# Loop representation of Quantum Gravity

Adrian P. C. Lim
Email<EMAIL_ADDRESS>
###### Abstract

A hyperlink is a finite set of non-intersecting simple closed curves in
$\mathbb{R}^{4}\equiv{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, each curve is
either a matter or geometric loop. We consider an equivalence class of such
hyperlinks, up to time-like isotopy, preserving time-ordering. Using an
equivalence class and after coloring each matter component loop with an
irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$, we can
define its Wilson Loop observable using an Einstein-Hilbert action, which is
now thought of as a functional acting on the set containing equivalence
classes of hyperlink.

Construct a vector space using these functionals, which we now term as quantum
states. To make it into a Hilbert space, we need to define a counting
probability measure on the space containing equivalence classes of hyperlinks.

In our previous work, we defined area, volume and curvature operators,
corresponding to given geometric objects like surface and a compact solid
spatial region. These operators act on the quantum states and by deliberate
construction of the Hilbert space, are self-adjoint and possibly unbounded
operators.

Using these operators and Einstein’s field equations, we can proceed to
construct a quantized stress operator and also a Hamiltonian constraint
operator for the quantum system. We will also use the area operator to derive
the Bekenstein entropy of a black hole.

In the concluding section, we will explain how Loop Quantum Gravity predicts
the existence of gravitons, implies causality and locality in quantum gravity,
and formulate the principle of equivalence mathematically in its framework.

MSC 2010:  83C45, 81T45, 57R56
Keywords: Loop Quantum gravity, Einstein-Hilbert, Quantum geometry, Causality,
Locality, Bekenstein entropy, Hamiltonian, Stress operator, Gravitons

##  1 Introduction

After Bernhard Riemann developed non-Euclidean geometry, or more commonly
known as Riemannian geometry, he tried to work on a geometric theory of
gravity before his death in 1866. He did not succeed, simply because he was
only working on 3-dimensional space. It was half a century later that Einstein
made use of Riemannian geometry and developed the theory of gravity on
4-dimensional space-time, which is now more commonly known as Generality
Relativity (GR).

The turn of the 20th century saw the birth of Quantum Mechanics (QM), which
was successful in explaining the orbits of electrons in an atom. The
mathematical framework for the formalism of Quantum Mechanics (QM) is based on
Functional Analysis, of which its rigorous mathematical foundation is
attributed to important researchers like Dirac, Hilbert, Neumann and Weyl.

At around 1927, physicists began to apply QM to fields, resulting in Quantum
Field Theories (QFT). This calls for making QFT mathematically consistent,
which leads to the emergence of a new branch of mathematics called
Constructive Quantum Field Theory (CQFT). The article [1] gives a very good
account of CQFT. See also [2]. A sub-branch of CQFT would be Gauge Quantum
Field Theory (GQFT), whereby one would quantize classical local gauge theories
using path integrals.

The next natural step would be to quantize gravity. Thiemann in [3, 4],
stressed the importance of a background free metric in a quantum theory of
gravity, especially in extreme cases such as near a black hole or near the big
bang singularity. When the effects of gravity can be ignored, it is reasonable
to use the Minkowski metric, which is the metric used in QFT. However, GR
resolves to do away with this background metric. Perturbative methods will no
longer work when trying to merge GR with QFT together.

The two theories, GR and QM, are incompatible in many ways, and by no means
are the reasons we cite below exhaustive. Firstly, time is absolute in QM, but
time is relative in GR. Causality is respected in relativity and on this note,
GR does not allow instantaneous communication of information. But whether QM
respects causality is debatable. A correct quantum theory of gravity should
treat time as relative and incorporate causality in its theory. Thus, the
notion of absolute time has to be abandoned. We will replace it with time-
ordering instead, to be discussed extensively throughout this article.

Secondly, GR gives us a deterministic geometric description of space-time,
whereas QM is a non-deterministic theory. We cannot keep both characteristics,
and it is obvious that a quantum theory of gravity must be a non-deterministic
description of space-time. Thus, we must have a topological theory to form the
mathematical foundation of a quantum theory of gravity. But bear in mind that
if we have a topological theory, then there is no distinction between space
and time, hence causality will not be respected. Our proposed idea would be to
consider an equivalence class of geometrical objects in the theory, in place
of topological objects. See Section 6.

Finally, GR is a local theory, while QM is a global theory. It would seem that
it is not possible to reconcile both together and we have to choose one in
favor of the other. But, we will see that it is possible to keep both aspects
in the theory. See subsection 10.3.

In [3], Thiemann explained that there are major inconsistencies in QFT and GR.
In QFT, when distances are short, GR predicted the conversion of virtual
particles into black holes. The quantization of volume will forbid such
phenomena from happening. See Section 7.2. Einstein’s equations will lead to
singularities in the metric, for example the metric in the Friedmann-
Robertson-Walker (FRW) universe model has a singular metric when we approach
beginning of time. Quantum gravity forbids this from happening. See Section
7.3.

Finally there is another inconsistency, which is in Einstein’s equations,
given by Equation (8.1). There are problems with this equation, as explained
again in [3]. The metric enters into the computation of the stress-energy
tensor, which in QFT, is computed using the Minkowski metric. However,
Einstein’s equations are used to solve for the metric, and when the space is
strongly curved, the correct metric is no longer the Minkowski metric. Hence
in such a regime, QFT is no longer valid. Furthermore, the LHS of the equation
is geometry or classical theory, whereas the RHS is matter interaction or
quantum theory. Mathematically, the field equations are also incompatible, in
the sense that the LHS is interpreted as a tensor, whereas the RHS should be
interpreted as an operator. Therefore, he called for writing the equation as
an operator equation and solve for a quantized metric. However, we cannot
concur with him on this. The metric is a dynamical variable, not an observable
to be quantized. To rectify this inconsistency, we will instead use Einstein’s
equations to quantize the stress-energy tensor. See Section 8.

QM and GR have indeed revolutionized physics as we know it and both have
changed the way we view the world. See [5]. To understand Planck scale physics
(See [6].), and with the reasons cited above, it is indeed necessary to have a
quantum theory of gravity.

But what would a theory of quantum gravity look like? A quantum theory of
gravity should be independent of background metric (See [7].), and also it
should be invariant under diffeomorphism of the ambient space-time manifold,
which we will choose to be ${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, in this
article. These are the main ingredients quantum gravity should have, as
explained by Thiemann.

A potential candidate for quantum gravity is M-theory, which is a topological
theory. This theory lives in ten dimensions and is necessarily supersymmetric.
Unfortunately, it is not background metric independent, as pointed out in [8].
Another issue with M-theory is that it is difficult to maintain locality and
causality together, as explained in [9]. Thiemann in [4] also gave reasons to
support his own reservations.

Witten argued in [10] that Quantized Chern-Simons Theory (QCS) is related to
knot theory. This may arguably be the first paper that gives rise to
Topological Quantum Field Theory (TQFT). Now, QCS is actually a GQFT, using
the classical Chern-Simons action, which is independent of any metric used, to
construct Wilson Loop observables. Indeed it does give us knot invariants that
one is familiar with. See [11]. We can use a path integral, but using the
Yang-Mills action, to quantize the Yang-Mills Theory into Quantized Yang-Mills
Theory (QYM). Unfortunately, it is not a TQFT, as its construction is metric
dependent.

The Einstein-Hilbert action in GR will yield Einstein’s field equations
through the principle of least action. See [12]. A quantized theory of
Einstein-Hilbert Theory, should also be a TQFT, as a successful quantization
of GR should be metric independent. This insight has already been discussed in
[13] and [14]. But is there such a quantized Einstein-Hilbert Theory? The
answer is yes, and is given by Loop Quantum Gravity (LQG), which has been
around for decades.

A path integral of the form Expression 5.5 given later, is independent of
background metric. And we will use this expression to construct a loop
representation for quantum gravity, first mooted in [15]. In this LQG model,
one can actually see how matter and space-time interact, on equal pedestal,
details to be given later. This is referred to as quantum geometry in [8].
Matter is represented by matter hyperlinks, to be defined in Section 5,
whereas space-time is represented by geometric hyperlinks, surfaces and
compact solid regions. This is also highlighted in [3]. Furthermore, the
theory implies causality and predicts the existence of gravitons, which has to
play an important role in quantum gravity. See Section 10.

##  2 Spin Networks

LQG has existed for decades and many authors have written good introductory
works, such as [5, 6, 16, 17], on this subject and giving reasons for the
necessity for a quantum theory of gravity. A mathematical formulation of
quantum gravity in a 3-manifold can be found in [4, 18]. We would also like to
mention an expository article [19] on $2+1$ dimensional quantum gravity, which
is related to Chern-Simons theory, as explained in [12].

By no means is this list exhaustive and we apologize if we missed out any
other suitable references. From this list of references, people will generally
associate LQG with spin networks in a 3-manifold and canonical quantization of
GR. It is also synonymous with Quantum Spin Dynamics (QSD) or Canonical
Quantum Gravity (CQG) (See [5].), which we think are more appropriate
designations.

The idea of a spin network was first conceived by Penrose. A spin network is
essentially a graph, each vertex has valency 3 and a spin is assigned to each
edge, satisfying a certain inequality for edges incident on a common vertex.
We refer the reader to [20, 21, 22, 23] for a more detailed description.

The role of a spin network in quantum gravity is that it discretizes space-
time and approximates the space of connections by treating it as a finite
product copies of a (compact) gauge group. This allows us to define a Haar
measure on this finite dimensional space. By taking the union of all possible
spin networks, one defines a Hilbert space of functionals on connection. This
construction was made rigorous in [22]. On a separate note, Ashtekar and
Lewandowski defined a generalized measure on the space of connections. Such a
measure is also defined using cylindrical measures or cylinder functions. See
[3, 4].

Now, the space of connections modulo gauge transformations is an infinite
dimensional space, so there is no notion of Lebesgue measure or even Haar
measure that can possibly be defined on it. Thus, the construction as
described earlier do not define a measure. As such, it does not describe the
loop representation as described in [15]. Furthermore, the spin networks do
not define a suitable set of functionals on the space of connections. Hence,
it does not describe the conjugate self-dual representation, also described in
[15].

Another argument against using the spin networks to quantize gravity is that
it does not take into account of the Lorentz group, or the Poincare group. The
spin numbers assigned to the edges only represent the angular momentum. In
addition, spin networks are good for describing gravity in 3-manifold; to
describe gravity in 4-manifold, one has to consider spin foam. See [20].

The Hamiltonian formulation of GR will lead to define physically admissible
states, defined by three constraints, namely the Gauss constraint, spatial
diffeomorphism constraint and the Hamiltonian constraint. See [8, 24, 25]. The
Gauss constraint is associated with ${\rm SU}(2)$ gauge transformations. The
diffeomorphism constraint is associated with the invariance of GR under
spatial diffeomorphisms. The Hamiltonian constraint is associated with the
invariance of GR under diffeomorphisms of the spatial surface (region) for 1+2
(1+3) dimensional quantum gravity.

Because our ambient space is
${{\mathbb{R}}}^{4}\equiv{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, we see that
we can foliate ${{\mathbb{R}}}^{4}$ into a set containing hypersurfaces
$\\{\Sigma_{t}:\ t\in{{\mathbb{R}}}\\}$. See [4]. Explicitly, we have a
diffeomorphism
$\phi:{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}\rightarrow{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$,
whereby $(x_{0},x)\mapsto\phi(x_{0},x)$, with $\Sigma_{t}:=\\{\phi(t,x):\
x\in{{\mathbb{R}}}^{3}\\}$. This diffeomorphism $\phi$ is completely arbitrary
and thus, the set of foliations is in one to one correspondence with the set
of diffeomorphisms of ${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$. Let
$\\{S_{a}(t,p):\ a=1,2,3\\}$ be a set of tangent vector fields at
$p\in\Sigma_{t}$ and by introducing a metric $g$, let $n$ be a unit normal
vector to $\Sigma_{t}$ at $p$. The vector
$\partial_{x_{0}}\equiv\partial/\partial x_{0}$ is being push forward by
$\phi$ to be $\phi_{\ast}\partial_{x_{0}}$ and we can write

$\phi_{\ast}\partial_{x_{0}}=Nn+U^{a}S_{a},\ N\neq 0.$

Now, $N$ and $U^{a}S_{a}$ are called the lapse function and shifted vector
fields respectively. The lapse function $N$ is associated with the Hamiltonian
constraint; the shifted vector fields will be associated with the spatial
diffeomorphism constraint, as explained in [3].

Together, the spatial diffeomorphism constraint and the Hamiltonian constraint
are consequences of the invariance of GR under four-dimensional
diffeomorphism. But there is a reason why we consider the Hamiltonian
constraint separately. The diffeomorphism in the normal direction $n$ or
orthogonal to the hypersurface means evolution in the time parameter.

Let us return back to the lapse function $N$ and since it cannot vanish, we
assume that it is always positive. By foliating the space, we see that there
is a well-defined notion of a flow of time, thus breaking the symmetry between
space and time. This is crucial in the quantization of gravity, because later
it will allow us to apply axial gauge fixing. As a consequence, it also allows
us to order the hypersurfaces, i.e. $\Sigma_{t}<\Sigma_{\tau}$ if $t<\tau$.

The usual metric canonical coordinates are used in the ADM formulation of GR.
Later, Ashtekar introduced (density-valued) soldering and
$\mathfrak{su}(2)$-valued one forms in 3-space, to replace the metric
variables. See [26, 27]. These replacement variables are henceforth referred
to as Ashtekar variables, and the Gauss and Hamiltonian constraints can be
rewritten in these new variables. One advantage in this approach, is that
Thiemann used a generalization of Ashtekar’s formalism and promote the
Hamiltonian constraint to a quantum operator in the LQG framework. See [28].
We will also define the Hamiltonian constraint operator later in Section 8.

In any quantum theory, operators corresponding to physical observables has to
be constructed. In [29], the authors used canonical quantization on Ashtekar
variables to quantize area and volume, for 3-dimensional quantum gravity. For
a good expository account of canonical quantization, the reader can refer to
[24]. This procedure will work fine, but it fails when applied to curvature.
The reason is because curvature involves derivatives of the Ashtekar variables
and one will run into problems trying to apply canonical quantization.

In the quantization of area and volume, the authors in the said article used
spin networks as the states for which these operators act on. The confusion
comes in as spin networks will then appear to play multiple roles in
connection and loop representations of quantum gravity. See the section on
mathematical foundations of quantum gravity in [4]. To resolve this issue, the
authors in [23] described a mapping from the connection representation to loop
representation.

When implementing canonical quantization, one has to ensure that the three
constraints as discussed above are all satisfied, which will lead to some
techinical difficulties. See [25]. Using the spin network description in QSD
or CQG, will not incorporate diffeomorphism invariance inside the theory. To
obtain diffeomorphism invariance, a diffeomorphism constraint has to be
imposed on this space of functionals. But there are some mathematical
difficulties trying to implement this procedure. See [3, 8, 30].

###### Remark 2.1

Such a problem does not exist if one uses path integral quantization.

One possible remedy would be to construct a set of loops from a spin network,
as was done in [23]. The reason for considering loops is due to [15], whereby
the authors described the connection between quantum gravity and link
invariants. Any quantized theory which yield topological invariants will be
invariant under diffeomorphism of the ambient space.

With all the above reasons as discussed above, it would be better to be
consistent, and work on the loop representation of quantum gravity in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$. The objects we work with are
hyperlinks, which is a set of non-intersecting, simple closed curves in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$. In [11], we showed how a link in
${{\mathbb{R}}}^{3}$ can be projected to form a planar graph, with each vertex
having valency 4, and hence compute the linking number from it. In [31], we
defined the hyperlinking number of a hyperlink, and showed how one can compute
Wilson Loop observables from it. In [32], we showed that the hyperlinking
number is equal to the linking number of the projection of a hyperlink to form
a planar graph, up to $\pm 1$, provided we define a time-ordering (Definition
4.5) for the component loops. Under time-like isotopy (Definition 4.3) and
time ordering (Definition 5.6), the hyperlinking number of a hyperlink will be
an invariant.

When one considers loop representation of quantum gravity, all three
constraints are satisfied, as explained in [24]. In [30], the authors
mentioned using an averaging process to impose the diffeomorphism constraint
on the quantum Hilbert space. For the case involving the Hamiltonian
constraint, the reader can also refer to [33].

###### Remark 2.2

We will explain later how we implement the Gauss constraint in Item 4 in
Remark 5.2, the diffeomorphism constraint in Item 4 and the Hamiltonian
constraint in Item 5, both in Remark 5.9.

A path integral is essentially, an averaging process over all possible
dynamical variables considered in the classical theory. Path integrals are
used extensively in CQFT. See [1]. We would like to mention the work by the
authors in [15], who were the first to write down a path integral for quantum
gravity. As pointed out in [4], path integrals has the advantage that they are
diffeomorphism invariant, but unfortunately they are ill-defined.

The path integrals involving area, volume and curvature, were all defined and
computed in [31], [34] and [35] respectively. Each of these path integrals can
be explicitly computed using topological invariants defined in [32], hence
consistent with the view point that geometric notions should play a central
role in LQG, as stated in [8]. Thus, we now have a consistent way of
quantizing physical observables into operators, which we will summarize the
results in Section 7.

