quantum gravity, cosmology and categori...

Application 01/04 to the Addendum Nu. 1

of the Protocol on Scientific Cooperation

between

the Austrian Academy of Sciences and

the National Academy of Sciences of Ukraine

concluded on February 7, 1996

Multilateral research project

Quantum Gravity, Cosmology and

Categorification

Contents

1 Status of Research 1

1.1 2D dilaton quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Non-commutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Topological quantum field theories . . . . . . . . . . . . . . . . . . . . . . 61.5 Categorification of quantum gravity . . . . . . . . . . . . . . . . . . . . . 11

2 Aim of the Project and Work Plan 16

3 Personnel, Rearch Institutes and Funds 17

i

1 Status of Research

At the moment, cosmology is one of the fastest changing fields in physics. This fact

might, on the one hand, be ascribed to the vast amount of new observational data

(cf [1–3] e.g.), on the other hand there are still fundamental open questions within what

is nowadays called the cosmological standard model [4,5]: Where does the inflaton field

come from? Is there something like the cosmological constant Λ which contributes to

the dark energy etc.?

From a theoretical viewpoint, one might divide efforts today within cosmology into

two broad subclasses.

Firstly, we have models which extend the standard model to a certain amount, inflation

[6, 7], e.g., can be viewed as an add-on for the classical Friedman-Lemaıtre-Robertson-

Walker (FLRW) model. All these models have in common that they do not affect the

structure of space-time itself, i.e., they are still bound to a four-dimensional Riemannian

space-time and, in addition, do not modify the underlying gravity theory, i.e., General

Relativity (GR).

Secondly, we have models which are no longer tied to Riemannian space-time and might

also modify the underlying gravity theory.

1.1 2D dilaton quantum gravity

One example for the latter is the generalised gravity theories in two dimensions. Models

of gravity in two dimensions have been largely studied in recent years as toy models

addressing issues that are too complex to be faced directly in four dimensions. However,

the dynamics of two-dimensional gravity is rather different from its four-dimensional

counterpart, since the Hilbert-Einstein action is a topological invariant in two dimensions

and hence gives rise to trivial field equations. In order to derive the field equations

from an action principle it is then necessary to introduce an auxiliary scalar field η

(which in the following will be called dilaton) [8]. Recent astrophysical data suggest

that the Hilbert-Einstein action may need to be modified by rather non-standard terms

1

(1/R and even ln R were considered as possibilities, see [9]). Dilaton theories provide

a natural framework for such modifications, and two dimensional models are, as usual,

a convenient test ground. Such models, the most prominent being the dilaton gravity

of Jackiw and Teitelboim (JT) [8, 10–13], represent linear gauge theories. An excellent

summary (containing also a more comprehensive list of references on literature before

1988) is contained in the textbook of Brown [14]. Among those models spherically

reduced gravity (SRG), the truncation of D = 4 gravity to its s-wave part, possesses

perhaps the most direct physical motivation. One can either treat this system directly

in D = 4 and impose spherical symmetry in the equations of motion (e.o.m.-s) [15]

or impose spherical symmetry already in the action [15–25], thus obtaining a dilaton

theory. The rekindled interest in generalised dilaton theories in D = 2 (henceforth

GDTs) started in the early 1990-s, triggered by the string inspired [26–33] dilaton black

hole model1, studied in the influential paper of Callan, Giddings, Harvey and Strominger

(CGHS) [35]. At approximately the same time it was realized that 2D dilaton gravity

can be treated as a non-linear gauge-theory [37,38]. As already suggested by earlier work,

all GDTs considered so far could be extracted from a second order dilaton action [39,40].

A common feature of these classical treatments of models with and without torsion is

the almost exclusive use2 of the gauge-fixing for the D = 2 metric familiar from string

theory, namely the conformal gauge. Then the e.o.m.-s become complicated partial

differential equations. The determination of the solutions, which turns out to be always

possible in the matterless case, for nontrivial dilaton field dependence usually requires

considerable mathematical effort. The same had been true for the first papers on theories

with torsion [42, 43]. However, in that context it was realized soon that gauge-fixing is

not necessary, because the invariant quantities R and T aTa themselves may be taken

as variables in the Katanaev-Volovich (KV)-model [44–47]. This approach has been

extended to general theories with torsion3. As a matter of fact, in GR many other gauge-

fixings for the metric have been well-known for a long time: the Eddington-Finkelstein

1A textbook-like discussion of this model can be found in refs. [34, 36].2A notable exception is Polyakov [41].3A recent review of this approach is provided by Hehl and Obukhov [48].

