Noncommutative Algebraic Geometry, Topology, and Physics · Mathematical models in physics, an...

Noncommutative Algebraic Geometry, Topology, andPhysics

Olav Arnfinn Laudal

November 1, 2016

Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 1 / 141

The relationship between algebraic geometry, topology, and physics, is welldocumented, and the field is very popular. I shall, in my talk (do my bestto) introduce an extension of the methods used up to now, to include myversion of non-commutative algebraic geometry.The interest would be, to explain the relationship between notions likeGhost fields, Chern-Simons classes, Maxwell and Bloch’s equations, forelectromagnetism, resp. time development of spin structures andentanglement, and Seiberg-Witten’s monopole equation. I shall, maybe,just vaguely touch upon the beautiful results in low-dimension algebraictopology, based upon these methods, due to Donaldson and his followers.


CONTENT OF TALK

• Mathematical models in physics, an Introduction

• Time and Dynamics

• Gauge Groups and Measurements

• Geometry of Representations

• Dynamics and the Dirac Derivation


• I Phase Spaces of Associative Algebras

• The Kodaira-Spencer class

• Hamiltonians and Connections

• Chern-Simons classes and Chern characters

• The iterated Phase Space functor as a co-simplicial resolution

• Relations to the de Rham complex

• Dynamics and the Dirac Derivation

• Preparations and Time-Evolution of Representations.

• Finite dimensional representations of Ph∞(A).


• II The generic dynamical structure, C (σg ), of a polynomial k-algebraC , induced by a metric g

• The Dirac Derivation in C (σg )

• Force Laws in C (σg ), General Relativity

• Classical Connections

• Chern Characters and Chern-Simons Classes


• III Quotients in Geometry

• Quotients in non-commutative algebraic geometry

• Gauge Groups and quotients in physics

• The classical case


• IV Local Gauge Groups and their Actions

• Lie-Cartan Pairs and Lie Algebroids

• The role of the Cartan subalgebra of a local Gauge Group in physics


• VI-VII Deformations of Associative Algebras

• The case of U, a singular point with a 3-dimensional tangent space

• The Versal family, U of U, the Toy Model, H

• Metric and Time, a Big Bang Model, Kepler and Newton


• VIII The Universal Local Gauge Group of the Standard Model

• Spin and Isospin

• Local Gauge Groups and their Actions

• Photones and Quarks


• IX-X-XI Elementary Particles, Weyl, Pauli and Dirac spinors, Chirality

• The Generic Time-Action

• Maxwell-, Bloch’s and Seiberg-Witten’s Equations.

• The Generalized Dirac Equation


• XII The Standard Model


0 Mathematical Models in Physics, an Introduction0 No 1

If we want to study a natural phenomenon, called P, we must in thepresent scientific situation, describe P in some mathematical terms, say asa mathematical object, X, depending upon some parameters, in such away that the changing aspects of P would correspond to alteredparameter-values for X. This object would be a model for P if, moreover,X with any choice of parameter-values, would correspond to some, possiblyoccurring, aspect of P.Two mathematical objects X(1), and X(2), corresponding to the sameaspect of P, would be called equivalent, and the set, P, of equivalenceclasses of the objects P, would correspond to (possibly a quotient of) themoduli space, M, of the models, X. The study of the natural phenomenaP, and its changing aspects, would then be equivalent to the study of thestructure of P, and therefore to the study of the geometry of the modulispace M.


Time and Dynamics0 No 2

In particular, the notion of time would, in agreement with Aristotle and St.Augustin, correspond to some metric on this space.It turns out that to obtain a complete theoretical framework for studyingthe phenomenon P, or the model X, together with its dynamics, we shouldintroduce the notion of dynamical structure, defined for the space, M.This is done via the construction of a universal non-commutative PhaseSpace-functor, Ph(−) : Algk → Algk . It extends to the category ofschemes, and its infinite iteration Ph∞(−), is outfitted with a universalDirac derivation, δ ∈ Derk(Ph∞(−),Ph∞(−)).A dynamical structure defined on an associative k-algebra A ∈ Algk is nowa δ-stable ideal σ ⊂ Ph∞(A), and its quotient A(σ) := Ph∞(A)/(σ), withits Dirac derivation. The structure we are interested in, is the spaceU := Ph∞(M)/σ, corresponding to an open affine covering by algebras ofthe type, A(σ).


Gauge Groups and Measurements0 No 3

But now we observe that there may be an action of a Lie algebra g, on U,such that the dynamics of P, really corresponds to that of the quotientU/g0.To any open subset U, of U, there would be associated a, not necessarilycommutative, affine k-algebra, A(σ) := OU(U), with an action of the Liealgebra g0, such that the non-commutative quotient U/g0, represented bythe system A(σ)/g0 of simple representations of the algebra, A(σ), with ag-connection. This system contains all the available information about thestructure of U, restricted to U. An element of A(σ) would be called alocal observable, and wishing to measure the value of an observable, leadsto the study of the eigenvectors, and their eigenvalues, of the g0-invariantrepresentations of this algebra, which as we shall see, is the same as therepresentations of the system A(σ)/g0 .


Geometry of Representations0 No 4

We may show that any (geometric:=finitely generated +++) k-algebra A,may be recovered from its finite dimensional simple representations, andthat there is an underlying quasi-affine (commutative) scheme-structure oneach component Simpn(A), parametrizing the simple representations ofdimension n. In fact, we have shown the following,There is a commutative k-algebra C (n) with an open subvarietyU(n) ⊆ Simp1(C (n)), an etale covering of Simpn(A), over which thereexists a versal representation V ' C (n)⊗k V , a vector bundle of rank ndefined on Simp1(C (n)), and a versal family, i.e. a morphism of algebras,

ρ : A −→ EndC(n)(V )→ EndU(n)(V ),

inducing all isoclasses of simple n-dimensional A-modules.


Dynamics and the Dirac Derivation.0 No 5

EndC(n)(V ) induces also a bundle, of operators, on the etale coveringU(n) of Simpn(A). Assume given a universal derivation, δ ∈ Derk(A).Pick any v ∈ Simpn(A) corresponding to the right A-module V , withstructure homomorphism ρv : A→ Endk(V ), then δ composed with ρv ,gives us an element,

δv ∈ Ext1A(V ,V )).

Therefore, δ defines a unique one-dimensional distribution in ΘSimpn(A),which, once we have fixed a versal family, defines a vector field,[δ] ∈ ΘSimpn(A), and in good cases, a (rational) derivation, the

Dirac Derivation : [δ] ∈ Derk(C (n)),

governing the dynamics of our model.Problem: How do we find this universal derivation?


I Phase spaces of associative algebrasI No 1

Given an associative k-algebra A, denote by A/k − alg the category wherethe objects are homomorphisms of k-algebras κ : A→ R, and themorphisms, ψ : κ→ κ′ are commutative diagrams,

Aκ

κ′

@@@

@@@@

Rψ

// R ′

and consider the functor,

Derk(A,−) : A/k − alg −→ Sets.

It is representable by a k-algebra-morphism, ι : A −→ Ph(A), with auniversal family, i.e. a universal derivation,

d : A −→ Ph(A).


A universal derivation associated to an A-moduleI No 2

Clearly we have the identities,

d∗ : Derk(A,A) = MorA(Ph(A),A),

and,d∗ : Derk(A,Ph(A)) = EndA(Ph(A)),

the last one associating d to the identity endomorphism of Ph. Let now Vbe a right A-module, with structure morphism

ρ : A→ Endk(V ).

We obtain another universal derivation,

c : A −→ Homk(V ,V ⊗A Ph(A)),

defined by, c(a)(v) = v ⊗ d(a).Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 18 / 141

The Kodaira-Spencer classI No 3

Using the long exact sequence, of Hochschild cohomology,

0→HomA(V ,V ⊗A Ph(A))→ Homk(V ,V ⊗A Ph(A))ι→ Derk(A,Homk(V ,V ⊗A Ph(A)))

κ→ Ext1A(V ,V ⊗A Ph(A))→ 0,

and,c ∈ Derk(A,Homk(V ,V ⊗A Ph(A))),

we obtain the non-commutative Kodaira-Spencer class,

c(V ) := κ(c) ∈ Ext1A(V ,V ⊗A Ph(A)),

inducing, via the identity d∗, the Kodaira-Spencer morphism,

g : ΘA := Derk(A,A) −→ Ext1A(V ,V ).


Hamiltonians and ConnectionsI No 4

Using again the long exact sequence, of Hochschild cohomology,

0→HomA(V ,V )→ Homk(V ,V )ι→ Derk(A,Homk(V ,V ))

κ→ Ext1A(V ,V )→ 0,

we prove,

Theorem

Let ρ : A→ Endk(V ), be an A-module, and let δ ∈ Derk(A,Homk(V ,V )),map to 0 in Ext1

A(V ,V ), i.e. assume κ(δ) = 0, then there exist anelement, Qδ ∈ Homk(V ,V ), the Hamiltonian, such that for all a ∈ A,

ρ(δ(a)) = [Qδ, ρ(a)].

If V is a simple A-module, ad(Qδ) is unique.


Chern CharactersI No 5

As is well known, in the commutative case, the Kodaira-Spencer classgives rise to a Chern character by putting,

chi (V ) := 1/i ! c i (V ) ∈ Ext iA(V ,V ⊗A Ph(A)),

and if c(V ) = 0, the curvature R(V ) of ∇, induces a curvature class,

R∇ ∈ H2(k ,A; ΘA,EndA(V )).


The iterated Phase Space functor, Ph∗

I No 7

The phase-space construction may, be iterated. Given the k-algebra A wemay form the sequence, Phn(A)0≤n, defined inductively by

Ph0(A) = A, Ph1(A) = Ph(A), ...,Phn+1(A) := Ph(Phn(A)).

Let in0 : Phn(A)→ Phn+1(A) be the canonical imbedding, and letdn : Phn(A)→ Phn+1(A) be the corresponding derivation. Since thecomposition of in0 and the derivation dn+1 is a derivationPhn(A)→ Phn+2(A), corresponding to the homomorphism,

Phn(A)→in0 Phn+1(A)→in+10 Phn+2(A)

there exist by universality a homomorphism in+11 : Phn+1(A)→ Phn+2(A),

such that,in0 in+1

1 = in0 in+10

and such that,dn in+1

1 = in0 dn+1.

Notice that we here compose functions and functors from left to right.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 22 / 141

The Semi-Cosimplicial Structure of Ph*I No 8

Clearly we may continue this process constructing new homomorphisms,

inj : Phn(A)→ Phn+1(A)0≤j≤n,

such that,inp in+1

0 = in0 in+1p+1

with the property,dn in+1

j+1 = inj dn+1.

We find the following identities,

inp in+1q = inq−1in+1

p , p < q

inp in+1p = inp in+1

p+1

inp in+1q = inq in+1

p+1 , q < p.


The Semi-Cosimplicial Structure of Ph*I No 9

To see this, compose with in−10 and dn−1, and use induction. Thus, the

Ph∗(A) is a semi-co-simplicial k-algebra with a co-section h0, onto A. Andit is easy to see that h0 together with the corresponding co-sectionshp : Php+1(A)→ Php(A), for Php(A) replacing A, form a trivialisinghomotopy for Ph∗(A). Thus, we have,

Hn(Ph∗(A)) = 0, n ≥ 0,

i.e.• Ph∗(A) is a cosimplicial resolution of the algebra A.