Diffeomorphism invariance in a theory is immediate once topological invariants
are obtained. Compare this with canonical quantization in QSD or CQG, whereby
one has to enforce a diffeomorphism constraint. In this article, we will see
that area, volume and curvature will be quantized into operators and their
respective eigenvalues have a physical interpretation, to be explained in the
later sections, something which canonical quantization is not able to achieve.

##  3 Spin representation

###### Remark 3.1

Throughout this article, we adopt Einstein’s summation convention, i.e. we sum
over repeated superscripts and subscripts. Indices such as $a,b,c,d$ and greek
indices such as $\mu,\gamma,\alpha,\beta$ will take values from 0 to 3;
indices labeled $i,j,k$, $\bar{i},\bar{j},\bar{k}$ will only take values from
1 to 3.

Let $\mathfrak{su}(2)$ be the Lie Algebra of ${\rm SU}(2)$. Here, we summarize
the representation of the following Lie Algebra
$\mathfrak{su}(2)\times\mathfrak{su}(2)$ as described in [36]. This direct
product of Lie Algebras inherit the Lie bracket from $\mathfrak{su}(2)$ in the
obvious way.

Let $\\{\breve{e}_{1},\breve{e}_{2},\breve{e}_{3}\\}$ be any basis for the
first copy of $\mathfrak{su}(2)$ and
$\\{\hat{e}_{1},\hat{e}_{2},\hat{e}_{3}\\}$ be any basis for the second copy
of $\mathfrak{su}(2)$, satisfying the conditions

$\displaystyle[\breve{e}_{1},\breve{e}_{2}]=$ $\displaystyle\breve{e}_{3},\ \
[\breve{e}_{2},\breve{e}_{3}]=\breve{e}_{1},\ \
[\breve{e}_{3},\breve{e}_{1}]=\breve{e}_{2},$
$\displaystyle[\hat{e}_{1},\hat{e}_{2}]=$ $\displaystyle\hat{e}_{3},\ \
[\hat{e}_{2},\hat{e}_{3}]=\hat{e}_{1},\ \
[\hat{e}_{3},\hat{e}_{1}]=\hat{e}_{2}.$

Using this basis, define

$\mathcal{E}^{+}=\sum_{i=1}^{3}\breve{e}_{i}\ ,\
\mathcal{E}^{-}=\sum_{i=1}^{3}\hat{e}_{i}.$

Let

$\hat{E}^{01}=(\breve{e}_{1},0),\ \hat{E}^{02}=(\breve{e}_{2},0),\
\hat{E}^{03}=(\breve{e}_{3},0)$

and

$\hat{E}^{23}=(0,\hat{e}_{1}),\ \hat{E}^{31}=(0,\hat{e}_{2}),\
\hat{E}^{12}=(0,\hat{e}_{3}).$

Do note that $\hat{E}^{\alpha\beta}=-\hat{E}^{\beta\alpha}$ and write

$\hat{E}^{\tau(1)}=\hat{E}^{23},\ \ \hat{E}^{\tau(2)}=\hat{E}^{31},\ \
\hat{E}^{\tau(3)}=\hat{E}^{12},$

all in $\mathfrak{su}(2)\times\mathfrak{su}(2)$. Finally, denote

$\mathcal{E}:=\left(\begin{array}[]{cc}\mathcal{E}^{+}&\ 0\\\ 0&\
-\mathcal{E}^{-}\\\
\end{array}\right)\cong(\mathcal{E}^{+},\mathcal{E}^{-})\in\mathfrak{su}(2)\times\mathfrak{su}(2).$

Let $\rho^{\pm}:\mathfrak{su}(2)\rightarrow{\rm End}(V^{\pm})$ be an
irreducible finite dimensional representation, indexed by half-integer and
integer values $j_{\rho^{\pm}}\geq 0$. The representation
$\rho:\mathfrak{su}(2)\times\mathfrak{su}(2)\rightarrow{\rm
End}(V^{+})\times{\rm End}(V^{-})$ will be given by
$\rho=(\rho^{+},\rho^{-})$, with

$\rho:\alpha_{i}\hat{E}^{0i}+\beta_{j}\hat{E}^{\tau(j)}\mapsto\left(\sum_{i=1}^{3}\alpha_{i}\rho^{+}(\breve{e}_{i}),\sum_{j=1}^{3}\beta_{j}\rho^{-}(\hat{e}_{j})\right).$

By abuse of notation, we will now write $\rho^{+}\equiv(\rho^{+},0)$ and
$\rho^{-}\equiv(0,\rho^{-})$ in future and thus
$\rho^{+}(\hat{E}^{0i})\equiv\rho^{+}(\breve{e}_{i})$,
$\rho^{-}(\hat{E}^{\tau(j)})\equiv\rho^{-}(\hat{e}_{j})$.

Without loss of generality, we assume that $\rho^{\pm}(\hat{E})$ is skew-
Hermitian for any $\hat{E}\in\mathfrak{su}(2)$ and that $V^{\pm}$ is a finite
dimensional inner product space. Note that the dimension of $V^{\pm}$ is given
by $2j_{\rho^{\pm}}+1$. Then it is known that the Casimir operator is

$\displaystyle\sum_{i=1}^{3}\rho^{+}(\hat{E}^{0i})\rho^{+}(\hat{E}^{0i})\equiv\sum_{i=1}^{3}\rho^{+}(\breve{e}_{i})\rho^{+}(\breve{e}_{i})=-\xi_{\rho^{+}}I_{\rho^{+}},$
$\displaystyle\sum_{i=1}^{3}\rho^{-}(\hat{E}^{\tau(i)})\rho^{-}(\hat{E}^{\tau(i)})\equiv\sum_{i=1}^{3}\rho^{-}(\hat{e}_{i})\rho^{-}(\hat{e}_{i})=-\xi_{\rho^{-}}I_{\rho^{-}},$

$I_{\rho^{\pm}}$ is the identity operator for $V^{\pm}$ and
$\xi_{\rho^{\pm}}:=j_{\rho^{\pm}}(j_{\rho^{\pm}}+1)$. In QM,
$\sqrt{\xi_{\rho^{\pm}}}$ is interpreted as the total momentum of a particle,
given the representation $\rho^{\pm}$.

Without any loss of generality, by choosing a suitable basis in $V^{+}$, we
will always assume that $\rho^{+}(i\mathcal{E}^{+})$ is diagonal, with the
real eigenvalues given by the set

$\left\\{\begin{array}[]{ll}\\{\pm\breve{\lambda}_{1},\pm\breve{\lambda}_{2},\cdots,\pm\breve{\lambda}_{(2j_{\rho^{+}}+1)/2}\\},&\hbox{$2j_{\rho^{+}}+1$
is even;}\\\
\\{\pm\breve{\lambda}_{1},\pm\breve{\lambda}_{2},\cdots,\pm\breve{\lambda}_{j_{\rho^{+}}},0\\},&\hbox{$2j_{\rho^{+}}+1$
is odd.}\end{array}\right.$

If we assume that $\breve{\lambda}_{k}>0$ and
$\breve{\lambda}_{1}<\breve{\lambda}_{2}<\cdots$ is increasing, then we have
$\breve{\lambda}_{k+1}-\breve{\lambda}_{k}=\sqrt{3}$ for each $k$. That is,
the eigenvalues are equally spaced apart by a constant $\sqrt{3}$. In QM,
these eigenvalues are related to the $z$-component of the momentum of a
particle, given the representation $\rho^{+}$.

Similarly, by choosing another suitable basis in $V^{-}$, we will always
assume that $\rho^{-}(i\mathcal{E}^{-})$ is diagonal, with the real
eigenvalues given by the set

$\left\\{\begin{array}[]{ll}\\{\pm\hat{\lambda}_{1},\pm\hat{\lambda}_{2},\cdots,\pm\hat{\lambda}_{(2j_{\rho^{-}}+1)/2}\\},&\hbox{$2j_{\rho^{-}}+1$
is even;}\\\
\\{\pm\hat{\lambda}_{1},\pm\hat{\lambda}_{2},\cdots,\pm\hat{\lambda}_{j_{\rho^{-}}},0\\},&\hbox{$2j_{\rho^{-}}+1$
is odd.}\end{array}\right.$

We can interpret these eigenvalues in a similar manner.

We have

$\displaystyle\begin{split}{{\rm{Tr}}}\
\rho(e^{i\mathcal{E}^{+}})=&\left\\{\begin{array}[]{ll}\sum_{v=1}^{(2j_{\rho^{+}}+1)/2}2\cosh(\breve{\lambda}_{v}),&\hbox{$2j_{\rho^{+}}+1$
is even;}\\\
1+\sum_{v=1}^{j_{\rho^{+}}}2\cosh(\breve{\lambda}_{v}),&\hbox{$2j_{\rho^{+}}+1$
is odd;}\end{array}\right.\\\ {{\rm{Tr}}}\
\rho(e^{i\mathcal{E}^{-}})=&\left\\{\begin{array}[]{ll}\sum_{v=1}^{(2j_{\rho^{-}}+1)/2}2\cosh(\hat{\lambda}_{v}),&\hbox{$2j_{\rho^{-}}+1$
is even;}\\\
1+\sum_{v=1}^{j_{\rho^{-}}}2\cosh(\hat{\lambda}_{v}),&\hbox{$2j_{\rho^{-}}+1$
is odd.}\end{array}\right.\end{split}$ (3.1)

In either case, we have ${{\rm{Tr}}}\ \rho^{\pm}(e^{i\mathcal{E}^{\pm}})\geq
1$ and hence $\log\ {{\rm{Tr}}}\ \rho^{\pm}(e^{i\mathcal{E}^{\pm}})\geq 0$,
for any irreducible representation. Finally, note that ${{\rm{Tr}}}\
\rho^{\pm}(e^{i\mathcal{E}^{\pm}})$ is well-defined, even though
$\mathcal{E}^{\pm}$ is not in general.

##  4 Time-like hyperlink

As stated in [3], according to a theorem by Geroch, any globally hyperbolic
4-manifold has to be of the form ${{\mathbb{R}}}\times M$, whereby $M$ is a
3-manifold. See [37]. The simplest but no doubt important 3-manifold to be
considered will be ${{\mathbb{R}}}^{3}$. Thus in this article, the 4-manifold
in consideration will be
${{\mathbb{R}}}^{4}\cong{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, whereby
${{\mathbb{R}}}$ will be referred to as the time axis and ${{\mathbb{R}}}^{3}$
is the spatial 3-dimensional Euclidean space. Fix the standard coordinates on
${{\mathbb{R}}}^{4}$, $(x_{0},x_{1},x_{2},x_{3})$, whereby $x_{0}$ will be
referred to as time.

In future, when we write ${{\mathbb{R}}}^{3}$, we refer to the spatial
subspace in ${{\mathbb{R}}}^{4}$. Let
$\pi_{0}:{{\mathbb{R}}}^{4}\rightarrow{{\mathbb{R}}}^{3}$ denote this
projection. Let $\\{e_{i}\\}_{i=1}^{3}$ be the standard basis in
${{\mathbb{R}}}^{3}$. And $\Sigma_{i}$ is the plane in ${{\mathbb{R}}}^{3}$,
containing the origin, whose normal is given by $e_{i}$. So, $\Sigma_{1}$ is
the $x_{2}-x_{3}$ plane, $\Sigma_{2}$ is the $x_{3}-x_{1}$ plane and finally
$\Sigma_{3}$ is the $x_{1}-x_{2}$ plane. Note that
${{\mathbb{R}}}\times\Sigma_{i}\cong{{\mathbb{R}}}^{3}$ is a 3-dimensional
subspace in ${{\mathbb{R}}}^{3}$ and let
$\pi_{i}:{{\mathbb{R}}}^{4}\rightarrow{{\mathbb{R}}}\times\Sigma_{i}$ denote
this projection.

For a finite set of non-intersecting simple closed curves in
${{\mathbb{R}}}^{3}$ or in ${{\mathbb{R}}}\times\Sigma_{i}$, we will refer to
it as a link. If it has only one component, then this link will be referred to
as a knot. A simple closed curve in ${{\mathbb{R}}}^{4}$ will be referred to
as a loop. A finite set of non-intersecting loops in ${{\mathbb{R}}}^{4}$ will
be referred to as a hyperlink in this article. If we assign an orientation to
each component loop, then the hyperlink is said to be oriented.

We need to consider the space of hyperlinks in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, which is too big for our
consideration. Given any hyperlink in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, it is ambient isotopic to the
trivial hyperlink, a disjoint union of ‘unlinked’ loops. Hence, the
equivalence class of ambient isotopic hyperlinks will give us only the trivial
hyperlink. Therefore, we will instead consider a special equivalence class of
hyperlinks.

###### Definition 4.1

(Time-like hyperlink)
Let $L$ be a hyperlink. We say it is a time-like hyperlink if given any 2
distinct points
$\vec{p}\equiv(x_{0},x_{1},x_{2},x_{3}),\vec{q}\equiv(y_{0},y_{1},y_{2},y_{3})\in
L$, $\vec{p}\neq\vec{q}$,

  1. 1.

(T1) $\sum_{i=1}^{3}(x_{i}-y_{i})^{2}>0$;

  2. 2.

(T2) if there exist $i,j$, $i\neq j$ such that $x_{i}=y_{i}$ and
$x_{j}=y_{j}$, then $x_{0}-y_{0}\neq 0$.

We make the following remarks, which is immediate from the definition.

###### Remark 4.2

  1. 1.

We refer the reader to [32] for the reason why we use the term ‘time-like’.

  2. 2.

In Condition T1, we insist that any 2 distinct points in a hyperlink are also
separated, when projected using $\pi_{0}$, in 3-dimensional spatial space
${{\mathbb{R}}}^{3}$. This is to ensure that when we project the hyperlink in
${{\mathbb{R}}}^{3}$, we obtain a link.

  3. 3.

Conditions T1 and T2 imply that given a hyperlink $L$, for each $i=1,2,3$,
$\pi_{i}(L)\in{{\mathbb{R}}}\times\Sigma_{i}$ is a link.

###### Definition 4.3

Two oriented time-like hyperlinks $L$ and $L^{\prime}$ in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ are time-like isotopic to each other
if there is an orientation preserving continuous map
$F:{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}\times[0,1]\rightarrow{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$,
such that

  1. 1.

$F_{0}$ is the identity map;

  2. 2.

$F_{t}$ is a homeomorphism from ${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ to
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$;

  3. 3.

$F_{1}(L)=L^{\prime}$;

  4. 4.

each $F_{t}(L)$ is a time-like hyperlink.

In other words, two time-like hyperlinks $L_{1}$ and $L_{2}$ in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ are time-like isotopic if $L_{1}$ can
be continuously deformed to $L_{2}$ while remaining time-like.

###### Remark 4.4

By definition, $L$ and $L^{\prime}$ in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ are time-like isotopic to each other
implies that

  1. 1.

$\pi_{0}(L)$ and $\pi_{0}(L^{\prime})$ are ambient isotopic to each other in
${{\mathbb{R}}}^{3}$;

  2. 2.

$\pi_{i}(L)$ and $\pi_{i}(L^{\prime})$ are ambient isotopic to each other in
${{\mathbb{R}}}\times\Sigma_{i}$, $i=1,2,3$.

Throughout this article, all our hyperlinks will be time-like and we consider
equivalence classes of such hyperlinks using Definition 4.3.

###### Definition 4.5

For any
$\vec{p}=(p_{0},p_{1},p_{2},p_{3})\in{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$,
we define $\tau(\vec{p})=p_{0}$. Given two loops $\overline{l}$ and
$\underline{l}$, we say that $\overline{l}<\underline{l}$ if for any
$\vec{p}\in\overline{l}$, any $\vec{q}\in\underline{l}$, we have that
$\tau(\vec{p})<\tau(\vec{q})$. When $\overline{l}<\underline{l}$, we say that
the loop $\overline{l}$ occurs before the loop $\underline{l}$. If
$\overline{l}>\underline{l}$, we say that the loop $\overline{l}$ occurs after
the loop $\underline{l}$.

##  5 Construction of a Vector space of functionals

Let $V\rightarrow{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ be a 4-dimensional
vector bundle, with structure group $SO(3,1)$. This implies that $V$ is
endowed with a metric, $\eta^{ab}$, of signature $(-,+,+,+)$, and a volume
form $\epsilon_{abcd}$. Here,
$\epsilon^{\mu\gamma\alpha\beta}\equiv\epsilon_{\mu\gamma\alpha\beta}$ is
equal to 1 if the number of transpositions required to permute $(0123)$ to
$(\mu\gamma\alpha\beta)$ is even; otherwise it takes the value -1.

