COSMOS AND ITS FURNITURE
OLAV ARNFINN LAUDAL
Abstract. In this note I shall continue the study of the geometry of the
moduli-space of pairs of points in 3 dimensions. I shall treat the general
relativistic interactions that are mathematically well defined, and thus the
comportment of objects, and fluids, the furniture of the title. Then I shall
show that a versal family of the general associative k -algebras in dimension 4,
furnishes a possible mathematical model for a Big Bang-scenario in cosmology,
i.e. for the beginning of it all. These subjects are all treated within the set-up
of [10]. The results are somewhat unexpected, at least to me. Among them
we find a generalized Kepler movement, a generalized Schrødinger, or NavierStokes, equation, possibly taking the place of Einstein’s Field Equations, and
also a very neat version of the Dirac-equation.
Contents
1. Introduction
1.1. Philosophy
1.2. Phase Spaces, and the Dirac Derivation
1.3. Non-commutative Algebraic Geometry, and Moduli of Simple Modules
1.4. Dynamical Structures
1.5. Quantum Fields and Dynamics
1.6. Gauge groups and Invariant Theory
2. The Generic Dynamical Structures associated to a Metric
3. Time-space and Space-times
3.1. The Notions of Time and Space
3.2. Newton’s and Kepler’s Laws
3.3. Thermodynamics, the Heat Equation and Navier-Stokes
4. Cosmology, Cosmos and Cosmological Time
4.1. Background, and some Remarks on Philosophy of Science.
4.2. Deformations of Associative Algebras
4.3. Spin and Isospin
4.4. The Universe as a Versal Base sSpace
References
1
1
3
10
10
11
13
13
20
20
21
25
29
29
30
31
34
35
1. Introduction
1.1. Philosophy. In a first paper on the problem of defining Time, see [6], see also
the book [10], we sketched a toy model in physics, where the space-time of classical
physics became a section of a universal fiber space Ẽ, defined on the moduli space,
H := Hilb(2) (E3 ), of the physical systems we chose to consider, in this case the
systems composed of an observer and an observed, both sitting in Euclidean 3space, E3 . This moduli space is easily computed, and has the form H = H̃/Z2 ,
Keywords: Representations, simple modules, phase spaces, Dirac derivations, time, quantum
theory, time-space, stress-mass, heat-equation, Navier-Stokes equation, cosmology and the primordial fluid.
1
2
OLAV ARNFINN LAUDAL
where H = k[t1 , ..., t6 ], k = R and H := Spec(H) is the space of all ordered pairs of
points in E3 , H̃ is the blow-up of the diagonal, and Z2 is the obvious group-action.
The space H, and by extension, H and H̃, was called the time-space of the model.
Measurable time, in this mathematical model, turned out to be a metric ρ on the
time-space, measuring all possible infinitesimal changes of the state of the objects
in the family we are studying. This implies that the notion of relative velocity may
be interpreted as an oriented line in the tangent space of a point of H̃. Thus the
space of velocities is compact.
This lead to a physics where there are no infinite velocities, and where the
principle of relativity comes for free. The Galilean group, acts on E3 , and therefore
on H̃. The Abelian Lie-algebra of translations defines a 3-dimensional distribution,
˜ in the tangent bundle of H̃, corresponding to 0-velocities. Given a metric on H̃,
∆
we define the distribution c̃, corresponding to light-velocities, as the normal space
˜ We explain how the classical space-time can be thought of as the universal
of ∆.
space restricted to a subspace M̃ (l) of H̃, defined by a fixed line l ⊂ E3 . Under
the section Time-Space and Space-Times, we shall also show how the generator
τ ∈ Z2 , above, is linked to the operators C, P, T in classical physics, such that
τ 2 = τ P T = id. Moreover, we observe that the three fundamental gauge groups of
current quantum theory U (1), SU (2) and SU (3) are part of the structure of the
fiber space,
Ẽ −→ H̃.
In fact, for any point t = (o, x) in H, outside the diagonal ∆, we may consider the
line l in E3 defined by the pair of points (o, x) ∈ E3 × E3 . We may also consider
the action of U (1) on the normal plane Bo (l), of this line, oriented by the normal
(o, x), and on the same plane Bx (l), oriented by the normal (x, o). Using parallel
transport in E3 , we find an isomorphisms of bundles,
Po,x : Bo → Bx , P : Bo ⊕ Bx → Bo ⊕ Bx ,
the partition isomorphism. Using P we may write, (v, v) for (v, Po,x (v) = P ((v, 0)).
We have also seen, in loc.cit., that the line l defines a unique sub scheme H(l) ⊂ H.
The corresponding tangent space at (o, x), is called A(o,x) . Together this define a
decomposition of the tangent space of H,
TH = Bo ⊕ Bx ⊕ A(o,x) .
If t = (o, o) ∈ ∆, and if we consider a point o0 in the exceptional fiber Eo of H̃ we
find that the tangent bundle decomposes into,
˜
TH̃,o0 = Co0 ⊕ Ao0 ⊕ ∆,
˜ is the
where Co0 is the tangent space of Eo , Ao0 is the light velocity defining o0 and ∆
0-velocities. Both Bo and Bx as well as the bundle C(o,x) := {(ψ, −ψ) ∈ Bo ⊕ Bx },
become complex line bundles on H −∆. C(o,x) extends to all of H̃, and its restriction
to Eo coincides with the tangent bundle. Tensorising with C(o,x) , we complexify all
bundles. In particular we find complex 2-bundles CBo and CBx , on H − ∆, and
we obtain a canonical decomposition of the complexified tangent bundle. Any real
metric on H will decompose the tangent space into the light-velocities c̃ and the
˜ and obviously,
0-velocities, ∆,
˜ CTH = Cc̃ ⊕ C∆.
˜
TH = c̃ ⊕ ∆,
This decomposition can also be extended to the complexified tangent bundle of H̃.
Clearly, U (1) acts on TH , and SU (2) and SU (3) acts naturally on CBo ⊕ CBx
˜ respectively. Moreover SU (2) acts on CCo0 , in such a way that their
and C∆
COSMOS AND ITS FURNITURE
3
actions should be physically irrelevant. U (1), SU (2), SU (3) are our elementary
gauge groups.
The above example should be considered as the most elementary one, seen from
the point of view of present day physics. In fact, whenever we try to make sense
of something happening in nature, we consider ourselves as observing something
else, i.e. we are working with an observer and an observed, in some sort of ambient
space, and the most intuitively acceptable such space, today, is obviously the 3dimensional Euclidean space.
However, the general philosophy behind this should be the following. If we
want to study a natural phenomenon, called P, we must in the present scientific
situation, describe P in some mathematical terms, say as a mathematical object,
X, depending upon some parameters, in such a way that the changing aspects of P
would correspond to altered parameter-values for X. This object would be a model
for P if, moreover, X with any choice of parameter-values, would correspond to
some, possibly occurring, aspect of P.
Two mathematical objects X(1), and X(2), corresponding to the same aspect of
P, would be called equivalent, and the set, P, of equivalence classes of these objects
would be a quotient of the moduli space, M, of the models, X. The study of the
natural phenomenon P, would then be equivalent to the study of the structure of
P. In particular, the notion of time would, in agreement with Aristotle and St.
Augustin, see, [18] and [8], be any metric on this space.
Any open subset U, of M, would have a, not necessarily commutative, affine
k-algebra, A := OM (U), containing the available information about the structure
of U. An element of this algebra would be called an observable, and wishing to
measure the value of an observable, leads to the study of representations of this
algebra. In particular, we are interested in those representations that are gauge
invariant. This means, in our situation, that we have identified an action of a Lie
algebras g, on M, such that P is a quotient of M/g. Here is where invariant theory
and non-commutative algebraic geometry really enters the play, see (1.6).
With this philosophy, and this toy-model in mind, we embarked on the study of
moduli spaces of representations (modules) of associative algebras in general.
It then turned out that to obtain a complete theoretical framework for studying
the phenomenon P, together with its dynamics, we need to introduce the notion
of dynamical structure, a structure defined on the space, M. This is done via the
construction of a general, universal Phase Space, P h(M).
1.2. Phase Spaces, and the Dirac Derivation. Given an associative k-algebra
A, denote by A/k − alg the category where the objects are homomorphisms of
k-algebras κ : A → R, and the morphisms, ψ : κ → κ0 are commutative diagrams,