Therefore, for any object,

κ : A→ R ∈ A/k − alg

the co-simplicial algebra above induces simplicial sets,

Mork(Ph?(A),R), MorA(Ph?(A),R),

and one should be interested in the homotopy.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 24 / 141

HomotopyI No 10

See also that this generalizes to a canonical functor,

Spec : (k − alg ∆)op −→ SPr(k)

where (k − alg)∆ is the category of co-simplicial k-algebras, and SPr(k) isthe category of simplicial presheves on the category of k-schemes enrichedby some Grothendieck topology.As usual, the imbedding of the category of k-algebras in the category ofco-simplicial algebras is defined simply by giving any k-algebra a constantco-simplicial structure. The fact that Ph?(A) is a resolution of A, istherefore simply saying that,

Spec(Ph?((A))→ Spec(A),

is a week equivalence in SPr(k).This might be a starting point for a theory of homotopy for(non-commutative) k-schemes.


CohomologyI No 11

We may also consider, for any k-algebra R, the simplicial k-vectorspace,

Derk(Ph?(A),R),

Consider this complex for R = A, i.e. MorA(Ph?(A),A). Clearly,MorA(Phn+1(A),A) = Derk(Phn(A),A), n ≥ 0, and we have,

MorA(Phn(A),A) = ξ0ξi1...ξir |0 ≤ il ≤ il+1 ≤ n|ξ0 = idA, ξi ∈ Derk(A), i ≥ 1Since Ph is a functor, and Ph∗+1 is a co-simplicial resolution of A. we mayapply this to any scheme X , given in terms of an affine covering U, andobtain an algebraic homology (or cohomology), with converging spectralsequences,

E 1pq = Hp(HU

−q(Derk(Ph?(A),A))),Eq,p = H−qU (Hp(Derk(Ph?(A),A)))

If we, in MorA(Phn(A),A), identify ξ ∼ αξ, α ∈ k∗, we obtain a rationalcohomology with converging spectral sequences,

Epq1 = Hp(Hq

U(MorA(Phn(A),A),Q)),Eq,p2 = Hq

U(Hp(MorA(Phn(A),A),Q))Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 26 / 141

Relation to the de Rham complex II No 12

Consider now the diagram,

Ai00 // Ph(A)

i1p // Ph2(A)

i2p // Ph3(A)

i3p //

m11

OO

i1p // m1

2

OO

i2p // m1

3

OO

i3p //

where, for each integer n, the symbol inp , for p = 0, 1, ..., n signify thefamily of A-morphisms between Phn(A) and Phn+1(A) defined above, andwhere m1

n is the ideal of Phn(A) generated by im(d), which is the same asthe ideal generated by the family, in−1

p (in−2p (...(i1

p (d(A))...)), for allpossible p. And, inductively, let mm

n be the ideal generated by m1nm

m−1n .


Relation to the de Rham complex III No 13

We find an extended diagram,

A

i00 // Ph(A)

i1p // Ph2(A)

i2p // Ph3(A)

i3p // ...

Ad

""EEE

EEEE

EEi00 // A

d

%%KKKKKKKKKKi1p // A

d

%%KKKKKKKKKKi2p // A

d

##FFF

FFFF

FFFF

i3p // ...

m11/m

21

d

$$IIIIIIIII

i1p // m1

2/m22

d

$$JJJJJJJJJ

i2p // m1

3/m23

d

""EEE

EEEE

EEE

i3p // ...

m21/m

31

i1p // m2

2/m32

i2p // m2

3/m31

i3p // ...

The diagonals are not necessarily complexes, but to kill all dn, n ≥ 2, itsuffices to kill d2, and for this it suffices to kill d1d0, as one easily see,applying the edge homomorphisms to, d1(d0(a)) for all a ∈ A.


CurvatureI No 14

Definition

The curvature R(A) of the associative k algebra, A, is the k-linear mapcomposition of d0 and d1,

R(A) = d0d1 : A→ m22/m

32.

Now, kill the curvature R(A), and all the terms under the first diagonal,beginning with m2

1/m31, together with all terms generated by the actions of

the edge homomorphisms on these terms, and let, Ωmn be the quotient of

mmn /m

m+1n , for n ≥ 0. Clearly, Ω0

n = A for all n ≥ 0, and we have got agraded semi co-simplicial A-module, with a k-differential d , such thatd2 = 0,


Generalized de Rham complex.I No 15

The diagram is now looking like,

A

i00 // Ph(A)

i1p // Ph2(A)

i2p // Ph3(A)

i3p //

Ad

""DDD

DDDD

DDi00 // A

d

%%JJJJJJJJJJJi1p // A

d

%%KKKKKKKKKKKKi2p // A

d

!!DDD

DDDD

DDD

i3p //

Ω11

d

$$IIIIIIIIIII

i1p // Ω1

2

d

$$JJJJJJJJJJJ

i2p // Ω1

3

d

!!BBB

BBBB

BBB

i3p //

Ω22

i2p // Ω2

3

i3p //

It is therefore a graded complex, in two ways. First as a complex inducedfrom the semi-cosimplicial structure, with differential of bidegree (1,0),and second, as complex with differential d , of bidegree (1,1).


The commutative caseI No 16

Consider now the complex,

A→d Ω11 →d Ω2

2 →d Ω33 →d ...

.

Theorem

Suppose A is commutative, then there is a natural morphism of complexesof A-modules,

Ω?A ⊂ Ω?

?,

with,Ωn

A ' Ωnn.


The proofI No 17

Let, ai ∈ A, i = 1, ..., r , and compute in Ωr? the value of, d r (a1a2...ar ). It

is clear that this gives the formula,∑di1(a1)di2(a2)..dir (ar ) = 0,

the sum being over all permutation (i1, i2, ..., ir ) of (0, 1, ..., r − 1). Herewe consider A as a subalgebra of Phn(A) via the unique compositions ofthe i s0 : Phs(A) ⊂ Phs+1(A). In particular, we have,

d0(a1)d1(a2) + d1(a1)d0(a2) = 0,

for all a1, a2 ∈ A. This relation and the relation d0(a)d1(b) = d1(b)d0(a),which follows from commutativity, d(a)b = bd(a), forcing the left andright A-action on ΩA to be equal, immediately give us,d0(a)d1(b) = −d0(b), d1(a). It is now clear that the map that sends theelement da1 ∧ da2 ∧ ... ∧ dar ∈ Ωr

A to d0(a1)d1(a2)..dr−1(ar ) ∈ Ωrr is an

isomorphism, and the rest should be clear.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 32 / 141

Generalization to modules II No 18

Let now, V be an A-module, and assume c(V ) = 0, and pick aconnection, ∇ ∈ Homk(V ,V ⊗A Ph(A)) with c = ι(∇). This imply thatfor a ∈ A and v ∈ V we have ∇(va) = ∇(v)a + v ⊗ d0(a). Composing ∇with the cosection, o : Ph(A)→ A, corresponding to the 0-derivation of A,we therefore obtain an A-linear homomorphism P : V → V , a potential.Since i0

0 : A→ Ph(A) is a section of o, we find a k-linear map,

∇0 := ∇− P : V → V ⊗m11

Using the property,dn in+1

j+1 = inj dn+1,

it is easy to find well defined k-linear maps,

∇1 : V → V ⊗ Ω22, ∇2 : V → V ⊗ Ω3

3, ...,∇n : V → V ⊗ Ωn+1n+1 ∀n ≥ 0,

given by ,∇n+1 := ∇n in+1

1 , n ≥ 0.


Generalization to modulesI No 19

Fix the connection ∇. For all, v ∈ V , ω ∈ Ωnn, the formula,

∇n(v ⊗ ω) = ∇n(v)ω + v ⊗ dn(ω).

makes sense, and defines derivations, also called d . We obtain a situationjust like above,

V

i00// V ⊗ Ph(A)

i1p // V ⊗ Ph2(A)

i2p // V ⊗ Ph3(A)

i3p //

Vd

$$JJJJJJJJJJi00 // V

d

''PPPPPPPPPPPPPi1p // V

d

((PPPPPPPPPPPPPi2p // V

d

$$IIIIIIIIIIII

i3p //

V ⊗ Ω11

d

''OOOOOOOOOOO

i1p // V ⊗ Ω1

2

d

''OOOOOOOOOOO

i2p // V ⊗ Ω1

3

d

##HHHHHHHHHHH

i3p //

V ⊗ Ω22

i2p // V ⊗ Ω2

3

i3p //


Generalization to modulesI No 20

In general, there are no reasons for these d ′s to define complexes, and weshall make the following definition,

Definition

The curvature R(V ,∇) of the connection ∇ defined on the right kA-module V , is the k-linear map composition of d0 and d1,

R(V ,∇) = d0d1 : V → V ⊗ Ω22.

The following result is then easily proved,

Theorem

Suppose A is commutative, and let ∇ : ΘA → Endk(V ) be the connectioncorresponding to ∇0. Suppose moreover that the curvature R of ∇ is 0,then R(V ) = 0, implying that d2 = 0, and so the diagonals in the diagramabove, are all complexes.


Dynamics and the Dirac DerivationI No 21

Consider now the co-simplicial algebra,

A→i00 Ph(A)→i1

p Ph2(A)→i2p Ph3(A)→i3

p ....

where, for each integer n, the symbol inp , for p = 0, 1, ..., n signify thefamily of A-morphisms between Phn(A) and Phn+1(A) defined above. Theinductive (direct) limit,

Ph∞(A) = lim−→n≥0

Phn(A), inj

comes with homomorphisms

in : Phn(A) −→ Ph∞(A), satisfyinginj in+1 = in, j = 0, 1, . . . , n

Moreover, the family of derivations, dn0≤n defines a uniqueDirac-derivation,

δ : Ph∞(A) −→ Ph∞(A),

such that in δ = dn in+1.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 36 / 141

Dynamics in Rep(A)I No 25

Since Ext1A(V ,V ) is the tangent space of the miniversal deformation space

of V as an A-module, we see that the non-commutative space Ph(A) alsoparametrizes the set of generalised momenta, i.e.the set of pairs of an A-module V , and a tangent vector of the formal moduli of V, at that point.Therefore the above implies that any representation,ρ : Ph∞(A)→ Endk(V ), corresponds to a family ofPhn(A)-module-structures on V , for n ≥ 1, i.e. to an A-moduleρ0 : (V0 := V ), an element η0 ∈ Ext1

A(V ,V ), i.e. a tangent of thedeformation functor of V0 := V , as A-module, an elementη1 ∈ Ext1

Ph(A)(V ,V ), i.e. a tangent of the deformation functor of V1 := V

as Ph(A)-module, an element η2 ∈ Ext1Ph2(A)(V ,V ), i.e. a tangent of the

deformation functor of V2 := V as Ph2(A)-module, etc.


Dynamics in Rep(A)I No 26

All this is just ρ0 : A→ Endk(V ), considered as an A-module, togetherwith a sequence ηn, 0 ≤ n, of a tangent, or a momentum, η0, anacceleration vector, η1, and any number of higher order momenta ηn.

• Thus, specifying a Ph∞(A)-representation V , implies specifying aformal curve through ρ0(v0), the base-point, of the miniversaldeformation space of the A-module V . Formally, this curve is givenby the composition of the homomorphism ε(τ) := exp(τδ) and ρ.


Preparation and Time Evolution of MeasurementsI No 27

It is, however, impossible to prepare a physical situation such that ameasurement, i.e. an object like ρ0, is given by an infinite sequence ηn,of dynamical data. We shall have to be satisfied with a finite number ofdata, and normally with just the first one, i.e. the momentum η0. This isthe Problem of Preparation and (of the) Time Evolution of arepresentation ρ, to be treated in the sequel.