Suppose $V$ has the same topological type as $T{{\mathbb{R}}}^{4}$, so that
isomorphisms between $V$ and $T{{\mathbb{R}}}^{4}$ exist. Hence we may assume
that $V$ is a trivial bundle over ${{\mathbb{R}}}^{4}$. Without loss of
generality, we will assume the Minkowski metric $\eta^{ab}$ is given by

$\eta=-dx_{0}\otimes dx_{0}+\sum_{i=1}^{3}dx_{i}\otimes dx_{i}.$

However, there is no natural choice of an isomorphism. A vierbein $e$ is a
choice of isomorphism between $T{{\mathbb{R}}}^{4}$ and $V$. It may be
regarded as a $V$-valued one form, obeying a certain condition of
invertibility. A spin connection $\omega_{\mu\gamma}^{a}$ on $V$, is anti-
symmetric in its indices $\mu$, $\gamma$. It takes values in $\Lambda^{2}(V)$,
whereby $\Lambda^{k}(V)$ denotes the $k$-th antisymmetric tensor power or
exterior power of $V$. The isomorphism $e$ and the connection $w$ can be
regarded as the dynamical variables of General Relativity.

The curvature tensor is defined as

$R_{\mu\gamma}^{ab}=\partial_{a}\omega_{\mu\gamma}^{b}-\partial_{b}\omega_{\mu\gamma}^{a}+[\omega^{a},\omega^{b}]_{\mu\gamma},\
\partial_{a}\equiv\partial/\partial x_{a},$

or as $R=d\omega+\omega\wedge\omega$. It can be regarded as a two form with
values in $\Lambda^{2}(V)$.

Using the above notations, the Einstein-Hilbert action is written as

$S_{{\rm EH}}(e,\omega)=\frac{1}{8}\int_{{{\mathbb{R}}}^{4}}e\wedge e\wedge
R=\frac{1}{8}\int_{{{\mathbb{R}}}^{4}}\epsilon^{\mu\gamma\alpha\beta}\epsilon_{abcd}\
e_{\mu}^{a}e_{\gamma}^{b}R_{\alpha\beta}^{cd}.$ (5.1)

The expression $e\wedge e\wedge R$ is a four form on ${{\mathbb{R}}}^{4}$
taking values in $V\otimes V\otimes\Lambda^{2}(V)$, which maps to
$\Lambda^{4}(V)$. But $V$ with the structure group $SO(3,1)$ has a natural
volume form, so a section of $\Lambda^{4}(V)$ may be canonically regarded as a
function. Thus Equation (5.1) is an invariantly defined integral. By varying
Equation (5.1) with respect to $e$, we will obtain the Einstein equations in
vacuum. See [12].

The metric $\eta^{ab}$ on $V$, together with the isomorphism $e$ between
$T{{\mathbb{R}}}^{4}$ and $V$, gives a (non-degenerate) metric
$g^{ab}=e_{\mu}^{a}e_{\gamma}^{b}\eta^{\mu\gamma}$ on $T{{\mathbb{R}}}^{4}$.
By varying Equation (5.1) with respect to the connection $\omega$, we will
obtain an equation that identifies $\omega$ as the Levi-Civita connection
associated with the metric $g^{ab}$.

Let $\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})\subset
L^{2}({{\mathbb{R}}}^{4})$ be a Schwartz space, which consists of functions of
the form $f=p\cdot\sqrt{\phi_{\kappa}}$, whereby $p$ is a polynomial on
${{\mathbb{R}}}^{4}$, and
$\phi_{\kappa}(\vec{x})=\frac{\kappa^{4}}{(2\pi)^{2}}e^{-\kappa^{2}|\vec{x}|^{2}/2}$
is the Gaussian function with variance $1/\kappa^{2}$.

Using the standard coordinates on ${{\mathbb{R}}}^{4}$, let
$\Lambda^{1}({{\mathbb{R}}}^{3})$ denote the subspace in
$\Lambda^{1}({{\mathbb{R}}}^{4})$ spanned by $\\{dx_{1},dx_{2},dx_{3}\\}$.
Define

$\displaystyle\begin{split}L_{\omega}:=&\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})\otimes\Lambda^{1}({{\mathbb{R}}}^{3})\otimes\mathfrak{su}(2)\times\mathfrak{su}(2),\\\
L_{e}:=&\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})\otimes\Lambda^{1}({{\mathbb{R}}}^{3})\otimes
V.\end{split}$ (5.2)

Any spin connection $\omega$ as above can be written as $\omega\equiv
A^{a}_{\alpha\beta}\otimes dx_{a}\otimes\hat{E}^{\alpha\beta}$, whereby
$A^{a}_{\alpha\beta}:{{\mathbb{R}}}^{4}\rightarrow{{\mathbb{R}}}$ is smooth
and we identify $\Lambda^{2}(V)$ with
$\mathfrak{su}(2)\times\mathfrak{su}(2)$. Considering that this space of
smooth spin connections is too big for our purpose, we need to ‘trim’ down
this space. Under axial gauge fixing, every spin connection can be gauge
transformed into

$\omega=A^{i}_{\alpha\beta}\otimes
dx_{i}\otimes\hat{E}^{\alpha\beta}\in\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})\otimes\Lambda^{1}({{\mathbb{R}}}^{3})\otimes\mathfrak{su}(2)\times\mathfrak{su}(2)\equiv
L_{\omega},$

$A^{i}_{\alpha\beta}:{{\mathbb{R}}}^{4}\rightarrow{{\mathbb{R}}}$ smooth,
subject to the conditions

$A^{1}_{\alpha\beta}(0,x^{1},0,0)=0,\ A^{2}_{\alpha\beta}(0,x^{1},x^{2},0)=0,\
A^{3}_{\alpha\beta}(0,x^{1},x^{2},x^{3})=0.$ (5.3)

There is an implied sum over repeated indices. We will drop all these
restrictions on $A^{i}_{\alpha\beta}$.

###### Remark 5.1

  1. 1.

Note that
$A^{i}_{\alpha\beta}=-A^{i}_{\beta\alpha}\in\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})$.

  2. 2.

A $\mathfrak{su}(2)$-valued spin connection was also considered in [38].

Recall $e$ is $V$-valued one form. Let $\\{E^{\gamma}\\}_{\gamma=0}^{3}$ be a
basis for $V$, using the Minkowski metric for the 4-dimensional vector space
$V$. Even though $e$ is not a gauge, we will still apply axial gauge fixing
argument as above. After applying gauge fixing on $e$, we consider
$e:T{{\mathbb{R}}}^{4}\rightarrow V$, written as

$e=B^{i}_{\gamma}\otimes dx_{i}\otimes
E^{\gamma}\in\overline{\mathcal{S}}_{\kappa}({{\mathbb{R}}}^{4})\otimes\Lambda^{1}({{\mathbb{R}}}^{3})\otimes
V\equiv L_{e}.$

There is an implied sum over repeated indices. Observe that $e$ is now no
longer invertible. We may interpret $E^{\alpha}$ as generator for translation
in the $E^{\alpha}$ direction, which corresponds to translation in the
Poincare group. Note that the Lorentz group is a Lie subgroup of the Poincare
Lie group. If we regard $\\{\hat{E}^{0i}:i=1,2,3\\}$ and
$\\{\hat{E}^{\tau(i)}:i=1,2,3\\}$ as generators of boost and rotation
respectively, then we may interpret $\\{\omega,e\\}$ as a connection with
values in the Poincare Lie Algebra.

After applying axial gauge fixing to $\omega$ and $e$, the Einstein-Hilbert
action is then computed as ($\partial_{0}\equiv\partial/\partial x_{0}$)

$\displaystyle S_{{\rm EH}}(e,\omega)=\frac{1}{8}$
$\displaystyle\int_{{{\mathbb{R}}}^{4}}\epsilon^{abcd}B^{1}_{\gamma}B^{2}_{\mu}[\hat{E}^{\gamma\mu}]_{ab}\cdot\partial_{0}A^{3}_{\alpha\beta}[\hat{E}^{\alpha\beta}]_{cd}dx_{1}\wedge
dx_{2}\wedge dx_{0}\wedge dx_{3}$ $\displaystyle+$
$\displaystyle\frac{1}{8}\int_{{{\mathbb{R}}}^{4}}\epsilon^{abcd}B^{2}_{\gamma}B^{3}_{\mu}[\hat{E}^{\gamma\mu}]_{ab}\cdot\partial_{0}A^{1}_{\alpha\beta}[\hat{E}^{\alpha\beta}]_{cd}dx_{2}\wedge
dx_{3}\wedge dx_{0}\wedge dx_{1}$ $\displaystyle+$
$\displaystyle\frac{1}{8}\int_{{{\mathbb{R}}}^{4}}\epsilon^{abcd}B^{3}_{\gamma}B^{1}_{\mu}[\hat{E}^{\gamma\mu}]_{ab}\cdot\partial_{0}A^{2}_{\alpha\beta}[\hat{E}^{\alpha\beta}]_{cd}dx_{3}\wedge
dx_{1}\wedge dx_{0}\wedge dx_{2}.$

We sum over repeated indices.

###### Remark 5.2

  1. 1.

Using the Minkowski metric $\eta$, we see that the metric
$g^{ab}=B_{\mu}^{a}\eta^{\mu\gamma}B_{\gamma}^{b}$ is in fact degenerate. LQG
only applies when we encounter a singularity in the metric. Quantization of
the metric is avoided. See [39].

  2. 2.

Hooft in [9] also mentioned about applying axial gauge fixing to the Einstein-
Hilbert action. By applying axial gauge fixing, we distinguish time from
space. Earlier on in Section 2, we also described the foliation of space,
which gives us the notion of a flow of time.

  3. 3.

The reader may object to apply axial gauge fixing to $e$; after all $e$ is not
a gauge and in General Relativity, $e$ defines a metric, which is non-
degenerate in classical General Relativity. However, as discussed in [12], we
must consider $e$ to be non-invertible to make sense of or develop 2+1 quantum
gravity. Likewise here, to develop a 3+1 quantum gravity, we have to consider
non-invertible $e$.

  4. 4.

The authors in [30] regarded 3-dimensional gravity as a ‘gauge theory’ and
thus they considered a Gauss constraint. This constraint is applied to
consider connections modulo gauge transformations. See also [3]. This is
analogous to the axial gauge fixing applied to $e$ and $\omega$. A similar
gauge fixing was also done in [38]. By considering only gauge invariant forms,
the Gauss constraint is automatically satisfied.

In [15], the authors described the loop representation of quantum gravity in a
3-manifold. The idea is to define a functional on an equivalence class of
links, which is invariant under diffeomorphism of the 3-manifold. In other
words, this functional should yield the same value for any two links, ambient
isotopic to each other, and thus the functionals should produce link
invariants.

But Einstein’s theory of gravitation is on a 4-manifold. Any set of non-
intersecting loops in ${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ is ambient
isotopic to a trivial hyperlink. Henceforth, we will consider an equivalence
class of such loops, termed time-like isotopic, as described in Section 4. We
will now proceed to define a loop representation of quantum gravity in
${{\mathbb{R}}}^{4}$, as described in [15].

Start with two hyperlinks $\overline{L}$ and $\underline{L}$, the former
termed as a matter hyperlink $\overline{L}=\\{\overline{l}^{u}:\
u=1,\ldots,\overline{n}\\}$ and the latter
$\underline{L}=\\{\underline{l}^{v}:\ v=1,\ldots,\underline{n}\\}$ as a
geometric hyperlink. Orient both hyperlinks and color the components of the
matter hyperlink with a representation $\rho$. This means choose a
representation $\rho_{u}:\mathfrak{su}(2)\times\mathfrak{su}(2)\rightarrow{\rm
End}(V_{u}^{+})\times{\rm End}(V_{u}^{-})$ for each component loop
$\overline{l}^{u}$, $u=1,\ldots,\overline{n}$, in the hyperlink
$\overline{L}$. Note that we do not color the components loops in
$\underline{L}$, i.e. we do not choose a representation for $\underline{L}$.

The spin connection $\omega$ is $\mathfrak{su}(2)\times\mathfrak{su}(2)$
valued one-form. The first copy of $\mathfrak{su}(2)$ is generated by
$\\{\hat{E}^{0i}\\}_{i=1}^{3}$, which corresponds to boost in the $x_{i}$
direction in the Lorentz group; the second copy of $\mathfrak{su}(2)$ is
generated by $\\{\hat{E}^{\tau(i)}\\}_{i=1}^{3}$, which corresponds to
rotation about the $x_{i}$-axis in the Lorentz group. When we give a
representation $\rho^{\pm}$ to a loop $\overline{l}$, which we interpret as
representing a particle, we are effectively assigning values to the
translational and angular momentum of this particle.

Given the 2 sets of hyperlinks
$\overline{L}\equiv\\{\overline{l}^{u}\\}_{u=1}^{\overline{n}}$ and
$\underline{L}\equiv\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}}$, there are
many (infinite) ways to tangle the 2 hyperlinks together to form another
hyperlink with $\overline{n}+\underline{n}$ components. The reason is because
we only consider equivalence classes of hyperlinks, so it does not make sense
to just consider the union of $\overline{L}$ with $\underline{L}$. Hence we
need and will make a choice as to how they are tangled together, and denote
this new hyperlink by
$\chi(\overline{L},\underline{L})\equiv\chi(\\{\overline{l}^{u}\\}_{u=1}^{\overline{n}},\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})$.
We will also refer it as a colored hyperlink, assumed to be time-like.

Let $q$ be some real constant, known as the charge. Define

$\displaystyle\begin{split}V(\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})(e):=&\exp\left[\sum_{v=1}^{\underline{n}}\int_{\underline{l}^{v}}\sum_{\gamma=0}^{3}B^{i}_{\gamma}\otimes
dx_{i}\right],\\\
W(q;\\{\overline{l}^{u},\rho_{u}\\}_{u=1}^{\overline{n}})(\omega):=&\prod_{u=1}^{\overline{n}}{{\rm{Tr}}}_{\rho_{u}}\mathcal{T}\exp\left[q\int_{\overline{l}^{u}}A^{i}_{\alpha\beta}\otimes
dx_{i}\otimes\hat{E}^{\alpha\beta}\right].\end{split}$ (5.4)

Here, $\mathcal{T}$ is the time-ordering operator as defined in [11] and
${{\rm{Tr}}}_{\rho}$ is the matrix trace in the representation $\rho$. We sum
over repeated indices, with $i$ taking values in 1, 2 and 3.

A loop representation is an Einstein-Hilbert path integral, of the form

$\frac{1}{Z}\int_{\omega\in L_{\omega},\ e\in
L_{e}}V(\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})(e)W(q;\\{\overline{l}^{u},\rho_{u}\\}_{u=1}^{\overline{n}})(\omega)e^{iS_{{\rm
EH}}(e,\omega)}\ DeD\omega,$ (5.5)

whereby $De$ and $D\omega$ are Lebesgue measures on $L_{e}$ and $L_{\omega}$
respectively and

$Z=\int_{\omega\in L_{\omega},\ e\in L_{e}}e^{iS_{{\rm EH}}(e,\omega)}\
DeD\omega.$ (5.6)

Note that it is indexed by a parameter $\kappa>0$. In [36], we made sense of
this path integral. When we take the limit as $\kappa$ goes to infinity, this
sequence of path integrals will define a functional, termed as a Wilson Loop
observable of a colored hyperlink $\chi(\overline{L},\underline{L})$.

###### Remark 5.3

A similar expression was also considered in [15], which was argued to give us
knot invariants, the knots being embedded in ${{\mathbb{R}}}^{3}$.

In [31] or [32], we defined the hyperlinking number ${\rm
sk}(\overline{l}^{u},\underline{L})$ between an oriented time-like loop
$\overline{l}^{u}$ and an oriented time-like hyperlink $\underline{L}$,
computed from $\chi(\overline{L},\underline{L})$. Suppose the matter hyperlink
$\overline{L}$ is colored with
$\\{\rho_{u}\\}_{u=1}^{\overline{n}}=\\{(\rho_{u}^{+},\rho_{u}^{-})\\}_{u=1}^{\overline{n}}$.
Then in [31], we defined the Wilson loop observable for the colored hyperlink
$\chi(\overline{L},\underline{L})$ as

$\displaystyle Z(q;$
$\displaystyle\chi(\overline{L},\underline{L})):=\prod_{u=1}^{\overline{n}}\left({{\rm{Tr}}}_{\rho^{+}_{u}}\
\exp[-\pi iq\ {\rm
sk}(\overline{l}^{u},\underline{L})\cdot\mathcal{E}^{+}]+{{\rm{Tr}}}_{\rho^{-}_{u}}\
\exp[\pi iq\ {\rm
sk}(\overline{l}^{u},\underline{L})\cdot\mathcal{E}^{-}]\right).$

This leads us to make the following definition.

