Quantum Gravity, Cosmology and Categori...
Transcript of 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
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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-
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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
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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
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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