 
κ
R
AA
AA 0
AAκ
AA
/ R0
ψ
and consider the functor,
Derk (A, −) : A/k − alg −→ Sets.
It is, see [9], et [10], representable by a k-algebra-morphism,
ι : A −→ P h(A),
with a universal family given by a universal derivation,
d : A −→ P h(A).
4
OLAV ARNFINN LAUDAL
Ph(A) is relatively easy to compute. In particular, if A = k[x1 , .., xn ] is the polynomial algebra, we have,
P h(A) = k < x1 , .., xn , dx1 , .., dxn > /([xi , xj ], [xi , dxj ] + [dxi , xj ]).
Clearly we have the identities,
d∗ : Derk (A, A) = M orA (P h(A), A),
and,
d∗ : Derk (A, P h(A)) = EndA (P h(A)),
the last one associating d to the identity endomorphism of P h. Let now V be a
right A-module, with structure morphism ρ : A → Endk (V ). We obtain a universal
derivation,
c : A −→ Homk (V, V ⊗A P h(A)),
defined by, c(a)(v) = v ⊗ d(a). Using the long exact sequence, of Hochschild
cohomology,
0 →HomA (V, V ⊗A P h(A)) → Homk (V, V ⊗A P h(A))
ι
κ
→ Derk (A, HomA (V, V ⊗A P h(A))) → Ext1A (V, V ⊗A P h(A)) → 0,
we obtain the non-commutative Kodaira-Spencer class,
c(V ) := κ(c) ∈ Ext1A (V, V ⊗A P h(A)),
inducing, via the identity d∗ , the Kodaira-Spencer morphism,
g : ΘA := Derk (A, A) −→ Ext1A (V, V ).
If c(V ) = 0, then the exact sequence above proves that there exist an element,
∇ ∈ Homk (V, V ⊗A P h(A)) such that δ̃ = ι(∇). This is just another way of
proving that δ̃ is given by a connection,
∇ : Derk (A, A) −→ Homk (V, V ).
As is well known, in the commutative case, the Kodaira-Spencer class gives rise to
a Chern character by putting,
chi (V ) := 1/i! ci (V ) ∈ ExtiA (V, V ⊗A P h(A)),
and if c(V ) = 0, the curvature R(V ) induces a curvature class,
R∇ ∈ H 2 (k, A; ΘA , EndA (V )).
Any P h(A)-module W , given by its structure map,
ρW : P h(A) −→ Endk (W )
corresponds bijectively to an induced A-module structure on W , and a derivation
δρ ∈ Derk (A, Endk (W )), defining an element,
[δρ ] ∈ Ext1A (W, W ).
Fixing this element we find that the set of P h(A)-module structures on the Amodule W is in one to one correspondence with,
Endk (W )/EndA (W ).
Conversely, starting with an A-module V and an element δ ∈ Derk (A, Endk (V )),
we obtain a P h(A)-module Vδ . It is then easy to see that the kernel of the natural
map,
Ext1P h(A) (Vδ , Vδ ) → Ext1A (V, V ),
induced by the linear map,
Derk (P h(A), Endk (Vδ )) → Derk (A, Endk (V ))
COSMOS AND ITS FURNITURE
5
is the quotient,
DerA (P h(A), Endk (Vδ ))/Endk (V ).
Remark 1.1. 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 P h(A) also
parametrizes the set of generalized 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.
The phase-space construction may, of course, be iterated. Given the k-algebra
A we may form the sequence, {P hn (A)}1≤n , defined inductively by
P h0 (A) = A, P h1 (A) = P h(A), ..., P hn+1 (A) := P h(P hn (A)).
Let in0 : P hn (A) → P hn+1 (A) be the canonical imbedding, and let dn : P hn (A) →
P hn+1 (A) be the corresponding derivation. Since the composition of in0 and the
derivation dn+1 is a derivation P hn (A) → P hn+2 (A), there exist by universality a
homomorphism in+1
: P hn+1 (A) → P hn+2 (A), such that,
1
dn ◦ in+1
= in0 ◦ dn+1 .
1
Notice that we here compose functions and functors from left to right. Clearly we
may continue this process constructing new homomorphisms,
{inj : P hn (A) → P hn+1 (A)}0≤j≤n ,
with the property,
n
dn ◦ in+1
j+1 = ij ◦ dn+1 .
It is easy to see, [9], that,
inp in+1
= inq−1 in+1
, p<q
q
p
inp ipn+1 = inp in+1
p+1
inp in+1
= inq in+1
q
p+1 , q < p,
i.e. the P h∗ A is a semi-cosimplicial algebra. The system of k-algebras and homomorphisms of k-algebras {P hn (A), inj }n,0≤j≤n has an inductive (direct) limit,
P h∞ A, together with homomorphisms,
in : P hn (A) −→ P h∞ (A)
satisfying,
inj ◦ in+1 = in , j = 0, 1, .., n.
Moreover, the family of derivations, {dn }0≤n define a unique derivation,
δ : P h∞ (A) −→ P h∞ (A),
such that,
in ◦ δ = dn ◦ in+1 ,
and it is easy to see that this is a universal construction, i.e. for any pair of a
morphism,
i : A −→ B
and a derivation ξ ∈ Derk (B), ι ◦ ξ factorizes via P h∞ (A), and δ. Put
P h(n) (A) := im in ⊆ P h∞ (A), .
The k-algebra P h∞ (A) has a descending filtration of two-sided ideals, {Fn }0≤n
given inductively by:
F1 = P h∞ (A) · im(δ) · P h∞ (A)
and,
δ(Fn ) ⊆ Fn+1 , Fn1 Fn2 ...Fnr ⊆ Fn , n1 + ... + nr = n
6
OLAV ARNFINN LAUDAL
such that the derivation δ induces derivations,
δn : Fn −→ Fn+1 .
Using the canonical homomorphism in : P hn (A) −→ P h∞ (A) we pull the filtration
{Fp }0≤p back to P hn (A), not bothering to change the notation.
Definition 1.2. Let D(A) := limn≥1 P h∞ (A)/Fn , be the completion of P h∞ (A) in
←−
the topology given by the filtration {Fn }0≤n . The k-algebra P h∞ (A) will be referred
to as the k-algebra of higher differentials, and D(A) will be called the k-algebra of
formalized higher differentials. Put,
Dn := Dn (A) := P h∞ (A)/Fn+1
Clearly δ defines a derivation on D(A), and an isomorphism of k-algebras,
:= exp(δ) : D(A) → D(A).
and in particular, an algebra homomorphism,
η̃ := exp(δ) : A → D(A),
inducing the algebra homomorphisms,
η̃n : A → Dn (A),
which, by killing, in the right hand algebra, the image of the maximal ideal, m(t), of
A corresponding to a point t ∈ Simp1 (A), induces a homomorphism of k-algebras,
η̃n (t) : A → Dn (A)(t) := Dn /(Dn m(t)Dn ),
and an injective homomorphism,
η̃(t) : A → lim Dn (A)(t).
←−
n≥1
Definition 1.3. The universal derivation, δ ∈ Derk (P h∞ (A)) will be called the
Dirac derivation.
Recall now that for any associative k-algebra B, and for any right B-modules V ,
W , there is an exact sequence,
HomB (V, W ) → Homk (V, W ) → Derk (B, Homk (V, W ) → Ext1B (V, W ) → 0,
where the image of,
η : Homk (V, W ) → Derk (B, Homk (V, W ))
is the sub-vectorspace of trivial (or inner) derivations.
Modulo the trivial (inner) derivations, any derivation δ ∈ Derk (B, Endk (V ))
therefore defines a class,
ξ(v) ∈ Ext1B (V, V ),
i.e. a tangent vector of the formal moduli of the representation V , at the unique
point.
Remark 1.4. The above implies that any representation, V , of B = P h∞ (A),
corresponds to a family of P hn (A)-module-structures on V , for n ≥ 1, i.e. to
an A-module V0 := V , an element ξ0 ∈ Ext1A (V, V ), i.e. a tangent of the deformation functor of V0 := V , as A-module, an element ξ1 ∈ Ext1P h(A) (V, V ), i.e.
a tangent of the deformation functor of V1 := V as P h(A)-module, an element
ξ2 ∈ Ext1P h2 (A) (V, V ), i.e. a tangent of the deformation functor of V2 := V as
P h2 (A)-module, etc. All this is just V , considered as an A-module, together with
a sequence {ξn }, 0 ≤ n, of a tangent, or a momentum, ξ0 , an acceleration vector,
ξ1 , and any number of higher order momenta ξn . Thus, specifying a P h∞ (A)) representation V , implies specifying a formal curve through v0 , the base-point, of the
miniversal deformation space of the A-module V .
COSMOS AND ITS FURNITURE
7
Now let A = k[t1 , ..., td ], and consider any representation of P h∞ (A) as kalgebra,
ρ : P h∞ (A) → Endk (V ),
V a k-vector space. There are corresponding homomorphisms of k-algebras,
ρ[τ ]
A → P h∞ (A) → Endk (C) ⊗k k[[τ ]],
defined by,
Dip := ρ(dp ti ), Xi := ρ(τ )(ti ) = ρ(exp(τ δ)) =
X
τ p /p!Dip .
p≥0
Since [ti , tj ] =P0, we must have [Xi , Xj ] = 0, and, since the relations in P h∞ (A)
are given by, p+q=n≥0 1/p!q! [dtpi , dqj ] = 0, this is exactly the condition
X
1/p!q! [Dip , Djq ] = 0,
p+q=n≥0
for the family of matrices {Dip , p ≥ 0, i = 1, ..., d} to define a homomorphism, ρ,
of k-algebras.
Clearly if dimV = 1 there are no conditions, and we may pick arbitrarly Dip ∈ k,
and obtain formal power series,
X
Xi =
Din /n!τ n ,
n
which, when convergent, gives the dynamics of the point with respect to the coupling
constants Dip .
Assume dimV = 2, and put,
0
xi (1)
0
αi (1)
0
0
ρ(ti ) = Di =
=:
,
0
xi (2)
0
αi0 (2)
and, αi0 (r, s) := xi (r) − xi (s), r, s = 1, 2. Let, for q ≥ 0,
q
α (1) riq (1, 2)
Diq = q i
,
ri (2, 1) αiq (2)
Put,
αil (r, s) : = αil (r) − αil (s), r, s = 1, 2,
κpi = αip (p, q)/αi0 (p, q)
k X
k
k
ri (r, s) =
σk−l αil (r, s), r, s = 1.
l
l=0
where the sequence {σl }, l = 0, 1, ... is an arbitrary sequence of coupling constants,
with σ0 = 0. Then this defines a representation,
ρ : P h∞ (A) → Endk (V ),
if κpi = κpj = κp , i, j = 1, ..., d.
The situation above arises when we consider two (different) points
P1 = (α10 (1), ..., αd0 (1)), P2 = (α10 (2), ..., αd0 (2)),
in space, with pre-described tangents, ξ1 = (α11 (1), ..., αd1 (1)) and ξ2 = (α11 (2), ..., αd1 (2)).
For these two points, considered as dimension 1 representations of P h(A), we
know that there is a 1-dimensional space of tangents between the points, i.e. the
Ext1P h(A) (k(P1 ), k(P2 )) = k. This leads to possibly non-zero elements ri1 (1, 2) =
−ri1 (2, 1)) in the matrix representation of the non-commutative deformation of the
family {k(P1 ), k(P2 )} of P h(A)-modules.
8
OLAV ARNFINN LAUDAL
We now have a much more complete picture of the situation. The dynamics of the
pair of points is described by the Dirac derivation. Assuming that for time τ = 0,
we know the position (α0 (1), α0 (2)), and the momenta α1 (1), α1 (2), the dynamics
is described, in terms of the time, τ , by the matrices, Xi = ρ(exp(τ δ))(ti ), where
we have the explicit formulas,
αi0 (1) + τ αi1 (1)
σ(αi0 (1, 2) + αi1 (1, 2))
Xi (τ ) =
, i = 1, ..., d.
σ(αi0 (2, 1) + αi1 (2, 1))
αi0 (2) + τ αi1 (2)
The trace, and determinant are,
tr(Xi ) = (αi0 (1) + αi0 (2)) + τ (αi1 (1) + αi1 (2))
det(Xi ) = (αi0 (1) + τ αi1 (1)) × (αi0 (2) + τ αi1 (2)) + σ 2 (αi0 (1, 2) + αi1 (1, 2))2 .
The spectrum of Xi , or the eigenvalues, are given as,
p
αi0 (τ ) = 1/2(tr(Xi ) ± tr(Xi )2 − 4det(Xi )).
From this we see that if all coupling constants vanish, i.e. if σ = 0, then we have
undisturbed linear motions of the two points, αi0 (τ ) = αi0 + τ αi1 . Writing out the
Taylor series of α0 (τ ), we find for i = 1, ..., d, p, q = 1, 2, p 6= q,
αi0 (p)(τ ) = αi0 (p) + τ αi1 (p) − (1/2τ 2 σ12 (1 + κ)2 αi0 (p, q) + ...
The corresponding spectrum for
αi1 (p)(τ )
=
dXi
dτ
αi1 (p)
= ρ(exp(τ δ))(dti ) is of course given by,
− τ σ12 (1 + κ)2 αi0 (p, q) + ...
In general, considering an Object, P in space, consisting of r points {Pp }p=1,...,r ,
we find that the dynamics is closely related to the interaction process, involving
non-commutative deformation of families of representations, described in [10], in
fact, we easily obtain the following theorem,
Theorem 1.5. Given an object, P in d-space, consisting of r points {Pp }p=1,...,r .
With the notations above, in particular, αil (r, s) = αil (r)−αil (s), consider the matrix,
 k