2

(EF) gauge, the Painleve-Gullstrand gauge, the Lemaitre gauge etc. . As compared to

the “diagonal” gauges like the conformal and the Schwarzschild type gauge, they possess

the advantage that coordinate singularities can be avoided, i.e. the singularities in those

metrics are essentially related to the “physical” ones in the curvature. It was shown for

the first time in [49] that the use of a temporal gauge for the Cartan variables in the

(matterless) KV-model made the solution extremely simple. This gauge corresponds

to the EF gauge for the metric. Soon afterwards it was realized that the solution

could be obtained even without previous gauge-fixing, either by guessing the Darboux

coordinates [50] or by direct solution of the e.o.m.-s [51]. Then the temporal gauge of [49]

merely represents the most natural gauge fixing within this gauge-independent setting.

The basis of these results had been a first order formulation of D = 2 covariant theories

by means of a covariant Hamiltonian action in terms of the Cartan variables and further

auxiliary fields Xa which (beside the dilaton field X) take the role of canonical momenta.

They cover a very general class of theories comprising not only the KV-model, but also

more general theories with torsion4. The most attractive feature of such theories is that

an important subclass of them is in a one-to-one correspondence with the GDT-s. This

dynamical equivalence, including the essential feature that also the global properties are

exactly identical, seems to have been noticed first in [52] and used extensively in studies

of the corresponding quantum theory [53–55]. In the latter the temporal gauge again

prevaricates complications from Faddeev-Popov ghosts [56] which are present otherwise.

Generalizing the first order formulation to the much more comprehensive class of

“Poisson-Sigma models” [57, 58] on the one hand helped to explain the deeper reasons

of the advantages from the use of the first oder version, on the other hand led to very

interesting applications in other fields [59], including especially also string theory [60,

61]. Recently this approach was shown to represent a very direct route to 2D dilaton

supergravity [62] without auxiliary fields. For more technical and historical details on

dilaton gravity the review [63] may be consulted.

4In that case there is the restriction that it must be possible to eliminate all auxiliary fields Xa and

X .

3

There are also purely two-dimensional reasons to look for generalisations of the dilaton

gravities. It has been demonstrated [64] that the exact string black hole [29] cannot be

embedded in generalised dilaton gravities. Later Strobl [65] suggested a very general

framework for topological gravity theories. However, the relation between these theories

and metric theories of gravity is not clear. The methods of non-perturbative treatment

of classical and quantum dilaton gravities have been developed for pseudo-Euclidean

spaces. The Euclidean regime differs considerably from the pseudo-Euclidean one since,

for example, different asymptotic conditions and different gauge conditions have to be

used. Another example is non-commutative gravity in two dimensions which leads to

the second main topic.

1.2 Non-commutative geometry

Over recent years, non-commutative geometry interacts fruitfully with theoretical physics.

We want to mention the Seiberg-Witten approach to non-commutative field theory

[61, 66, 67], especially. There, matter and gauge fields are replaced by Seiberg-Witten

maps of the commutative fields and variables, the pointwise product by the Weyl-Moyal

product. The Seiberg-Witten approach provides a systematic way to introduce Lorentz

violating operators into the Lagrangian. It also enables one to use arbitrary gauge

groups. The action can be expanded in the non-commutativity parameter. The ze-

roth order term resembles the commutative action. The additional Lorentz violating

terms are not put in by hand, but they represent the effect of the non-commutative

space-time structure [66–71]. Therefore, also the standard model with gauge group

SU(3)C × SU(2)F × U(1)Y can be attacked by these means [68]. However, also an al-

ternative approach to the standard model exists [69]; additional degrees of freedom are

introduced, which they have to get rid of at the end. Quantisation in the θ-expanded

theory, as presented here, seems to be straight forward. Feynman rules can be extracted

from the Lagrangian directly. No problems with unitarity are expected to be encoun-

tered. However, problems with unitarity occur in the non-expanded theory. These prob-

lems can be solved by a consequent analysis of perturbation theory in a Hamiltonian

4

approach, cf. [72–76] for scalar field theory.