Finite Dimensional Representations of Ph∞(k[t])I No 28

Theorem

Given an r-dimensional k[t] := k[t1, ..., td ]-module, consisting of r pointsPp = (α0

1(p), α02(p), ..., α0

d(p))p=1,...,r . Assume given,αn

i (p) ∈ k , i = 1, ..., d , n ≥ 0, p = 1, ..., r , and arbitrary couplingconstants, σm(p, q) ∈ k with σ0 = 0. Putαn

i (p, q) = αni (p)− αn

i (q), i = 1, ..., d, and assume, for all n ≥ 1,

∑h

(n

h

)σn−h(p, q)(αh

i (p, q)α0j (p, q)− α0

i (p, q)αhj (p, q))

=∑

k,l ,m,s

n!σn−k−m(p, s)σk−l(s, q)

l!m!(k − l)!(n − k −m)!(αm

j (p, s)αli (s, q)− αl

i (p, s)αmj (s, q)),


Representations of Ph∞(k[t])I No 29

Consider the matrix,

Dni :=

αn

i (1) rni (1, 2) ... rn

i (1, r)rni (2, 1) αn

i (2) ... rni (2, r)

. . ... .rni (r , 1) rn

i (r , 2) ... αni (r))

with the relations,

a : r 0i (p, q) = 0, rn

i (p, q) =n∑

l=0

(n

l

)αl

i (p, q)σn−l(p, q),

Then ρ(dnti ) = Dni define a representation,

ρ : Ph∞(k[t]))→ Mr (k)


ProofI No 30

Let us, as above, consider the matrix,

Xi = ρ(exp(τδ))(ti ) =∑n≥0

τn/n!Dni

Putting

αi (p) =∞∑

n=0

τn/n! αni (p), αi (p, q) =

∞∑n=0

τn/n! αni (p, q),

and, σ(p, q) =∑∞

n=0 τn/n! σn(p, q), we find the explicite formulas,

Xi =

αi (1) σ(1, 2)αi (1, 2) ... σ(1, r)αi (1, r)

σ(2, 1)αi (2, 1) αi (2) ... σ(2, r)αi (2, r)... ... ... ...

σ(r , 1)αi (r , 1) σ(r , 2)αi (r , 2) ... αi (r)

, i = 1, ..., d .

Now, compute, and put, [Xi ,Xj ] = 0 and see that the condition of thetheorem emerges.


Formal Moduli of Finite Representations of Ph∞(k[t]))I No 31

We may consider the space

A(r) = k[αni (p), σn(p, q)]/a,

with coordinates αni (p), σn(p, q), i = 1, ..., d , n > 0, p, q = 1, ..., r, and

where the ideal a is generated by the equations above, as the versal basespace for the versal family of the non-commutative deformation theoryapplied to the family of Ph∞(k[t])) modules defined by the object P.In dimensions d = 3, r = 3, and order 1, the condition above reads:∀p, s, q = 1, 2, 3

σ1(p, q)(α1(p, q)× α0(p, q)) = −σ1(p, s)σ1(s, q)(α0(p, s)× α0(s, q)).

This says that for any two of the three points in space, the relativemomentum must sit in the plane defined by the three points, the lengthbeing determined by the 3 coupling constants. Moreover, the obvious sumof all three relative momenta must be 0.


II The generic dynamical structure induced by a metricII No 1

LetC := k[t1, ..., tn]

Then,

Ph(C ) = k < t1, ..., tn, dt1, ..., dtn > /([ti , tj ], [dti , tj ] + [ti , dtj ]).

A non-degenerate metric, g = 1/2∑d

i=1 gi ,jdtidtj ∈ Ph(C ) induces anisomorphism of C -modules

ΘC = HomC (ΩC ,C ) ' ΩC .

Consider the bilateral ideal (σg ) of Ph(C ) generated by

(σg ) = ([dti , tj ]− g i ,j),

and put,C (σg ) := Ph(C )/(σg ).


The Dirac Derivation in C (σg )II No 2

Let, moreover,

T : =∑

j

Tj = −1/2∑i ,j ,l

δtl (gi ,j)g l ,idtj (1)

= −1/2(∑k,l

Γkk,ldtl +

∑k,p,q

gk,qΓpk,qgp,ldtl), (2)

and consider the inner derivation of C (σg ), defined by,

δ := ad(g − T ).

After a dull computation, we obtain, in C (σg ),

δ(ti ) = dti , i = 1, ..., d .

Therefore, by universality, we have a well-defined dynamical structure, i.e.a δ stable ideal (σg ) ⊂ Ph∞(C ), with Dirac derivation, δ = ad(g − T ). Itis also easy to see that (σg ) is invariant w.r.t. isometries.


Force Laws in C (σg )II No 3

There are in Ph(C ) force laws, one is given by,

(∗) d2ti =−∑p,q

Γip,qdtpdtq − 1/2

∑p,q

gp,q(Fi ,pdtq + dtpFi ,q)

+ 1/2∑l ,p,q

gp,q[dtp, (Γi ,ql − Γq,i

l )]dtl + [dti ,T ]

=−∑p,q

Γip,qdtpdtq −

∑p,q

gp,qFi ,pdtq + 1/2∑l ,p,q

gp,q[Fi ,q, dtp]

+ 1/2∑l ,p,q


l )]dtl + [dti ,T ]

and consistent with the Dirac derivation in C (σg ), as well as with theclassical equation for geodesics in C (σ0), where we haveσ0 =< [dti , tj ], [dti , dtj ]] >, so that C (σ0) = Ph(C )com, thecommutative classical phase space.


Consequences of the Force Laws, General RelativityII No 4

We observe that we have got:A Field Theory for connections on C -representations V , i.e. forrepresentations,

ρ : C (σg )→ Endk(V )

with Dirac derivation for representations, [δ] = 0 and HamiltonianQ = ρ(g − T ).We have also a model for General Relativity, for the schemeSpec(Ph(C )com), i.e. for representations,

ρ : C (σ0)→ EndC(σ0)(C (σ0))→ k

with Dirac derivation,

δ =∑i ,p,q

(−Γip,qvpvpδvi + viδti )

Here vi are the ”vertical” coordinates in the phase space ofC = k[t1, .., tn], i.e. the momenta. The Hamiltonian Q is, obviously,trivial.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 47 / 141

Further consequences of the Force LawsII No 5

For the Euclidean or the Minkowski metric, the Force Law, or the equationof motion, reduces to,

d2ti = −∑p,q


gp,q[Fi ,q, dtp].

Given a representation,

ρ : Ph(C )→ Endk(ΘC ),

this implies the following simple evolution equation,

ρ(dti ) = ρ(d2ti ) = −∑p

(Fi ,pρ(dp) + 1/2∇δs (Fi ,p)).

which, when we have introduced our favourite Toy Model, will give us theBloch and the Maxwell equations.


The Space of PotentialsII No 6

Choose a particular momentum, ρ∗, and putξi :=

∑nl=1 g ilδl ∈ Derk(C ), ρ ∗ (dti ) =: ∇ξi . Then the space of

representations, ρ of C (σ) is given as above, by

ρψ(ti ) = ti , ρψ(dti ) = ∇ξi + ψi ,

where ψi ∈ EndC (V ).( On a free C -module V , one usually write ξi for∇ξi .)The set of iso-classes is identified with the space of equivalence classes ofthe corresponding set of potentials, ψ := (ψ1, ψ2, ..., ψn), naturallyisomorphic to,

P = EndC (V )n.

It does not form an algebraic variety, but it has a nice structure.


The Tangent Space of P ,II No 7

The tangent space Tρ, of P at ρψ : C (σ)→ EndC (V ), represented byψ ∈ P, may also be identified with a quotient of P. In fact,

Ext1C(σg )(ρψ, ρψ) = Derk(C (σg ),Endk(V ))/Triv .

Any derivation η ∈ Derk(C (σg ),Endk(V )), maps the relations of C (σg ) tozero, so we shall have,

[η(dti ), tj ] + ρ(dti )η(tj)− η(tj)ρ(dti ) = η(g i ,j).

If V is a projective C -module, such that Ext1C (V ,V ) = 0, there exists a

linear map, Φ0 ∈ Endk(V ), such that η(tj) = tjΦ0 − Φ0tj , for all j . Wemay therefore, assume all η(ti ) = 0, and then the derivation η isdetermined by the family of elements, η(dti ) ∈ EndC (V )), i = 1, ..., n.


The Classical Gauge InvarianceII No 8

For ρ := ρψ, corresponding to ψ ∈ P, we have seen that,

Tρ = Ext1C(σ)(ρ, ρ) = P/Triv

where the the elements in Triv , corresponding to trivial derivations alsomapping ti to 0, are exactly those given by the n-tuples,

((ξ1(Φ) + [ψ1,Φ]), ..., (ξn(Φ) + [ψn,Φ])),

for some Φ ∈ EndC (V ).


The Gauge Group EndC (V )II No 9

The expression,

Φ(ψ) := (ξ1(Φ) + [ψ1,Φ], ..., ξn(Φ) + [ψn,Φ]),

therefore corresponds to an infinitesimal gauge transformation,

Φ ∈ Derk(P) =: h

of the space, P, of representations of C (σg ), acting linearly like

Φ(ρψ) = (ρψ) + Φ(ψ)

In particular, the tangent space,

Tρψ = EndC (V )n/h.

The physical relevant space is therefore the quotient ,

P = P/h

of P with respect to the action of the Lie algebra h.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 52 / 141

The Action of the Dirac Derivation on PII No 10

As in the finite dimensional situation, the Dirac derivation induces a vectorfield

[δ] ∈ ΘP,

as long as we, by vector field, understand any map which to an element ψin P associate an element in its tangent space, i.e. in Ext1

C(σg )(Vρ,Vρ), forρ = ρ0 = ψ. It must, however, vanish at ρ, since the Dirac derivationδ = ad(g − T ), necessarily must be mapped to a trivial derivation inDerk(C (σ),Endk(V )), therefore to 0 in Ext1

C(σ)(Vρ,Vρ). But then itcorresponds to an infinitesimal transformation of V ,

ρψ(g − T ) = 1/2∑

gi ,jρψ(dti )ρψ(dtj) + 1/2∑

l

(Γli ,l + Γl

i ,l)ρψ(dti )

This may be interpreted as saying that time acts within eachrepresentation, ρ : C (σg )→ Endk(V ).


Classical NotationsII No 11

The physicists usually write δψ := Φ(ψ), not caring to mention Φ, takingfor granted that δψ := δΦ(ψ) stands for an infinitesimal movement of ψ inthe direction of Φ, and call the transformation above, an infinitesimalgauge transformation. The literature on gauge theory, and its relation tonon-commutativity of space, and to quantisation of gravity, is huge. Ithink that the introduction of the non-commutative phase space, and inthe metric case, the generic dynamical system,

(σg ) = ([dti , tj ]− g i ,j),

can, to some degree, elucidate the philosophy behind this effort. See e.g.the papers, [?], and [?], where the authors initially introducenon-commutativity in the ring of observables generated by coordinates, xν ,by imposing,

[xν , xµ] = Θν,µ,

where Θi ,j are constants.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 54 / 141

A RemarkII No 12

The above treatment of the notion of gauge groups and gaugetransformations may also explain why, in physics, one considers potentialsas interaction carriers, thus as particles mediating force upon otherparticles. And maybe one can also see why the notion of Ghost Fields orParticles of Faddeev and Popov, comes in. It seems to me that theintroduction of ghost particles is linked to working with a particular sectionof the quotient map, P → P.The Dirac derivation, which is entirely dependent upon the notion ofnon-commutative phase space, is not (explicitely) found in present dayphysics. The parsimony principle is therefore, normally, introduced via theconstruction of a Lagrangian, and an Action Principle, i.e. a function ofthe (assumed physically significant) variables, the fields and theirderivatives, defined in P, assumed to to be invariant under the gaugetransformations, so really defined in P, and supposed to stay stable duringtime development, see [?].