###### Definition 5.4

Consider the representation $\\{(\rho_{u}^{+},0)\\}_{u=1}^{\overline{n}}$,
whereby $\overline{l}^{u}$ is colored with the representation
$(\rho_{u}^{+},0)$. Using the setup as above, define Expression 5.5 as

$\displaystyle Z^{+}(q;\chi(\overline{L},\underline{L}))$
$\displaystyle:=\prod_{u=1}^{\overline{n}}\ {{\rm{Tr}}}_{\rho^{+}_{u}}\
\exp[-\pi iq\ {\rm sk}(\overline{l}^{u},\underline{L})\cdot\mathcal{E}^{+}].$

Now consider the representation $\\{(0,\rho_{u}^{-})\\}_{u=1}^{\overline{n}}$,
whereby $\overline{l}^{u}$ is colored with the representation
$(0,\rho_{u}^{-})$. Similarly, we define Expression 5.5 as

$\displaystyle Z^{-}(q;\chi(\overline{L},\underline{L}))$
$\displaystyle:=\prod_{u=1}^{\overline{n}}\ {{\rm{Tr}}}_{\rho^{-}_{u}}\
\exp[\pi iq\ {\rm sk}(\overline{l}^{u},\underline{L})\cdot\mathcal{E}^{-}].$

###### Remark 5.5

When $\rho_{u}\equiv(\rho_{u}^{+},0)$, then $Z$ reduces to $Z^{+}$. Similarly
for $\rho_{u}\equiv(0,\rho_{u}^{-})$, $Z$ reduces to $Z^{-}$.

Even though the hyperlinking number is well-defined for a pair of time-like
hyperlinks $\\{\overline{L},\underline{L}\\}$, tangled to form a time-like
hyperlink $\chi(\overline{L},\underline{L})$, it is however not an invariant
under the equivalence relation given in Definition 4.3. To obtain an
equivalence invariant, it is necessary to time-order the loops, as was done in
[32].

###### Definition 5.6

(Time-ordered pair of hyperlinks)
Suppose we have a time-like hyperlink, denoted as
$\chi(\overline{L},\underline{L})$, consisting of 2 non-empty sets of time-
like hyperlink,
$\overline{L}=\\{\overline{l}^{1},\cdots,\overline{l}^{\overline{n}}\\}$ and
$\underline{L}=\\{\underline{l}^{1},\cdots,\underline{l}^{\underline{n}}\\}$.

Refer to Definition 4.5. Pick a component loop
$\overline{l}^{u}\subset\overline{L}$ and another component loop
$\underline{l}^{v}\subset\underline{L}$. We require that either
$\overline{l}^{u}<\underline{l}^{v}$ or $\overline{l}^{u}>\underline{l}^{v}$.
In this case, we say that the time-like hyperlink
$\chi(\overline{L},\underline{L})$ consists of a time-ordered pair of time-
like hyperlinks.

Let $u=1,\ldots,\overline{n}$. We write $\overline{l}^{u}<\underline{L}$ if we
have $\overline{l}^{u}<\underline{l}^{v}$ for every
$v=1,\ldots,\underline{n}$; $\overline{l}^{u}>\underline{L}$ if
$\overline{l}^{u}>\underline{l}^{v}$ for every $v=1,\ldots,\underline{n}$.

###### Remark 5.7

  1. 1.

Suppose we have a matter loop $\overline{l}$ and a geometric loop
$\underline{l}$, both tangled together to form a time-like hyperlink
$\chi(\overline{l},\underline{l})$. Suppose that $\overline{l}<\underline{l}$.
In [32], we showed that the hyperlinking number ${\rm
sk}(\overline{l},\underline{l})$ will be equal to $3\times{\rm
lk}(\pi_{0}(\overline{l}),\pi_{0}(\underline{l}))$; we will add a negative
sign in front if $\overline{l}>\underline{l}$. Here, ${\rm
lk}(\pi_{0}(\overline{l}),\pi_{0}(\underline{l}))$ is the linking number
between knots $\pi_{0}(\overline{l})$ and $\pi_{0}(\underline{l})$.

  2. 2.

Because both $Z^{+}(q;\chi(\overline{L},\underline{L}))$ and
$Z^{-}(q;\chi(\overline{L},\underline{L}))$ defined earlier, depend on the
hyperlinking numbers between $\overline{l}^{u}$ and $\underline{L}$, we see
that it is actually dependent on the time-ordering between each matter loop
$\overline{l}^{u}$ and each component geometric loop in $\underline{L}$.

###### Notation 5.8

Let $L({{\mathbb{R}}}^{4})$ denote the space of (oriented) hyperlinks in
${{\mathbb{R}}}^{4}$. Now the space of hyperlinks is too big for our use.
Recall we have the equivalence relation defined in Definition 4.3, which we
will denote by $\sim$. Thus we can consider the quotient space
$L({{\mathbb{R}}}^{4})/\sim$.

Create 2 copies of this quotient space $L({{\mathbb{R}}}^{4})/\sim$. The first
copy will be denoted by $\overline{L}({{\mathbb{R}}}^{4})$, termed as the
space of matter hyperlinks; the second copy will be denoted by
$\underline{L}({{\mathbb{R}}}^{4})$, termed as the space of geometric
hyperlinks.

By abuse of notation, we will write $\overline{L}$ to denote an equivalence
class, using the time-like hyperlink $\overline{L}$ as a representative.
Similarly, $\underline{L}$ will also denote an equivalence class, representing
all time-like hyperlinks time-like isotopic to $\underline{L}$.

From $\overline{L}({{\mathbb{R}}}^{4})$, we choose a sequence of oriented
hyperlinks, which we will denote this sequence by
$\\{\overline{L}_{m}\\}_{m=1}^{\infty}$. In the sequence
$\\{\overline{L}_{m}\\}_{m=1}^{\infty}$, for each
$\overline{L}_{m}=\\{\overline{l}^{1}_{m},\ldots,\overline{l}^{\overline{n}(m)}_{m}\\}$,
we let $\overline{n}(m)$ denote the number of components in $\overline{L}_{m}$
and color each component $\overline{l}^{u}_{m}$ in the hyperlink
$\overline{L}_{m}$ with a representation
$\rho_{m,u}:\mathfrak{su}(2)\times\mathfrak{su}(2)\rightarrow{\rm
End}(V_{m,u}^{+})\times{\rm End}(V_{m,u}^{-})$. It shall be noted that matter
hyperlinks will always be colored.

For each matter hyperlink $\overline{L}_{m}$, we choose an oriented geometric
hyperlink
$\underline{L}_{m}\equiv\\{\underline{l}_{m}^{1},\ldots,\underline{l}_{m}^{\underline{n}(m)}\\}\in\underline{L}({{\mathbb{R}}}^{4})$,
and decide how they are tangled together, denoted by
$\chi(\overline{L}_{m},\underline{L}_{m})$. Hence,

$\chi(\overline{L}_{m},\underline{L}_{m})=\\{\overline{l}^{1},\ldots,\overline{l}^{\overline{n}(m)},\underline{l}^{1},\ldots,\underline{l}^{\underline{n}(m)}\\}$

is a new colored time-like oriented hyperlink formed from
$\overline{L}_{m}=\\{\overline{l}^{1},\ldots,\overline{l}^{\overline{n}(m)}\\}$
and
$\underline{L}=\\{\underline{l}^{1},\ldots,\underline{l}^{\underline{n}(m)}\\}$,
with each component loop of $\overline{L}_{m}$ tangled with
$\underline{L}_{m}$, represented by a time-like equivalence class, whereby
each $\overline{l}^{k}\in\overline{L}_{m}$ is time-ordered with
$\underline{l}^{r}\in\underline{L}_{m}$, for each
$r=1,2,\cdots,\underline{n}(m)$, as described in Definition 5.6. Note that
$\underline{n}(m)$ is the number of components in $\underline{L}_{m}$.

We will call the sequence $\\{\underline{L}_{m}\\}_{m=1}^{\infty}$ as the
configuration space.

###### Remark 5.9

  1. 1.

The sequence $\\{\overline{L}_{m}\\}_{m=1}^{\infty}$ represents all the
possible quantum states of the particles.

  2. 2.

Recall we have $\overline{L}$ and $\underline{L}$ as two separate oriented
hyperlinks. Note that $\overline{L}$ should represent a set of particles in
${{\mathbb{R}}}^{4}$, and from QFT, we have a discrete number of possible
particle states. We will also interpret the projected component matter loop
inside ${{\mathbb{R}}}^{3}$, as representing the orbit of a particle. The
other hyperlink, $\underline{L}$, should be thought of as the dual to the
hyperlink $\overline{L}$.

  3. 3.

When we write $\chi(\overline{L}_{m},\underline{L}_{m})$, it is understood
that it means an equivalence class
$[\chi(\overline{L}_{m},\underline{L}_{m})]$. We say that
$\chi(\overline{L}_{m},\underline{L}_{m})$ is equivalent to
$\chi(\overline{L}_{m}^{\ast},\underline{L}_{m}^{\ast})$ if

     * •

$\chi(\overline{L}_{m},\underline{L}_{m})$ is time-like isotopic to
$\chi(\overline{L}_{m}^{\ast},\underline{L}_{m}^{\ast})$;

     * •

during the time-like isotopic process, the time-ordering between each
component loop in the matter hyperlink and each component loop in the
geometric hyperlink is unchanged.

  4. 4.

By considering equivalence classes of time-like isotopic hyperlinks, we are in
fact imposing the spatial diffeomorphism constraint.

  5. 5.

For some foliation of space, we can always consider the matter loop as
embedded inside $\bigcup_{\epsilon_{0}<t<\epsilon_{1}}\Sigma_{t}$, and the
geometric loop as embedded inside
$\bigcup_{\delta_{0}<t<\delta_{1}}\Sigma_{t}$, whereby either
$\epsilon_{1}<\delta_{0}$, or $\delta_{1}<\epsilon_{0}$. This gives us a time-
ordering between the two loops, which gives rise to causality, as we will
discuss in greater detail in the rest of the article. By imposing time-
ordering on the matter and geometric loops, we are in fact imposing the
Hamiltonian constraint.

Represent each oriented hyperlink $\overline{L}_{n}$ as a function
$\overline{\Psi}_{n}^{\ \pm}\equiv\overline{\Psi}_{\overline{L}_{n}}^{\ \pm}$,
whose domain is the configuration space, denoted by
$\\{\underline{L}_{m}\\}_{m=1}^{\infty}$, whereby for each given $m$,
$\underline{L}_{m}=\\{\underline{l}_{m}^{v}\\}_{v=1}^{\underline{n}(m)}$,
$\underline{n}(m)\in\mathbb{N}$ is an oriented hyperlink. The function
$\overline{\Psi}_{n}^{\ \pm}$ at $\underline{L}_{m}$ is defined as

$\overline{\Psi}_{n}^{\
\pm}(\underline{L}_{m}):=\left\\{\begin{array}[]{ll}Z^{\pm}(q;\chi(\overline{L}_{n},\underline{L}_{n})),&\hbox{$m=n$;}\\\
0,&\hbox{$m\neq n$,}\end{array}\right.$

$Z^{\pm}(q;\cdot)$ was defined in Definition 5.4. In our previous work [32],
we showed that the hyperlinking number is an equivalence invariant for a time-
like oriented hyperlink $\chi(\overline{L}_{n},\underline{L}_{n})$, as long as
during the time-like isotopy, the time-ordering and orientation is preserved.
Thus, $Z^{\pm}(q;\cdot)$ is well-defined. See also Remark 5.7.

Written as an Einstein-Hilbert path integral, we have

$\displaystyle\overline{\Psi}_{m}^{\ +}(\underline{L}_{m}):=$
$\displaystyle\lim_{\kappa\rightarrow\infty}\int_{\omega\in L_{\omega},\ e\in
L_{e}}V(\underline{L}_{m})(e)W(q;\\{\overline{l}_{m}^{u},(\rho_{m,u}^{+},0)\\}_{u=1}^{\overline{n}(m)})(\omega)e^{iS_{{\rm
EH}}(e,\omega)}\ DeD\omega,$ $\displaystyle\overline{\Psi}_{m}^{\
-}(\underline{L}_{m}):=$
$\displaystyle\lim_{\kappa\rightarrow\infty}\int_{\omega\in L_{\omega},\ e\in
L_{e}}V(\underline{L}_{m})(e)W(q;\\{\overline{l}_{m}^{u},(0,\rho_{m,u}^{-})\\}_{u=1}^{\overline{n}(m)})(\omega)e^{iS_{{\rm
EH}}(e,\omega)}\ DeD\omega,$

computed using the colored hyperlink
$\chi(\overline{L}_{m},\underline{L}_{m})$. Note that $L_{\omega}$ and $L_{e}$
were defined by Equations (5.2), and dependent on $\kappa$.

On the flip side, we can consider $\\{\overline{L}_{n}\\}_{n=1}^{\infty}$ as
the dual configuration space. And we represent each oriented hyperlink
$\underline{L}_{m}$ now as a function $\underline{\Psi}_{m}^{\
\pm}\equiv\underline{\Psi}_{\underline{L}_{m}}^{\ \pm}$, acting on this dual
space. It is no surprise that we can define this functional as

$\underline{\Psi}_{m}^{\
\pm}(\overline{L}_{n}):=\left\\{\begin{array}[]{ll}Z^{\pm}(q;\chi(\overline{L}_{m},\underline{L}_{m})),&\hbox{$n=m$;}\\\
0,&\hbox{$n\neq m$.}\end{array}\right.$

###### Definition 5.10

(Phase space)
Let $\\{(\underline{L}_{m},\overline{L}_{m})\\}_{m=1}^{\infty}$ be known as
the phase space.

Let $\overline{V}^{\ \pm}$ be the vector space, spanned by the basis
$\\{\overline{\Psi}_{n}^{\ \pm}\\}_{n=1}^{\infty}$. This will be the vector
space of real-valued functionals acting on the configuration space.

Let $\underline{V}^{\ \pm}$ be the vector space, spanned by the basis
$\\{\underline{\Psi}_{m}^{\ \pm}\\}_{m=1}^{\infty}$. This will be the vector
space of real-valued functionals acting on the dual configuration space.

Let $V^{\pm}$ be the subspace inside $\overline{V}^{\
\pm}\otimes\underline{V}^{\ \pm}$, spanned by the basis
$\\{\overline{\Psi}_{m}^{\ \pm}\otimes\underline{\Psi}_{m}^{\
\pm}\\}_{m=1}^{\infty}$. This is a vector space of functionals acting on the
phase space $\\{(\underline{L}_{m},\overline{L}_{m})\\}_{m=1}^{\infty}$, by

$\left(\overline{\Psi}_{n}^{\ \pm}\otimes\underline{\Psi}_{n}^{\
\pm}\right)(\underline{L}_{m},\overline{L}_{m})=\overline{\Psi}_{n}^{\
\pm}(\underline{L}_{m})\underline{\Psi}_{n}^{\ \pm}(\overline{L}_{m}).$

###### Definition 5.11

(Construction of Quantum Loop Space)
The set $\overline{L}:=\\{\overline{L}_{m}\\}_{m=1}^{\infty}$ is a discrete
space, and we will equip it with a probability counting measure, denoted by
$N:\mathbb{N}\rightarrow[0,1]$, such that $\sum_{m=1}^{\infty}N(m)=1$.

Define an inner product on $\overline{V}^{\ \pm}$ by

$\left\langle\overline{\Psi}_{m}^{\ \pm},\overline{\Psi}_{\bar{m}}^{\
\pm}\right\rangle:=\sum_{n=1}^{\infty}\overline{\Psi}_{m}^{\
\pm}(\underline{L}_{n})\overline{\Psi}_{\bar{m}}^{\
\pm}(\underline{L}_{n})N(n).$

Complete $\overline{V}^{\ \pm}$ into a Hilbert space
$\mathcal{H}(\overline{V}^{\ \pm})$.

Similarly, complete $\underline{V}^{\ \pm}$ into a Hilbert space
$\mathcal{H}(\underline{V}^{\ \pm})$ with the inner product

$\left\langle\underline{\Psi}_{m}^{\ \pm},\underline{\Psi}_{\bar{m}}^{\
\pm}\right\rangle:=\sum_{n=1}^{\infty}\underline{\Psi}_{m}^{\
\pm}(\overline{L}_{n})\underline{\Psi}_{\bar{m}}^{\
\pm}(\overline{L}_{n})N(n).$

Finally, complete $V^{\pm}$ into a Hilbert space $\mathcal{H}(V^{\pm})$ using
the inner product

$\left\langle\overline{\Psi}_{m}^{\ \pm}\otimes\underline{\Psi}_{m}^{\
\pm},\overline{\Psi}_{\bar{m}}^{\ \pm}\otimes\underline{\Psi}_{\bar{m}}^{\
\pm}\right\rangle:=\sum_{n=1}^{\infty}\overline{\Psi}_{m}^{\
\pm}(\underline{L}_{n})\overline{\Psi}_{\bar{m}}^{\
\pm}(\underline{L}_{n})\underline{\Psi}_{m}^{\
\pm}(\overline{L}_{n})\underline{\Psi}_{\bar{m}}^{\
\pm}(\overline{L}_{n})N(n).$

###### Remark 5.12

  1. 1.

The reader should think of $N(m)$ as the probability for a set of particles to
be represented by $\overline{L}_{m}$. Note that the number of particles is not
conserved in the dual configuration space.

  2. 2.

In the construction, we see that both $\\{\overline{\Psi}_{n}^{\
\pm}\\}_{n=1}^{\infty}$ and $\\{\underline{\Psi}_{m}^{\
\pm}\\}_{m=1}^{\infty}$ are bases for the Hilbert spaces
$\mathcal{H}(\overline{V}^{\ \pm})$ and $\mathcal{H}(\underline{V}^{\ \pm})$
respectively. Thus, all the Hilbert spaces we have constructed are separable.
By their construction, both are orthogonal bases for the chosen counting
measure.