The Seiberg-Witten approach to non-commutative field theory does not only work

for constant non-commutativity parameter θ, but can also be generalised to space-time

dependent θ(x) [77–81].

Of special interest is the so called κ-deformation. Similar to q-deformation, space-time

acquires a quantum group symmetry. Poincare covariance is not broken but deformed

to κ-Poincare covariance [78, 79, 82–84]. The connection of κ-deformed field theory, or

deformed field theories in general, to quantum gravity has to be explored thoroughly.

First steps have been done in [85].

Since no satisfactory non-commutative gravity exists so far, two-dimensional theories

(cf. sect. 1.1) may be again a good starting point. Indeed, some progress in this direc-

tion already exists (see [86, 87] where a non-commutative version of the JT model was

constructed).

1.3 Cosmology

One of the main topics in the theory of gravitation is the study of cosmological models.

In [88] have been studied two-dimensional cosmologies in the context of the JT-model,

in the case of minimally coupled and conformally coupled matter. On the one hand, the

main reason to consider more general structures within cosmology is the idea that new

geometrical quantities might shed light on the problems of the cosmological standard

model, e.g. provide an explanation for the rather artificial introduction of an additional

scalar field, like the inflaton. Especially the inclusion of torsion, and possibly non-

metricity, may be a good starting point for extending the standard model of cosmology

by means of new geometrical quantities. First promising results by including torsion are

in [89]. The new quantities couple to the anisotropic space-time [90, 91], spin, shear,

and dilaton current of matter, which are supposed to come into play at high energy

densities, i.e., at early stages of the universe. An example for the latter is the assertion

that quantum effects of the electromagnetic field (EMF) in the external gravitational

field in the anisotropic Bianchi I model give a contribution to the degree of polarisa-

5

tion of the EMF in the quadrupole harmonics. It is known that the size of this effect

parametrically depends on the moment of time starting from which the vacuum of EMF

became unstable. On the assumption of the observational limits on the quantity of the

degree of polarisation of the cosmic microwave background (CMB) one can determine

the limits on the amount of red shift beyond which quantum effects started to play a

role. According to the results of the papers [90,91], the moment of time when the quan-

tum effects of photons switch on can correspond to the rising of the anisotropy on the

background of the initially isotropic matter.

1.4 Topological quantum field theories

The topics of 1.1-1.3, although connected by the use of special (2D) models, seem quite

separated. Part of the difficulty of combining general relativity and quantum theory is

that they use different sorts of mathematics: one is based on objects such as manifolds,

the other on objects such as Hilbert spaces. As “sets equipped with extra structure”,

these look like very different things, so combining them in a single theory has always led

to difficulties. However, work on topological quantum field theory has uncovered a deep

analogy between the two. Moreover, this analogy operates at the level of categories and

here the modern methods of category theory may provide further relations [92–96]. In

refs. [97–100] is has been attempted to formulate the method of additional structures as

a set of axioms for a category, which would be sufficient for an abstract expression of

the basic concepts of the theory of structures on objects of a category. Then all main

properties of a structure are properties of its forgetful functor. Additional (external)

structures on objects of a category provide the possibility to construct new categories

for physics [101].

In physics, interest in categories was sparked by developments relating topology and

quantum field theory. In 1985, Jones [102] came across an invariant of knots, which

could be systematically derived from quantum groups, invented in exactly soluble 2-

dimensional field theories. In a next step, 3-dimensional gravity was introduced into

modern physics by Deser, Jackiw and ’t Hooft [103, 104], then Witten arrived at a

6

manifestly 3-dimensional approach to the new knot invariants, deriving them from a

quantum field theory in 3-dimensional space-time (Chern-Simons theory) [105]. This

approach also gave invariants of 3-dimensional manifolds.

Atiyah formulated in 1989 an axiomatic setup for topological quantum field theories

(TQFTs) [106]. Independently and at about the same time G. Segal gave a mathematical

definition of conformal field theories (CFTs) [107], which is very similarly based on

categories and functors. In 1993 J. Frohlich and T. Kerler demonstrated that tensor

categories play a central role in mathematical formulation of quantum groups and TQFT

[108].