αi (1) rik (1, 2) ... rik (1, r)
rk (2, 1) αk (2) ... rk (2, r)
i
i
i

Dik := 
 .
.
...
. 
rik (r, 1) rik (r, 2) ... αik (r))
with,
ri0 (p, q) = 0, rik (p, q) =
k X
k
l=1
l
αil (p, q)σk−l (p, q), p,
where σm (p, q) ∈ k are arbitrary coupling constants, with σ0 = 0. Then these
operators define a representation,
ρ : P h∞ (A) → M r (k),
if, for all n ≥ k,
X n
X n
h
0
α (p, q)αj (p, q)σn−h (p, q) −
α0 (p, q)αjh (p, q)σn−h (p, q)
h i
h i
h
h
X X nn − k X k m
=
αj (p, s)σn−k−m (p, s)
αil (s, q)σk−l (s, q)
k
m
l
k,m s
l
X X nk X n − k −
αil (p, s)σk−l (p, s)
αjm (s, q)σn−k−m (s, q)
k
l
m
s
m
k,l
COSMOS AND ITS FURNITURE
9
We have already looked at the case r = 2, and seen that the result makes physical
sense. For r = 3, n = 2 we find,
σ1 (p, q)(αi1 (p, q)αj0 (p, q) − αi0 (p, q)αj1 (p, q)) =
σ1 (p, s)σ1 (s, q)(αj0 (p, s)αi0 (s, q) − αi0 (p, s)αj0 (s, q)).
In dimension d = 3 this has a particularly nice interpretation. Let α0 (i, j) be
the vector starting at Pi and ending at Pj , and let ξi be a tangent vector at Pi for
i = 1, 2, 3. Put α1 (i, j) = ξi − ξj , then 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 relative momentum
must sit in the plane defined by the three points, the length being determined by
the 3 coupling constants. Moreover, the sum of all three momenta must be 0. All
this is just relativity, since the three points are all there is in our 3-dimensional
space.
In this case, the spectrum of Xi will give us very complex trajectories, complete
with all kinds of turbulence, where we no longer may talk about different trajectories. The points may merge. One should also ask the question whether we ever
may talk about points or particles with just a position and a momentum associated? We shall come back to this situation in section 3 in relation with the theory
of relativity.
With the same notations, the general formula above takes the form,
X n
σn−h (p, q)(αih (p, q)αj0 (p, q) − αi0 (p, q)αjh (p, q))
h
h
X n!σn−k−m (p, s)σk−l (s, q)
(αm (p, s)αil (s, q) − αil (p, s)αjm (s, q)),
=
l!m!(k − l)!(n − k − m)! j
k,l,m,s
and we recognize the formula defining the higher order Massey products, see [3],
[4], and also [10], chapter 5. We may use these explicit formulas to show that
there is a more general set-up, related to Grothendieck’s generalized differential
algebra, appearing as infinite limits of these finite dimensional representations. In
fact, consider an A-module E, and an extension of this representation to,
ρ : P h∞ (A) → Endk (E).
We have seen that ρ must be given in terms of operators,
Dip := ρ(dp ti ) ∈ Endk (E),
satisfying the conditions,
∀n ≥ 0,
X
1/p!q![Dip Djq ] = 0.
p+q=n
There is an obvious family of solutions of these equations, given by any differential
operator,
Q ∈ Dif f (E),
p
0
with, Di = ρ(ti ) = ti , Di := ad(Q)p (ti ) ∈ Dif f (C), p ≥ 1. If E = ΘA , the
formulas above,
αi0 (p)(τ ) = αi0 (p) + τ αi1 (p) − 1/2τ 2 σ12 (1 + κ)2 αi0 (p, q) + ...
αi1 (p)(τ ) = αi1 (p) + τ 2 σ12 (1 + κ)2 αi0 (p, q) + ...,
may be interpreted as follows. Given a vector field ψ ∈ ΘA , and a pair of points
P, Q close to each other, the dynamics transports the point P = α0 (p) to Q =
α0 (p)+τ α1 (p), and it transport the tangent vector ψ(P ) = α1 (p) to α1 (q) = α1 (p)−
10
OLAV ARNFINN LAUDAL
τ σ12 (1 + κ)2 α0 (p, q), i.e. we have a difference operator, obviously the discretization
of a derivation operator acting on the vector field, along itself!
This is consistent with the dynamics of a limit of finite representations being
the time derivative of a connection, acting on vector fields, see section 2, and (3.3).
Any connection may in this way, be considered limit of finite dimensional representations, extending a very special result about the harmonic oscillator, see [10]. The
Levi-Civita connection, becomes a special case, for which the differential operator,
Q, is the Laplace-Beltrami operator, and the interaction is given by gravitation,
represented by the metric, see the sections 2 and 3.
1.3. Non-commutative Algebraic Geometry, and Moduli of Simple Modules. The basic notions of affine non-commutative algebraic geometry related to a
(not necessarily commutative) associative k-algebra, for k an arbitrary field, have
been treated by many authors in several texts, see e.g.[13], [14], [4], [5], [6]. Given a
finitely generated algebra A, we prove the existence of a non-commutative schemestructure on the set of isomorphism classes of simple finite dimensional representations, i.e. right modules, Simp<∞ (A).
We show, in [6], that any geometric k-algebra A, see also [10], may be recovered from the (non-commutative) structure of Simp<∞ (A), and that there
is an underlying quasi-affine (commutative) scheme-structure on each component
Simpn (A) ⊂ Simp<∞ (A), parametrizing the simple representations of dimension
n. In fact, we have shown the following,
Theorem 1.6. There is a commutative algebra C(n) with an open subvariety
U (n) ⊆ Simp1 (C(n)), an étale covering of Simpn (A), over which there exists
a versal representation Ṽ ' C(n) ⊗k V , a vector bundle of rank n defined on
Simp1 (C(n)), and a versal family, i.e. a morphism of algebras,
ρ̃ : A −→ EndC(n) (Ṽ ) → EndU (n) (Ṽ ),
inducing all isoclasses of simple n-dimensional A-modules.
Suppose, in line with our Philosophy that we have uncovered the moduli space,
M, of the mathematical models X, of our phenomena P, and that A is the affine
k-algebra of this space, assumed to contain all the parameters of our interest. Then
the above construction furnishes the Geometric Landscape on which our Quantum
Theory will be based.
Notice that, EndC(n) (Ṽ ) ' Mn (C(n)), and we shall, in the sequel, use this
isomorphism without further warning.
1.4. Dynamical Structures. We would now like to use this theory for the kalgebra P h∞ (A). However, P h∞ (A) is rarely of finite type. We shall therefore introduce the notion of dynamical structure, and the order of a dynamical structure,
to reduce the problem to a situation we can handle. This is also what physicists do,
they invoke a parsimony principle, or an action principle, originally proposed by
Fermat, and later by Maupertuis, with exactly this purpose, reducing the preparation needed, to be able to see ahead.
Definition 1.7. A dynamical structure, σ, is a two-sided δ-stable ideal (σ) ⊂
P h∞ (A), such that
A(σ) := P h∞ (A)/(σ),
the corresponding, dynamical system, is of finite type. A dynamical structure, or
system, is of order n if the canonical morphism,
σ : P h(n−1) (A) → A(σ)
COSMOS AND ITS FURNITURE
11
is surjective. If A is generated by the coordinate functions, {ti }i=1,2,...,d a dynamical
system of order n may be defined by a force law, i.e. by a system of equations,
δ n tp = Γp (ti , dtj , d2 tk , .., dn−1 tl ), p = 1, 2, ..., d.
Put,
A(σ) := P h∞ (A)/(δ n tp − Γp )
where σ := (δ n tp −Γp ) is the two-sided δ-ideal generated by the defining equations of
σ. Obviously δ induces a derivation δσ ∈ Derk (A(σ), A(σ)), also called the Dirac
derivation, and usually just denoted δ.
Notice that if σi , i = 1, 2, are two different order n dynamical systems, then we
may well have,
A(σ1 ) ' A(σ2 ) ' P h(n−1) (A)/(σ∗ ),
as k-algebras.
1.5. Quantum Fields and Dynamics. Given any dynamical structure σ, any
family of components of Simp(A(σ)), with its versal family Ṽ , will, in the sequel,
be called a family of particles. A state of the particle, i.e. a section φ ∈ Ṽ , is now
a function on the moduli space Simp(A), not just a function on the configuration
space, Simp1 (A), nor Simp1 (A(σ)). The value φ(v) ∈ Ṽ (v) of φ, at some point
v ∈ Simpn (A), will be called a state of the particle, at the event v.
EndC(n) (Ṽ ) induces also a bundle, of operators, on the étale covering U (n) of
Simpn (A(σ)). A section, ψ of this bundle will be called a quantum field. In particular, any element a ∈ A(σ) will, via the versal family map, ρ̃, define a quantum
field, and the set of quantum fields form a k-algebra.
Physicists will tend to be uncomfortable with this use of their language. A
classical quantum field for any traditional physicist is, usually, a function ψ, defined
on some configuration space, (which is not our Simpn (A(σ)), with values in the
polynomial algebra generated by certain creation and annihilation-operators in a
Fock-space.
As we have shown in the book, [10], this interpretation may be viewed as a special
case of our general set-up. There we have also introduced Planck’s constant(s),
Fock-space, Bosons, Fermions, and Super-symmetry, etc. Since these problems will
not play any part in the following, we just refer to [10]. We shall, however, come
back to the notions spin, hyper-spin, and quarks in the next subsection, and finally
in section 4. Pick any v ∈ Simpn (A(σ)) corresponding to the right A(σ)-module
V , with structure homomorphism ρv : A(σ) → Endk (V ), then the Dirac derivation,
as we have seen, δ composed with ρv , gives us an element,
δv ∈ Ext1A(σ) (V, V )).
The Dirac derivation δ therefore defines a unique one-dimensional distribution
in ΘSimpn (A(σ)) , which, once we have fixed a versal family, defines a vector field,
ξ ∈ ΘSimpn (A(σ)) ,
and, in good cases, a (rational) derivation,
ξ ∈ Derk (C(n)).
Notice now that any right A(σ)-module V is also a P h∞ (A)-module, and therefore corresponds to a family of P hn (A)-module-structures on V , for n ≥ 1, i.e.
to an A-module V0 := V , an element ξ0 ∈ Ext1A (V, V ), i.e. a tangent of the deformation functor of V0 := V , as A-module, an element ξ1 ∈ Ext1P h(A) (V, V ), i.e.
a tangent of the deformation functor of V1 := V as P h(A)-module, an element
ξ2 ∈ Ext1P h2 (A) (V, V ), i.e. a tangent of the deformation functor of V2 := V as
P h2 (A)-module, etc. All this is just V , considered as an A-module, together with
12
OLAV ARNFINN LAUDAL
a sequence {ξn }, 0 ≤ n, of a tangent, or a momentum, ξ0 , an acceleration vector, ξ1 , and any number of higher order momenta ξn . Thus, specifying a point
v ∈ Simpn (A(σ)) implies specifying a formal curve through v0 , the base-point, of
the miniversal deformation space of the A-module V .
Knowing the dynamical structure, (σ), and the state of our object V at a time
τ0 , i.e. knowing the structure of our representation V of the algebra A(σ), at that
time (which is a problem that we shall return to), the above makes it reasonable
to believe that we, from this, may deduce the state of V at any later time τ1 .
This assumption, on which all of science is based, is taken for granted in most
textbooks in modern physics. This paper is, in fact, another attempt to give this
basic assumption a reasonable basis. The mystery is, of course, why Nature seems
to be parsimonious, in the sense of Fermat and Maupertuis, giving us a chance of
guessing dynamical structures.
We now have some fundamental results, proved in [10].
Theorem 1.8. Formally, at any point v ∈ U (n), with local ring Ĉ(n)v , there is a
ˆ ), and an Hamiltonian Q ∈ End ˆ (Ṽv ), such that,
derivation [δ] ∈ Derk (C(n)
v
C(n)
v
as operators on Ṽv , we must have,
δ = [δ] + [Q, −].
ˆ ),
This means that for every a ∈ A(σ), considered as an element ρ̃(a) ∈ Mn (C(n)
v
δ(a) acts on Ṽv as
ρ̃(δ(a)) = [δ](ρ̃(a)) + [Q, ρ̃(a)].
As pointed out above, there are local (or even global) extensions of this result,
where [δ] and Q may be assumed to be defined (rationally) on C(n), see[10].
This Q ∈ Endk (V ), the Hamiltonian of the system, is in the singular case, when
[δ](v) = ξ(v) = 0, also called the Dirac operator, and sometimes denoted δ-slashed,
see e.g. [16], or other texts on Connes’ spectral tripples. In fact, a spectral tripple
is composed of a vector space like V , together with a Dirac operator, like Q, and a
complexification etc.
If [δ](v) = 0, it is also easy to see that what we have observed implies that Heisenberg’s and Schrödinger’s way of doing quantum mechanics, are strictly equivalent.
In line with our general philosophy, we shall consider ξ, or [δ] as measuring time
in Simpn (A(σ)), respectively in Spec(C(n)).
Assume for a while that k = R, the real numbers, and that our constructions go
through, as if k were algebraically closed. Let v(τ 0 ) ∈ Simpn (A(σ)) be an element,
an event. Suppose there exist an integral curve γ of ξ through v(τ0 ) ∈ Simp1 (C(n)),
ending at v(τ1 ) ∈ Simp1 (C(n)), given by the automorphisms e(τ ) := exp(τ [δ]), for
τ ∈ [τ0 , τ1 ] ⊂ R. The supremum of τ for which the corresponding point, v(τ ), of γ
is in Simpn (A(σ)) should be called the lifetime of the particle.It is relatively easy
to compute these lifetimes, when the fundamental vector field ξ has been computed.
Let now φ(τ0 ) ∈ Ṽ (v0 ) ' V be a (classically considered) state of our quantum
system, at the time τ0 , and consider the (uni)versal family,
ρ̃ : A(σ) −→ EndC(n) (Ṽ )
restricted to U (n) ⊆ Simp1 (C(n)), the étale covering of Simpn (A(σ)). We shall
consider A(σ) as our ring of observables.
What happens to φ(τ0 ) ∈ V (0) when time passes from τ0 to τ , along γ? Notice
that, in the classical quantum-theoretical case, one works with one fixed representation, corresponding to what we have called a singular point of ξ. This implies
that we are looking at a representation V with ξ(v) = 0, and so we have no time.
What we call time is then the parameter of the one-parameter automorphism group
COSMOS AND ITS FURNITURE
13
u(τ ) := exp(τ Q) acting on V . This also leads to a Schrödinger equation, and to
the next result, proving that ψ is completely determined, along any integral curve
γ by the value of ψ(τ0 ), for any τ0 ∈ γ.