  3. 3.

The reason why we made the bases orthogonal is because later on, we will show
that they will form an eigenbasis for the operators we are interested in.
Because operators corresponding to observables must be Hermitian operators by
a postulate in QM, hence we made the bases orthogonal.

In QM, one would define a time-evolution operator, which is unitary, hence the
probability is conserved as the states evolve with time. But in LQG, there is
no absolute notion of time, so one should not attempt to define a time-
evolution operator. One way to remedy this situation, as proposed by Hooft,
would be to define an equivalence relation, whereby two states are deemed to
be equivalent if they evolve into a same state in the future. One then
constructs a Hilbert space, whereby each element in a basis is identified with
such an equivalence class. For details, refer to [9].

In our construction, time-like equivalence and time-ordering ensure that a
hyperlink $\chi(\overline{L},\underline{L})$, will be equivalent to the same
hyperlink, be it translated forward or backward in time. This means that our
equivalence relation implies the relation suggested by Hooft, but not
conversely. And the preceding probability measure defined on the equivalence
class $[\overline{L}_{m}]$ or $[\underline{L}_{m}]$, represented by a
hyperlink $\overline{L}_{m}$ or $\underline{L}_{m}$ respectively, will not
change under translation in time.

In the absence of a space-time metric, time and space are ambiguous. However,
from our definition of the hyperlinking number, one sees that time is
distinguished from space. Earlier on, we mentioned that the hyperlinking
number will be an invariant, if we consider an equivalence class of a pair of
time-like hyperlinks $\chi(\overline{L},\underline{L})$, with the additional
condition that each component matter loop is time-ordered with a component
geometric loop. One should then interpret causality as a consequence of time-
ordering. Thus, we only consider diffeomorphisms of space-time that preserves
causality. In a topological theory on ${{\mathbb{R}}}^{4}$, there is no
distinguished axis that is labelled as time, hence there will be no notion of
causality. Considering time-like and time-ordered pair of loops, is not a flaw
in the LQG theory, but rather it preserves the notion of time as an external
parameter in QM, and causality in Special Relativity. And despite this
restriction, it still keeps the malleability of space-time, as dictated in GR.

##  6 Observables

In the previous section, we constructed the phase space. Notice that we used
(time-like) hyperlinks to construct the phase space, which are geometrical
objects. In loop quantum gravity, there is no preferred metric, so we need to
consider equivalence classes of a hyperlink. Hence, when we choose two
equivalence classes of hyperlink $\overline{L}$ and $\underline{L}$, we need
to make a choice as to how they are tangled. Furthermore, we time-ordered a
pair of matter and geometric component loops in Definition 5.6.

If a matter loop occurs before a geometric loop, one can interpret physically
as saying that the matter loop $\overline{l}$ will produce a geometric loop
$\underline{l}$, that tangles with it. Furthermore, each matter hyperlink
$\overline{L}$ can only produce one geometric hyperlink, up to time-like
equivalence. The tangled hyperlink $\chi(\overline{L},\underline{L})$ is an
equivalence class of hyperlink, consisting of time-ordered pair of hyperlinks.
By definition of the equivalence class, if a matter loop precedes before a
geometric loop, then any ambient isotopy of these loops must maintain that the
transformed matter loop is the cause, the transformed geometric loop is the
effect. Thus, causality is preserved under the equivalence relation.

In this section, we would like to discuss the observables that one can define
on the phase space. Interestingly enough and to no surprises, the observables
we are going to consider will be geometrical objects.

###### Definition 6.1

Let $l$ be a time-like loop and $S=\bigcup_{k=1}^{n}S_{k}$, whereby $S_{k}$ is
a connected compact surface with boundary $\partial S_{k}$. We say that $l$ is
time-ordered with $S$, if $\tau(\vec{x})<\tau(\vec{y})$ or
$\tau(\vec{x})>\tau(\vec{y})$, for every $\vec{x}\in l$ and $\vec{y}\in
S_{k}$. We will denote this relation as $l<S_{k}$ ($l>S_{k}$) for the former
(latter), by abuse of notation.

On the configuration space $\\{\underline{L}_{m}\\}_{m=1}^{\infty}$, choose an
orientable compact surface,
$\overline{S}\subset\\{0\\}\times{{\mathbb{R}}}^{3}$, thus $\overline{S}$ is a
Seifert surface for $\partial\overline{S}$, if $\overline{S}$ has non-empty
boundary. The surface $\overline{S}$ is represented up to ambient isotopy in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ and disjoint from the geometric
hyperlinks in the configuration space. We also assume that its boundary
$\partial\overline{S}$, if any, is a time-like hyperlink.

Pick a geometric hyperlink $\underline{L}_{m}$ from the configuration space,
disjoint from $\overline{S}$, and we need to make a choice as to how
$\overline{S}$ and $\underline{L}_{m}$ are ‘linked together’. Each component
geometric loop must be time-ordered with each connected component surface with
boundary in $\overline{S}$. If $\overline{S}$ has no boundary, then time-
ordering is not required.

Write $\\{\overline{S},\emptyset,\chi(\overline{L}_{m},\underline{L}_{m})\\}$
to denote a time-like triple, as defined in the last section of [32]. If we
orientate the hyperlink and the surface, then this oriented time-like triple
represents the equivalence class of these oriented geometrical objects in
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$. On this time-like oriented triple,
the linking number between $\overline{S}$ and $\underline{L}_{m}$, denoted as
${\rm lk}(\underline{L}_{m},\overline{S})$, as defined in [32], and the
hyperlinking number between $\overline{L}_{m}$ and $\underline{L}_{m}$, are
time-like equivalence invariants, preserving the time-ordering.

A piercing is formed when an arc in the projected loop $\pi_{0}(l)$ intersects
the projected surface $\pi_{0}(S)$, without being tangent to the surface. The
linking number is then computed by adding up all the algebraic numbers of
these piercings.

In [32], we showed that when the surface has no boundary, then the linking
number between the oriented closed surface and an oriented loop is invariant
under ambient isotopy. But, if the surface has a boundary, then the linking
number between the oriented surface and the oriented loop, is not a
topological invariant. Under the above equivalence relation, a time-ordering
needs to be imposed and suppose $l<S$ for a loop $l$ and a connected surface
$S$. Then, we only consider ambient isotopy whereby during the ambient
isotopic process, time-ordering remains unchanged and it further shows that

  * •

the time-like oriented hyperlink formed from $l$ and $\partial S$, is time-
like isotopic to the time-like oriented hyperlink formed from $l^{\prime}$ and
$\partial S^{\prime}$;

  * •

$S$ is ambient isotopic to $S^{\prime}$.

Thus, causality is respected under the equivalence relation.

When $S$ has no boundary, no time-ordering is required under the equivalence
relation. But, we saw in [32] that indeed a time-ordering is implicitly
implied and preserved, under ambient isotopy of the loop and a connected
surface. When a piercing is formed before (after) the time duration for which
the surface is formed, we will refer it as a left (right) piercing. Using
ambient isotopy, this implicitly defined time-ordering looks at whether all
projected oriented arcs going from a left to a right piercings, lie either in
the interior or exterior of $\pi_{0}(S)=S\subset{{\mathbb{R}}}^{3}$, during
the time duration for which the connected surface is formed. By simply
reversing the orientation of the loop, we can change this time-ordering. Thus,
the linking number between a connected closed surface and a loop is dependent
on this implicitly defined time-ordering. This time-ordering is of course
preserved during any ambient isotopy.

On the dual configuration space $\\{\overline{L}_{n}\\}_{n=1}^{\infty}$, we
will again choose 2 geometrical objects as observables; an orientable surface
$\underline{S}\subset{{\mathbb{R}}}^{3}$ and a compact (3-dimensional) region
$R\subset{{\mathbb{R}}}^{3}$. We will identify both geometrical objects inside
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$ by $\\{0\\}\times\underline{S}$ and
$\\{0\\}\times R$ respectively and note that we choose representatives for
this surface and solid region, up to ambient isotopy in
$\\{0\\}\times{{\mathbb{R}}}^{3}$ and ${{\mathbb{R}}}^{3}$ respectively.

Each matter hyperlink $\overline{L}_{m}$ from the dual configuration space
must be disjoint from $\underline{S}$ and $R$. For a matter hyperlink, we need
to choose a frame for each component loop in $\overline{L}_{m}$, so that
$\pi_{0}(\overline{L}_{m})$ is a framed link in ${{\mathbb{R}}}^{3}$. The
framing on the link $\pi_{0}(\overline{L}_{m})$ will give rise to nodes on it.
See [32].

We now need to make a choice as to how the given oriented surface
$\underline{S}$ and the oriented matter hyperlink $\overline{L}_{m}$ are
linked. Again, time-ordering needs to be imposed, when $\underline{S}$ has an
non-empty boundary, assumed to be a time-like hyperlink. As for $R$, we need
to decide how many nodes on $\pi_{0}(\overline{L}_{m})$ lie inside $R$.

Once decided, form an oriented time-like triple
$\\{\underline{S},R,\chi(\overline{L}_{m},\underline{L}_{m})\\}$, as defined
in [32]. Note that $\underline{S}$ and $R$ are represented as an ambient
isotopic class in ${{\mathbb{R}}}^{3}$. This oriented time-like triple
represents an equivalence class of these geometrical objects. The piercing
number $\nu_{\underline{S}}(\overline{L}_{m})$ counts the number of piercings
formed between $\pi_{0}(\overline{L}_{m})$ and $\pi_{0}(\underline{S})$. And
the confinement number, denoted by $\nu_{R}(\overline{L}_{m})$, counts how
many nodes on $\pi_{0}(\overline{L}_{m})$, lie in the interior of $R$. On this
triple, the piercing and confinement numbers are time-like, preserving time-
ordering equivalence invariants. For more details, refer to [32].

###### Remark 6.2

Let $\vec{x}=(x_{0},x)$ be a point on a loop in ${{\mathbb{R}}}^{4}$, which
projects down to a node $\pi_{0}(\vec{x})$ on a framed knot in
${{\mathbb{R}}}^{3}$. Assume that this node lies inside the region $R$.
Because the loop does not intersect with $\\{0\\}\times R$, we have either
$x_{0}>0$ or $x_{0}<0$. Suppose $x_{0}>0$. Under the equivalence relation,
when the framed loop is transformed under time-like isotopy, then $\vec{x}$
will be mapped to a point $\vec{y}$, such that

  * •

$\pi_{0}(\vec{y})$ will be a node on the transformed knot, and it must remain
in the interior of the region $R$;

  * •

during the time-like isotopy, any transformed loop must remain disjoint from
the region $R$ and nodes on projection, must always remain disjoint from the
boundary of $R$.

Thus, we also must have $y_{0}>0$. In other words, those nodes on the
projected loop with time component positive, will be mapped to nodes on the
transformed knot, with time component positive. Therefore, one can say that
the nodes are implicitly time-ordered and under the equivalence relation, this
time-ordering does not change, again implying that causality is preserved.

Our description of the geometrical objects in consideration is now complete
and we remind the reader, that they are represented, inside
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, up to time-like isotopy, preserving
time-ordering. We will introduce physical quantities that can be computed
using the above described oriented time-like triple. Because there is no
preferred metric, the quantities that one is able to compute from these
geometrical objects must be invariants under the equivalence relation.

In the next section, we will introduce observables, which will in turn define
quantum operators, whose eigenvalues are computed using these invariants.

##  7 Quantized operators

To develop a quantum theory of gravity, we need to quantize certain
observables. Area and volume are observables that depend on the vierbein $e$;
the curvature depends on the spin connection $\omega$.

We need to find a basis for $\mathcal{H}(\overline{V}^{\ \pm})$, such that
they form an eigenbasis for the area operator $\hat{A}$ and for the volume
operator $\hat{V}$. Then, the respective eigenvalues will be the quantized
values for area and volume. Similarly, we need to find a basis for
$\mathcal{H}(\underline{V}^{\ \pm})$, such that they form an eigenbasis for
the quantized curvature operator $\hat{F}$, and its eigenvalues will be the
quantized values for curvature.

Recall in Section 5, we wrote down Expression 5.5 using the functionals $V$
and $W$. We can now proceed to quantize the observables using the Einstein-
Hilbert path integral.

###  7.1 Area operator

Let $S$ be an orientable compact surface inside the spatial subspace
${{\mathbb{R}}}^{3}\hookrightarrow{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$,
disjoint from the matter hyperlink. Because we can consider ambient isotopy of
$S$ in ${{\mathbb{R}}}^{3}$, we assume that $S$ is inside $x_{2}-x_{3}$ plane.
Furthermore, we insist that $\pi_{0}(\overline{L})$ intersect the surface $S$
at most finitely many points. Using the dynamical variables
$\\{B_{\mu}^{a}\\}$ and the Minkowski metric $\eta^{ab}$, we see that the
metric $g^{ab}\equiv B^{a}_{\mu}\eta^{\mu\gamma}B^{b}_{\gamma}$ and the area
is given by

${\rm Area\ of}\ S(e):=A_{S}(e)\equiv\int_{S}\sqrt{g^{22}g^{33}-(g^{23})^{2}}\
dA.$

In [31], we quantized the area of $S$ into an operator $\hat{A}_{S}$ using the
following path integral expression (indexed by a parameter $\kappa$)

$\frac{1}{Z}\int_{\omega\in L_{\omega},\ e\in
L_{e}}V(\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})(e)W(q;\\{\overline{l}^{u},\rho_{u}\\}_{u=1}^{\overline{n}})(\omega)A_{S}(e)\
e^{iS_{{\rm EH}}(e,\omega)}\ DeD\omega,$ (7.1)

$V$ and $W$ as defined in Equation (5.4) and $Z$ is a normalization constant
given by Equation (5.6). Note that $\\{\rho_{u}^{\pm}\\}_{u=1}^{\overline{n}}$
is a representation for the hyperlink $\overline{L}$.

The main theorem in [31] says that the limit of this sequence of path
integrals in Expression 7.1 as $\kappa$ goes to infinity, can be computed from
the number of times the projected hyperlink $\pi_{0}(\overline{L})$ pierce the
surface $S$, weighted by the momentum corresponding to the representation
$\rho_{u}$. When $\rho_{u}=\rho_{u}^{\pm}$, we will write this limit as
$\hat{A}_{S}[Z^{\pm}(q;\chi(\overline{L},\underline{L}))]$, for which the
reader may recall, we defined $Z^{\pm}(q;\chi(\overline{L},\underline{L}))$ in
Definition 5.4.

Recall in Section 5, for a colored hyperlink $\overline{L}$, we can define a
functional $\overline{\Psi}_{\overline{L}}^{\ \pm}$ acting on the
configuration space. In [32], we defined the piercing number $\nu_{S}(l)$
between a compact surface $S$ in $\\{0\\}\times{{\mathbb{R}}}^{3}$, with or
without boundary, and a loop $l$. This is a well-defined invariant, up to
time-like isotopy, preserving time-ordering. It counts the number of times
$\pi_{0}(l)$ intersects the surface $S$ in ${{\mathbb{R}}}^{3}$, the
intersection points termed as piercings. We always choose a representative of
$l$ and $S$ such that the $\pi_{0}(l)$ and $S$ in ${{\mathbb{R}}}^{3}$ have
the minimum number of piercings, which gives us $\nu_{S}(l)$.

We have the following corollary from the main theorem in [31].

###### Corollary 7.1

Given an orientable compact surface $S\subset\\{0\\}\times{{\mathbb{R}}}^{3}$,
quantization of the area of $S$, gives an operator $\hat{A}_{S}$, which acts
on functionals in $\overline{V}^{\ \pm}$.

Suppose $\overline{\Psi}_{\overline{L}}^{\ \pm}\in\overline{V}^{\ \pm}$, with
$\overline{L}=\\{\overline{l}^{u}\\}_{u=1}^{\overline{n}}$. Then

$\hat{A}_{S}\overline{\Psi}_{\overline{L}}^{\
+}=\frac{|q|\sqrt{\pi}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{+}}}\right]\overline{\Psi}_{\overline{L}}^{\
+}$

and

$\hat{A}_{S}\overline{\Psi}_{\overline{L}}^{\
-}=i\frac{|q|\sqrt{\pi}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{-}}}\right]\overline{\Psi}_{\overline{L}}^{\
-}.$

Hence, the canonical basis $\\{\overline{\Psi}_{n}^{\ \pm}\\}_{n=1}^{\infty}$
is an eigenbasis for $\hat{A}_{S}$. Furthermore, the basis is orthogonal using
the counting measure $N$.

###### Remark 7.2

The eigenvalues are either purely real or purely imaginary.

Note that $\nu_{S}(\overline{l}^{u})$ is computed from the oriented time-like
triple $\\{S,\emptyset,\chi(\overline{L},\underline{L})\\}$ and the
eigenvalues of $\hat{A}_{S}$ are invariant under diffeomorphism of
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, provided it respects time-like
isotopy and also preserves time-ordering.