We shall focus on two categories in this project. One is the category CKS whose

objects are Cayley-Klein spaces (CKS) [109–111] and whose morphisms are linear op-

erators between these. The other is the category nCKG whose objects are (n − 1)-

dimensional Cayley-Klein geometries (CKG) [112–116] and whose morphisms are n-

dimensional Cayley-Klein geometry going between these. This plays an important role

in relativistic theories where spacetime is assumed to be n-dimensional: in these theo-

ries the objects of nCKG represents possible choices of “space geometries”, while the

morphisms – called “cobordism” – represent possible choices of “space-time geometries”.

While an individual manifold does not resembles very much like a Cayley-Klein space,

the category nCKG turns out to have many structural similarities to the category CKS.

The goal of this project is to explain these similarities and show that the most puzzling

features of quantum theory all arise from ways in which CKS resembles nCKG more

than the category Set, whose objects are sets and whose morphisms are functions. In

quantum field theory on curved space-time, space and space-time are not just mani-

folds: they come with fixed “Cayley-Klein metrics” that allow us to measure distances

and times. In this context, S and S ′ are Cayley-Klein manifolds, and M : S → S ′ is a

Cayley-Klein cobordism from S to S ′. A topological quantum field theory then consists

of a map Z assigning a Hilbert space of states Z(S) to any (n−1)-manifold S and a linear

operator Z(M) : Z(S) → Z(S ′) to any cobordism between such manifolds. This map

cannot be arbitrary, though: it must be a functor from the category of n-dimensional

7

cobordisms to the category of Hilbert spaces. A functor between categories is a map

sending objects to objects and morphisms to morphisms, preserving composition and

identities. Our main point is that treating a TQFT as a functor from the category of

n-dimensional cobordisms to the category of Hilbert spaces is a way of making very

precise some of the analogies between general relavity and quantum theory. However,

we can go further! A TQFT is more than just a functor. It must also be compatible with

the “monoidal category” structure of the category of n-dimensional cobordisms and the

category of Hilbert spaces.

So, a n-dimensional TQFT is defined as a monoidal functor from the category Cob(n+1)

of oriented (n+1)-cobordism with disjoint union as tensor product to the category Vect

of complex finite dimensional vector spaces with the usual tensor product of vector

spaces.

In recent years, there has been also an increasing interest in algebraic structures on

a modular category motivated by coherence problems arising from TQFT [117, 118].

The categories of representations of Cayley-Klein quantum groups are braided monoidal

Cayley-Klein categories [119, 120]. Another motivation comes from developments in

homotopy theory, in particular, models for the stable homotopy category. Monoidal

categories correspond to loop spaces, and the group completion of the classifying space

of a braided monoidal category is a two-fold loop space [121]. Modular categories are

monoidal categories with additional structure (braiding, twist, duality, a finite set of

dominating simple objects satisfying a non-degeneracy axiom). If we remove the last

axiom, we get a pre-modular category. A pre-modular category provides invariants of

links, tangles, and sometimes of 3-manifolds. Any modular category yields a TQFT

[122–124].

There are modifications of the definition of TQFTs by modifying the cobordism cate-

gory by using additional structures on manifolds. For instance, we can specify a framing

of the tangent bundle of the cobordisms and of a formal neighborhood of the closed

manifolds. Another possibility is to include insertions of submanifolds in the manifolds

and matching insertions in the cobordisms. Tensor and duality preserving functors from

8

such modified cobordism categories to Vect are called TQFTs too.

Any modification in the cobordism category may lead to a modification in TQFT.

This modification can be thought of as an extended version of TQFT. For example in

Chern-Simons TQFT, cobordisms are supplied with some additional structures.

The role of higher-dimensional algebras is clear from the various constructions of

extended TQFTs. Baez and Dolan [125] outline a program in which n-dimensional

TQFTs are described as n-category representations. They described a n-dimensional

extended TQFT as a weak n-functor from the free stable weak n-category with duals

of one objects to n-Hilb , the category of n-Hilbert spaces, which preserve all levels of

duality. Homotopy theory methods were used to build examples of TQFTs. Homotopy

quantum field theories (HQFT) are defined as topological quantum field theories for

manifolds endowed with additional structure in the form of a map into some background

space X, it is a theory of objects over X. All these theories do is to fix a background

space X and to compute a weighted sum over homotopy classes of maps f : M → X for

a closed manifold M .