Theorem 1.9. The evolution operator u(τ0 , τ1 ) that changes the state ψ(τ0 ) ∈
Ṽ (v0 ) into the state φ(τ1 ) ∈ Ṽ (v1 ), where τ1 − τ0 is the length of the integral curve
γ connecting the two points v0 and v1 , i.e. the time passed, is given by,
Z
ψ(τ1 ) = u(τ0 , τ1 )(φ(τ0 )) = exp[ Q(τ )dτ ] (φ(τ0 )),
γ
R
where exp γ is the non-commutative version of the ordinary action integral, essentially defined by the equation,
Z
Z
Z
exp[ Q(τ )dt] = exp[
Q(τ )dτ ] ◦ exp[
Q(τ )dτ ]
γ
γ2
γ1
where γ is γ1 followed by γ2 .
1.6. Gauge groups and Invariant Theory. We may use the above in an attempt
to make precise the notion of gauge group, and gauge invariance, and thus be able
to understand why the physicists define their objects, the particles, the way they
do.
Suppose, with the notations of (1.5), that we have identified a Lie algebra g, of
infinitesimal automorphisms, i.e. derivations, of A(σ), the ”moduli space” of our
mathematical objects X, leaving invariant the physical properties of our phenomena
P. We would then be led to restrict our representations, V , to those that are
invariant under g, i .e. to those for which, [ξ]V = 0, for all ξ ∈ g, see Theorem
(1.8). This would then imply that the corresponding Hamiltonians, q(ξ) := Qξ
define a representation,
q : g −→ Endk (V ),
such that, ρ(ξ(a)) = [q(ξ), ρ(a)]. Put,
Rep(A(σ), g) := {V ∈ Rep(A(σ))| [ξ]V = 0, ∀ξ ∈ g},
and consider the obvious sub-quotient of iso-classes of simple (A(σ), g)-representations,
Simp(A(σ), g) ⊂ Rep(A(σ), g)/ ∼ .
This space, Simp(A(σ), g), together with its non-commutative structure sheaf,
would in [6] have been called the quotient space of the space A(σ) with respect
to the action of g.
As we shall see, the gauge ”group” of the standard model,
g := u(1) × su(2) × su(3),
which is part of our toy model, see [10], also pops up in our cosmological model,
section 4, and now as a real gauge ”group” in the above sense. The elementary
particles in that model should therefore, in line with the usage of present quantum
physics, be characterised by certain simple representations and their eigenvectors.
See section 4.
2. The Generic Dynamical Structures associated to a Metric
Now,
P let C = k[t1 , ..., tn ] be a commutative polynomial k-algebra, and let g =
1/2 i,j=1,..,r gi,j dti dtj ∈ P h(C), be a Riemannian metric. Recall the formula for
the Levi-Civita connection,
X
∂gj,k
∂gi,j
∂gk,i
+
−
).
gl,k Γlj,i = 1/2(
∂tj
∂ti
∂tk
l
14
OLAV ARNFINN LAUDAL
Since in P h∞ (C), we have,
X
X
∂gi,j
δ(g) =
dtk dti dtj +
gi,j (d2 ti dtj + dti d2 tj ),
∂tk
i,j,=1,..,r
i,j,k=1,..,r
we may plug in the formula,
δ 2 tl = −Γl := −
X
Γli,j dti dtj .
on the right hand side, and see that we have got a solution of the Lagrange equation,
δ(g) = 0,
in the commutative situation. This solution has the form of a force law,
X
d2 tl = −Γl := −
Γli,j dti dtj ,
generating a dynamical structure (σ) := (σ(g)) of order 2. The dynamic system is,
of course, as an algebra,
C(σ) = k[t, ξ]
where ξj is the class of dtj . The Dirac derivation now takes the form,
X
∂
∂
− Γl
),
δ=
(ξl
∂tl
∂ξl
l
and the fundamental vector field [δ] in Simp1 (C(σ)) = Spec(k[ti , ξj ]), is, of course,
the same. Therefore [δ](g) = 0, which means that g is constant along the integral
curves of [δ] in Simp1 (P h(C)), which projects onto Simp1 (C) to give the geodesics
of the metric g, with equations,
X
ẗl = −
Γli,j ṫi ṫj .
i,j
We may also look at this from another point of view. Suppose given any dynamical structure with Dirac derivation δ on P h(C). Consider Simp1 (P h(C)). It is
obviously represented by C(1) := k[t, ξ], and the Dirac derivation induces a derivation [δ] ∈ Derk (C(1)), and the Hamiltonian must vanish. Therefore we have two
options for the same notion of time in the picture, g and [δ]. The last derivation
must therefore be a Killing vector field, i.e. we must have a solution of Lagrange
equation,
[δ](g) = 0.
As a contrast to this 1-dimensional representation case, let us consider infinite
dimensional representations, for which the Dirac derivation [δ] vanishes, and the
notion of time is taken care of by the Hamiltonian Q. We have already seen, in
(1.2), that there are for any commutative k-algebra C, and any differential operator
Q ∈ Dif f (C) a representation, ρ : P h∞ (C) → Endk (C), that let ρ(f ) for f ∈ C
act as multiplication in C, and otherwise given by the formula, ρ(da) = [Q, a]. Now
we shall show how this can be generalized to representations on C-free modules, i.e.
bundles on Spec(C), coupled with the existence of a metric g in P h(C), representing
time in Spec(C).
For this purpose, consider first the Levi-Civita-connection,
∇ : ΘC −→ Endk (θC )
expressed in coordinates as,
∇δi (δj ) =
X
l
Γlj,i δl ,
COSMOS AND ITS FURNITURE
15
∂
where we have put δi :=
. Classsically we define the curvature tensor Ri,j (δk ) =
∂t
i
P l
l Ri,j,k δl , of a connection ∇, as the obstruction for ∇ to be a Lie-algebra homomorphism. We find,
X
l
([∇δi , ∇δj ] − ∇[δi ,δj ] )(δk ) =
Ri,j,k
δl .
l
This, we shall see, is a commutative version of the more precise notion of curvature,
related to a more general dynamic system, to be studied below.
Recall that the Ricci tensor is given as,
X j
Rici,k (g) =
Ri,j,k
j
and that, assuming the metric is non-degenerate with inverse g k,i , one defines the
scalar curvature of g, as,
X
S(g) :=
g k,i Rici,k .
k,i
These are fundamental metric invariants, components of Einstein’s Field Eqations,
Ric − 1/2S(g)g = U,
where U is the stress-mass tensor.
Remark 2.1. One purpose of this paper is to make an effort to understand the
nature of this Stress-Mass Tensor, U . In fact we question this equation, and will
propose an alternative one, better adapted to the quantum-theoretical scenario developped above. It is the replacement for U , in this alternative equation, that we
have termed the Furniture of the space under consideration. In particular this may
be of interest when we talk about the Universe, i.e. about Cosmology, see the last
section of this note.
Pd
A non-degenerate metric, g = 1/2 i=1 gi,j dti dtj ∈ P h(C) induces an isomorphism of C-modules
ΘC = HomC (ΩC , C) ' ΩC ,
and a corresponding dynamical system,
(σ) = ([dti , tj ] − g i,j )
It is easy to see that this is not violating the relations, [dti , tj ] + [ti , dtj ] = 0 of
P h(C). Notice also that in C(σ) we have,
∂g i,p
∂g j,p
− g jk
,
∂tl
∂tk
meaning that the curvature,
Ri,j = [dti , dtj ], does not commute with the action of
¯ i := P gi,p dtp , we find that [dt
¯ i , tj ] = δi,j . Moreover, if we let,
C. Introducing dt
[[dti , dtj ], tk ] = g il
g := 1/2
d
X
¯ 2,
dt
i
i=1
¯ i . Using the above, we find that there is a one-to-one
we find, ad(g)(ti ) = dt
correspondence between connections ∇ on a C-module V and morphisms,
ρ∇ : C(σ) → Endk (V ),
defined by,
ρ∇ (dti ) =
X
j
where ξi :=
i,j
j g δj is the dual to dti .
P
g i,j ∇δj = ∇ξi ,
16
OLAV ARNFINN LAUDAL
Consider now the Levi-Civita connection ∇δi =
∂
+ ∇i , where,
∂ti
∇i ∈ EndC (ΘC ),
is given by the matrix formula,
∇i := (∇i (p, q)) = (Γqp,i ).
Put,
T := 1/2
X ∂g j,k
∂tj
j,k
dtk = 1/2
X ∂g j,k
gk,l dtl
∂tj
j,k,l
and consider the inner derivation of C(σ),
δ := ad(g − T ).
After a dull computation, using the well known formula for Levi-Civita connection,
X
∂gi,j
=
(Γlk,i gl,j + Γlk,j gi,l )
∂tk
l
X
∂g r,j
(g r,l Γjk,l + g l,j Γrk,l ),
=−
∂tk
l
we obtain, in C(σ),
T : = −1/2(
X
Γkk,l dtl +
k,l
X
g k,q Γpk,q gp,l dtl )
k,p,q
δ(ti ) : = ad(g − T )(ti ) = dti , i = 1, ..., d.
Therefore we have a well-defined dynamical structure (σ), with Dirac derivation
δ := ad(g − T ). It is easy to see that (σ) is invariant w.r.t. isometries.
Moreover, the representation, ρ = ρΘ of C(σ), defined on ΘC , by the Levi-Civita
connection, has a Hamiltonian,
X
Q := ρ(g − T ) = 1/2
g ij ∇δi ∇δj ,
i,j
i.e. the generalized Laplace-Beltrami operator, which is also invariant w.r.t. isometries, although the proof demands some algebra.
Again we have two options for the notion of time. In line with classical Quantum
Theory, we would, for a state ψ ∈ ΘC , consider the Schrødinger equation,
dξ
= Q(ξ),
dτ
where τ would be an ad hoc chosen time parameter. In our case there are just
two natural choices, namely ξ itself, or the metric g, measuring the time t, see the
discussion at the end of (1.2). Since we have,
Dξ (ξ) = µ
dξ
,
dt
where µ = g(ξ, ξ)1/2 , it seems reasonable to replace the classical Schrødinger equation, in our situation, by the following,
dξ
= Q(ξ).
dt
We shall, return to this last equation, in the next section.
Now, put,
X
¯ l := (Γ̄j )
Γ̄ip,q :=
g r,i Γlr,p gl,q , ∇
i,l
l,r
COSMOS AND ITS FURNITURE
17
then,
T =
X
Tl dtl
l
X
¯ l ).
Tl = −1/2( (Γjj,l + Γ̄jj,l ) = −1/2(trace∇l + trace∇
j
Since δ(ti ) := ad(g − T )(ti ) = dti , the general force law, in C(σ), looks like,
X
d2 ti = [g − T, dti ] = −1/2
(Γ̄ip,q + Γ̄iq,p )dtp dtq
p,q
+ 1/2
X
gp,q (Rp,i dtq + dtp Rq,i )
p,q
+ [dti , T ],
where, as above, Ri,j = [dti , dtj ]. Put,
X
X
i,j
Γj,i
g j,k Γik,p , Fi,j := Ri,j −
(Γj,i
p =
p − Γp )dtp ,
p
k
then we find, see [10], for a proof.
Theorem 2.2 (General Force Law). In C(σ) we have the following force law,
d2 ti = −
X
Γip,q dtp dtq − 1/2
p,q
X
gp,q (Fi,p dtq + dtp Fi,q )
p,q
+ 1/2
X
q,i
gp,q [dtp , (Γi,q
l − Γl )]dtl + [dti , T ].
l,p,q
We shall consider the above formula as a general Force Law, in P h(C), induced
by the metric g. As explained before, this means the following: Let c be the δ- stable
ideal generated by this equation in P h∞ (C). Since the force law above holds in the
dynamical system defined by (σ), we obviously have c ⊂ (σ), and we might hope
this new dynamical system leads to a Quantum Field Theory, as defined above,
with new and interesting properties.
Remark 2.3. Connections and Yang-Mills Theory. Let A = k[t1 ,P
..., tn ], be any
plynomial algebra over a field k. Consider a metric g ∈ P h(A), g = i,j gi,j dti dtj ,
and as above, the corresponding generic dynamical system,
A(σ) = P h(A)/([dti , tj ] − g i,j ),
There are no finite dimensional representations of such algebras, and we are led
to consider another method to obtain a real dynamical structure in this situation.
This is, in fact, an example of a singular situation, where the introduction of a
Lagrangian is important. We shall not elaborate on this here, the relevant theory, in
our tapping, is found in the book [10]. But the treatment of the notion of interaction,
as we see it, demands a little comment on this situation. The space of iso-classes of
representations, ρ, of A(σ) on the A-module ΘA , is identified with the space of isoclases of the corresponding connections ∇. It does not form an algebraic variety,
but it has a nice structure.
Fix a representation ρ0 , corresponding to the connection ∇0 , say the trivial connection, depending on some chosen coordinates, and define for every representation
ρ corresponding to a connection ∇, an A-linear map,
(1)
P := ∇ − ∇0 : ΘA → EndA (ΘA )
18
OLAV ARNFINN LAUDAL
The set of such maps, the potentials, is a vector space P. It is now easy to
prove that the set of iso-classes of representations of A(σ), on the A-module ΘA , is
the quotient of the infinite dimensional affine space P via the action of the group
G := AutA (ΘA ), defined for Φ ∈ G, and ρ ∈ P, by
µ : G → Autk (P), µ(Φ)(ρ) = Φ−1 ρΦ.
At the Lie algebra level, there is an action κ of the Lie algebra, EndA (ΘA ) of G, on
P, such that for φ ∈ EndA (ΘA ), and for P = (P1 , ..., Pn ), Pi := P (dti ), we have,
X
∂φ
κ(φ)(Pi ) =
g i,j (
+ [Pj , φ])
∂tj
j=1,...,n
In particular the tangent space of P at a representation ρ, corresponding to a connection with potential P, is identified with the quotient vector space
∂φ
∂φ
+ [P1 , φ], ...,
+ [Pn , φ]).
P/(
∂t1
∂tn
This is an easy consequence of the fact that, g being a non-degenerate metric implies
that the linear map,
(gi,j ) : P −→ P,
is an isomorphism. Notice also that we may, in an invariant way, consider the
Levi-Civita connection, and use it as a base-point ρ0 of the moduli space of representations.
The tangent space between any two representations, ρi , i = 1, 2, represented by
elements of P, may also be identified with a quotient of P. In fact,
Ext1A(σ) (ρ1 , ρ2 ) = Derk (A(σ), Endk (ΘA ))/T rivials.
Any derivation ξ ∈ Derk (A(σ), Endk (ΘA )), maps the relations of A(σ) to zero, so
we shall have,
[ξ(dti ), tj ] + ρ1 (dti )ξ(tj ) − ξ(tj )ρ2 (dti ) = ξ(g i,j ).
Since ΘA is a free A-module, such that Ext1A (ΘA , ΘA ) = 0, there exists a linear
map, Φ ∈ Endk (ΘA ), such that ξ(tj ) = tj Φ − Φtj , for all j. We may therefore
assume all ξ(ti ) = 0, and it follows from the above equation, that we have, ξ(dti ) ∈
EndA (ΘA ). In case ρ1 = ρ2 , corresponding to P , we see that the trivial derivations
are exactly those given by the n-tuples,
X
X
∂φ
∂φ
(
g 1,j (
+ [Pj , φ]), ...,
g n,j (
+ [Pj , φ])).
∂t
∂t
j
j
j=1,...,n
j=1,...,n
For any φ ∈ EndA (Θ), the expression,
∂φ
∂φ
+ [P1 , φ], ...,
+ [Pn , φ]),
∂t1
∂tn
therefore corresponds to an infinitesimal automorphism of the representation ρ, of
the algebra A(σg ), corresponding to P . But note that the dynamical structure, σ,
depends upon the choice of a metric g. Therefore φ ∈ EndA (Θ), acts on the particle type represented by ρ. This may, to some degree, explains why, in physics, one
considers potentials as interaction carriers, particles mediating force upon other
particles. This is, however, not the whole story, see the general definition of interaction, [10].
(
Notice that we also know that, in the representation-dimension 1 case, i.e. in
the commutative case, this dynamical structure reduces to the equation of motion,