Chern Characters and Chern-Simons ClassesII No 13

A non trivial element in the toolbox of the physicists, helping them toguess the Lagrangian, is the Chern-Simons functional, that we now spellout in some generality. Since everything we have done above is functorial,we may work on nonsingular schemes, instead of commutative k-algebras.Let us, as above, assume given a a metric g on some scheme X , and thatwe have an affine covering, given in terms of a family of commutativek-algebras Cα and a bundle V , defined on X , corresponding to a set ofrepresentations, ρ0 : C = Cα → Endk(Vα). Let ρ : OX (σg )→ Endk(V ) bea momentum at ρ0, i.e. an extension of ρ0 to OX (σg ). Then we knowthat ρ induces a connection on V , and we have denoted byP := EndOX

(V )d , the set of such connections, or if we want to, the set ofrepresentations ρ : OX (σg )→ Endk(V ). Recall that we may pick a0-object ρ∗ in this set, such that all other representations ρ, is ρ∗ plus apotential ψ ∈ EndOX

(V )d .


Chern Characters and Chern-Simons ClassesII No 14

Consider now for any OX -bundle ρ,V the class in HHn(OX ,Endk(V ))defined by the Hochschild co-chain, chn(ρ)the k-linear map, defined locallyby,

chn : C⊗nα → Endk(Vα),

where,chn(c1 ⊗ c2...,⊗cn) = ρ(dc1dc2...dcn) ∈ Endk(Vα).

We know that chn is a Hochschild co-cycle, since,

δ(chn)((c1 ⊗ c2...⊗ cn+1)) = c1ρ1((dc2dc3...dcn+1)) (3)

+n∑1

(−1)ρ1((dc1...d(cici+1)...dcn+1)) + (−1)n+1ρ1(dc1...dcn)cn+1 = 0

(4)


Invariance of the Chern-Simons ClassesII No 15

The Generalized Chern-Simons Class of ρψ, is then the class in the obviousdouble complex, defined by the covering Cα, and the classes,

chn(ρ) ∈ HHn(Cα,Endk(Vα)),

or, usually, by 1/n! chn. Let Φ ∈ h := EndC (V ), and consider Φ as aHochschild 0-cocycle,

Φ : k = C 0 → Endk(V ),Φ(α) = αΦ,

then,

δΦ(ti ) = ρψ(dti )Φ− Φρψ(dti ) = [∇ξi + ψi ,Φ] = ξi (Φ) + [ψi ,Φ].

This proves thatch1(ρψ) = ch1(ρψ + Φ(ψ))

and implies that the Ghost Fields have well defined Chern-Simons classes.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 58 / 141

The Moduli Space of MetricsII No 16

The relationship between the two different notions of time, the metric ofour moduli space C of our models, and the Dirac derivative of Ph∞(C ) istight.Let M be the space (of isomorphism classes) of metrics on C . For everypoint g ∈M consider the diagram,

Ph(C )

//

ρ1

%%JJJJJJJJJJPh2(C )

ρ2

// Ph3(C ) . . .

ρ3xxqqqqqqqqqqq// Ph∞(C )δ

Ci//

d==zzzzzzzzz

C (σg ) ρ// Endk(V )

Here ρ1 is the representation of Ph(C ) induced by a representation ρ ofC (σg ).


The Moduli Space of MetricsII No 17

Consider the family of k-algebras,

µ : C(σ)→M

indexed by the possible metrics of C , such that g ∈M corresponds toC (σg ).Let,

TM,g = (hi ,j), hi ,j = hj ,i ∈ C .

be the tangent space to M, at g . Define, hi ,j by,

hi ,j = −∑p,q

gi ,php,qgq,j , h = hi ,j ∈ TM.

Consider now the first order deformation g + εh, of the metric g , and thecorresponding Dirac derivation, ad(g + εh − T ′) in C (σ(g+εh)). Then wefind, in C (σ(g+εh)),

[d ′ti , tj ] = g i ,j − hi ,jε = (g + εh)i ,j .


MetricsII No 18

Moreover, the derivation η : Ph(C )→ Endk(V ) defined by,

η(ti ) = 0, η(dti ) =∑l ,q

hi ,lρ1(gl ,qdtq) =∑

l

hi ,l∇δl

corresponding to the 2.-order derivative of ρ, i.e. to the morphism,

η(ρ) := ρ2 : Ph2(C )→ Endk(V ),

for which, ρ2(d2ti ) = η(dti ), induces an element,

η(h) ∈ Ext1Ph(C)(V ,V )

producing an injective map,

η : TM,g → Ext1Ph(C)(V ,V ).


Gravitational WavesII No 19

Thus, any non-trivial deformation of the metric g induces a non-trivialdeformation of the Ph(C )-representation (ρ1,V ), and any first ordernon-trivial deformation of the Ph(C )-representation (ρ1,V ) induces anon-trivial deformation of the metric. Given any non-zero tangenth ∈ TMg , we see that there corresponds a non-zero second-order tangentof the representation ρ0, given by the element η(h) ∈ Ext1

Ph(C)(V ,V ),determined by the ρ1-derivation η.This η(h) is the measure of an acceleration of the representation ρ0, andshould correspond to a ”cataclysmic change” of any massy ”particleρ0 : C → End(V )”, given by

..ρ0 = h, the solution of which would be a

”wave”, ρ0. Combined with the structure of our ”Toy Model”, see [?], thisfits well with the present understanding of gravitational waves. It also fitreasonably well with the present cosmological theory.



Moreover, let us, for every metric g ∈M consider the scalar curvature Ras a function on the space C := Spec(C ), and fix a ”compact” subsetΩ ⊂ C . Since R ∼ 1/rd , where r is the ”radius of curvature” of Ω at thecorresponding point, it is not unreasonable to consider

S(g) :=

∫Ω

Rdvg ,

where dvg is the volume element in C defined by the metric g , as the”furniture”, i.e. the ”material” content of the the part of space, Ω,represented by all representations of C (σg ). But then one could look at S ,as a functional,

S : M→ R

the stability of which would give us a unique vector field on M,

G ∈ ΘM, G(g) = Gi ,j := Ri ,j − 1/2Rgi ,j ∈ TM,g .



Then the injective map,

η : TM,g → Ext1Ph(C)(V ,V ),

would be a Field Equation, of Einstein-Hilbert type.An ”increment” h, of the metric g (in Ω) would correspond to the elementof η(h) ∈ Ext1

Ph(C)(V ,V ), of first order, for all representations ρ,corresponding to an increment of energy of the total furniture of ourUniverse. So, again we find a Schrødinger type equation; equatingderivation with respect to time, with G, and seeing that ρ2 corresponds toan Hamiltonian operator on V , we find,

δt(ρ1) = ρ2.

This is our generalised Einstein field equation, η(h) for all ρ, beingidentified with the mass, energy, stress-tensor of the general relativitytheory.


III Quotients in GeometryNo 1

Let g ⊂ Derk(A), be a sub Lie-module, and consider first, in thecommutative case, the scheme (algebraic variety), X = Spec(A), and theLie algebra g as a Lie algebra of vector fields defined on X . The set ofmaximal integral subschemes,

X/g := M ⊂ X |ΘM = g|M,

is called the quotient of X by g, and coincides, in good cases with thequotient of X by the group of automorphisms G acting on X , withlieG = g, when this exists. In the classical, commutative case, one wouldidentify,

X/g = Spec(Ag),

where, Ag := a ∈ A|∀γ ∈ g, γ(a) = 0. This is, however, only reasonablewhen g is reductive and/or all orbits of G are closed, which they rarely are.


Quotients in Non-commutative Algebraic GeometryIII No 2

In the non-commutative situation, this last definition of a quotient, has nomeaning. The algebra A may have no 1-dimensional representations at all.The solution is to define the relevant points Simp(A), of the geometry A,defined by A, and then to seek out those points that should correspond tothe points of the quotient A/g.Consider a representation ρ : A→ Endk(V ), and let for every γ ∈ g thederivation, γ ρ ∈ Derk(A,Homk(V ,V )), map to 0 in Ext1

A(V ,V ). Thismeans that the representation ρ, is not moved by the action of g. Asabove, it follows from, κ(γ ρ) = 0, that there exist an element,Qγ ∈ Homk(V ,V ), the Hamiltonian, such that for all γ ∈ g and all a ∈ A,

ρ(δ(a)) = Qγ ρ(a)− ρ(a) Qγ .

which can be written as,

Qγ(av) = γ(a)v + aQγ(v), ∀v ∈ V ,

i.e. Q is a g-connection on V .Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 66 / 141

Quotients in Non-commutative Algebraic GeometryIII No 3

Now, we define,

A/g := Simp(ρ|δ := γ ρ = 0 ∈ Ext1A(V ,V ))

where Simp of course picks out those representations of this sort, with nosuch sub-representations.The aim of, my kind, of non-commutative algebraic geometry is to toassociate to any family of representations V, of the algebra A, the optimalextension-algebra,

η : A→ O(V)

for which the family V becomes the set of simple O(V)-representations.Then V should be called: A Scheme for O(V).


Quotients in Geometry and Physics: Gauge GroupsIII No 4

The notion of Gauge Group in physics, is intimately related to thisnon-commutative quotient structure.We shall, in fact, show that there is an algebra H, representing ourCosmos, a Lie algebra, G acting on H(σ) := Ph(H)/([h, dh]|h ∈ H), thepartially commutativization of Ph(H), such that the main ingredients ofthe Standard Model, including representations like the Weyl and the DiracSpinors pop up as points of the quotient,

H(σ)/G,

suggesting that the standard Quantum Theory should be thought of aspart of non-commutative algebraic geometry.


Global Gauge GroupsIII No 5

Suppose, we have identified a k-Lie algebra g0 ⊂ Derk(A), of infinitesimalautomorphisms, i.e. of derivations of A, a global gauge groupe, leavinginvariant the physical properties of our phenomena P. We would then beled to consider the quotient space M/g0, which in our non-commutativegeometry, is equivalent to restricting our representations,ρ : A→ Endk(V ), to those representations V for which, g0 ⊂ gV . Thiswould then imply that the corresponding Hamiltonians, Qγ define ag0-connection on V ,

Q : g0 −→ Endk(V ),

such that, for all c ∈ A, and for all γ ∈ g0, ρ(γ(c)) = [Qγ , ρ(c)].


Global Gauge Groups and ForcesIII No 6

This is usually written,ρ(γ(c)) = [γ, ρ(c)].