###### Remark 7.3

Time-ordering between $S$ and $\overline{L}$ is only necessary when $S$ has a
boundary. When a connected surface $S$ has no boundary, a time-ordering is
implicitly defined, as discussed in Section 6. In the event that we have to
time-order the surface $S$ with the component matter loops, note that the
piercing number is independent of any time-ordering, as it only counts the
total number of piercings, without taking into account of their algebraic
sign. So the Area operator is actually independent of any time-ordering
imposed or implicitly defined.

The term $\sqrt{\xi_{\rho}}$ appears in the expression and it is interpreted
as total momentum for representation $\rho$. So, the eigenvalues of the area
operator gives us the total momentum impact by the matter hyperlink on a
surface $S$, taking into account momentum coming from either boost or
rotation. As an application of this result, we will derive the Bekenstein-
Hawking expression for the entropy of a Schwarzschild black hole in Section 9.

In the derivation of the path integral for the area, it was necessary to write
the surface $S$ as a finite disjoint union of smaller surfaces. In [31], we
showed that the main result was independent of how we partition the surface.
Suppose we write $S=\bigcup_{m}S_{m}$. As the result says that the area is
given by counting the number of piercings between $\pi_{0}(l)$ and $S$, we see
that each smaller surface $S_{m}$, either has one quanta of area, or zero
quanta of area, which corresponds to the projected loop piercing the surface
once, or there is no piercing respectively. This means that if $S_{m}$
contains one piercing, any local surface in $S_{m}$ containing the piercing,
will have the same quanta of area as $S_{m}$.

###  7.2 Volume operator

Fix a closed and bounded 3-manifold $R\subset{{\mathbb{R}}}^{3}$, possibly
disconnected with finite number of components, disjoint from the matter
hyperlink. Henceforth, we will refer to $R$ as a compact solid region. Its
boundary is a closed (compact without boundary) surface. Using the dynamical
variables $\\{B_{\mu}^{a}\\}$ and the Minkowski metric $\eta^{ab}$, we see
that the metric $g^{ab}\equiv B^{a}_{\mu}\eta^{\mu\gamma}B^{b}_{\gamma}$ and
the volume $V_{R}$ is given by

$V_{R}(e):=\int_{R}\sqrt{\epsilon_{ijk}\epsilon_{\bar{i}\bar{j}\bar{k}}g^{i\bar{i}}g^{j\bar{j}}g^{k\bar{k}}}.$

In [34], we quantized the volume of $R$ into an operator $\hat{V}_{R}$ using
the following path integral expression (indexed by a parameter $\kappa$)

$\frac{1}{Z}\int_{\omega\in L_{\omega},\ e\in
L_{e}}V(\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})(e)W(q;\\{\overline{l}^{u},\rho_{u}\\}_{u=1}^{\overline{n}})(\omega)V_{R}(e)\
e^{iS_{{\rm EH}}(e,\omega)}\ DeD\omega,$ (7.2)

whereby $V$ and $W$ were defined in Equation (5.4) and $Z$ is a normalization
constant given by Equation (5.6).

A framed hyperlink when projected in ${{\mathbb{R}}}^{3}$ using
$\pi_{0}:{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}\rightarrow{{\mathbb{R}}}^{3}$,
gives us a framed link. Imagine adding nodes to each component knot inside the
framed link. Each node has an algebraic number $\\{\pm 1\\}$ assigned to it
and we always assume that all the nodes on a component knot have the same
sign. Equivalence class of framed loop $l$ allows us to move the set of nodes
in the knot $\pi_{0}(l)$ and we denote this equivalence class of finite points
by ${\rm Nd}(\pi_{0}(l))$.

The main theorem in [34] says that the limit of the sequence of path integrals
in Expression 7.2 as $\kappa$ goes to infinity, is computed by adding up all
the nodes on a projected framed hyperlink, which lie in the interior of $R$,
weighted by the Casimir operator corresponding to the representation
$\rho_{u}$. If we choose the representation
$\\{\rho_{u}^{\pm}\\}_{u=1}^{\overline{n}}$ for the hyperlink $\overline{L}$,
then we will write this limit as
$\hat{V}_{R}[Z^{\pm}(q;\chi(\overline{L},\underline{L}))]$,
$Z^{\pm}(q;\chi(\overline{L},\underline{L}))$ was defined in Definition 5.4.

In [32], we defined the confinement number $\nu_{R}(l)$, which counts the
number of nodes in ${\rm Nd}(\pi_{0}(l))$ which are in the interior of $R$.
Under the time-like isotopy, as defined in [32], $\nu_{R}(l)$ is an invariant.

Recall for a colored hyperlink $\overline{L}$, we can define a functional
$\overline{\Psi}_{\overline{L}}^{\ \pm}$ acting on a configuration space. As a
consequence, we have the following corollary from the main theorem in [34].

###### Corollary 7.4

Let $R\subset{{\mathbb{R}}}^{3}$ be a compact solid region, possibly
disconnected with finite components and let $\partial R$ denote its boundary,
which is closed. Quantization of the volume of the region $R$, will give us an
operator $\hat{V}_{R}$, which acts on functionals in $\overline{V}^{\ \pm}$.

Suppose $\overline{\Psi}_{\overline{L}}^{\ \pm}\in\overline{V}^{\ \pm}$. Then,

$\hat{V}_{R}\overline{\Psi}_{\overline{L}}^{\
+}=\frac{q^{2}\pi^{3/2}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{R}(\overline{l}^{u})\xi_{\rho_{u}^{+}}\right]\overline{\Psi}_{\overline{L}}^{\
+}$

and

$\hat{V}_{R}\overline{\Psi}_{\overline{L}}^{\
-}=\frac{q^{2}\pi^{3/2}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{R}(\overline{l}^{u})\xi_{\rho_{u}^{-}}\right]\overline{\Psi}_{\overline{L}}^{\
-}.$

Hence, the canonical basis $\\{\overline{\Psi}_{n}^{\ \pm}\\}_{n=1}^{\infty}$
is an eigenbasis for $\hat{V}_{R}$. Furthermore, the basis is orthogonal using
the counting measure $N$.

Note that $\nu_{R}(\overline{l}^{u})$ is computed from the oriented time-like
triple $\\{\emptyset,R,\chi(\overline{L},\underline{L})\\}$ and the
eigenvalues of $\hat{V}_{R}$ are invariant under diffeomorphism of
${{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, provided it respects time-like
isotopy, preserving time-ordering.

###### Remark 7.5

Remark 6.2 said that each node $\pi_{0}(\vec{x})$ has a definite sign from its
time-component $\tau(\vec{x})$, but it does not affect the confinement number.
Thus, the confinement number and hence the eigenvalues of the volume operator
are actually independent of any time-ordering implicitly defined.

If one looks at the eigenvalues of $\hat{V}_{R}$, notice that we have terms
involving the square of the total momentum $\sqrt{\xi_{\rho_{u}^{\pm}}}$,
which in classical mechanics, is interpreted as the kinetic energy of the
particle represented by the colored hyperlink $\overline{l}^{u}$, with
representation $\rho_{u}^{\pm}$. The Casimir operator $\xi_{\rho_{u}^{+}}$
($\xi_{\rho_{u}^{-}}$) will represent translational (rotational) kinetic
energy. So, one can view the volume operator as measuring the total kinetic
energy of the matter hyperlinks, in a region $R$. Volume operator can be
interpreted as energy operator is also evident in [3].

We can use the eigenvalues of the volume operator to resolve some
inconsistency in QFT, as raised by Thiemann in [3]. A particle’s momentum $p$
is inversely proportional to its Compton length and its energy $E$ is
proportional to its Schwarzschild radius. When its Compton length is
comparable to its Schwarzschild radius, then GR predicts that this particle
will turn into a black hole, implying that Hawking radiation and all sorts of
virtual particles with large momentum and energy will henceforth be emitted.

This will happen at short distances, in the order of Planck’s distance (around
$10^{-33}$ cm). At such short distances, quantum gravity should come into
play. The main theorem in [34] says that the (kinetic) energy of the particle
is discretized. This means that QFT and GR are no longer applicable and the
above qualitative picture does not apply.

In the derivation of the volume path integral, it was necessary to write the
region $R=\bigcup_{m}R_{m}$ as a disjoint union of smaller regions. The
computations in [34] showed that the final answer is independent of this
partition. As the volume of $R$ is given by counting the number of nodes in
its interior, we may assume that $R_{m}$ has either zero or one quanta of
volume, which is equivalent to $R_{m}$ containing zero or one node in its
interior respectively. Thus, any smaller region $R^{\prime}\subset R_{m}$,
will have the same quanta of volume, as long as it contains the node, if any.
In other words, any local region inside $R_{m}$ will contain the same quanta
of volume.

###  7.3 Curvature operator

Choose an orientable, compact surface
$S\subset{{\mathbb{R}}}\times{{\mathbb{R}}}^{3}$, with or without boundary,
which is disjoint from the geometric hyperlink $\underline{L}$. If it has no
boundary, we term $S$ as closed. Furthermore, we insist the projected
hyperlink $\pi_{a}(\underline{L})$ intersect $\pi_{a}(S)$ at most finitely
many points, $a=0,1,2,3$. If $S$ has a boundary $C$, then $C$ tangles with
$\underline{L}$ to form a time-like hyperlink. And we will also time-order the
geometric loop with the surface $S$ if there is a boundary, according to
Definition 6.1.

Define

$\displaystyle F_{S}(\omega):=\frac{1}{2}\int_{S}$
$\displaystyle\frac{\partial A_{\alpha\beta}^{i}}{\partial x_{0}}\otimes
dx_{0}\wedge dx_{i}\otimes\hat{E}^{\alpha\beta}+\frac{\partial
A_{\alpha\beta}^{j}}{\partial x_{i}}\otimes dx_{i}\wedge
dx_{j}\otimes\hat{E}^{\alpha\beta}$
$\displaystyle+A_{\alpha\beta}^{i}A_{\gamma\mu}^{j}\otimes dx_{i}\wedge
dx_{j}\otimes[\hat{E}^{\alpha\beta},\hat{E}^{\gamma\mu}].$

In [35], we quantized the curvature of $S$ into an operator $\hat{F}_{S}$
using the following path integral expression (indexed by a parameter $\kappa$)

$\frac{1}{Z}\int_{\omega\in L_{\omega},\ e\in L_{e}}F_{S}(\omega)\cdot
V(\\{\underline{l}^{v}\\}_{v=1}^{\underline{n}})(e)W(q;\\{\overline{l}^{u},\rho_{u}\\}_{u=1}^{\overline{n}})(\omega)e^{iS_{{\rm
EH}}(e,\omega)}\ DeD\omega,$ (7.3)

whereby $V$ and $W$ were defined in Equation (5.4) and $Z$ is a normalization
constant given by Equation (5.6).

###### Remark 7.6

  1. 1.

A quantized curvature operator at each point $p\in{{\mathbb{R}}}^{4}$ is not
defined, as commented in [38].

  2. 2.

If we regard the curvature as a field, then it was explained in [40] that the
curvature $R$ do not yield a well-defined operator; rather only smeared fields
will yield quantized operators. In this case, we smear the curvature over a
surface $S$, given by $F_{S}$.

The main theorem in [35] says that the limit of this sequence of path
integrals in Expression 7.3 as $\kappa$ goes to infinity, is computed from the
linking number between $\underline{L}$ and the surface $S$, denoted as ${\rm
lk}(\underline{L},S)$. See [32]. If we choose the representation
$\\{\rho_{u}^{\pm}\\}_{u=1}^{\overline{n}}$ for the hyperlink $\overline{L}$,
then we will write this limit as
$\hat{F}_{S}[Z^{\pm}(q;\chi(\overline{L},\underline{L}))]$,
$Z^{\pm}(q;\chi(\overline{L},\underline{L}))$ given in Definition 5.4.

Recall for a hyperlink $\underline{L}$, we can define a functional
$\underline{\Psi}_{\underline{L}}^{\ \pm}$ acting on the dual configuration
space. As a consequence, we have the following corollary from the main theorem
in [35].

###### Corollary 7.7

Let $S\subset{{\mathbb{R}}}^{4}$ be an oriented closed connected surface. The
spin curvature integrated over the oriented surface $S$ will give us an
operator $\hat{F}_{S}$, which acts on functionals in $\underline{V}^{\ \pm}$.

Suppose $\underline{\Psi}_{\overline{L}}^{\ \pm}\in\underline{V}^{\ \pm}$.
Then,

$\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ +}=-i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\ +}\otimes\mathcal{E}$

and

$\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ -}=-i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\ -}\otimes\mathcal{E}.$

If $S$ is a compact oriented connected surface with boundary, then the
calculations in [35] and using the definition of linking number given in [32],
we see that

$\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ +}=-\frac{i\sqrt{4\pi}}{4}\
{\rm lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\
+}\otimes\mathcal{E}$

and

$\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ -}=-\frac{i\sqrt{4\pi}}{4}\
{\rm lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\
-}\otimes\mathcal{E}.$

The quantized curvature operator is actually
$\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued. Since
$\underline{\Psi}_{\underline{L}}^{\ \pm}$ is a scalar-valued linear
functional on the dual configuration space, we see that
$\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ \pm}$ is actually
$\mathcal{E}$-valued. Note that $\mathcal{E}^{\pm}$ depends on our earlier
choice of $\\{\breve{e}_{i}\\}_{i=1}^{3}$ and $\\{\hat{e}_{i}\\}_{i=1}^{3}$
respectively, so it is not unique. However, its eigenvalues, given by $\pm
i\sqrt{3}/2$ are well-defined, independent of any choice of
$\\{\breve{e}_{i}\\}_{i=1}^{3}$ or $\\{\hat{e}_{i}\\}_{i=1}^{3}$.

Thus, the eigenvalues of $\hat{F}_{S}\underline{\Psi}_{\underline{L}}^{\ \pm}$
will be

$\frac{\sqrt{12\pi}}{2}\ {\rm
lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\ \pm}\quad{\rm
and}\quad-\frac{\sqrt{12\pi}}{2}\ {\rm
lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\ \pm},$ (7.4)

each with algebraic multiplicity 2, when $S$ has no boundary. When $S$ has a
boundary, we replace the denominator 2 with an 8 instead.

Each such pair of eigenvalues correspond to the eigenvector
$\underline{\Psi}_{\underline{L}}^{\ \pm}$, so we will say that the canonical
basis $\\{\underline{\Psi}_{n}^{\ \pm}\\}_{n=1}^{\infty}$ is an eigenbasis for
the operator $\hat{F}_{S}$, with corresponding eigenvalues given by Equation
(7.4). Furthermore, the basis is orthogonal using the counting measure $N$.

Note that the linking number is computed from the oriented time-like triple
$\\{S,\emptyset,\chi(\overline{L},\underline{L})\\}$ and the quantum Hilbert
space $\\{\hat{F}_{S},\mathcal{H}(\underline{V}^{\ \pm})\\}$, so it is
invariant under the equivalence relation. When $S$ has a boundary, then we
need time-like isotopy and time-ordering to ensure that the linking number is
proportional to the hyperlinking number between $\partial S$ and
$\underline{L}$, so it is invariant under the equivalence relation. In both
cases, the linking number is not computed locally in space, unlike area and
volume observables.

###### Remark 7.8

  1. 1.

Unlike the piercing and confinement numbers, the linking number between a
geometric hyperlink and a compact surface $S$ with boundary, is actually
dependent on any time-ordering between the surface and component geometric
loops, when imposed.

  2. 2.

When the surface is closed, no time-ordering between the surface and the loop
needs to be imposed. But, as explained in Section 6, a time-ordering is
actually implicitly defined when the surface is connected, dependent on the
orientation of the loop, which will affect the sign of the linking number
accordingly.

Quantized curvature has an important application to cosmology, as explained in
[3]. In the Friedmann-Robertson-Walker (FRW) universe model, it is predicted
that as time goes down to 0, the universe will be so warped up that curvature
is infinity and the metric will be singular. However, in such a regime, volume
would be so small that quantum gravity has to take over as the main physics
theory. Furthermore, LQG is the correct theory for gravity when metric becomes
singular. See Remark 5.2 and also [41]. The curvature predicted by LQG would
always be finite, so the FRW model no longer holds in quantum gravity regime.
We would also like to highlight the work done by the authors in [42], who
proposed replacing the big bang, with a ‘quantum bounce’, which will also
resolve the singularity predicted by the FRW model.

##  8 Hamiltonian constraint and Stress operators

If one looks at Einstein’s field equations in GR, one side of the equation is
geometric in nature; the other side is not. Except for trivial cases, it is
almost impossible to write down the stress-energy tensor. This indeed taints
the beauty of the theory of GR. See [43].

In [3], Thiemann also explained that Einstein’s equation is flawed in the
sense that one side of the equation is classical theory, whereas the other
side of the equation is quantum theory. The variable to be solved in
Einstein’s equation is the metric $g$. He proposed to convert Einstein’s
equation into an operator-valued equation and solve for a quantized metric.
His proposed quantum Einstein’s equation can be found in [4].

In our humble opinion, this is not correct. Rather, we will use Einstein’s
equation to define a stress operator, using the area and curvature operators
defined in [31] and [35] respectively. Also see Section 7.