X
d2 ti = −
Γ̄ip,q dtp dtq
p,q
COSMOS AND ITS FURNITURE
19
See next paragraph for a computed example.
An easy calculation in C(σ), shows that,
X
X ∂Tj
[T, dti ] = 1/2
Tj Rj,i − 1/2
g l,i dtj =: qi .
∂t
l
j
j,l
But, be careful, these charges, qi , see [10],(4.7), no longer vanish in the classical
phase-space, i.e. in the commutativization of P h(C).
Now, choose a representation
ρE : C(σ) → Endk (E),
i.e. a connection ∇, on a C-module E.
The generalized curvature Fi,j ∈ C(σ) maps to the classical one. In fact,
X
i,j
ρ(Fi,j ) = [ρ(dti ), ρ(dtj )] −
(Γj,i
p − Γp )ρ(dtp ) = [∇ξi , ∇ξj ] − ∇[ξi ,ξj ] .
p
Put, for short,
F (ξi , ξj ) = Fi,j := ρ(Fi,j ) ∈ EndC (E).
In EndC (E), our Force Law above will now take the form,
ρE (d2 ti ) +
X
Γip,q ∇ξp ∇ξq
p,q
= 1/2
X
Fp,i ∇δp + 1/2
X
p
∇δp Fp,i + 1/2
p
X
q,i
δq (Γi,q
l − Γl )∇ξl + [∇ξi , ρE (T )].
l,q
Notice also, that for the Levi-Civita connection, there is a possible relationship
between this formula and the Einstein field equation. See [15], Proposition 4.2.2.,
p.114. If, above we assume that we are in a geodesic reference frame, i.e. along a geodesic γ in our
Pspace Simp1 (C), then an average of the excess-relative-acceleration,
i.e. of d2 ti + p,q Γip,q dtp dtq , evaluated in ΘC |γ, is proved to be given by the Ric
tensor. But, this relative-acceleration is, for any representation corresponding to a
connection ∇, equal to
X
X
X
q,i
1/2
Fp,i ∇δp + 1/2
∇δp Fp,i + 1/2
δq (Γi,q
l − Γl )dtl + [∇ξi , ρE (T )].
p
p
l,q
Since this excess-relative-acceleration, representing a tidal force, should be a
measure of the inertial mass present, it is tempting to consider this force law as
a generalized, quantized, Maxwell-Einstein’s equation. The reference to Maxwell
here is natural, since if the bundle E = ΘC above is the tangent bundle, and we
consider the connection, given by the potential A = (A1 , ..., An ), Ai ∈ C, then the
resulting curvature is the electro-magnetic force field. See [10], Example (4.13)
for the notion of Charge, and the Example (4.12), where the problem of Mass is
addressed.
In this generality, it is not really meaningful to ask for invariance of this general
Force Law, w.r.t. isometries. This is linked to the fact that, in general, this
force law, considered as a dynamical structure on C, may have non-singular finitedimensional representations, and then invariance under isometries of Simp1 (C) is
not the proper question to pose. We shall come back to this later, see the section
(3), on Newton and Kepler’s laws.
Notice that applying ρΘ , corresponding to the Levi-Civita connection, the above
translate into,
ρ(dti ) = [Q, ti ], ρ(d2 ti ) =
d
X
X
[Q, ρ(dti )], ρ(dti ) =
g i,j ∇δj =: ξi .
j=1
j
20
OLAV ARNFINN LAUDAL
where Q is the Laplace-Beltrami operator.
Given any observable f ∈ P h(C), then in line with (1.2), we would expect the
the dynamics of the future values of f to be the spectrum of the operator,
F (τ ) := exp(τ × ad(Q))(f ).
3. Time-space and Space-times
3.1. The Notions of Time and Space. Go back to our basic model, Hilb2 (E3 ) :=
H = H̃/Z2 , see the Introduction and [10], classifying the family of pairs of points
(o, x) of the Euclidean 3-space, E3 . As above we shall first consider the structure
of H̃, before we extend the results to H. Recall from the Introduction, that given
a metric g on H̃, there are two 3-dimensional distributions, normal to each other,
˜ and the other being the light velocities c̃.
one being the canonical 0-velocities, ∆,
˜
By definition, g is the time, and it is easy to see that its restriction g∆
˜ , to ∆, is the
proper time of Einstein’s relativity theory. The group of isometries of H̃, leaving
˜ stable, does not contain the Lorentz boosts, K = J 0,1 , J 0,2 , J 0,3 . This, together
∆
with the results of the section Connections and the Generic Dynamical Structure
explains the fact that K is not conserved, and why one does not use the eigenvalues
of K to label physical states, see [17], I, 2.4, p. 61.
Moreover, the tangent bundle T (H̃), outside of ∆ is decomposed into the sum
of the basic tangent bundles Bo , Bx , Ao,x , each of rank 2. Recall also that Ao,x is
decomposed into a unique 0-velocity, dual to < dt0 > and a light velocity dual to
< dt3 >. The sub-bundle So,x , given by the triples (ψ, −ψ, φ) ∈ Bo ⊕ Bx ⊕ Ao,x , in
which the pair (ψ, −ψ) corresponds to a light-velocity, is at each point of H − ∆ a
4-dimensional tangent sub-space, with a unique 0-velocity.
Consider the symmetry in H̃, induced by the generator τ ∈ Z2 . The tangent
space of H̃ at the point t ∈
/ ∆ is represented by the vector space of pairs {(ξo , ξx )}
where ξo is a tangent vector in the Euclidean 3-space at the point o, and ξx is a
tangent vector at the point x. Any such pair may be written as,
(ξo , ξx ) = (1/2(ξo + ξx ), 1/2(ξo + ξx )) + (1/2(ξo − ξx ), 1/2(−ξo + ξx )),
˜ and, if the metric is trivial, the second in c̃. Clearly
where the first vector is in ∆,
˜ fixed and is the multiplication by −1 on c̃. τ also inverts chirality,
τ leaves ∆
and spin, since the orientation of Bo , defined by (o, x) is the the inverse of the
orientation of Bx defined by (x, o).
Classically one defines the symmetry operators in Minkowski space, the parity
operator P , by multiplying the space-coordinates with −1, the time inversion operator T , by multiplying the time coordinate with −1, and the charge conjugation
operator C, by multiplying the spin σ by −1. Identifying c̃ with the past light-cone
in Minkowski space, we see that τ corresponds to the transform (xo , x1 , x2 , x3 ) →
(−xo , −x1 , −x2 , −x3 ), i.e. τ corresponds to P T , so τ 2 = τ P T = id. This suggests
that τ is the charge conjugation operator, but as we have seen it inverts both spin
and the momenta in light-direction, so it is (slightly?) different from the classically
defined C, see [17], I, (3.3), p.131.
Choosing a line l ⊂ E3 , the subscheme H(l) ⊂ H, has a much simpler structure
than H. The sub-bundle So,x , restricted to H(l) can be integrated in H, and
we obtain a 4-dimensional subspace S(l) ⊂ H, in which we choose coordinates
t0 , dt1 , dt2 , t3 , where dt0 and dt3 are as above, and dt1 , dt2 are dual coordinates for
the transverse bundle, Bo , (isomorphic to the inversely oriented bundle Bx ), normal
to l in E3 .
This subspace S(l) of H, may be identified with a natural moduli-subspace
M (l) ⊂ H̃. In fact, let M (l) = {(o, x) ∈ H̃| 1/2(o + x) ∈ l}. Since o = 1/2(o + x) +
1/2(o − x), x = 1/2(o + x) − 1/2(o − x), it is of dimension 4, and contains H(l). At
COSMOS AND ITS FURNITURE
21
every point (o, x) ∈ H(l) ⊂ M (l) the tangent space is easily identified with So,x ,
therefore identifying S(l) and M (l), as spaces. However, the structure of M (l) is
much richer than that of S(l) ' H(l) × A2 . In particular, the actions of the gauge
group are different, see the section Cosmos, Cosmology and Cosmological Time, for
another characterization of M (l).
Moreover, the usual Minkowski space, is recovered as the restriction, U (l), of
the universal family U , to M (l). The metric is deduced from the energy-function
of M (l).
With this moduli space of an observer observing another point in E3 we have
seen, in a general situation, that we may introduce dynamics, and therefore now,
relativistic dynamics, by introducing time as a metric, g, in H̃. The equation of
motion for a 1-point object, is then given as a consequence of the Dirac derivation
on the Simp1 (P h(H̃)), that takes the form as a force law,
X
d2 ti = −
Γip,q dtp dtq ,
p,q
for any given system of coordinates {ti } parametrizing the of the part of H̃, that
we are interested in.
In the next section we shall see that this leads to the classical equations of motion
of Newton and Kepler.
More generally, if we are given an object composed of n points in E3 , interacting
internally, and also with the gravitational field, then we may go back to (1.2) and
use the hints there to deduce equations of motion for the combined system.
3.2. Newton’s and Kepler’s Laws. Let us study the geometry of H. Recall
that H̃ → H, is the (real) blow up of the diagonal ∆ ⊂ H, where H is the space of
pairs of points in E3 . Clearly any point t ∈ H outside the diagonal, determines a
vector ξ(o, x) and an oriented line l(o, x) ⊂ E3 , on which both the observer o and
the observed x sits. This line also determines a subscheme H(l) ⊂ H, see above
and [8], and in H(l) there is unique light velocity curve l(t), through t, an integral
curve of the distribution c̃, and this curve cuts the diagonal ∆ in a unique point
c(o, x), the center of gravity of the observer and the observed, and thus defines a
unique point ξ(t), of the blow-up of the diagonal, in the fiber of H̃ → H, above
c(o, x). Any tangent η := (η1 , η2 ), η2 = −η1 , of H in c̃, at t = (o, x), normal to l(t),
corresponds to a light velocity, to a spin vector, η1 × ξ(o, x), in E3 , with spin axis,
the corresponding oriented line. The length of the spin vector is called the spin of
η. We have shown in [8] that there exists a metric on H which, restricted to every
3-space c(t) − {c(o, x)}, has the form,
ds2 = dt23 + (1 + r−2 )dt21 + (1 + r−2 )dt22 − r−4 (t1 dt1 + t2 dt2 )2 ,
where we have chosen the coordinates such that dt3 corresponds to the oriented
line l(o, x), and dti , i = 1, 2, correspond to the spin-momenta, assuming t3 6= 0.
The nice property of this metric is the following. Consider a spin momentum η :=
(η1 , −η1 ) and its corresponding spin vector η × ξ(o, x) := (η1 × ξ(o, x), −η1 × ξ(o, x))
along the line l(t). Clearly the length of this vector, when r = t3 is large, is just the
classical spin. When r tends to zero, η defines a tangent vector of the exceptional
fiber of the blow up at c(o, x), i.e. of the projective 2-space, and of the covering
2-sphere. And we see that the length of η × ξ(o, x) tends to the length of this
tangent vector in the Fubini-Studi metric of P2 .
To see what this may lead us to, we need a convenient parametrization of H̃.
Consider, as above, for each t ∈ H̃ the length ρ, in E3 , the Euclidean space, of the
vector (o, x). Given a point λ ∈ ∆, and a point ξ ∈ E(λ) = π −1 (λ), the fiber of,
π : H̃ → H,
22
OLAV ARNFINN LAUDAL
at the point λ, for o = x. Since E(λ) is isomorphic to S 2 , parametrized by φ, any
element of H̃ is now uniquely determined in terms of the triple t = (λ, φ, ρ), such
that c(t) = c(o, x) = λ, and such that ξ is defined by the line ox. Here ρ ≥ 0,
see also the section Cosmology, Cosmos and Cosmological Time. Notice also that,
at the exceptional fiber, i.e. for ρ = 0, the momentum corresponding to dρ is not
defined.
Consider any metric on H̃, of 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 S 2 = E(λ).
Denote the covariant derivations by,
∂
∂
∂
∂
Dρ :=
+ ∇ρ , Dφ :=
+ ∇φ , Dθ :=
+ ∇θ , φ := (φ, θ), Dλ :=
+ ∇λ
∂ρ
∂φ
∂θ
∂λ
It is reasonable to believe that the geometry of (H̃, g), might explain the notions
like energy, mass, charge, etc.. In fact, we tentatively propose that the source
of mass and charge etc. is located in the black holes E(λ). This would imply
that mass, charge, etc. are properties of the 5-dimensional superstructure of our
usual 3-dimensional Euclidean space, essentially given by a density, h(λ, φ, θ). This
might bring to mind Kaluza-Klein-theory. However, it seems to me that there are
important differences, making comparison very difficult.
Let us first treat the following simple case,
ρ−h 2
hρ = (
) , hφ = (ρ − h)2 , hλ = 1,
ρ
where h is a positive real number. This metric is everywhere defined in the subspace
of M (l), where we have reduced the spherical coordinates φ, to just φ. Notice that
for ρ = 0, there are no tangent vectors in dρ direction.
It clearly reduces to the Euclidean metric far away from ∆, and it is singular
on the horizon of the black hole, given by ρ = h, which in H is simply a sphere
in the light-space, of radius h. Moreover it is clear that h is also the radius of the
exceptional fibre, since the length of the circumference of ρ = 0, is 2πh. Clearly, the
exceptional fiber, the black hole itself, is not visible, and does not bound anything.
However, the horizon bounds a piece of space. Moreover, if we reduce the horizon to
a point in H, then the circumference, or area of the exceptional fiber, as measured
using the above metric, reduces to zero, and the metric becomes the usual Euclidean
metric. We shall reduce to a plane in the light directions, i.e. we shall just assume
that S 2 = E(λ), is reduced to a circle, with coordinate φ. This is actually no
restriction made, as is easily seen. Consider the Lagrangian L = g, see (Example
4.1), we find the Euler-Lagrange equations,
∂L
∂L
ρ−h 2 2
ρ−h h
)−
= 2(
) d ρ + 2(
)( 2 )dρ2 − 2(ρ − h)dφ2
∂dρ
∂ρ
ρ
ρ
ρ
∂L
∂L
2 2
0 = δ(
)−
= 2(ρ − h) d φ + 4(ρ − h)dρdφ
∂dφ
∂φ
∂L
∂L
0 = δ(
)−
= 2d2 λ
∂dλ
∂λ
where λ = λ, as above. We solve these equations, and find,
0 = δ(
d2 ρ = −(
h
ρ2
)dρ2 + (
)dφ2
ρ(ρ − h)
(ρ − h)
d2 φ = −2/(ρ − h)dρdφ
d2 λ = 0,
COSMOS AND ITS FURNITURE
23
which give us immediately the following formulas,
Γ11,1 = h/ρ(ρ − h), Γ12,2 = −ρ2 /(ρ − h)
Γ21,2 = 1/(ρ − h), Γ22,1 = 1/(ρ − h)
Γ3i,j = 0,
and,
Γ̄11,1 = Γ11,1 , Γ12,2 = −Γ12,2
Γ̄21,2 = Γ21,2 = Γ̄22,1
Γ̄3i,j = 0,
All other components vanish. From this we find the following formula, for the
covariant derivations,
∂
+ ∇ρ ,
∂ρ
∂
+ ∇φ ,
Dφ := ∇δ2 =
∂φ
∂
Dλ := ∇δ3 =
+ ∇λ
∂λ
3
X
Q=
1/hi ∇2δi
Dρ := ∇δ1 =
i=1
2
ρ(δ (ti )) = [Q, ρ(dti )] = 1/hi [Q, ∇δi ].
Here the hi is the function defined above, i.e. gi,i in our metric, and,