The curvature,

R(γ1, γ2) := [Qγ1 ,Qγ2 ]− Q[γ1,γ2] ∈ EndA(V ),

corresponds to a global force acting on the representation ρ. These forces,mediated by the gauge-particles, λ ∈ g0, will be the first to be studied insome details. Put,

Rep(A, g0) := ρ ∈ Rep(A)|, κ(γρ) = 0, ∀γ ∈ g0 = ρ ∈ Rep(A)|g0 ⊂ gρ,where Rep(A) is the category of all representations of C , and notice that,in the commutative situation, if we consider the case where the gaugegroup g0 = Derk(A) then Rep(A, g0) is the category of A-Connections, forwhich the space of isomorphism classes is discrete. Notice that this is thesituation in the classical quantum theory, where the Hilbert Space isalways considered as the unique state space of interest.


Simple ObjectsIII No 7

An object V ∈ Rep(A, g0) is called simple if there are no non-trivialsub-objects of V in Rep(A, g0). The generalized quotient Simp(A)/g0, isby definition, the set, Simp(A : g0), of iso-classes of simple objects inRep(A, g0).If the curvature also vanish, there is a canonical homomorphism,

φ : U(g0)→ Endk(V ).

where U(g0) is the universal algebra of the Lie algebra g0.


Gauge RepresentationsIII No 8

In the general case let,A′(g0) ⊂ Endk(A),

be the sub-algebra generated by A and g0. Then we put,

A(g0) = A′(g0)/(γa− aγ − γ(a))

and we have an identification between the set of g0-connections on V , andthe set of k-algebra homomorphisms,

ρg : A(g0)→ Endk(V ),

since any such would respect the relation above, such that,

ρg0(γa) = ρg0(γ)ρg(a) = ρg0(a)ρg0(γ) + ρg0(γ(a)).


Gauge Representations and QuotientsIII No 9

In the above situation, we have the following isomorphisms,

Simp(A)/g0 := Simp(A : g0) ' Simp(A(g0)).

Notice that the commutant in A(g0), of g0 is the subring,

Ag0 := a ∈ A|∀γ ∈ g0, γ(a) = 0 ⊂ A.

Notice also that the commutativisation, A(g0)com, of A(g0) is the quotientof A(g0) by an ideal containing γ(a)|a ∈ A, γ ∈ g0. Therefore there is anatural map,

Ag0 → A(g0)com.

However this map may not be injective, so we cannot, in general, identifythe rank 1 points of Simp(A, g0), with Simp1(Ag0).


The Commutative Classical CaseIII No 10

If A = C is assumed commutative, the classical invariant theory identifiesthe two schemes, Spec(C )/g0 and Spec(C g0), which in the above light, isnot entirely kosher. However, if a ⊂ C is an ideal, stable under the actionof g0, then since any derivation γ of C acts on the multiplicative operatorsa ∈ C as γ(a) = γa− aγ, it is clear that the quotient C/a is an(C , g0)-representation. Moreover,

C/a ∈ Simp1(C )/g0

if and only if the subset Simp1(C/a) ⊂ Simp1(C ) is the closure of amaximal integral subvariety for g0. The space of such integral subvarietiesis what we, previously have termed the non-commutative quotient,Spec(C )/g0.


IV Local Gauge GroupsNo 1

Suppose now there is a C -Lie algebra g1, acting C -linearly on thoseC -modules V , which we would consider of physical interest. g1 would thenbe called a local gauge group. One may then want to know wether thegiven action of g0 moves g1 in its formal moduli as C -Lie algebra. If so, theaction of g1 would not be invariant under the the gauge transformationsinduced by g0, and we should not consider (ρ, g1) as physically kosher.If, on the other hand, the action of g0 does not move g1 in its formalmoduli, it should follow that there is a relation between the g0-action (i.e.the connection) on V , and the action of g1. Now, in the caseg0 ⊂ Derk(C ), it follows from the Kodaira-Spencer map,

ks : Derk(C )→ A1(C , g1 : g1),

see Lemma(2.3), of [6], that the following holds,


Quotients of Local Gauge GroupsIV No 2

Assume g1 as a C -module is such that g0 ⊂ gg1 , and let cki ,j ∈ C be the

structural constants of g1 with respect to some C -basis xi, and letδ : F→ g1 be a surjective morphism of a free C -Lie algebra F onto g1,mapping the generators xi of F onto xi . LetFi ,j = [xi , xj ]−

∑k cki ,jxk ∈ kerδ, and let γ ∈ Derk(C ). Then, ks(γ) is the

element of A1(C , g1 : g1) determined by the element of HomF(ker(δ), g1),given by the map,

Fi ,j → −∑k

γ(cki ,j)xk .

For ks(γ) to be 0, there must exist a C - derivation Dγ : F→ g1, (apotential), such that

Dγ(Fi ,j) = Dγ([xi , xj ]−∑k

cki ,jxk) = −∑k

γ(cki ,j)xk


Quotients of Local Gauge GroupsIV No 3

Let now ∇ : g0 → Derk(g1) be a connection on the C -module g1. Then,

ks(γ) = 0

iff there exist a g0-Lie-connection of the form D = ∇− D on g1, i.e. ak-linear map,

D : g0 → Derk(g1),

such that, for γ ∈ g0, c ∈ C , κ ∈ g1,

Dγ(c · κ) = cDγ(κ) + γ(c) · κ.


Lie-Cartan PairsIV No 4

If the curvature,

R(γ1, γ2) := [Dγ1,Dγ2]−D[γ1, γ2]

representing the 2. order action of g0 on g1, (the force, as a physicistmight have said), vanish, the map,

D : g0 → Derk(g1),

would be a Lie-algebra morphism. And then this notion is related to thenotion of a Lie-Cartan pair. In fact, the structure given by D defines a Liealgebra structure on the sum

g := g0 ⊕ g1,

with Lie-products of the sum defined as the product in each Lie algebra,and the cross-products defined for, γ ∈ g0, ξ ∈ g1, as,

[γ, ξ] = Dγ(ξ),


Lie AlgebroidsIV No 5

A structure like this, a Lie-Cartan pair, is now often called a Lie algebroid,and the operation of the C -Lie algebra g1 is, when it is supposed to bephysically interesting, called a local gauge group.The category of representations ρ : C → Endk(V ) with this property vis avis the Lie algebra g, should be called g-insensitive, and the space ofsimple g-insensitives shall be denoted.

Simp(A)/g.

This is going to be our central object of study.


V The Role of Cartan sub-Algebras in PhysicsIV No 6

Physicists have a way of classifying, or naming states, i.e. the elements ofthe representation vector space V , according to certain numbersassociated to them, like spin, charge, hyperspin, etc. We find this in thesituation above, as follows.Consider the Cartan algebra h ⊂ g1. It will operate on the aboverepresentation space V as diagonal matrices, and the eigenvectors may belabeled by the corresponding eigenvalues.Notice also that if Vi ∈ Rep(C , g), i = 1, 2, then it follows from (1.15) thatan extension of the C (g0)-module V1 with V2 will also sit in Rep(C , g).Put, as above, A := C (σ) where σ is a dynamical structure, and assumegiven an action of the Lie algebra g0 on C . Notice that since Ph() is afunctor in the category of algebras and algebra morphisms, the action ofg0 extends to Ph(C ), but not necessarily to a dynamical system of thetype C (σ).This will turn out to be important for our version of the Standard Model.


VI Deformations of associative algebrasV No 1

We fix a field k. All algebras occurring, will be associative k-algebras. Adeformation of an algebra A parametrized by the (commutative) k-algebra,B, is a flat k-algebra homomorphism, B → AB , such that there is a”point”, i.e. a homomorphism, ρ : B → k, such that k ⊗B AB ' A.For any associative k-algebra A, there is a formal moduli, i.e. a completelocal k-algebra, H(A), and a deformation, a versal family µ : H(A)→ A,of associative algebras, such that for any other deformation B → AB , withB a finite dimensional local k-algebra, there is a diagram,

µ : H(A) //

A

B // AB

with AB ' B ⊗H(A) A.


The tangent space T ? of the formal moduli of anassociative algebraVI No 2

The tangent space of H(A), i.e. the dual k vectorspace of m/m2, where m

is the maximal ideal of H(A), is calculated as:

T ? = A1(k ,A; A) = HomF (ker(ρ),A)/Der

where, ρ : F → A, is a surjective homomorphism of a free k-algebra Fonto the given algebra A, HomF is the set of F -bilinear maps, andDer ⊂ HomF , denotes the restrictions to ker(ρ), of the derivationsDerk(F ,A).


Deformations of U as associative algebrasVI No 3

• A = k[x1, x2, x3] is the commutative coordinate ring of the affine3-space.

• U = A/(x)2 is, geometrically, a thick point in affine 3-space, but

• U is also a quotient of the free associative k-algebra,F = k < x1, x2, x3 >,

• Let ρ : F → U be the quotient map, then the kernel,ker(ρ) = (xixj), i , j = 1, 2, 3

A deformation of U parametrized by the (commutative) k-algebra, B, is aflat k-algebra homomorphism,B → B < x1, x2, x3 > /(xixj +

∑bli ,jxl + b0

i ,j)It is easy to compute the tangent space of the formal moduli of U,

dimkA1(k ,U; U) = dimkHomF (ker(ρ),U)/Der = 27.


The restricted formal moduli of UVI No 4

Pick any two 3-vectors,

o := (o1, o2, o3), p := (p1, p2, p3).

The bi-linear homomorphisms,

κ : ker(rho) = (xixj)→ U, κ(xixj) := oixj + xipj

represents linearly independent elements in A1(k,U; U), and span asubspace of the tangent space of the formal moduli of U generating aquotient of the versal family,

µ : H(U) //

U

H(U) // U


The restricted versal familyVI No 5

• Let o := (o1, o2, o3), and p := (p1, p2, p3) be sets of independentcoordinates, and let,

• H(U) =: H = k[o1, o2, o3, p1, p2, p3]

• Put, H := Spec(H) = A3×A3, and we have got a deformation of U,

ρ : H → U := H < x1, x2, x3 > /(xixj − oixj − xipj + oipj)

parametrized by the 6-dimensional scheme H.

• Let o := (o1, o2, o3), and p := (p1, p2, p3) be two vectors, thenU(o, p) := k < x1, x2, x3 > /(xixj − oixj − xipj + oipj)

• Put, B = k[t], bli ,j = εi ,j ,l t, b0

i ,j = δi ,j then the deformation of Ualong t for t 6= 0 is constant, equal to the Quaternions.


VIIThe Toy ModelNo 1

My favourite ”Toy Model”, of General Relativity, and Quantum Theory isthe philosophically reasonable (?) Physical Model, of an Observer and anObserved in 3-dimensional space, mathematically modelled by the Hilbertscheme H of length 2 sub-schemes in A3. Consider the diagonal,∆ ⊂ A3 × A3 = H, and let H be the blow up of H in ∆. We find adiagram,

H

Z2

// H ⊃ ∆ ∆ ⊂ Hoo

H

where H = Hilb2(A3) = H/Z2. Call he generator, P of Z2, the Parityoperator.Coordinates for H: (λ, ω, ρ), with λ ∈ ∆, ω a spherical coordinate of theblow-up of ∆ in H, and, ρ the length from ∆ along the line defined by ω.


Coordinates of HVII No 2

This is a convenient parametrisation of H. Consider, as above, for eacht ∈ H the length ρ, in E3, the Euclidean space, of the vector (o, p). Givena point λ ∈ ∆, and a point ξ ∈ E (λ) = π−1(λ), of the fiber of,

π : H → H,

at the point λ, for o = p. Since E (λ) is isomorphic to S2, parametrized byω = (φ, θ), any element of H is now uniquely determined in terms of thetriple t = (λ, ω, ρ), such that c(t) := c(o, p) = λ, and such that ξ isdefined by the line op, and the action of θ keeps ξ fixed. Here ρ ≥ 0, andnotice that, at the exceptional fiber, i.e. for ρ = 0, the momentumcorresponding to dρ is not defined.