We know that Ricci curvature ${{\rm{Ric}}}$ and scalar curvature ${\rm Scal}$
are defined as

${{\rm{Ric}}}_{\alpha}^{\beta}=R_{\alpha\gamma}^{\gamma\beta}\quad{\rm
and}\quad{\rm Scal}:={\rm Ric}_{\alpha}^{\alpha}\equiv
R_{\alpha\gamma}^{\gamma\alpha},$

whereby $R_{\alpha\beta}^{\gamma\delta}$ are the components of the Riemann
curvature tensor.

Einstein’s equations are given by (${{\rm{Ric}}}_{\mu\gamma}\equiv
g_{\mu\beta}{{\rm{Ric}}}_{\gamma}^{\beta}$)

${{\rm{Ric}}}_{\mu\gamma}-\frac{1}{2}{\rm Scal}\ g_{\mu\gamma}=\kappa
T_{\mu\gamma},\ \kappa=8\pi G,$ (8.1)

whereby $T$ is the stress-energy tensor and $G$ is Newton’s gravitational
constant. We are going to quantize $T$ into an operator $\hat{T}_{S}^{\pm}$,
which depends on a surface $S$, using the Einstein’s equation.

Pick a closed surface $S\subset\\{0\\}\times{{\mathbb{R}}}^{3}$. From our
quantization of the curvature tensor, we see that we have to integrate $R$
over a given surface. And from Remark 7.6, we see that it is not possible to
define a quantized Ricci or Scalar curvature using a surface $S$, as the
surface integral would require specifying a metric.

So we replace Ricci curvature with the operator $\hat{F}_{S}$ as defined in
Corollary 7.7. But notice that $\hat{F}_{S}$ is actually a multiple of
$\mathcal{E}\equiv(\mathcal{E}^{+},-\mathcal{E}^{-})$.

###### Notation 8.1

Write

$\hat{F}_{S}^{+}\underline{\Psi}_{\underline{L}}^{\ +}=-i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\
+}\otimes\mathcal{E}^{+},\quad\hat{F}_{S}^{-}\underline{\Psi}_{\underline{L}}^{\
-}=i\sqrt{4\pi}\ {\rm lk}(\underline{L},S)\underline{\Psi}_{\underline{L}}^{\
-}\otimes\mathcal{E}^{-}.$

Thus, $\hat{F}_{S}=\left(\begin{array}[]{cc}\hat{F}_{S}^{+}&\ 0\\\ 0&\
\hat{F}_{S}^{-}\\\ \end{array}\right)$, which acts on
$\left(\begin{array}[]{c}\underline{\Psi}_{\underline{L}}^{\ +}\\\
\underline{\Psi}_{\underline{L}}^{\ -}\\\ \end{array}\right)$.

As for the metric $g$, we replace it with the Area operator $\hat{A}_{S}$
defined in Corollary 7.1. But the Area operator has the real and imaginary
eigenvalues, which we can view them as components in a 2-vector.

###### Notation 8.2

Write

$\hat{A}_{S}^{+}\overline{\Psi}_{\overline{L}}^{\
+}:=\frac{|q|\sqrt{\pi}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{+}}}\right]\overline{\Psi}_{\overline{L}}^{\
+},\quad\hat{A}_{S}^{-}\overline{\Psi}_{\overline{L}}^{\
-}:=\frac{|q|\sqrt{\pi}}{2}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{-}}}\right]\overline{\Psi}_{\overline{L}}^{\
-}.$

By abuse of notation, we will now write $\hat{A}_{S}$ in matrix form,

$\hat{A}_{S}=\left(\begin{array}[]{cc}\hat{A}_{S}^{+}&\ 0\\\ 0&\
\hat{A}_{S}^{-}\\\ \end{array}\right),\ {\rm which\ acts\ on}\
\left(\begin{array}[]{c}\overline{\Psi}_{\overline{L}}^{\ +}\\\
\overline{\Psi}_{\overline{L}}^{\ -}\\\ \end{array}\right).$

###### Definition 8.3

(Stress Operator)
Choose a closed surface $S\subset\\{0\\}\times{{\mathbb{R}}}^{3}$. Recall we
defined phase spaces

$V^{+}\subset\overline{V}^{\ +}\otimes\underline{V}^{\ +},\quad
V^{-}\subset\overline{V}^{\ -}\otimes\underline{V}^{\ -}.$

Define a stress operator $\hat{T}_{S}$, which acts on the direct product
$V^{+}\times V^{-}$,

$\displaystyle\hat{T}_{S}$
$\displaystyle\equiv\left(\begin{array}[]{cc}\hat{T}_{S}^{+}&\ 0\\\ 0&\
\hat{T}_{S}^{-}\\\
\end{array}\right):=\frac{1}{\kappa}\left(\hat{F}_{S}-\frac{1}{2}\hat{A}_{S}\right).$

Suppose $\overline{L}=\\{\overline{l}^{u}\\}_{u=1}^{\overline{n}}$, each
matter loop component colored with representation $\rho_{u}^{+}$ and
$\rho_{u}^{-}$. Then on $V^{+}\subset\overline{V}^{\ +}\otimes\underline{V}^{\
+}$, we have

$\displaystyle\hat{T}_{S}^{+}\left(\overline{\Psi}_{\overline{L}}^{\
+}\otimes\underline{\Psi}_{\underline{L}}^{\ +}\right)=$
$\displaystyle\frac{1}{\kappa}\left(\overline{\Psi}_{\overline{L}}^{\
+}\otimes\hat{F}_{S}^{+}\underline{\Psi}_{\underline{L}}^{\
+}-\frac{1}{2}\hat{A}_{S}^{+}\overline{\Psi}_{\overline{L}}^{\
+}\otimes\underline{\Psi}_{\underline{L}}^{\ +}\right),$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\kappa}\left(-i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\mathcal{E}^{+}-\frac{|q|\sqrt{\pi}}{4}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{+}}}\right]\right)\overline{\Psi}_{\overline{L}}^{\
+}\otimes\underline{\Psi}_{\underline{L}}^{\ +},$

whereby $\kappa=8\pi G$.

Similarly, on $V^{-}\subset\overline{V}^{\ -}\otimes\underline{V}^{\ -}$, we
have

$\displaystyle\hat{T}_{S}^{-}\left(\overline{\Psi}_{\overline{L}}^{\
-}\otimes\underline{\Psi}_{\underline{L}}^{\ -}\right)=$
$\displaystyle\frac{1}{\kappa}\left(\overline{\Psi}_{\overline{L}}^{\
-}\otimes\hat{F}_{S}^{-}\underline{\Psi}_{\underline{L}}^{\
-}-\frac{1}{2}\hat{A}_{S}^{-}\overline{\Psi}_{\overline{L}}^{\
-}\otimes\underline{\Psi}_{\underline{L}}^{\ -}\right),$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\kappa}\left(i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\mathcal{E}^{-}-\frac{|q|\sqrt{\pi}}{4}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{-}}}\right]\right)\overline{\Psi}_{\overline{L}}^{\
-}\otimes\underline{\Psi}_{\underline{L}}^{\ -}.$

Note that $V^{\pm}$ is a vector space containing functionals acting on the
phase space $\\{(\underline{L}_{m},\overline{L}_{m})\\}_{m=1}^{\infty}$. And
we are using a time-like triple
$\\{S,\emptyset,\chi(\overline{L},\underline{L})\\}$ to compute the linking
and piercing numbers of $S$ with $\underline{L}$ and $\overline{L}$
respectively.

###### Remark 8.4

We can state a similar definition for a compact connected surface with
boundary. Then

$\hat{T}_{S}^{+}\left(\overline{\Psi}_{\overline{L}}^{\
+}\otimes\underline{\Psi}_{\underline{L}}^{\
+}\right)=\frac{1}{4\kappa}\left(-i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\mathcal{E}^{+}-|q|\sqrt{\pi}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{+}}}\right]\right)\overline{\Psi}_{\overline{L}}^{\
+}\otimes\underline{\Psi}_{\underline{L}}^{\ +},$

and

$\hat{T}_{S}^{-}\left(\overline{\Psi}_{\overline{L}}^{\
-}\otimes\underline{\Psi}_{\underline{L}}^{\
-}\right)=\frac{1}{4\kappa}\left(i\sqrt{4\pi}\ {\rm
lk}(\underline{L},S)\mathcal{E}^{-}-|q|\sqrt{\pi}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{-}}}\right]\right)\overline{\Psi}_{\overline{L}}^{\
-}\otimes\underline{\Psi}_{\underline{L}}^{\ -}.$

Note that $\hat{T}_{S}^{\pm}\left(\overline{\Psi}_{\overline{L}}^{\
\pm}\otimes\underline{\Psi}_{\underline{L}}^{\ \pm}\right)$ is a matrix-valued
linear functional on the phase space. Now $\mathcal{E}^{\pm}$ has purely
imaginary eigenvalues, hence the operator
$\hat{T}_{S}^{\pm}\left(\overline{\Psi}_{\overline{L}}^{\
\pm}\otimes\underline{\Psi}_{\underline{L}}^{\ \pm}\right)$ has real
eigenvalues. For $(\underline{\mathcal{L}},\overline{\mathcal{L}})$ in the
phase space, the two real eigenvalues for
$\hat{T}_{S}^{+}\left(\overline{\Psi}_{\overline{L}}^{\
+}(\underline{\mathcal{L}})\otimes\underline{\Psi}_{\underline{L}}^{\
+}(\overline{\mathcal{L}})\right)$ are given by

$\frac{1}{\kappa}\left(-\frac{|q|\sqrt{\pi}}{4}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{+}}}\right]\pm\frac{\sqrt{12\pi}}{2}\
{\rm lk}(\underline{L},S)\right)\overline{\Psi}_{\overline{L}}^{\
+}(\underline{\mathcal{L}})\otimes\underline{\Psi}_{\underline{L}}^{\
+}(\overline{\mathcal{L}}),$

and for $\hat{T}_{S}^{-}\left(\overline{\Psi}_{\overline{L}}^{\
-}(\underline{\mathcal{L}})\otimes\underline{\Psi}_{\underline{L}}^{\
-}(\overline{\mathcal{L}})\right)$, given by

$\frac{1}{\kappa}\left(-\frac{|q|\sqrt{\pi}}{4}\left[\sum_{u=1}^{\overline{n}}\nu_{S}(\overline{l}^{u})\sqrt{\xi_{\rho_{u}^{-}}}\right]\mp\frac{\sqrt{12\pi}}{2}\
{\rm lk}(\underline{L},S)\right)\overline{\Psi}_{\overline{L}}^{\
-}(\underline{\mathcal{L}})\otimes\underline{\Psi}_{\underline{L}}^{\
-}(\overline{\mathcal{L}}).$

These eigenvalues depend on how the matter and geometric hyperlinks are linked
with the closed surface $S$. Furthermore, we can see quantum fluctuations of
the area eigenvalues, due to the geometric hyperlink. We can write a similar
expression when $S$ has a boundary.

From the discussion, we will say that the basis vectors
$\\{\overline{\Psi}_{n}^{\ \pm}\otimes\underline{\Psi}_{n}^{\
\pm}\\}_{n=1}^{\infty}$ for $V^{\pm}$ form an eigenbasis for the operators
$\hat{T}_{S}^{\pm}$. Furthermore, this basis is orthogonal using the counting
measure $N$.

The eigenvalues from the area operator were interpreted as total momentum of
the matter hyperlink, coming from boost and rotation separately. Curvature of
a field is usually interpreted as the field strength and since we integrate
the curvature over a surface $S$ and quantize it via a path integral, the
eigenvalues of $\hat{F}_{S}$ should be interpreted as the quantized flux over
the surface $S$. See [44]. Hence, the eigenvalues of the stress operator
measure the total quanta of matter and geometric hyperlinks passing through
the surface $S$.

###### Remark 8.5

From Remark 7.8, the linking number is dependent on the imposed time-ordering,
when the surface has a boundary. Even when $S$ is connected and has no
boundary, the linking number is dependent on the implicitly defined time-
ordering, as explained in Section 6. Hence, the stress operator is actually
dependent on time-ordering.

Observe that the stress operators do not contain any information of energy. So
where is the Hamiltonian? Recall we explained that the volume operator gives
us the kinetic energy of matter hyperlinks. But in the construction of the
stress operator, we see that the two stress operators were defined to be
acting on the tensor product space $V^{\pm}\subset\overline{V}^{\
\pm}\otimes\underline{V}^{\ \pm}$. The volume operator acts only on
$\overline{V}^{\ \pm}$, so we must have a second operator that acts on
$\underline{V}^{\ \pm}$ and it must describe energy. The missing operator is
the potential operator $\hat{U}$, which we will now define.

Modanese in [45] talked about potential energy in quantum gravity. Borrowing
ideas from Quantum Chromodynamics (See [46].), he explained how to compute the
energy eigenvalues from a path integral expression. In a background
independent environment, time and length do not make sense and the potential
energy should refer to geometrical invariant quantities. This motivates the
following definition.

###### Definition 8.6

(Potential operator)
Recall in our construction of the phase space, for each matter hyperlink
$\overline{L}_{m}=\\{\overline{l}_{m}^{1},\ldots,\overline{l}_{m}^{\overline{n}(m)}\\}$,
we chose how it is tangled with a corresponding geometric hyperlink
$\underline{L}_{m}$. We denoted this hyperlink as
$\chi(\overline{L}_{m},\underline{L}_{m})$. Now for each component matter loop
from $\overline{L}_{m}$, we will write
$\chi(\overline{l}_{m}^{u},\underline{L}_{m})$ to denote the hyperlink inside
$\chi(\overline{L}_{m},\underline{L}_{m})$, formed using
$\overline{l}_{m}^{u}$ and $\underline{L}_{m}$. Note that
$\overline{l}_{m}^{u}$ is colored with a representation $\rho_{m,u}$.

We define the potential operator, denoted by $\hat{U}$, as

$\left[\hat{U}\underline{\Psi}_{m}^{\
\pm}\right](\overline{L}_{n}):=\left\\{\begin{array}[]{ll}\left[\sum_{u=1}^{\overline{n}(m)}\log\
Z^{\pm}(q;\chi(\\{\overline{l}_{m}^{u},\rho_{m,u}\\},\underline{L}_{m}))\right]\underline{\Psi}_{m}^{\
\pm}(\overline{L}_{m}),&\hbox{$n=m$;}\\\ 0,&\hbox{$n\neq
m$.}\end{array}\right.$

###### Remark 8.7

  1. 1.

So far, all the observables defined in Section 7 were related to a geometric
object independent from the hyperlink. For the potential operator however, we
are using the geometric hyperlink to define it.

  2. 2.

The eigenvalues of $\hat{U}$ are non-negative.

  3. 3.

The hyperlinking number is an equivalence invariant, if we impose time-
ordering between matter and geometric loops. Hence the eigenvalues of the
potential operator are well-defined for this equivalence class. Remark 5.7
says that $Z^{\pm}$ is actually dependent of this time-ordering, thus the
potential operator $\hat{U}$ is actually independent of time-ordering.

Refer to Section 3. From Equation (3.1) and Definition 5.4, we leave to the
reader to check that indeed,

$\log\
Z^{\pm}(q;\chi(\\{\overline{l}^{u},\rho_{u}\\},\underline{L}))=\left\\{\begin{array}[]{ll}\log\left[\sum_{v=1}^{(2j_{\rho_{u}^{\pm}}+1)/2}2\cosh\left(\pi
q\ {\rm
sk}(\overline{l}^{u},\underline{L})\lambda_{v}^{\pm}\right)\right],&\hbox{$2j_{\rho_{u}^{\pm}}+1$
is even;}\\\ \ \log\left[1+\sum_{v=1}^{j_{\rho_{u}^{\pm}}}2\cosh\left(\pi q\
{\rm
sk}(\overline{l}^{u},\underline{L})\lambda_{v}^{\pm}\right)\right],&\hbox{$2j_{\rho_{u}^{\pm}}+1$
is odd.}\end{array}\right.$ (8.2)

Here, $\\{\pm\lambda_{v}^{\pm}\\}_{v=1}^{(2j_{\rho^{\pm}}+1)/2}$ and
$\\{\pm\lambda_{v}^{\pm}\\}_{v=1}^{j_{\rho^{\pm}}}$ are the set of non-zero
real eigenvalues of $\rho^{\pm}(i\mathcal{E}^{\pm})$, $\mathcal{E}^{\pm}$ as
defined in Section 3, corresponding to $2j_{\rho}^{\pm}+1$ being even and odd
respectively.

Hence, we see that $\\{\underline{\Psi}_{m}^{\ \pm}\\}_{m=1}^{\infty}$ forms
an eigenbasis for $\hat{U}$, the corresponding eigenvalues given by Equation
(8.2). Note that the eigenvalues are all greater than or equal to 0.

So what does the potential energy operator measure? If one assumes $q$ to be
small and we do a Taylor’s series expansion, then we have

$\sum_{u=1}^{\overline{n}}\log\
Z^{\pm}(q;\chi(\\{\overline{l}^{u},\rho_{u}\\},\underline{L}))=\sum_{u=1}^{\overline{n}}\log(2j_{\rho_{u}^{\pm}}+1)+q^{2}\sum_{u=1}^{\overline{n}}\frac{\pi^{2}}{(2j_{\rho_{u}^{\pm}}+1)}\sum_{v\geq
1}\left({\rm
sk}(\overline{l}^{u},\underline{L})\lambda_{v}^{\pm}\right)^{2}+\cdots,$

in both cases.