h/ρ(ρ − h)
0
0
0
1/(ρ − h) 0
∇ρ = 
0
0
0


2
0
−ρ /(ρ − h) 0
0
0
∇φ = 1/(ρ − h)
0
0
0
∇λ = 0


0
−1/ρ(ρ − h) 0
0
0
[∇ρ , ∇φ ] = −ρ/(ρ − h)
0
0
0

−2
−1
−hρ (ρ − h) − hρ−1 (ρ − h)−2
0
∂
2
−2 
0
−(ρ − h)−2
∇ρ = ρ (ρ − h)
∂ρ
0
0


0 2hρ 0
∂
0
0
∇φ = (ρ − h)−2 −1
∂ρ
0
0
0
The corresponding equations for the geodesics in H̃ are,
d2 ρ
h
dρ 2
ρ2
dφ
=
−(
)(
)
+
(
)( )2 ,
2
dt
ρ(ρ − h) dt
(ρ − h) dt
d2 φ
dρ dφ
= −2/(ρ − h)
dt2
dt dt
d2 λ
= 0.
dt2

0
0
0
24
OLAV ARNFINN LAUDAL
where t is time. But time is, by definition, the distance function in H̃, so we must
have,
dφ
dλ
ρ − h 2 dρ 2
) ( ) + (ρ − h)2 ( )2 + ( )2 = 1,
(
ρ
dt
dt
dt
from which we find,
(
dρ 2
dλ
dφ
) = ρ2 (ρ − h)−2 (1 − ( )2 ) − ρ2 ( )2 .
dt
dt
dt
dλ
From the third equation, we find that dtj , j = 1, 2, 3, are constants, and | dλ
dt | is
2
the rest-mass of the system. Put K 2 = (1 − | dλ
|
),
then
K
is
the
kinetic
energy
of
dt
the system. The definition of time therefore give us,
ρ−2 (
dρ 2
dφ
) = (ρ − h)−2 K 2 − ( )2 .
dt
dt
Put this into the first equation above, and obtain,
d2 ρ
ρ
1
ρ+h
dφ
= −hK 2 (
)
+(
)ρ( )2 .
2
2
dt
ρ − h (ρ − h)
ρ−h
dt
Assume now r := ρ − h ≈ ρ, we find,
d2 r
hK 2
dφ
=
−
+ r( )2 ,
2
2
dt
r
dt
i.e. Keplers first law. The constant h, i.e. the radius of the exceptional fiber, is
thus also related to mass. In fact, this suggests that mass, is a property of the
space H̃. In this case it is a function of the surface of the exceptional fiber, i.e. the
black hole, associated with the point λ in the ordinary 3-space ∆. In the same way,
the second equation above gives us Keplers second law,
r(
d2 φ
dr dφ
) + 2( )( ) = 0.
2
dt
dt dt
Notice that with the chosen metric, time, in light velocity direction, is standing still
on the horizon ρ = h, of the black hole at λ ∈ ∆. Therefore no light can escape from
the black hole. In fact, no geodesics can pass through ρ = h. Notice also that, for
a photon with light velocity, we have K = 1, so we may measure h, by measuring
the trajectories of photons in the neighborhood of the black hole. Finally, see that
if the distance between the two interacting points is close to constant, i.e. if we
have a circular movement, the left side of the time-equation becomes zero, and we
therefore have the following equation,
ρdφ = Kdt + hdφ,
which may be related to the perihelion precicion.
Notice now that if we had used the force law resulting from the general force law
of our metric, i.e.
X
d2 ti = −
Γ̄ip,q dtp dtq ,
p,q
we would have got the same Kepler’s laws, except that the first law would have had
the form,
d2 r
hK 2
dφ
=
−
− r( )2 ,
2
2
dt
r
dt
which is curious, since it corresponds to an attractive centrifugal force.
COSMOS AND ITS FURNITURE
25
3.3. Thermodynamics, the Heat Equation and Navier-Stokes. Let us now
go back to Section 2, and consider the generic dynamical structure (σ), related to
the metric on H̃, of the form,
g = h1 (λ, φ, ρ)dρ2 + h2 (λ, φ, ρ)dφ2 + h3 (λ, φ, ρ)dλ2 ,
where dφ2 is the natural metric in S 2 = E(λ).
Recall
for C = H = k[t1 , ..., t6 ], and a non-singular Riemannian metric g =
P
1/2 i,j=1,..,r gi,j dti dtj ∈ P h(C), the notations,
X
X
i,j
Γj,i
g j,k Γik,p , Ri,j := [dti , dtj ], Fi,j := Ri,j −
(Γj,i
p =
p − Γp )dtp ,
p
k
and the general force Law in P h(C),
d2 ti = −
X
Γip,q dtp dtq − 1/2
p,q
+ 1/2
X
gp,q (Fi,p dtq + dtp Fi,q )
p,q
X
q,i
gp,q [dtp , (Γi,q
l − Γl )]dtl + [dti , T ],
l,p,q
generating the dynamical structure c := c(g).
Remark 3.1. In principle, according to our philosophy, the natural common quantization of classical general relativity and Yang-Mills theory would be based on
the the dynamical properties of Simp≤∞ (H̃(c)), with respect to the versal family,
ρ̃ : OH̃(c) → EndH̃(c) (Ṽ ), where we have to consider OH̃(c) as a presheaf of associative k-algebras, defined in H̃. As a first try, we shall concentrate on singular
situations, and in particular, on the structure of the Levi-Civita representation.
It is reasonable to believe that the geometry of (H̃, g), might explain the notions
like energy, mass, charge, etc. In fact, we tentatively propose that the source of
mass and charge etc. is located in the black holes E(λ). This would imply that
mass, charge, etc. are properties of the 5-dimensional metric superstructure of our
usual 3-dimensional Euclidean space, essentially given by a density, h(λ, φ, θ). This
might bring to mind Kaluza-Klein-theory. However, it seems to me that there are
important differences, making comparison very difficult.
Recall that at a point t = (o, x) ∈ H −∆, the tangent space, ΘH̃ (t), is represented
by the space of all pairs of 3-vectors, ξ(t) = (ξo , ξx ), ξo , fixed at o, and ξx , fixed at
the point x in E3 . Moreover, any such tangent vector may, depending only upon
˜ and
the choice of metric, be decomposed into the sum ξ = ξ1 + ξ2 , with ξ1 ∈ ∆,
ξ2 ∈ c̃.
Recall also that the center of gravity of the observer and the observed, c(o, x) ∈ ∆,
defined in terms of a Euclidean structure on our 3-dimensional space, defines a
unique point ξ(t), of the blow-up of the diagonal, in the fiber of H̃ → H, above
c(o, x).
Now, consider the Levi-Civita connection,
ρΘ : H̃(σ) → EndR (ΘH̃ ),
together with the Hamiltonian, i.e. the Laplace-Beltrami operator, Q ∈ EndH̃ (ΘH̃ ).
Any state ξ ∈ ΘH̃ , may be interpreted as a relative momentum (ξo , ξx ) of the
pair of points (o, x) ∈ E3 , defined for all pairs of points in the domain of definition
for ξ. Write, as above,
˜
ξ = p + m, p = (p1 , p2 , p3 , 0, 0, 0) ∈ c̃, m = (0, 0, 0, m1 , m2 , m3 ) ∈ ∆,
26
OLAV ARNFINN LAUDAL
where we have introduced local coordinates,
˜ and x = (x1 , x2 , x3 ) for c̃.
λ = (λ1 , λ2 , λ3 ), for ∆,
The norm µ := |ξ| is then the energy-density of the system defined by ξ, and the
norms,
ρ := |m|, κ := |p|
are the density of mass, respectively the kinetic momentum density. Then we
find that (p1 , p2 , p3 ) is a classical, relative momentum-vector, and, (v1 , v2 , v3 ) =
µ−1 (p1 , p2 , p3 ), is a classical velocity vector.
Consider now the corresponding Schrødinger equation,
d
(ξ) = Q(ξ),
dt
where, since measurable time is the metric, we must have,
d
Dξ = µ .
dt
Let us compute the left hand side of the Schrødinger equation. It is clear that,
d
d
d
(ξ) = (m) + (p)
dt
dt
dt
3
3
3
3
X
X
X
X
=
µ−1 mi Dλi (m) +
µ−1 mi Dλi (p) +
vi Dxi (m) +
vi Dxi (p).
1
1
1
1
Now, assume the classical velocities vi are small compared to the velocity of light
which in this model is 1. Consequently the norm of m must be close to µ, and the
∂p
∂p
P3 −1
term,
, where τ is the relativistic proper
mi
may be compared to
1µ
∂λi
∂τ
time.
The outcome of this, when we reduce to the subscheme M (l) ⊂ H̃, corresponding
to a chosen line l ⊂ E3 , is that the Schrødinger equation is, in a realistic classical
situation, the coupled equation, containing the general relativistic Heat Equation,
dρ
= Q(ρ),
dt
and a generalized Navier-Stokes Equation,
dp
= Q(p),
dt
which, when the metric is close to Euclidean, and the momentum-vector is small,
takes the form
∂v
µ(κ
+ (∇.v)v) = Q(v),
∂τ
with,
∂
Q(v) = ∆(v) + ( )2 (v)
∂τ
In general, we might hope that knowing ξ, i.e. the 6 functions defined in H̃,
locally defining the vector ξ, the Schrødinger equation would determine the metric,
g, i.e. the 6 functions hρ , hφ , hλ . This would presumably lead to time-developments
ξ(T ) and g(T ), determined by any given ground-state, ξ? , and clocked by some
parameter T . This would again have as a consequence, that any cyclic behavior of
the phenomenon modeled by ξ(T ), would lead to a gravitational wave defined by
g(T ).
In particular, the collapsing of a star, or the Big Bang, both usually modeled as
a fluid depending on pressure, temperature, energy density, viscosity etc., would in
the above scenario, define a generalized gravitational wave, g(T ).
COSMOS AND ITS FURNITURE
27
Notice that, in the trivial metric case, the Schrødinger equation identifies ρ = |m|
as a combined temperature and pressure. In general, however, there is a kind of
deviatoric stress tensor, (σi,j ), defined for any element λ ∈ ∆ by,
3
X
d
(ξ)(λ, ω i ) =
σi,j dωj
dt
j=1
where ωi are directional coordinates of the exceptional fibre E(λ). It is tempting
to define pressure, as p = (1/3)T r(σi,j ).
Remark 3.2. The vector field ξ ∈ ΘH̃ may be interpreted as a description of the
relative state of the space, everywhere, a kind of mass-stress-tensor. This is the
Furniture of our cosmos, referred to in the title of this paper. If we know ξ, in
the neighborhood of a star, in the situation of a collapsing star, or even in the
neighborhood of a black hole, then we would be tempted to consider the Schrødinger
equation,
d
(ξ)(= Q(ξ),
dt
as our Field Equation, replacing the Einstein Field Equation. A solution would be a
metric g determining the dynamics of the past and the future of our space. To make
this reasonably understandable, we need a mathematical model of the beginning of
it all, the Big Bang. This is, however, the subject of the last section in this paper.
As a first example, consider the very special case of the metric of the last section,
defined by,
hρ = (
ρ−h 2
) , hφ = (ρ − h)2 , h3 = 1,
ρ
where h is a positive real number.
Put ρ = t1 , φ = t2 , λ = t3 , then the we found the following formulas,
Γ11,1 = h/ρ(ρ − h), Γ12,2 = −ρ2 /(ρ − h)
Γ21,2 = 1/(ρ − h), Γ22,1 = 1/(ρ − h)
Γ3i,j = 0
All other components vanish. From this we find the following formula,
∂
+ ∇ρ ,
∂ρ
∂
Dφ := ∇δ2 =
+ ∇φ ,
∂φ
∂
Dλ := ∇δ3 =
+ ∇λ
∂λ
3
X
Q=
1/hi ∇2δi
Dρ := ∇δ1 =
i=1
ρ(δ 2 (ti )) = [Q, ρ(dti )] = 1/hi [Q, ∇δi ].
28
OLAV ARNFINN LAUDAL
Here the hi is the function defined above, i.e. gi,i in our metric, and,