Metrics or TimeVII No 3

Any metric, and therefore also the notion of Time of the Model H, can begiven the form,

g = hρ(λ, φ, ρ)dρ2 + hφ(λ, φ, ρ)dφ2 + hλ(λ, φ, ρ)dλ2,

where dφ2 is the natural metric in S2 = E (λ). Pick the simplified space,in which ω is reduced to the angle φ, and the coordinates λ is reduced toone parameter λ = |λ|.

g = (ρ− h(λ)

ρ)2dρ2 + (ρ− h(λ))2dφ2 + κ(λ)dλ2,

This correspond to considering the sub-universe of M(BB), parametrizedby (λ, φ, ρ).


Big Bang ModelVII No 4

Computing the geodesics (the Force Law), taking into account that Timeis the metric, we find,

d2λ

dt2= −(

ρ− h(λ)

ρ)(

1

κ(λ))(

dh

dλ)(

dρ

dt)2

− (ρ− h(λ))(1

κ(λ))(

dh

dλ)(

dφ

dt)2

+ (dlog(κ)

dλ)(

dλ

dt)2


Big Bang ModelVII No 5

Moreover we find,

d2ρ

dt2= −(

h(λ)

ρ(ρ− h(λ)))(

dρ

dt)2

+ (2

(ρ− h(λ)))(

dh

dλ)(

dρ

dt)(

dλ

dt) + (

ρ2

(ρ− h(λ)))(

dφ

dt)2,

d2φ

dt2= −2/(ρ− h(λ))

dρ

dt

dφ

dt+ 2/(ρ− h(λ))(

dh

dλ)(

dφ

dt)(

dλ

dt)

where t, is the time parameter of the model. From these formulas we seethat the Gravitation is expanding inside the Horizon and contractingoutside. Conservation of mass implies, h(λ) = h0/λ. From this followsthat the Horizon at the BB, i.e. for λ = 0, is all of space. Interpreting λas Cosmological time we find a striking cosmological model, complete withInflation and Hubble formulas, v = r/t and v/

√1− v 2 = r/λ.


KeplerVII No 6

Now let h(λ) = h be constant, then the geodesics have the equations,

d2ρ

dt2= −(

h

ρ(ρ− h))(

dρ

dt)2 + (

ρ2

(ρ− h))(

dφ

dt)2,

d2φ

dt2= −2/(ρ− h)

dρ

dt

dφ

dt, Kepler ′s 2.Law .

d2λ

dt2= 0,

The definition of time gives us,

ρ−2(dρ

dt)2 = (ρ− h)−2K 2 − (

dφ

dt)2.

where, K 2 = (1− (dλdt )2), is the kinetic energy.


Kepler and NewtonVII No 7

Put this into the first equation above, and obtain,

d2ρ

dt2= −hK 2(

ρ

ρ− h)

1

(ρ− h)2+ (

ρ+ h

ρ− h)ρ(

dφ

dt)2.

Assume now r := ρ− h ≈ ρ, we find,

d2r

dt2= −hK 2

r 2+ r(

dφ

dt)2, Kepler ′s 1.Law .

The constant h, the radius of the exceptional fibre, is thus also related tomass. Recall that the Schwarzschild radius, the Einstein equivalent to h, isassumed to be,

rs = 2GM/c2,

where, G = Newton’s gravitational constant, M = mass, c = speed oflight, which here, of course, is put equal to 1.


VIII U extends to HNo 1

For every (o, p) ∈ A3 × A3 = H, there is an associative 4-dimensionalk-algebra U(o, p), the fibre of U at (o, p). Moreover, if o = p, thenU(o, p) ' U. We obtain the extended diagram,

H

Z2

// H ∆oo ∆ ⊂ Hoo

H Uoo

OO

Uoo

OO

This follows from the following elementary relations: For (o, p) ∈ H, forevery c ∈ A3 and for any non-zero κ ∈ k , we have,

• U(κo, κp) ' U(o, p)

• U(o, p) ' U(o − c , p − c)

• U(−p,−o) ' U(o, p)


A Universal Gauge GroupVIII No 2

Consider the bundle of Lie algebras, defined on H by,• g := Der H(U)• g(o, p) = Derk(o,p)(U(o, p)), (o, p) ∈ H.

Any element δ ∈ g(o, p) must have the form,δ(xi ) = δ0

i + δ1i x1 + δ2

i x2 + δ3i x3. Consider the 4-vectors,

δi = (δ0i , δ

1i , δ

2i , δ

3i ), o = (1, o1, o2, o3), p = (1, p1, p2, p3)

Theorem

• δ ∈ g(o, p) if and only if δi .o = δi .p = 0,

• If o 6= p, then, g(o, p) '

0 ? ?0 ? ?0 ? ?

• rad(g) = u, r1, r2, g/rad ' sl(2) = h, e, f ⊂ g

• h ∈ h ⊂ g; the generator of the Cartan algebra.


Canonical Basis for ΘHVIII No 3

Notice also that, in this case, the unique 0 (resp light)-velocity tangentline at the point t0 = (o, p), o = (0, 0, 0), p = (1, 0, 0), killed by g, isrepresented by,

• d3 := ((1, 0, 0), (1, 0, 0)), the unique 0-velocity, resp.

• c3 := ((1, 0, 0), (−1, 0, 0)), the unique light-velocity.

Let, d1 := ((0, 1, , 0), (0, 1, 0)), d2 := ((0, 0, 1), (0, 0, 1)) and let,c1 := ((0, 1, , 0), (0,−1, 0)), c2 := ((0, 0, 1), (0, 0,−1)),.

• Then c1, c2, c3, d1, d2, d3 is a basis for the tangent space Θt0,

c1, c2, c3 for ct0, and d1, d2, d3 for ∆t0

.

Put, Bo :=< c1, d1 > and Bp :=< c2, d2 >.


Spin and IsospinVIII No 4

Denote by ∆ ⊂ ΘH the 3-dimensional distribution, generated by the

translations (o, p)→ (o + c , p + c), c ∈ A3, and complexify all bundles.Introduce a Riemanian metric g on the space H, and see that we have thefollowing structures uniquely defined,

• ∆C, with the action of su(3)

• ΘH ' Bo ⊕ Bp ⊕ Ao,p ' ∆⊕ c ; resp. 0-and light-velocities.

• There exists a canonical action of gC on ΘH,C, respecting thedecomposition, such that,

• g(o, p) acts on the tangent space TH,(o,p) = TA3,o × TA3,p killing thevector p − o, in both factors.

The canonical action of these principal Lie bundles on the complexifiedtangent bundle ΘH ”determins all Fields”.


Action of the Gauge groupVIII No 5

The generators, h, e, f ∈ sl(2) ⊂ g act, in the above basis, like,

h =

1 0 0 0 0 00 −1 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 −1 00 0 0 0 0 0

e =

0 1 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 00 0 0 0 0 0



f =

0 0 0 0 0 01 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 0 0

.

The generators, u, r1, r2 ∈ rad(g) act, in the above basis, like,

u =

1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 0

.



r1 =

0 0 0 0 0 00 0 0 0 0 01 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 0

.

r2 =

0 0 0 0 0 00 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1 0

.



We observe here that the rotations (spin) around the 3 different axes in c ,respectively in ∆, are given by, (e − f ), (e − r1), (f − r2), about the axedefined by (op) = c3, c2, c1, respectively (op) = d3, d2, d1,Recall that ∆ ⊂ ΘH , is the sub-bundle defined, at the point t as the space

of tangents of the form (ξ, ξ). Given a metric on H, we may look at theaction of su(3) on ∆⊗ C. Knowing that Θ = c ⊕ ∆, su(3) acts in theobvious way on the lower right corner, like

γ =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 ∗ ∗ ∗0 0 0 ∗ ∗ ∗0 0 0 ∗ ∗ ∗

.



Since the 0-velocity direction defined at (o, p), is (o − p, o − p), whichhere is d3, we may in an essential unique way decompose the Cartansubalgebra h ⊂ su(3), into the Cartan subalgebra h1, for the su(2)-component leaving δ3 invariant, and the part h2 ⊂ h perpendicular, in theKilling metric, to h1.

h1 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1/2 0 00 0 0 0 −1/2 00 0 0 0 0 0

, h2 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1/3 0 00 0 0 0 1/3 00 0 0 0 0 −2/3

,

Classically one denotes the other 6 base elements of su(3)⊗ C, as,ei±, i = 1, 2, 3.



The restriction to ∆ of these operators, in the basis d1, d2, d3 are givenby,

e1+ =

0 1 00 0 00 0 0

, e2+ =

0 0 00 0 10 0 0

, e3+ =

0 0 10 0 00 0 0

.

and their duals,

e1− =

0 0 01 0 00 0 0

, e2− =

0 0 00 0 00 1 0

, e3− =

0 0 00 0 01 0 0

.


Commutators in su(3)VIII No 11

We need the commutators, [h1, h2] = 0, and,

[h1, e1±] = ±ei±, [h1, e

2±] = ∓1/2e2

±, [h1, e3±] = ±1/2e3

±

together with the following ones,

[h2, e1±] = 0, [h2, e

2±] = ±e2

±, [h2, e3±] = ±e3

±

Notice, for later use, that the only non-zero products of the ei± are,

e1+ · e2

+ = e3+, e

2− · e1

− = e3−.


IX Elementary ParticlesNo 1

Together these formulas show that the quotients of of theg∗-representation ΘH are the following,

• c and therefore also the photon c1, c2 and a singleton, c3, bothsimple.

• ∆ and therefore the electron d1, d2 and a singleton, d3, bothsimple.

• Weyl spinors, Bo ,Bp, and Dirac spinors, Bo ⊕ Bp

The non-trivial simple quotients of of the su(3)-representation ΘH arereduced to,

• The quarks, ∆


PhotonsIX No 2

We observe that the action of φ ∈ u(1) is, exp(ıφ · (e − f )), on the the(transverse) light-wave is given by,

A(φ) =

cosφ −sinφ 0 0 0 0sinφ cosφ 0 0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

.

It is clear that this, together with the formulas above give good reasons tobelieve that there is a relation between this model, and the StandardModel (and so also to the 8-fold way of Gell-Mann). Moreover, here allingredients are universally given by the information contained in thesingularity U, the Big Bang, in my tapping. The choice of metric depends,however, on the nature of what I have called the Furniture of the model.


The Pauli MatricesIX No 3

It is now easy to see that the the Pauli matrices are found as follows,

σ1 = e + f =

(0 11 0

),

σ2 = −ie + if =

(0 i−i 0

),

σ3 = h =

(1 00 −1

)Moreover, the parity operator P, the generator γ of the symmetry groupZ2, operating on H, acts on c , as multiplication by (-1), see WS.Therefore, it maps the basis (c1 + d1), (c2 + d2) into(−c1 + d1), (−c2 + d2). Consequently, we have an isomorphism,

P : Bo → Bp,


ChiralityIX No 4

Chirality, in the physicists language, is explained as follows. The morphismP, extended to Bo ⊕ Bp, in the basis chosen above, is given by the matrix,

P =

(0 idid 0

),

which turns left-handedness to right-handedness, with respect to thedirection (o, p), resp. (p, o).