There are the important theorems by Reshetikhin and Turaev saying that HQFTs

only depended on the n-homotopy type of X [123, 126, 127]. We suggest that TQFTs

can be considered as a first approximation to full-blown quantum gravity, HQFTs are a

first approximation to gravity coupled with matter.

In n-categorical set up, one of the examples of monoidal 2-categories is the category

nCob , which has 0-manifolds as 0-cells, 1-manifolds with corners, i.e., cobordism between

0-manifolds as 1-cells, and 2-manifolds with corners as 2-cells.

Instead of taking 0-cells as 0-manifolds, one can also start with objects as 1-manifolds

with or without corners to get Atiyah-Segal-style TQFT.

A 2-dimensional TQFT is a particular case of the construction. Here the category

Cob 1+1 or Cob 2 has compact oriented 1-manifolds as objects and compact oriented

cobordism between them as morphisms.

Extended TQFTs constructed by Kerler and Lyubashenko [128] involves higher cat-

egory theory, namely double categories and double functors. Their construction of ex-

9

tended version of TQFTs is quite different from the n-categorical version of extended

TQFTs proposed by Baez and Dolan. It is not a generalized version of Turaev’s construc-

tion of TQFT functor, actually both constructions are different because of the different

base categories.

Baez’ and Dolan’s hypothesis for extended TFQTs [129] shows that the TQFT func-

tor which produce 2-dimensional extended TQFTs cannot be generalised easily to a

3-dimensional extended TQFT functor. Either they do not have a nice structure in

higher dimensions or their structure is very complicated, e.g. the enriched n-categorical

version of Vect is not very clear in dimension n ≥ 2.

For n = 2, one can think 2-vector spaces as a vector space over the category Vect k of

vector spaces over k.

Thus, we have a monoidal category M with tensor product ⊗ and a functor ⊗ :

Vect k ×M → M satisfying various conditions. One needs to construct different TQFT

functors at different dimensional levels. This suggests that in most of the higher dimen-

sional cases these TQFTs functors will be independent of each other.

For the higher dimensional extended TQFTs, one needs to generalise internal categor-

ical structures for higher dimensions in such a way that existing base category structures

remain preserved, e.g. as in the case of 2Vect , which contains ordinary vector spaces as

objects. If we consider 3Vect to be the category having objects as internal categories of

2Vect and arrows are internal functors, then under suitable conditions 3Vect can give

a higher category version of 2Vect which also contains 2-vector spaces as objects.

The n-category structure is a result of iteration using the ordinary categorical structure

with weaker modified coherence conditions. Our deformation of the category structure is

similar. We modify diagonal comultiplication, but save all diagrams from the categorical

axioms.

We describe a deformation of categories which gives new structures. But their theory

is similar to the category theory because we deform only comultiplications which are in

compositions on all levels in n-category. Our deformation can be applied to n-categories

on different levels. Such a deformation of j-level induces a deformation of the structure

10

on all higher levels.

On the other hand, weak n-categories (as far as the notion is properly developed)

have been put to use in extended TQFTs, pointing towards applications in the field of

quantum gravity. In a n-dimensional TQFT, a functor is given from the n-dimensional

cobordism category (with objects compact, closed, oriented (n-1)-dimensional manifolds

and morphisms the n-dimensional cobordisms between them) to the category Vect of

finite-dimensional vector spaces. Roughly speaking, in extended TQFTs singularities

of different codimensions are allowed which lead to a structure of a higher dimensional

category instead of the simple cobordism category and, consequently, a corresponding

notion of higher functor on this. An especially interesting point is discussed in [125,130,

131]. The idea emerged that categorification (i.e., lifting mathematical structures from

sets to higher categories) is related to quantisation.

1.5 Categorification of quantum gravity

The construction of a quantum theory of gravity remains probably the issue left unsolved

in the last century, in spite of a lot of efforts and many important results obtained during

an indeed long history of works. The problem of a complete formulation of quantum

gravity is still quite far from being solved. With this relationship in mind, there is a

strong motivation to explore the use of categorical methods in approaches to quantum

gravity. We should mention the recent work on Categorical State Sum (CSS) models

for quantum gravity (see [129] and the literature cited therein) where also methods of

category theory (and of higher categories in the case of open spin foams) are involved.

The CSS models seem to offer a promising road to a quantum geometry of space-time

and to a possible understanding of the relationship between quantum gravity in the more

conventional sense and string theoretical approaches.

