford
Geometric dualities and model theory
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