h/ρ(ρ − h)
0
0
0
1/(ρ − h) 0
∇ρ = 
0
0
0


2
0
−ρ /(ρ − h) 0
0
0
∇φ = 1/(ρ − h)
0
0
0
∇λ = 0


0
−1/ρ(ρ − h) 0
0
0
[∇ρ , ∇φ ] = −ρ/(ρ − h)
0
0
0

−hρ−2 (ρ − h)−1 − hρ−1 (ρ − h)−2
0
∂
0
−(ρ − h)−2
∇ρ = ρ2 (ρ − h)−2 
∂ρ
0
0


0 2hρ 0
∂
0
0
∇φ = (ρ − h)−2 −1
∂ρ
0
0
0

0
0
0
The left hand side of the Schrødinger equation,
∂ξ
= Q(ξ),
∂t
for a general vector field, ξ = (f1 , f2 , f3 ), takes the form,
∂f1
∂f2
∂f3
, f1
, f1
) + f1 (f1 h(ρ − h))−1 , f2 h(ρ − h)−1 , 0)
∂ρ
∂ρ
∂ρ
∂f2
∂f3
∂f1
+(f2
, f2 (
, f2
) + f2 (−f2 ρ2 (ρ − h)−1 , f1 h(ρ − h)−1 , 0)
∂φ
∂φ
∂φ
∂f1
∂f2
∂f3
+(f3
, f3
, f3
),
∂λ
∂λ
∂λ
and the right hand side becomes, with the obvious simplified notations,
Dξ (ξ) =(f1
Q(ξ) = − (ρ − h)−4 (ρ4 f1:ρρ + h2 f1 − ρ2 f1 + ρ3 f2:ρ + ρ4 f1:ρ,ρ + h4 f1:λ,λ − 2ρ3 f2:φ
+ρ2 f1:φ,φ + h2 f 1 : φ, φ − 4hρ3 f1:λ,λ + 6h2 ρ2 f1:λ,λ − 2hρf1 − h2 ρf1:ρ
−2hρ3 f1:ρ,ρ + h2 ρ2 f1:ρ,ρ − 4h3 ρf1:λ,λ + 2hρ2 f2:φ − 2hρf1:φ,φ )Dρ
−(ρ − h)−4 (3ρ3 f3:ρ + 2ρf1:φ − 2hf1:φ + ρ2 f 2 : φ, φ + ρ4 f2:λ,λ + h4 f2:λ,λ
+ρ4 f2:ρ,ρ h2 f2:φ,φ − 4hρ2 f2:ρ − 2hρf2:φ,φ + 6h2 ρ2 f2:λ,λ − 2hρf2 + h2 ρf2:ρ
+h2 ρ2 f2:ρ,ρ − 2hρ3 f2:ρ,ρ − 4h3 ρf2:λ,λ − 4hρ3 f2:λ,λ )Dφ
+ρ−1 (ρ − h)−1 (ρ2 f3:ρ + ρ3 f3:ρ,ρ + ρf3:φ,φ + ρ3 f3:λ,λ
−2hρ2 f3:λ,λ + h2 ρf3:λ,λ )Dλ
Put,
ξ = (0, f (ρ), 0),
Then we find Dξ (ξ) = (0, 0, 0), and the second order differential equation,
ρ2 (ρ − h)2
df
d2 f
+ ρ(ρ − h)(3ρ − h) − 2hρf = 0,
2
dρ
dρ
with the easy solution,
f = (ρ − h)−2 ,
COSMOS AND ITS FURNITURE
29
which means that the fluid, the content of the space, rotates about the Black Hole
ρ = 0 with speed (ρ − h)−1 , so with infinite speed close to the horizon, almost
standing still at great distances, and therefore with lots of shear.
Notice that,
ξ = (0, (ρ − h)−2 , p), p constant,
is also a solution.
4. Cosmology, Cosmos and Cosmological Time
4.1. Background, and some Remarks on Philosophy of Science. In the
paper [8], we discussed the possibility of including a cosmological model in our toymodel of Time-Space. The 1-dimensional model we presented there was created
by the deformations of the trivial singularity, O := k[x]/(x)2 . Using elementary
deformation theory for algebras, we obtained amusing results, depending upon some
rather bold mathematical interpretations of the, more or less accepted, cosmological
vernacular. Here we shall go one step further on, and show that our toy-model, i.e.
the moduli space, H, of two points in the Euclidean 3-space, or its étale covering,
H̃, is created by the (non-commutative) deformations of the obvious singularity in
3-dimensions, U := k < x1 , x2 , x3 > /(x1 , x2 , x3 )2 .
The main axiom of the leading branches of quantum gravitation, and cosmology, seems to be that the space-time of the existing universe can be described via
a General Relativistic model, somehow given by Einstein’s equation with respect
to some mass-stress tensor, mass and energy being homogeneously and isotropically distributed in space. This leads to the assumption that the universe is a
4-dimensional space-time of a form commonly called a Friedman-Robertson-Walker
model. In particular the space has an open ended time-coordinate, leaving out the
Big Bang, but still assuming that the point-like Big Bang is in the closure of even
the shortest complete history of the universe.
There are a lot of assumptions here. One, not immediately seen, is that the spacetime is capable of containing something, and that these things can be described as
independent upon the space, even though they curve space, and otherwise intervene
in the dynamical process. For example, even in the very start of the universe, spin is
assumed to be present. So, in mathematical terms, the space-time must be outfitted
with a su(2), or a complexified sl(2), tangent structure, obviously determined by
the Big Bang event. Moreover, since the space-time of the model does not contain
the prime event, and the jump between that supposedly point-like event and the
mathematically well defined space-time is not part of the model, we do not have a
model of the big bang itself, but rather of what may have happened in our usual
space, a long time ago, with respect to a rather artificially chosen time parameter.
Above, and in my book, [10], I produced a model for time and space, H :=
H̃/Z2 , a time-space, of dimension 6, outfitted with gauge group actions of the Lie
algebras, u(1), su(2), and su(3), on the tangent space, easily seen to contain the
properties needed to formulate general relativity, with a 3-dimensional observable
subspace, identified with the visible world, and the whole structure not presenting
any obstructions to quantization.
This space, H̃, the Hilbert scheme of two points in 3-space, is the moduli space
for an observer observing an observed in 3-space, see the Introduction.
In this chapter I propose to show how this time-space can be thought of as an
immediate product of a mathematical scenario incorporating a Big Bang event,
making this event mathematically sound, and quite well known to people doing
deformation theory in algebraic geometry.
Starting with the pure notion of 3-dimensionality, i.e. a k-scheme U = Spec(U )
with only one point and a 3-dimensional tangent space, in algebraical terms, the
30
OLAV ARNFINN LAUDAL
singularity,
U := k[x1 , x2 , x3 ]/(x1 , x2 , x3 )2 ,
we shall see, in the next subsection, that we may construct, in a canonical way, a
versal deformation base space, M, and a corresponding versal family U? , containing
all isomorphism classes of deformations of U , as associative algebra. This is, of
course, different from what we usually do in classical algebraic geometry, where the
interesting objects are the commutative singularities close to the given one.
The technique for this general deformation theory, can be found in [2], see also
[11]. In the last book we introduced the notion of moduli-suite. This correspond
here to a partition of the space M, in a series of rooms, containing an inner room,
composed of just one point ?, corresponding to the singularity we start with, U ,
a very special component that turns out to be H, and where the family U, the
restriction of U? to H has, corresponding to a point (o, p) ∈ H − ∆, the fibre,
U (o, p) := k < x1 , x2 , x3 > /(xi xj − oi xj − pj xi + oi pj ),
where we have used the coordinates x1 , x2 , x3 , to express the two points, o and p
in 3-space A3 , by coordinates, {oi }, {pj }, i, j = 1, 2, 3.
If o = x, U (o, p) is isomorphic to U . But U has, never the less, a unique extension
to all of H̃, and the Z2 -action also extends.
There is also a special room in the moduli suite, corresponding to the Quatornions, as deformation of U .
4.2. Deformations of Associative Algebras. The tangent space of the formal
moduli of the singularity U , as an associative k-algebra is,
T? := A1 (k, U ; U ) = HomF (kerρ, U )/Der,
where, ρ : F → U is any surjective homomorphism of a free k-algebra F , onto U ,
HomF means the F -bilinear maps, and Der denotes the subset of the restrictions to
kerρ of the derivations from F to U . In our case we may use F = k < x1 , x2 , x3 >,
and the obvious surjection, making kerρ = (x)2 , generated as F bimodule by the
family {xi,j := xi xj }.
Any F -bilinear morphism φ : (x)2 → U , must be of the form,
φ(xi,j ) = a0i,j +
3
X
ali,j xl
l=1
a0i,j
and the bilinearity is seen to imply that
= 0. Thus, the dimension of HomF (kerρ, U )
is 27.
Any derivation δ ∈ Der, must be given by,
δ(xi ) = b0i +
3
X
bli xl
l=1
and the restriction onto the generators of (x)2 , must have the form,
δ(xi,j ) = b0j xi + b0i xj ,
therefore determined by the b0i s . It follows that the tangent space T? is of dimension
27-3=24.
Let o, p ∈ A3 , with coordinates o = (o1 , o2 , o3 ), p = (p1 , p2 , p3 ), with respect to
the coordinate system, x, and put,
φo,p (xi,j ) = pj xi + oi xj ,
then it is easy to see that the maps {φo,p } generate a 6-dimensional sub vector
space H of T? . Notice that, if o = p then φo,p , is a derivation, thus 0 in T? .
COSMOS AND ITS FURNITURE
31
Now, let us consider the Lie algebra of infinitesimal automorphisms of U , g(U ) :=
Lie(Aut(U )) = gl3 (k). By deformation theory, see e.g.[11], g(U ) acts on the
tangent space T? , and a simple calculation gives the result: α ∈ g(U ), with
P3
α(xi ) = l=1 αil xl , acts on φ ∈ T? , given in terms of its coordinates {ali,j }, as,
α(ali,j ) =
3
X
p=1
αip alp,j +
3
X
p=1
αjp ali,p +
3
X
αpl api,j .
p=1
In particular,
α(φo,p ) = φα(o),α(p) ,
and the origin of the tangent space T? is the only fix-point of the Lie algebra g(U ).
This implies, see [11], that the modular stratum of the versal base space M, of U is
reduced to the base point, ? = [U ].
Now, the rather unexpected happens. We may integrate the tangent subspace
H, and obtain a family of flat deformations of U . In fact, it is easy to see that,
U (o, p) := k < x1 , x2 , x3 > /(xi xj − oi xj − pj xi + oi pj ),
is an associative k-algebra of dimension 4, and a deformation of U , in a direction
of H. The corresponding Lie algebra is given by,
[xi , xj ] = (pj − oj )xi + (oi − pi )xj − (oi pj − oj pi ).
Notice that if o = p then U (o, p) is isomorphic to U , as it should, and that, U (o, p) '
U (−o, −p). Moreover, for any 3-vector c ∈ A3 ,
U (o, p) ' U (o − c, p − c).
Choosing c = 1/2(p + o), we find o0 := o − c = −(p − c) =: −p0 , and it is
easy to see that if o0 6= 0 the sub Lie algebra above generated by {x1 , x2 , x3 }
in U (o0 , p0 ), is isomorphic to the standard 3-dimensional Lie algebra with relations,
[y1 , y2 ] = y2 , [y1 , y3 ] = y3 , [y2 , y3 ] = 0. Moreover, choosing c = (p + o), we find an
isomorphism,
U (o, p) ' U (−p, −o) ' U (p, o),
which should be related to the action of Z2 on H, and thus, according to our
philosophy, to the mathematical reason for the CPT-equivalence, see [10], (4.9).
Notice also that the algebra,
Q := k < x1 , x2 , x3 > /(xi xj − i,j,k xk + δi,j ),
where i,j,k and δi,j are the usual indices, the first one nonzero only for {i, j, k} =
{1, 2, 3}, and the last one the usual delta function, is isomorphic to the quaternions,
which therefore is another non-trivial deformation of U .
Consider now the restriction to the subscheme H − ∆, of the versal family of U ,
denoted by,
ν 0 : U0 → H − ∆.
Since for all non-zero κ ∈ R, we have U (p, −p) ' U (κp, κ(−p)), this family extends
uniquely to a family,
ν : U → H̃.
4.3. Spin and Isospin. Let t = (o, p) ∈ H − ∆, and compute the Lie algebra
Derk (U (t)).
Any element δ ∈ Derk (U (t)) has the form,
δ(xi ) = δi0 + δi1 x1 + δi2 x2 + δi3 x3 .
Put,
o = (1, o1 , o2 , o3 ), p = (1, p1 , p2 , p3 ),
and consider the 4-vectors δi = (δi0 , δi1 , δi2 , δi3 ), i = 1, 2, 3.
32
OLAV ARNFINN LAUDAL
The relation, δ(xi xj − oi xj − pj xi − oi pj ) = 0, leads to the following conditions,
δi · o = δi · p = 0, i = 1, 2, 3.
If o 6= p, it follows that o and p, are linearly independent, in a 4-dimensional vector
space, therefore each vector δi , i = 1, 2, 3 is confined to a 2-dimensional vector
space. Consequently, g(t) := Derk (U (t)) is of dimension 6. Using the isomorphism
above, we may choose coordinates such that o = (0, 0, 0), p = (1, 0, 0). Then the
fundamental vector op = (1, 0, 0). With this it is easy to see that, δ ∈ g(t) imply,
δi0 = δi1 = 0, i = 1, 2, 3.
From this follows that the Lie algebra g(t) is isomorphic to the Lie algebra of
matrices of the form,


0 δ12 δ13
0 δ22 δ23 
0 δ32 δ33
The radical r, is generated by 3 elements, {e, r1 , r2 },





0 0 0
0 1 0
0
e = 0 1 0 , r1 = 0 0 0 , r2 = 0
0 0 1
0 0 0
0
0
0
0

1
0 .
0
where e ∈
/ [g, g], and the quotient,
g(t)/r = sl(2).
with the usual generators u0 , u1 , u2 ,