Categorical State Sum (spin foam) models for quantum gravity are obtained by trans-

lating the geometric information on a (usually triangulated) manifold into the language

of combinatorics and category theory, so that the usual concepts of a metric and of

metric properties are somehow emerging from them, and are not regarded as fundamen-

11

tal. Moreover, a spin foam model implements in a precise way the idea of a sum over

geometries, so that we can envisage here a renaissance of the covariant and path integral

program for the quantisation of gravity [132], but now we are summing over labelled

2-complexes (spin foams), i.e., collections of faces, edges and vertices combined together

and labelled by representations of a group (or a quantum group). The space-time geom-

etry results from these elements only, i.e., from the fields of algebra and combinatorics

(and piecewise linear topology).

Spin foam models were developed also for topological field theories in different dimen-

sions, including 3d quantum gravity [133–137], and this represents a completely different

line of research arriving at the same formalism. In these models, category theory plays a

major role, since their whole construction can be rephrased in terms of operations in the

category of a Lie group (or quantum group), making the algebraic nature of the models

manifest.

The framework of spin foam models is very versatile. Spin foam models for many

different kinds of theories exist. There exists a spin foam formulation for topological

field theories [133–135], lattice gauge theories, both abelian and non-abelian [138, 139],

and gravity [140, 143–147].

We now turn to an analysis of a specific spin foam model for Euclidean quantum

gravity, trying to be as complete as possible, and pointing out the basic ideas as well as

the connections of the model with classical gravity. An appealing way to look at a spin

foam, about which we will say a bit more in the following, is as a kind of ”Feynman

diagram” occurring at the vertices [148].

So in a spin foam model the crucial element is the vertex amplitude which encodes

all the information about the interactions and the dynamical content of the theory.

Thus, the question to answer for a construction of a spin foam model for quantum

gravity is: What is the correct form of the vertex amplitude? Thinking of the spin foam

as embedded in a triangulated 4-manifold, as a coloured dual 2-skeleton of it, this is

translated into: What is the quantum amplitude for a 4-simplex? or how to describe

a quantum 4-simplex. More precisely, the problem one has to solve is how to translate

12

the geometrical information necessary to completely characterise a 4-simplex at the

classical level into the quantum domain of algebra and representation theory, obtaining a

characterisation of a quantum 4-simplex. Barrett and Crane [140] answered this question

precisely. A geometric 4-simplex in Euclidean space is given by the embedding of an

ordered set of 5 points (0, 1, 2, 3, 4) in R4 (its subsimplices are given by subsets of this

set) with embedding determined by the 5 position coordinates x0, x1, x2, x3, x4 ∈ R4

and required to be non-degenerate (the points should not lie in any hyperplane). Each

triangle in it determines a bivector (i.e., an element of ∧2R

4) constructed out of the

displacement vectors for the edges, taking the wedge product of two of them. Barrett and

Crane proved that, classically, a geometric 4-simplex in Euclidean space is completely

and uniquely characterised (up to parallel translation and inversion through the origin)

by a set of 10 bivectors bi, each corresponding to a triangle in the 4-simplex and satisfying

some properties.

Now, the problem is to find the corresponding quantum description, i.e., a charac-

terisation of a quantum 4-simplex. The crucial observation is that bivectors can be

considered as elements of the Lie Algebra so(4), because of the isomorphism between

∧2R

4 and so(4). We associate to each triangle in the triangulation a so(4) element.

Then, we turn these elements into operators choosing a (different) representation of

so(4) for each of them, i.e., considering the splitting so(4) ' su(2) ⊕ su(2), a pair of

spins (j, k), so that we obtain bivector operators acting on the Hilbert space given by the

representation space chosen. Each tetrahedron in the triangulation is then associated

with a tensor in the product of the four spaces of its triangles.

We can evaluate it using the graphical calculus developed for representation group

[116, 149–151] and quantum group theory [152] or, equivalently, the spin networks.

As we have seen, the field theory over group formalism provides a way to obtain this

sum over 2-complexes in a straightforward manner, and also gives a prescription for the

calculation of the weight to be assigned to each 2-complex. But the geometrical and

physical meaning of this approach has still to be fully investigated and understood. The

development of alternative approaches would also be of much interest in resolving of

13

cosmological problems.

