Geometric dualities and model theoryGeometric dualities and model theory Dualities in logic and...

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

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

Tarskian duality

Theory T ←→ Class of models M(T )

For a κ-categorical T

Theory T ←→ Model MT (of cardinality κ)

Tarskian duality

Theory T ←→ Class of models M(T )

For a κ-categorical T

Theory T ←→ Model MT (of cardinality κ)

Geometric dualitiesAffine commutative C-algebra

R = C[X1, . . . ,Xn]/I

Commutative unital C∗-algebra

Affine k -scheme

R = k [X1, . . . ,Xn]/I

k -scheme of finite type

Complex algebraic variety

Compact topological space

The geometry of k -definablepoints, curves etc of an algebraicvariety VR

The geometry of k -definablepoints, curves etc of a “Zariskigeometry” VS

Claim A

These are syntax – semantics dualities.

The dualities can be recast in the form of Tarskian dualities.

Claim A

The dualities can be recast in the form of Tarskian dualities.

From the scheme – language – theory to the structure

Claim A

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.

Claim A

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.

Claim A

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.

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.

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.

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.

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.

Illustration: sort V and a section

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.

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.

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.

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.

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

Affine k -scheme R

The integers Z

Zariski geometry VA

shut up and calculate!

Affine k -scheme R

The integers Z

Zariski geometry VA

Affine k -scheme R

The integers Z

Zariski geometry VA

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.

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.

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.

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 =⟨

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 =⟨

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.

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 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 .

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×.

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.

The space of states.

We construct a Vfin-projective limit object VA, corresponding tothe algebra

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×.

By construction

How noncommutative VA deforms into VA.

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).

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).

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 .

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 .

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.

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

The Hamiltonian:H =

12(P2 + Q2)

~ t , t ∈ R.

2 Ua sin tV a cos t

The Hamiltonian:H =

12(P2 + Q2)

~ t , t ∈ R.

2 Ua sin tV a cos t

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.

~ 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.

~ 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.

~ 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.

Tr(K t) =

∫R〈x |K tx〉 = 1

sin t2.

Note that in terms of conventional mathematical physics wehave calculated

Tr(K t) =∞∑

e−it(n+ 12 ),

a non-convergent infinite sum.

Tr(K t) =

∫R〈x |K tx〉 = 1

sin t2.

Note that in terms of conventional mathematical physics wehave calculated

Tr(K t) =∞∑

e−it(n+ 12 ),

a non-convergent infinite sum.

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.

Which formulas the scheme of calculations isapplicable to?

In a precise form this is equivalent to the Feynman path integralhypothesis (for quantum mechanics).

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