0 0 0
0
√
u0 = 1/2 2 0 1 0  , u1 = 0
0 0 −1
0
0
0
0


0
0
1 , u2 = 0
0
0
0
0
−1

0
0 .
0
In particular, we find that sl(2) ⊂ g(t).
Given a point t ∈ H, the tangent space at this point is, of course, nicely represented by the space of all pairs of 3-vectors, (ξ, µ), and it is easy to compute the
action of g(t) on this 6-dimensional vector space. Just as in the case of the action of
Derk (U ) on H, any δ ∈ g(t) acts as δ(ξ, µ) = (δ(ξ), δ(µ)), and in each coordinate,
the action is that of the matrix algebra above.
The Lie algebra sl2 (t) acts as follows. There are natural 3-dimensional subundles
Θo , Θp of the tangent bundle of H 0 := H − ∆, such that ΘH 0 = Θo ⊕ Θp . We may
find a natural basis for both components, for Θo as well as for Θp , {l, ν1 , ν2 }, where l
is the special tangent vector given by po, i.e. the tangent direction in our Euclidean
3-space, in which we are looking. It is obvious from the above matrix-bases of g(t),
that g(t) kills l. Therefore, in this basis, sl(2) acts on the planes normal to l = po.
As a consequence, if we pick any line l ⊂ A3 , then the tangent space of H(l) is
killed by g(t), for all t ∈ H(l).
We have therefore seen that for any point t ∈ H̃ the sl(2) component of the Lie
algebra of infinitesimal automorphisms of the universal algebra U (t), act on H̃ in
a particular nice way. The generators u0 , u1 , u2 , acting as vector fields, δ0 , δ1 , δ2 ,
sections of the sub bundle Bo ⊕ Bp of the tangent bundle.
Now, any derivation δ, of a k-algebra A, induces a derivation δ̄ of P h(A), since the
corresponding algebra-homomorphism (1 + δ) of A[] extends uniquely to P h(A[])
and therefore to P h(A[])/(d) = P h(A)[].
Let δ̄0 , δ̄1 , δ̄2 be the vector fields defined, in this way, on P h(H̃), and see that we
have,
δ̄0 = σ 0 δ0 , δ̄1 = σ 1 δ1 , δ̄2 = σ 2 δ2 ,
COSMOS AND ITS FURNITURE
33
where σ i := σ(ui ) for the representation,
σ : sl(2) → EndH̃ (Bo ) ' EndH̃ (Bp ),
corresponding to the linear action of sl(2) on the vector space generated by {dti }, i =
1..., 6 with respect to the chosen coordinate system, in the neighborhood of the point
t ∈ H̃.
From this, and the canonical decomposition of the tangent space of H̃ at the
point t ∈
/ ∆, represented by the vector space of pairs {(ξo , ξp )} where ξo is a
tangent vector in the Euclidean 3-space at the point o, and ξp is a tangent vector
at the point p. Any such pair may be written as,
(ξo , ξp ) = (1/2(ξo + ξp ), 1/2(ξo + ξp )) + (1/2(ξo − ξp ), 1/2(−ξo + ξp )),
˜ and, depending on the metric, the second in c̃. Clearly
where the first vector is in ∆,
˜ fixed and is the multiplication by −1 on c̃. τ also inverts chirality, and
τ leaves ∆
so also the spin, since the orientation of Bo , defined by (o, x) is the the inverse of
the orientation of Bx defined by (x, o). This decomposition, applied to the sum,
(Bo ⊕ Bp ) give us an isomorphism, (Bo ⊕ Bp ) ' (B∆
˜ ⊕ Bc̃ ). Noticing that the
matrices {v0 := −u0 , v2 := −u1 , v1 := −u2 } generate the same Lie algebra sl2 , with
orientation reversed, it is easy to see that the action of sl2 has the form,
0
ui
γi =
, i = 0, 1, 2,
−ui 0
and the derivation e becomes,
γ3 =
1
0
0
.
−1
The corresponding derivation of the local ring, δ 3 , is, classically, considered as the
time derivative. Replacing ui , i = 0, 1, 2, by the Pauli 2 by 2 matrices, these
operators are then used, by physicists, to formulate the Dirac equation.
(
3
X
γ µ pµ − mc)ψ = 0.
µ=0
which is assumed to take care of the dynamics of the electron. See for more information, [12], [17], and also [1].
We have a Killing metric k in sl(2). It should, in our situation, correspond to some
invariants of spin and energy of the objects with states in (Bo ⊕Bp ) ' (Bδ̃ ⊕Bc̃ ), the
Dirac spinors, defined by the representation of the invariant space, (P h(H̃), sl(2)),
see [6]. In particular we have the Casimir element of sl(2),
1/2u20 + u1 u2 + u2 u1 ,
and the relation,
√
( 2/2u0 + u1 + u2 )2 = 1/2u20 + u1 u2 + u2 u1 ,
The classical Dirac equation has therefore a mathematical analogy, as follows,
√
( 2/2δ̄0 + δ̄1 + δ̄2 − µ)ψ = 0.
The problem, of integrating the energy due to spin, as defined above, and mass,
and maybe, the effects of some fields, electromagnetic or whatever, is not obvious.
However, one might be tempted to express the content of this subsection, by
saying that the spin structure turns out to be an immediate consequence of the Big
Bang!
34
OLAV ARNFINN LAUDAL
4.4. The Universe as a Versal Base sSpace. So, where was the Big Bang,
in relation to our Time-Space, and what on earth is the meaning of the terms;
cosmological time, expansion of the universe, read-shift? How can one fill into this
geometric picture the more down to earth notions like; matter, stress, pressure,
charge, and forces, like; gravitation, electromagnetism, weak and strong forces,
acting on; elementary particles, quarks and their multiple combinations?
We should not have to goose-feed the Big Bang-created geometric picture, with
this additional structure. It should all be part of the Creation! Otherwise it must
be difficult to believe in the existence of this prime event.
Going back to the constructed family, the universal family of the Hilbert scheme
of sub-schemes of length 2 in A3 ,
π : E −→ H,
we have just proved that this family may be complemented with another family, no
longer a universal one, but just a versal family,
ν̃ : Ũ −→ H,
deduced from the family of 4-dimensional associative algebras, constructed above,
ν : U −→ H − ∆.
The 3-dimensional space ∆ is not a subspace of H, in fact, any point of this
ghost space correspond to the same 4-dimensional algebra, namely to U , the Big
Bang (BB) itself. A metric defined on ∆ therefore measures time at BB, before the
creation of the universe!
So let us fix a point ∗ ∈ ∆, the origin of the coordinate system (x1 , x2 , x3 ), used
to define U , thereby fixing the whereabouts of BB, clearly outside of our universe,
even though time is already there, as the metric in ∆, measuring 0-velocities of U .
Now, as we have seen, g(∗) = Derk (U ) = gl3 (k) acts on the tangentspace of the
versal base space, and in fact on the sub space identified with H. There is a very
special derivation, the Dirac derivation in this situation,


1 0 0
δ ∈ g(∗), δ = 0 1 0
0 0 1
the unit element.
Recall now that we know what cosmological time is, call it T , and look at some
function ρ(T ) ∈ k, and the corresponding action of,
δ(T ) := ρ(T )δ
on the tangent space of H, induced by the action on H, as part of the tangent space
of ∗ in the versal deformation space of U . At a point t = (o, p) ∈ H, the Dirac
derivation will induce a tangent,
δ(T )(o, p) = (ρ(T )o, ρ(T )p).
Now, as in [8] , consider the kernel of the Kodaira-Spencer map, see also [11].
Look at the family,
ν : U −→ H − ∆,
as parametrized by the k-algebra H = k[o1 , o2 , o3 , p1 , p2 , p3 ], outside the diagonal,
then the Kodaira Spencer map,
η : ΘH = Derk (H, H) → A1 (H, U, U),
is defined by,
∂
∂
η(
) = {ri,j = (xi xj − oi xj − pj xi + oi pj ) 7→
(ri,j ) = −xj + pj }
∂oi
∂oi
COSMOS AND ITS FURNITURE
35
∂
∂
) = {ri,j = (xi xj − oi xj − pj xi + oi pj ) 7→
(ri,j ) = −xi + oi }.
∂pi
∂pi
˜ is a linear combination of the standard vector fields of the form,
Any element δ ∈ ∆
X
X
∂
∂
ξ :=
ξi
+
ξj
,
∂oi j=1,2,3 ∂pj
i=1,2,3
η(
which is is easily seen to be an element in ker(η). But ker(η) also contains the
vector field,
X
X
∂
∂
δ :=
oi
+
pj
,
∂oi j=1,2,3 ∂pj
i=1,2,3
which is the same as the one induced by the Dirac derivation above.
Put ρ := ρ(T ), and see that,
(ρo, ρp) = (1/2(ρ · op), −1/2(ρ · op)) + (1/2(ρ · (o + p), 1/2(ρ · (o + p)).
˜ i.e. it is a 0-velocity.
The last tangent is in ∆,
Now assume we, to start, concentrate on the first question of this sub-section,
and assume the metric is the trivial one, so that mass, stress, and charge etc. is
neglected.
Then we find that the velocity associated to the direction of the tangent vector
(ρo, ρp) at t is given as v = sin(θ), where,
tg(θ) = |1/2(ρ · op), −1/2(ρ · op))|/|1/2(ρ · (o + p), 1/2(ρ · (o + p))|.
From this we deduce,
p
v/ 1 − v 2 = |op|/|(o + p)| = r/T,
or,
p
v = |op|/ |op|2 + |(o + p)|2 = r/t,
where T is cosmological time, and t is real time since the BB.
We have up to now, neglected the geometry of Cosmos, since to be able to say
something no-nonsensical about the metric, or the corresponding Laplace-Beltrami
operator Q, defining the gravitation of Cosmos, one should have to guess about a
content of the Universe, about the furniture, call it ξ, and deduce the metric from,
dξ
= Q(ξ).
dt
This seems to be what cosmologists are trying out, and I shall return to the
question in a later paper.
References
[1] O. Gravir Imenes (2011) The concept of charge in a model based on non-commutative algebraic geometry. Dissertation presented for the degree of Philosophiae Doctor. Faculty of
Mathematics and Natural Sciences, University of Oslo, September 2011.
[2] O. A. Laudal (1979) Formal moduli of algebraic structures Lecture Notes in Math.754,
Springer Verlag, 1979.
[3] O. A. Laudal (1986) Matric Massey products and formal moduli I in (Roos, J.E.ed.) Algebra,
Algebraic Topology and their interactions Lecture Notes in Mathematics, Springer Verlag,
vol 1183, (1986) pp. 218–240.
[4] O. A. Laudal (2002) Noncommutative deformations of modules Special Issue in Honor of
Jan-Erik Roos, Homology, Homotopy, and Applications, Ed. Hvedri Inassaridze. International Press, (2002).See also: Homology, Homotopy, Appl. 4(2002), pp. 357-396,
[5] O. A. Laudal (2000) Noncommutative Algebraic Geometry Max-Planck-Institut für Mathematik, Bonn, 2000 (115).
[6] O. A. Laudal (2001) Noncommutative algebraic geometry Proceedings of the International
Conference in honor of Prof. Jos Luis Vicente Cordoba, Sevilla 2001. Revista Matematica
Iberoamericana.19 (2003), 1-72.
36
OLAV ARNFINN LAUDAL
[7] O. A. Laudal (2003) The structure of Simpn (A) (Preprint, Institut Mittag-Leffler, 200304.) Proceedings of NATO Advanced Research Workshop, Computational Commutative
and Non-Commutative Algebraic Geometry. Chisinau, Moldova, June 2004.
[8] O. A. Laudal (2005) Time-space and Space-times Conference on Noncommutative Geometry
and Representatioon Theory in Mathematical Physics. Karlstad, 5-10 July 2004. Ed. Jürgen
Fuchs, et al. American Mathematical Society, Contemporary Mathematics, Vol. 391, 2005.
ISSN: 0271-4132.
[9] O. A. Laudal (2007) Phase Spaces and Deformation Theory Preprint, Institut Mittag-Leffler,
2006-07. See also the part of the paper published in: Acta Applicanda Mathematicae, 25
January 2008.
[10] O. A. Laudal (2007) Geometry of Time Spaces World Scientific, (2011).
[11] O. A. Laudal and G. Pfister (1988) Local moduli and singularities Lecture Notes in Mathematics, Springer Verlag, vol. 1310, (1988)
[12] F.Mandl and G. Shaw (1984) Quantum field theory A Wiley-Interscience publication. John
Wiley and Sons Ltd. (1984)
[13] C. Procesi (1967) Non-commutative affine rings Atti Accad. Naz. Lincei Rend.Cl. Sci. Fis.
Mat. Natur. (8)(1967), 239-255
[14] C. Procesi (1973): Rings with polynomial identities. Marcel Dekker, Inc. New York, (1973).
[15] R.K.Sachs and H.Wu (1977) General Relativity for Mathematicians Springer Verlag, (1977)
[16] T. Schücker (2002) Forces from Connes’ geometry arXiv:hep-th/0111236v2, 7 June 2002.
[17] S. Weinberg (1995) The Quantum Theory of Fields. Vol I, II, III Cambridge University
Press, (1995).
[18] St. Augustin Les Confessions de Saint Augustin par Paul Janet. Charpentier, LibraireÉditeur, Paris (1861).
Matematisk institutt,
University of Oslo,
Pb. 1053, Blindern,
N-0316 Oslo, Norway
E-mail address: arnfinnl@math.uio.no