The Barrett-Crane (BC) model [140,141] is a constrained topological state sum model

for quantum gravity. Recently [142], it was proposed that this model might incorporate

matter and gauge interactions if the condition on triangulations to be manifolds were

relaxed. That is, conical singularities would act as seeds of matter in the quantum

geometry of the state sum. The purpose of this Project is to examine the consequences

of this proposal, and of spin foam models in general, for early universe phenomenology.

The CKGs are quantized by using representation theory to obtain Hilbert spaces on

which geometric quantities act as operators. Thus, a categorical state sum is a discretized

version of a Feynman vacuum, where the fields and vertices correspond to CKG.

In other words, quantum CKGs in this approach are represented by families of Hilbert

spaces on which the sort of quantities typically measured in classical CKGs act as oper-

ators. The most basic geometric quantities are bivectors, i.e., skew symmetric rank two

tensors, which describe oriented area elements. Utilizing their expedient quantisation,

we define the other geometric quantities in terms of these bivectors, which are attached

to the faces of a triangulation.

In the quantisation procedure of the BC model [140], bivectors are represented by

representations of the Cayley-Klein algebras (CKAs) [109–115]. The bivectors on faces

and tetrahedra are constrained to be simple, i.e., to correspond to oriented area elements.

This has a natural quantisation in the restriction to the balanced unitary representations.

The conical matter proposal (CMP) is to consider

a) the conical singularities on edges as generating particles which propagate through

space and

b) the conical singularities on vertices as interaction vertices.

In comparison to other fundamental physical theories involving matter, the CMP has

one advantage: matter is naturally included in the theory of quantum gravity, rather than

added by hand. There is no new element; neither a gauge group, nor extra dimensions,

nor a topology on a compact manifold. The surfaces are not insertions into space-time;

they are only descriptions of part of its topological structure. It is therefore highly

14

remarkable, as we will explain, that the most natural approximation scheme available

suggests that the Standard Model may emerge from it.

Let us summarise the picture of the history of the universe which our model seems to

suggest. There would be an early (or rather sub-Planckian) phase, in which the universe

would be modelled by a topological quantum field theory, but with conic singularities

included in the manifold. This phase would be a substitute for the initial singularity of

Standard Big Bang (SBB) model. It would be followed by a phase of quantum grav-

ity, in which genus 1 and higher genus conic singularities would be interacting, while

the universe expanded and cooled. Next would come a decoupling, in which further

interactions involving higher genus singularities would be suppressed by topological ob-

structions, leaving an interacting world composed of genus 1 singularities.

Since it is very natural to pass from classical to quantum groups in constructing

categorical state sums, and in particular since the quantum BC model is well behaved

[140], our discussion above easily accommodates a cosmological constant, and indeed

may even require it. The representation theory of the quantum CKA seems to give a

quantum geometry of space with constant curvature very similar to the quantum CKG.

The phase transition might arise as a result of coarse graining of the topological

universe, which at the origin of time fluctuated into a combination of quantum variables

corresponding to a sufficiently large “size”. The question of the phase transition is

the point which most strongly suggests to us that still deeper theoretical constructions

will ultimately be needed in the pre-big bang scenario within string cosmology theory

[154, 155].

We have some speculative ideas about the emergence of a deeper theory. One aesthetic

drawback to the Project we are proposing is that we begin with space-time, and produce

matter as a sort of pinch within it. One might wish, rather, that space-time and matter

play dual roles. A hint that such a model might be possible is that the 2D modular

functor is a geometric realization of the braided monoidal categories, which reproduce

the category of representation of quantum groups. This is suggestive of a deeper model

with matter—space-time duality.

15

2 Aim of the Project and Work Plan

The main aim of the project is to construct a self–consistent categorical version of the

Topological Quantum Field Theory (TQFT) and Quantum Gravity and apply it in

resolution of cosmological problems. For the realization of this aim we are planning to

address as many of the following issues as possible within the 3 years period:

• First year (01.07.2004 – 31.12.2004):

to get restrictions on equivalence classes of generalisations of the Hilbert-Einstein

theories;

to construct modular Cayley-Klein categories as multiplicative structures of TQFT;

• First year (01.01.2005 – 30.06.2005):

to obtain an action for the exact string black hole which should facilitate further

study of string beta functions and non-perturbative effects;

to study applications of category theory to the problems of string cosmology;