The Dirac MatricesIX No 5

This shows that it is meaningful to consider the representations given bythe Dirac matrices,

γ0 =

(1 00 −1

), γk =

(0 σk

−σk 0

), k = 1, 2, 3.

as well as the new operators,

γk+3 =

(σk 00 −σk

), k = 1, 2, 3,

acting on, Bo ⊕ Bp, such that,

∀p 6= q, γpγq = −γqγp, γpγp = 1, p, q = 1, 2, 3, 4, 5, 6.


X The Generic Time-ActionNo 1

Fix a metric g , for H. Consider a representation,

ρ0 : H → Endk(B), ρ ∈ H/g

and an extension ρψ : H(σg )→ Endk(B), as a momentum of ρ0, in thetangent direction ψ ∈ Tρ0 = EndH(B)n, then

ρψ(dti ) = ξi + ψi , ψi ∈ EndH(B)

corresponds to a ρ0-derivation,

η ∈ Derk(H,Endk(B)),mappingto : 0 ∈ Ext1H

(B,B)

Then ad(ρψ(g − T )) = ad(Qh + [ψ] + Qv ) where

[ψ] :=∑

i

ψiδti , Qh = 1/2∑

gi ,jξiξj , Qv = 1/2∑

gi ,jψiψj


The Generic Time-ActionX No 2

Time is then given by,

δ = ad(ρψ(g − T )) = ad(Qh + [ψ] + Qv )

meaning that,

ρψ(dn+1ti ) = [Qh + [ψ] + Qv , ρψ(dnti )].

This takes care of the complete time development of an operator in V , aswell as a state vector φ ∈ B.The second order time-development is also given in terms of the ForceLaw valid in (H)(σg ),

d2ti =−∑p,q

Γip,qdtpdtq −

∑p,q


gp,q[Fi ,q, dtp]

+ 1/2∑l ,p,q


l )]dtl + [dti ,T ]. Maxwell


The Generic Time-ActionX No 3

Given the gauge group g = g0 ⊕ g1, there is a particular tangent direction,v = vi, with vi =

∑j γjg

j ,i , and γi ∈ g1, ”normal” to C in V , naturallyrelated to a derivation, D : g0 → Derk(g1). Put,

[v] :=∑

i

γi∇ξi ,Qh = 1/2∑

gi ,jξiξj , Qv = 1/2∑

gi ,jνiνj

then with this choice of momentum, the time operator becomes,

δ = ad(Qh + [v] + Qv ) ∈ Der(Endk(V ))

with the corresponding first order time-action in the state-space beingdefined by,

δ = [v] + Q ∈ Endk(V ).

and the curvature equal to,

Fi ,j = ad([vi , vj ]) Bloch


XI Time and FurnitureNo 1

Let M be the space of (isomorphism classes of) metrics on H. For everypoint g ∈M consider the diagram,

Ph(H)

//

ρ1

%%JJJJJJJJJJPh2(H)

ρ2

// Ph3(H) . . .

ρ3xxqqqqqqqqqqq

// Ph∞(H)δ

H i//

d==

H(σg ) ρ// Endk(B)

Where ρ1 is the representation of Ph(H) induced by the representation ρof H(σg ). Any extension ρ2 of ρ1 correspond to a tangent of ρ1, thereforeto an acceleration of ρ0 := i ρ. Let, TM,g be the tangent space to M, atg, i.e. TM,g = (hi ,j, hi ,j = hj ,i ∈ C ). This implies that,

TM,g ⊂ Ext1Ph(H)

(B,B)

More precisely, any first order deformation of the Ph(H)-representation(ρ1,B), induces a deformation of the metric g .


X General Relativity and Quantum Field TheoryNo 1

Let us pick a basis, γi, i = 1, ..., 6., in the finite dimensional Lie algebrag, and consider the operator,

[δ] :=∑p,q

gp,qγpδtq : Endk(Θ)→ Endk(Θ).

We may consider a family of representations,

ρ : Ph(H)→ EndH(Θ)

defined by,

ρ(ti ) = ti , ρ(dti ) =∑

j

g i ,jγj .


Dynamics

The dynamics comes out as,

ρ(d(a)) = [δ](ρ(a)) + [Q, ρ(a)].

where the Hamiltonian, Q := ρ(g − T ). Moreover, using the definition (1)above, we find after a short computation,

ρ(d2ti ) = [δ](ρ(dti )) + [Q, ρ(dti )].

In fact this follows from the well known formula,δtq g i ,r = −(

∑l g i ,lΓr

q,l + g i ,lΓrq,l), using the physically equivalent to the

Force Law of H(σg ), given by the formula (1) mentioned above,

d2ti = −1/2∑p,q

(Γip,q + Γi

q,p)dtpdtq (5)

+ 1/2∑p,q

gp,q(Rp,idtq + dtpRq,i ) + [dti ,T ] (6)


The Generalised Dirac EquationNo 19

The energy-mass-equation looks very much like an equations of Dirac type,and in fact, assuming the metric is trivial, and having picked constantgauge fields γi properly, we find that Q =

∑i γ

2i is the identity, and

then Theorem (1.9) actually produces the Weyl equation, for Weyl-spinors,

[δ](ξ) :=∑

i

σiδti (ξ) = Eξ, ξ ∈ Bo ,

and the Dirac equation, for Dirac-spinors,

[δ](ξ) :=∑

i

γ iδti (ξ) = Eξ, ξ ∈ Bo ,⊕Bp.

But, beware, the parity operator,

P =

(0 11 0

),

does not preserve the Weyl spinors.Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 115 / 141

Quantum Klein-Gordon

Put,κ :=

∑p,q=1,2,3

gp,qγpδtq , µ :=∑

p,q=4,5,6

gp,qγpδtq ,

and notice that since, by definition c and ∆ are normal, we have∑p gp,qgp,q′

= 0, for q ∈ c , q′ ∈ ∆.Then we find,

κ, µ = 0, (κ+ µ)2 =6∑

i=1

ξ2i , ξi :=

6∑q=1

g i ,qδtq .

proving that, [κ2, µ2] = 0, and (κ+ µ)2 = κ2 + µ2. So if ω is anonsingular eigenvector for both κ and µ, then,

κ(ω) =∑

p,q=1,2,3

gp,qγpδtq (ω) = Kω, µ(ω) =∑

p,q=4,5,6

gp,qγpδtq (ω) = mω.


Laplace-Beltrami

Consequently,,6∑

i=1

ξ2i (ω) = E 2ω = (K 2 + m2)ω.

This is, in our model, the combined Einstein, Klein-Gordon and Diracequation. Notice that, ∑

p

gi ,jξiξj =∑p,q

gp,qδtpδtq ,

is the Laplace-Beltrami operator.


Summing UpNo 20

Summarizing, we observe that our Toy Model has furnished,

• a Field Theory (including a Y-M model) for connections on H-bundlesB, i.e. for representations,

ρ : Ph(H)/(σg ) =: H(σg )→ Endk(B)

with Dirac derivation for representations, [δ] = 0 and HamiltonianQ = ρ(g − T ).

• a Quantum Field Theory for gauge fields, i.e. for representations,

ρ : Ph(H)/([dti , tj ]) =: Ph(H)part → EndH(ΘH)

with Dirac derivation [δ] =∑

p,q gp,qγpδtq and HamiltonianQ := ρ(g − T ).


Summing Up

Both of these models are compatible with,

• a General Relativistic model, for the scheme Spec(Ph(H)com), i.e. forrepresentations,

ρ : H(σ0) := Ph(H)com → EndH(σ0)(H(σ0)) ' H(σ0)

with Dirac derivation, [δ] =∑

(−Γi )δξi + ξiδti , and HamiltonianQ = 0,


Exterior Forces

The exterior force fields in physics are in our Field Theory Modelrepresentations like ρ : H(σg )→ Endk(B). The dynamics of ρ is given bythe time derivative ρ of ρ, being defined by,

ρ(dti ) = ρ(d2ti ) = −∑p,q

Γip,q∇p∇q − 1/2

∑p,q

gp,q(Fi ,p∇q +∇pFi ,q)

+ 1/2∑l ,p,q

gp,q[∇p, (Γi ,ql − Γq,i

l )]∇l + [∇i ,T ].

For trivial metric (or for the Minkowski metric), the equation reduces to,

ρ(dti ) = ρ(d2ti ) = −∑p

(Fi ,p∇p + 1/2δtp (Fi ,p)),

and the Hamiltonian Q = ad(g − T ) = ∆, reduces to the Laplacedifferential operator.


In this talk I shall sketch a mathematical model for a Big Bang scenario,based on relatively simple non commutative algebraic geometry.Let U be the 4-dimensional real affine algebra of a point with a3-dimensional tangent space, in A3

R. The versal family of deformation ofU, as associative R-algebras, contains, in a natural way, the 6-dimensionalmoduli space, H := Hilb2(R3), which is my Toy Model for the BigBang-scenario in cosmology, introduced in (WS).The study of the corresponding family of derivations of the 4-dimensionalalgebras, leads to a natural way of introducing an action of the gauge Liealgebras of the Standard Model, in ΘH .Introducing the notion of quotient spaces in non-commutative algebraicgeometry, we obtain a geometry that seems to fit well with the set-up ofthe Standard Model.These subjects are all treated within the set-up of (WS).


Standard Model

Fixing the Riemanian metric g on the space H, we have the followingstructures uniquely defined,

• Fermions: ∆C, with the action of su(3)

• Bosons: gC, with the regular representation of g

Ingredients of SM

• ΘH ' Bo ⊕ Bp ⊕ Ao,p ' ∆⊕ c ; resp. 0-and light-velocities.

• u(1) ' Bo ' Bp.

• sl(2) ⊂ g = DerH(U)

• su(3) := sug (∆C)

The canonical action of these principal Lie bundles on the complexifiedtangent bundle ΘH gives us all Fields.


SUSY 1

Choose a point BB ∈ ∆, considered as the point (0, 0) ∈ A3 × A3, andtreat H as a vector space. Recall that ∆, and therefore the point BB, arenot contained in H. Let, moreover M(BB) be the 4-dimensionalsub-scheme of H, consisting of the set (o, p) where o and p are lineardependent. Consider the diagram,

g

κ

AAA

AAAA

A

M(BB) // H ΘH ∆oo

Theorem

• There exists an H-linear map κ, defined by, κ(δ) =∑3

i=1 δ0i ( ∂∂oi

+ ∂∂pi

)

• If o and p, are linear independent, i.e. if (o, p) is not in M(BB), thensl(2) = g/ker(κ) ' ∆, as H-modules.


SUSY 2

Define: Qi : sl(2)C ⊕ ∆→ sl(2)C ⊕ ∆, Q1 = (κ, 0), Q2 = (0, κ−1) Thenwe have a N=1, SUSY: with Q1,Q2 = 1, [Qi ,Pj ] = 0The graded Lie algebra L ⊂ EndH(sl(2)C ⊕ ∆), generated by Qi and su(3)and g, give us a SUSY-like duality between the Bosons and the Fermionsof the model.And one recognises the ingredients of the Standard Model, in the followingcanonical representations of, sl(2) ⊂ g = DerH(U), and su(3) := sug (∆C),respectively,

• BoP' Bp ⊂ ΘH ; Weyl spinors.

• Bo ⊕ Bp ⊂ ΘH ; Dirac spinors.

• ∆ ⊂ ΘH ; Gell-Mann, 8-fold way, colour.

• d3 ∈ ∆ ∩ Ao,p; up-quark

• d1, d2 ∈ ∆, < d1, d2 >⊥ d3; left-right down-quark.