• Second year (01.07.2005 – 31.12.2005):

to develop non-perturbative calculations of correlation functions in Liouville field

theory;

to develop the graphical calculus for categorical spin foam models of quantum

gravity;

• Second year (01.01.2006 – 30.06.2006):

to develop a non-commutative generalisation of general dilaton gravity theories in

2D;

to look for quantum effects of the generation of polarisation of the cosmic mi-

crowave background (CMB) in the case of the Cayley-Klein-Cartan model;

16

• Third year (01.07.2006 – 31.12.2006):

to study quantum effects and Hawking radiation in non-commutative theories;

to study finite temperature effects for black holes and the early Universe;

to study the topological phase transitions in the early Universe;

• Third year (01.01.2007 – 30.06.2007):

to look for physical effects near non-commutative black holes;

to study the pre-big bang scenario within string cosmology theory;

to develop methods of group-theoretical & category-theoretic analysis of physical

effects in cosmological physics.

We are aware of the fact that this is an extremely ambitious plan. But even partial

results would represent important progress within the respective fields involved and may

contribute to the large coherent picture we have in mind.

3 Personnel, Rearch Institutes and Funds

The investigations should be done by

• Univ.-Prof. Dr. W. Kummer (Austria),

• Dr. D. Grumiller (Austria),

• Dr. M. Wohlgenannt (Germany),

• PhD-student C. Bohmer (Germany),

• Dr. D. V. Vassilevich (Ukraine),

• Dr. A.T.Vlassov (Ukraine),

• Dr. S.S. Moskaliuk (Ukraine),

• Dr. I. Burban (Ukraine),

17

• Univ.-Prof. Dr. A.G. Zagorodny (Ukraine),

and other experts and PhD-students from Austria, Czech Republic, Germany, Slovakia

and Ukraine.

The research of the proposed project will be carried out at the existing Institutes and

Departments of the AAS and the NASU together with the Austro-Ukrainian Institute

for Science and Technology (AUI) and other structures as follows:

• the Bogoliubov Institute for Theoretical Physics of NASU (Amount of support is

equal to EUR 12 000, which includes personal costs for Ukrainian researchers and

monographs’ printing expenses);

• the W. Thirring Institute for Mathematical Physics, Astrophysics and Nuclear In-

vestigations (Ukraine) (Amount of support is equal to EUR 15 000, which includes

visas support for Austrian and German researchers and travel costs to Austria for

Ukrainian researchers);

• the Bratislava Innovation Centre for Technology, Re-engineering and Business

(Amount of support is equal to EUR 15 000, which includes visas support and

accommodation in Slovakia for Ukrainian researchers);

• the Czech Research Centre (Amount of support is equal to EUR 15 000, which

includes visas support and accommodation in Czech Republic for Ukrainian re-

searchers);

• the Austro-Ukrainian Institute for Science and Technology (Amount of support is

equal to EUR 15 000, which includes visas support for Ukrainian researchers and

book’s printing expenses);

• the Austrian Science Fund – FWF: P15463-N08 (M. Wohlgenannt), Erwin Schrodinger

fellowship Project J2330-N08 (D. Grumiller), Osterreichischer Akademischer Aus-

tauschdienst – OEAD: 798-1/2003 (C. Bohmer) (Amount of support is equal to

EUR 72 000, which includes personal costs for German and Austrian researchers);

18

• the Austrian Academy of Sciences (Amount of support is equal to 72 000, which

includes travel costs to Ukraine for Austrian and German researchers and accom-

modation during visit Vienna for Ukrainian researchers).

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33

For and on behalf For and on behalf of the National

of the Austrian Academy of Sciences Academy of Sciences of Ukraine

(AAS) (NASU)

DIPL.-ING. DR. TECHN. A. VOGEL Univ.-Prof. Dr. A. P. SHPAK

Executive Director of AAS Chief Scientific Secretary of NASU

Univ.-Prof. Dr. W. KUMMER Univ.-Prof. Dr. A. G. Zagorodny

Member of AAS Director of the Bogoliubov Institute

Vice-president of the AUI for Theoretical Physics (BITP) of NASU

Vienna, 17.05.2004 Dr. S. S. MOSKALIUK

Kyiv, Scientific Researcher of BITP of NASU

President of the AUI

34

quantum gravity, cosmology and categori...

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