Consider the list of markers, or labels, h1, h2, I±3 = 3/4h2 ± 1/2h1, andYW = 1/2h2 ± h1.

Markers 1/2 · h h1 h2 I3 YW

d1 1/2 1/2 −1/3 0 −2/3

d2 −1/2 −1/2 −1/3 −1/2 1/3

d3 0 0 2/3 1/2 1/3

p1 = d1d3d3 1/2 1/2 1 1 0

p2 = d2d3d3 −1/2 −1/2 1 1 0Olav Arnfinn Laudal () Noncommutative Algebraic Geometry, Topology, and PhysicsNovember 1, 2016 125 / 141


n1 = d1d1d3 1 1 0 1/2 −1

n2 = d2d2d3 −1 −1 0 −1/2 1

n1,2 = d1d2d3 0 0 0 0 0

eL = d1d1d1 3/2 3/2 −1 0 −2

eR = d1d1d2 1/2 1/2 −1 −1/2 −1

W1 = d1d2d2 −1/2 −1/2 −1 −1 0

νL = d−11 d2 −1 −1 0 −1/2 1



W1 = d1d2d2 −1/2 −1/2 −1 −1 0

νL = d−11 d2 −1 −1 0 −1/2 1

To go from left to right handedness comes out by just exchanging d1 andd2. Here one may see that, eL + νL = eR , and one easily find reasons forthe decays,

n→ p+e+νe , p+ → e++π0 → e++2γ, d1 → d3+w−, p+ = e− → n+νe .

Notice also that we may, in an obvious way, identify νL with the elemente + f in one coordinate, swaping d1, d2.


Further consequences of the Force Laws, Maxwell

It is not difficult to see that the evolution equation,

ρ(dti ) = ρ(d2ti ) = −∑p

(Fi ,pρ(dp) + 1/2∇δs (Fi ,p)).

restricted to the subspace M(B) contains Maxwells equation. Interpreting,

C = M(B), vp = tp := ∇p = ∇δp (Ap)(t) ∈ EndM(B)(Θ), vp := E ·ρ(d2ti )(t)

we find that the observed electromagnetic fact, i.e. the Lorentz force law,

vi = −∑p

Fi ,pvp

implies that the tangent to ρ, given by the potential(1/2

∑p∇δp (Fi ,p)), i = 1, ..., 4, must be zero, (or as physicists might

say, gauge). This means, that there exist Φ ∈ EndM(B)(ΘM(B)), such that,

1/2∑p

∇δp (Fi ,p), i = 1, ..., 4 = ∇δi (Φ)+[1/2∑p

∇δp (Fi ,p)),Φ], i = 1, ., 4


Further consequences of the Force Laws, Maxwell

Assuming that the electromagnetic field potentials, Ai are scalars, thecurvatures, Fi ,p = [∇δp ,Ai ]− [∇δi ,Ap] are also scalars, so all commutators[1/2

∑p[∇δp ,Fi ,p],Φ] vanish, implying,

1/2∑p

∇δp (Fi ,p) = ∇δi (Φ), i = 1, ..4.

Now, let us choose coordinates such that t4 =: t0 = λ (the proper time forphysicists) is the only 0-velocity coordinate of M(B), and use the classicalnotations, Simple computation then shows, where ∇s is the (light-) spacepart of ∇. From this, and from the vanishing of the tangent to ρ, i.e. theequation above, the Maxwell equations follows, with Φ = ∇ · A, electriccurrent J = ∇s(Φ), and electric charge ρ = ∇t0(Φ). Together, (ρ, J) is a0-tangent to the representation ρ, the first component in c , i.e. in space(or light) direction, and the second in ∆, or in 0-velocity direction.


Conclusions and Apology

The physical interpretation of the mathematical models presented here,and in the literature referred to in the next slide, should not, at themoment, be considered as a serious proposition for a new physics. Withrespect to the Big Bang, for example, I think any existing model, must beapproached with utmost care.However, I think the idea of defining time as a metric on the moduli spaceof the iso-classes of the mathematical models used to specify/define thephysical phenomenon of interest, may be of some interest. I thereforeproposed to name the study of the geometry of reasonable moduli spacesof this kind:Geometry of Time-Spaces.


XII InteractionXII No 1

Assuming given a dynamical structure and a Dirac derivation δ, we haveassociated to any representation ρ : A(σ)→ Homk(V ) its uniqueextension, or force acting on V via,

ψ(a) : V → V , ψ = δ ρ.For gauge fields, given as above by a representation of H of the form B,where B is a representation of the principal Lie-bundle g⊕ su(3) definedon H, we have studied self-interactions of representations,

ρ : Ph(H)(σg )→ EndH(B),

and the force fields created by the derivations H → Endk(B) induced byρ. However, there are, in this case, other forces to be considered. The Hmodule B may be a free, and sl(2) and su(3) being semi-simple, do nothave nontrivial extensions, but the rad(g) part of g = sl(2)⊕ rad(g), hasnon-trivial extension modules, and as we shall see, this give us a model forthe Weak Interaction.


InteractionXII No 2

Consider the action of g = sl(2)⊕ rad(g), on c and ∆ then there is anisomorphisme,

Ext1H(g)

(Θ,Θ) ' Ext1g (Θ,Θ) ' Ext1

rad(g)(Θ,Θ)

and,

Ext1rad(g)(Θ,Θ) = Ext1

rad(g)(c, c)⊕Ext1rad(g)(∆, ∆)⊕Ext1

rad(g)(c , ∆)⊕Ext1rad(g)(∆, c)

Moreover,Ext1

rad(g)(c , c) ' Ext1rad(g)(∆, ∆)

and, any ψ ∈ Ext1rad(g)(∆, ∆) is given by the derivation,

ψ : rad(g)→ EndH(∆, ∆)

in the bases, d1, d2, d3 of ∆, the quarks, and u, r1, r2 of rad(g),modulo the trivial derivations, given by an elementa := (ai ,j) ∈ EndH(∆, ∆).


InteractionXII No 3

We find,

ψ(u) =

α1,1 0 α1,3

0 α2,2 α2,3

α3,1 α3,2 α3,3

mod

0 0 −a1,3

0 0 −a2,3

a3,1 a3,2 0

where, α1,1 = α2,2 = α3,3, and

ψ(r1) =

−α1,3 0 0−α2,3 0 0α1

3,1 α13,2 α1,3

mod

a1,3 0 0a2,3 0 0

a3,3 − a1,1 −a1,2 −a1,3

ψ(r2) =

0 −α1,3 00 −α2,3 0α2

3,1 α23,2 α2,3

mod

0 a1,3 00 a2,3 0−a2,1 a3,3 − a2,2 −a2,3

for some matrix (ai ,j).


InteractionXII No 4

Recall the structure of rad(g). We have [u, ri ] = ±ri , [r1, r2] = 0, and theactions on Θ, in the basis c1, c2, c3, d1, d2, d3, given by,

u =

1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 0

.

r1 =

0 0 0 0 0 00 0 0 0 0 01 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 0

r2 =

0 0 0 0 0 00 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1 0

.


InteractionXII No 5

Now, Ext1rad(g)(∆, ∆) = Der(rad(g),Endk(∆, ∆))/Triv . The rest is just

computation. In case (1), choose a = α.Then we find,

ψ(u) =

α1,1 0 00 α2,2 0

2α3,1 2α3,2 α3,3

where, α1,1 = α2,2 = α3,3, and

ψ(r1) =

0 0 00 0 0α1

3,1 α13,2 0

ψ(r2) =

0 0 00 0 0α2

3,1 α23,2 0

since α1,2 = α2,1 = 0.


InteractionXII No 6

If, in case (2), we choose a = −α, we find,

ψ(u) =

α1,1 0 2α1,3

0 α2,2 2α2,3

0 0 α3,3

where, α1,1 = α2,2 = α3,3, and

ψ(r1) =

−2α1,3 0 0−2α3,2 0 0α1

3,1 α13,2 2α1,3

ψ(r2) =

0 −2α1,3 00 −2α2,3 0α2

3,1 α23,2 2α2,3

since α1,2 = α2,1 = 0.


InteractionXII No 7

Case (1) can be interpreted as follows: Z0 := ψ(u)− αid , α = αi ,i

mediates a force, mapping d1, d2 to d3, W− := ψ(r1) and W+ := ψ(r2)do the same, but do not permute (d1, d2). In case (2), Z0 maps d3 tod1, d2, and W−,W+ sends d1, d2 to d3 and permute (d1, d2), i.e. theychange the orientation, see (6.3), and the relationship to left-andright-handedness, that is, to chirality.This seems to be very close to the weak interaction on quarks, mediatedby the weak interaction bosons, Z ,W+,W−, identified above with theBosons u, r1, r2 ∈ g, respectively.


A Canonical FiltrationI No 22

PutPh(n)(A) := im in ⊆ Ph∞(A)

The k-algebra Ph∞(A) has a descending filtration of two-sided ideals,Fn0≤n given inductively by

F1 = Ph∞(A) · im(δ) · Ph∞(A)

and

δFn ⊆ Fn+1, Fn1Fn2 . . .Fnr ⊆ Fn, n1 + . . .+ nr = n

such that the derivation δ induces derivations δn : Fn −→ Fn+1. Using thecanonical homomorphism in : Phn(A) −→ Ph∞(A) we pull the filtrationFp0≤p back to Phn(A), obtaining a filtration of each Phn(A) with,

Fn1 = Phn(A) · im(δ) · Phn(A)


A Canonical FiltrationI No 23

And inductively,

δFnp ⊆ Fn+1

p+1, Fnp1

Fnp2. . .Fn

pr⊆ Fn

p, p1 + . . .+ pr = p.


The Algebras of Higher DifferentialsI No 24

Let D(A) := lim←−n≥1Ph∞(A)/Fn, the completion of Ph∞(A) in the

topology given by the filtration Fn0≤n. The k-algebra Ph∞(A) will bereferred to as the k-algebra of higher differentials, and D(A) will be calledthe k-algebra of formalized higher differentials. Put

Dn := Dn(A) := Ph∞(A)/Fn+1

Clearly δ defines a derivation on D(A), and an isomorphism of k-algebras

ε := exp(δ) : D(A)→ D(A)

and in particular, an algebra homomorphism,

η := exp(δ) : A→ D(A)

inducing the algebra homomorphisms

ηn : A→ Dn(A)


References

1 O.A.Laudal: Cosmos and its Furniture. Cartier, Choudary, andWaldschmidt: Mathematics in the 21st Century: 6st WorldConference, Lahore, March 2013, Springer?

2 O. A. Laudal: Time-space and Space-times. Conference onNoncommutative Geometry and Representation Theory inMathematical Physics. Karlstad, 5-10 July 2004. Ed. Jurgen Fuchs,et al. American Mathematical Society, Contemporary Mathematics,Vol. 391, (2005). ISSN: 0271-4132.

3 O. A. Laudal: Geometry of Time Spaces, World Scientific, (2011).

4 O. A. Laudal and G. Pfister (1988):Local moduli and singularities.Lecture Notes in Mathematics, Springer Verlag, vol. 1310, (1988)

5 F. Mandl and G. Shaw: Quantum field theory. A Wiley-Intersciencepublication. John Wiley and Sons Ltd. (1984)

6 Harald Bjar and O. A. Laudal (1990): Deformations of Lie algebras,and Lie algebras of Deformations Comp. Math.75:69-111,1990.


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