If one compares the above series expansion with the Expression 28 in [45],
then one sees that the charge $q$ and its coefficient should be interpreted as
the mass of a particle and interaction energy respectively. The coefficient is
computed using the geometrical invariant hyperlinking number and we can
interpret physically as the matter hyperlink
$\\{l^{u}\\}_{u=1}^{\overline{n}}$ is held together by a geometric hyperlink
$\underline{L}$. Thus, gravitational attraction can now be visualized as
matter loops being held together by a geometric hyperlink.

We are now ready to define the Hamiltonian of the phase space, which measures
pockets of energy in compact regions in space.

###### Definition 8.8

(Hamiltonian constraint operator)
Choose a compact solid region $R\subset{{\mathbb{R}}}^{3}$. We define the
Hamiltonian constraint operator, $\hat{H}_{R}$ as

$\hat{H}_{R}=\hat{V}_{R}\otimes\underline{1}+\overline{1}\otimes\hat{U},$

where $\underline{1}$ and $\overline{1}$ are the identities on
$\underline{V}^{\ \pm}$ and $\overline{V}^{\ \pm}$ respectively. This operator
clearly acts on $V^{\pm}$, which is a vector space of functionals acting on
the phase space $\\{(\underline{L}_{m},\overline{L}_{m})\\}_{m=1}^{\infty}$.

###### Remark 8.9

From Remark 7.5, note that the eigenvalues of the volume operator
$\hat{V}_{R}$ for a solid compact region $R\subset{{\mathbb{R}}}^{3}$, is
independent of any time-ordering. Because of Remark 5.7, we see that the
operator $\hat{U}$ is dependent on how we time-order the matter and geometric
loops.

Note that $\hat{V}_{R}$ acts on $\overline{\Psi}_{m}^{\ \pm}$, which depends
on matter hyperlinks; whereas $\hat{U}$ acts on $\underline{\Psi}_{m}^{\
\pm}$, which depends on geometric hyperlinks. Furthermore, we are using the
time-like triple $\\{\emptyset,R,\chi(\overline{L},\underline{L})\\}$ to
compute the eigenvalues. Hence we have the following nice result.

###### Theorem 8.10

For a compact solid region $R\subset{{\mathbb{R}}}^{3}$, the basis vectors
$\\{\overline{\Psi}_{m}^{\ \pm}\otimes\underline{\Psi}_{m}^{\
\pm}\\}_{m=1}^{\infty}$ in vector space $V^{\pm}$ forms an eigenbasis for the
Hamiltonian constraint operator $\hat{H}_{R}$. Furthermore, it is an
orthogonal basis using the counting measure $N$.

Thiemann expressed the Hamiltonian constraint as $H(N)$, whereby $N$ is some
test function defined on spatial ${{\mathbb{R}}}^{3}$ and $H(N)$ is a volume
integral of $N$ in ${{\mathbb{R}}}^{3}$, independent of time. See Equation
(4.26) in [25].

By writing the ‘regularization version’ of the said volume integral as a
sequence of Riemann sums, Thiemann ‘promoted’ the integrands using rules of
canonical quantization, to be operators, hence defining a sequence of
operators. To complete the construction, one has to show that it converges to
some operator, denoted as $\hat{H}(N)$.

###### Remark 8.11

Obviously, there is ambiguity as to how one order the terms in the integrands,
and this is a problem that one has to face, when using canonical quantization.

We did not use any test function $N$ to define the Hamiltonian constraint
operator. This is because in our construction of observables given in Section
6, it does not make sense to define a test function, compactly supported in
regions $R\subset{{\mathbb{R}}}^{3}$, which should be thought of as an
equivalence class in ${{\mathbb{R}}}^{3}$, up to spatial diffeomorphism. In
[25], it was also acknowledged that $H(N)$ cannot be spatially diffeomorphism
invariant. Thiemann’s Hamiltonian constraint operator was also criticized for
being ‘too local’ in [47].

In contrast, we have chosen the Hamiltonian constraint operator to be
dependent on the choice of the compact solid region
$R\subset{{\mathbb{R}}}^{3}$, up to diffeomorphism equivalence. But the
potential operator which we have added into Definition 8.8, is actually
independent of it. As computation of the hyperlinking number of a hyperlink
requires the global topology of the hyperlink, the potential operator cannot
be considered a ‘local’ operator. Later in subsection 10.3, we will explain
why the volume operator should be considered as a ‘local’ operator. Thus, the
Hamiltonian constraint operator in Definition 8.8 is a non-local operator.

From Corollary 7.4, we see that the eigenvalues of $\hat{V}_{R}$ are always
non-negative. We also said that the eigenvalues of $\hat{U}$ are always non-
negative. Hence, the Hamiltonian constraint operator defined in Definition 8.8
will have non-negative eigenvalues. The eigenstate with zero eigenvalue will
correspond to the ground state. Geometrically, we can represent the ground
state as an empty set, which means there is no matter and geometric hyperlink.
If we choose $R=\emptyset$, then for a non-trivial quantum state, we see that
the Hamiltonian constraint operator will yield a strictly positive eigenvalue,
which is the eigenvalue of the potential operator. Hence we interpret this as
there will always be positive energy in the background of space-time.

The potential operator depends on the hyperlinking number between the matter
loop and geometric hyperlink. Together with the volume operator, which is
defined using a compact region in spatial ${{\mathbb{R}}}^{3}$, we see that
the Hamiltonian constraint operator we have defined earlier is dependent on
any time-ordering, as discussed above. This is similar to the quantized
curvature operator and the stress operator, both are also dependent on time-
ordering. See Remark 7.8.

###### Remark 8.12

Suppose for each $u=1,\ldots,\overline{n}$, we have the time-ordering
$\overline{l}^{u}<\underline{L}$ or $\overline{l}^{u}>\underline{L}$. Thus, we
will have

$\sum_{v\geq 1}\cosh\left(\pi q\ {\rm
sk}(\overline{l}^{u},\underline{L})\lambda_{v}^{\pm}\right)=\sum_{v\geq
1}\cosh\left(3\pi q\ {\rm
lk}(\pi_{0}(\overline{l}^{u}),\pi_{0}(\underline{L}))\lambda_{v}^{\pm}\right),$

from Remark 5.7. In such a scenerio, we see that the potential operator and
the Hamiltonian constraint operator, will be independent of time-ordering.

In Section 10, we will discuss how Einstein’s equations might allow us to
consider such a scenerio.

##  9 Entropy of a black hole

It was shown by Bekenstein and Hawking that the entropy of a black hole should
be finite, which was proved within the framework of QFT on curved space-times
and should therefore be valid in a semi-classical regime in which quantum
fluctuations of the gravitational field are negligible, for example in a large
black hole. As remarked by Thiemann in [3], any successful quantum theory of
gravity must obtain this same result.

It was shown in [48, 49] by Bekenstein that a black hole with area $A$ in
space-time should have a Bekenstein entropy

$S_{BH}=\frac{k}{4}\frac{c^{3}A}{G\hbar},$ (9.1)

for a two-dimensional event horizon of the black hole. Here, $G$ is Newton
Gravitational’s constant, $\hbar$ is Planck’s constant, $k$ is Boltzmann’s
constant and $c$ is speed of light, all set to 1. See also [50]. Using our
construction of the quantum Hilbert space, we are now going to show that
entropy is indeed proportional to area.

Let $\Lambda=\frac{\sqrt{\pi}}{4}\frac{G\hbar}{c^{3}}$. Krasnov in [51] asked,
given $A_{0}>0$, how many quantum states are there such that the eigenvalues
of the area operator $\hat{A}_{H}$ lie within the interval
$J:=[A_{0}-\bar{\delta}\Lambda,A_{0}+\bar{\delta}\Lambda]$? Here,
$A_{0}>>\Lambda$ and $\bar{\delta}$ is some small constant, to be determined
later. Mathematically, we are asking for the direct sum of the eigenspaces
whose eigenvalues lie in the closed interval.

To answer this question, we will make use of the area operator $\hat{A}_{S}$
defined in [31], of which we stated its eigenvalues in Corollary 7.1. But note
that each eigenstate will give us an eigenvalue $\rho^{+}$ or $\rho^{-}$, from
eigenstates $\overline{\Psi}_{n}^{\ +}$ and $\overline{\Psi}_{m}^{\ -}$
respectively. Thus, we will reformulate the above question and ask how many
states are there such that

$\rho^{\pm}\in\left[A_{0}^{\pm}-\bar{\delta}\Lambda,A_{0}^{\pm}+\bar{\delta}\Lambda\right].$

We will specify $\bar{\delta}$ shortly.

The eigenvalues are obtained by counting how many times a projected matter
hyperlink pierce a surface $S\subset{{\mathbb{R}}}^{3}$. However, we can
maintain how a projected hyperlink pierce the surface $S$, but change its
topology far away from the surface. Furthermore, we can add in more trivial
knots which does not pierce the surface $S$, while the piercing number remains
unchanged. In other words, there will be an infinite number of such states.
Therefore, to argue that there should only be a finite number of states
corresponding to eigenvalues in a compact interval, we need to make some
reasonable assumptions on the event horizon $H$ of a black hole.

In [52], some assumptions were made, of which only 2 will be useful for our
discussion.

A1

First assumption is that only the configurations of the hole itself, and not
the configurations of the surrounding geometry, affects the hole entropy. This
means that it suffices to consider only trivial matter hyperlinks, since how a
matter hyperlink is tangled will not affect the entropy. It also means that
all matter hyperlinks which does not pierce the hole when projected, will and
should not be considered as a quantum state.

A2

Secondly, since we are considering the thermodynamical behaviour of a system
containing the hole, we do not have to consider the black hole’s interior.
This means that if the event horizon is a closed surface, then any link in the
interior of the black hole surface, that does not pierce the surface, should
not be considered as a quantum state.

Consider an oriented surface $S$, which models the event horizon of a
blackhole. For simplicity, we assume that $S$ is connected and compact,
possibly with or without boundary. From Assumption A1, the matter hyperlink
representing any set of particles emitted out from the event horizon, is the
trivial hyperlink. From Assumption A2, we do not consider any loop strictly
inside the interior of a closed surface $S$. Any matter loop in consideration
should have a non-trivial representation. For each matter hyperlink, the
component matter loop must have either non-trivial translational or angular
momentum, i.e. $j_{\pm}\neq 0$. Finally, we assume that the piercing number
between each trivial loop and the event horizon is $\beta>0$. In summary, any
loop in a matter hyperlink must have a piercing number $\beta$ with $S$,
colored with a non-zero representation.

The states we need to consider in the dual configuration space will be
$\\{\overline{L}_{m}\\}_{m=1}^{\infty}$, each $\overline{L}_{m}$ will be a
trivial hyperlink. Orientate each of the hyperlink in a particular manner,
which will not affect the eigenvalue of $\hat{A}_{S}$. Color each matter
hyperlink $\overline{L}_{m}$ with a set of representations
$\\{\rho_{m,u}\equiv(\rho_{m,u}^{+},\rho_{m,u}^{-})\\}_{u\geq 1}$. Construct a
Quantum Loop Space using $\\{\overline{L}_{m}\\}_{m=1}^{\infty}$, denoted by
$\mathcal{H}(\overline{V}^{\ \pm})$, as defined in Definition 5.11. Each
quantum state $\overline{L}_{m}$ is represented by a set of unlinked trivial
loops, each loop colored by a representation, indexed by $(j^{+},j^{-})$, with
$j^{\pm}>0$.

Using Assumptions A1 and A2, each quantum state $\overline{L}_{m}$ in
$\mathcal{H}(\overline{V}^{\ \pm})$ is then represented by a $n$-tuple
$\vec{p}^{\pm}=(p_{1}^{\pm},\ldots,p_{n}^{\pm})$, with arbitrary $n$, denoting
the number of trivial loops in $\overline{L}_{m}$. Each
$0<p_{u}^{\pm}:=2j_{u}^{\pm}$, whereby $j_{u}^{\pm}$ is the half-integer
representation of the loop $\overline{l}^{u}$. States labelled with different
orderings of the same unordered $n$-tuples of integers $\vec{p}^{\pm}$ are
distinguishable for an external observable.

Now the eigenvalues for the area operator we obtained in Corollary 7.1 are
either purely real or purely imaginary. We will now consider the real case.
The imaginary case is similar. We will also drop the superscript ‘$+$’ and
ask, how many states in $\mathcal{H}(\overline{V}^{\ +})$ have real
eigenvalues in the interval
$\left[A_{0}-\bar{\delta}\Lambda,A_{0}+\bar{\delta}\Lambda\right]$?

By putting in all the physical constants, our eigenvalue of the area operator
for a particular quantum state
$\\{\overline{L}_{m},\rho_{m}\\}\in\mathcal{H}(\overline{V}^{\ +})$ is

$|q|\beta\frac{\sqrt{\pi}}{2}\frac{\hbar G}{c^{3}}\sum_{u\geq
1}\sqrt{j_{u}(j_{u}+1)}\equiv|q|\beta\frac{\sqrt{\pi}}{4}\frac{\hbar
G}{c^{3}}\sum_{u\geq 1}\sqrt{p_{u}(p_{u}+2)},\ p_{u}=2j_{u}.$

Now $j_{u}>0$ is a half-integer, so $p_{u}$ is a positive integer.

In [53], the area operator is actually scaled by some undetermined constant
$\gamma$. One can find a similar expression in [8]. Here, our undetermined
constant is actually the charge $q$. In Section 8, the charge $q$ has a nice
interpretation as mass, which is intrinsic to the matter hyperlink and has
nothing to do with the quantum states. Thus, we will drop $q$ from the area
operator in the rest of this section.

The proof we are presenting here, is taken from [52], but with some
modifications. Let $\delta:=0.55$. Note that
$\bigcup_{M\in\mathbb{N}}[M-\delta,M+\delta]\supset[1,\infty)$. For each
natural number $M>0$, define the following sets,

$\displaystyle A(M):=$ $\displaystyle\left\\{\vec{p}=(p_{1},\ldots,p_{n}):\
p_{u}\in\mathbb{N}\ {\rm and}\
M-\delta\leq\sum_{u=1}^{n}\sqrt{p_{u}(p_{u}+2)}\leq M+\delta\right\\},$
$\displaystyle A^{+}(M):=$
$\displaystyle\left\\{\vec{p}=(p_{1},\ldots,p_{n}):\ p_{u}\in\mathbb{N}\ {\rm
and}\ \sum_{u=1}^{n}p_{u}=M\right\\},$ $\displaystyle A^{-}(M):=$
$\displaystyle\left\\{\vec{p}=(p_{1},\ldots,p_{n}):\ p_{u}\in\mathbb{N}\ {\rm
and}\ \sum_{u=1}^{n}(p_{u}+1)=M\right\\}.$

Let $N(M)$ be the number of ordered $n$-tuples in $A(M)$; let $N_{\pm}(M)$ be
the number of ordered $n$-tuples in $A^{\pm}(M)$.

###### Lemma 9.1

Let $p_{+}=p+1$, $p$ is a natural number. Then we have

$1\leq\sqrt{(p_{+}+1)^{2}-1}-\sqrt{(p+1)^{2}-1}\leq\sqrt{8}-\sqrt{3}<1.1=2\delta.$
(9.2)

Proof. Note that $p_{u}(p_{u}+2)=(p_{u}+1)^{2}-1$. Suppose
$(p_{1},\cdots,p_{n})\in A(m)$ for some $m\leq M$. Define

$f(p)=\sqrt{(p_{+}+1)^{2}-1}-\sqrt{(p+1)^{2}-1}>0,\ p\geq 1.$

For the lower bound, note that

$\displaystyle\sqrt{(p_{+}+1)^{2}-1}-\sqrt{(p+1)^{2}-1}=$
$\displaystyle\frac{(p_{+}+1)^{2}-1-(p+1)^{2}+1}{\sqrt{(p_{+}+1)^{2}-1}+\sqrt{(p+1)^{2}-1}}$
$\displaystyle\geq$ $\displaystyle\frac{p^{2}+4p+4-p^{2}-2p-1}{p_{+}+1+p+1}$
$\displaystyle=$ $\displaystyle\frac{2p+3}{2p+3}\geq 1.$

Take the derivative,

$\displaystyle f^{\prime}(p)=$
$\displaystyle\frac{p_{+}+1}{\sqrt{(p_{+}+1)^{2}-1}}-\frac{p+1}{\sqrt{(p+1)^{2}-1}}$
$\displaystyle=$
$\displaystyle\frac{(p_{+}+1)\sqrt{(p+1)^{2}-1}-(p+1)\sqrt{(p_{+}+1)^{2}-1}}{\sqrt{(p_{+}+1)^{2}-1}\sqrt{(p+1)^{2}-1}}.$

Using the lower bound $f(p)\geq 1$, we have

$\displaystyle(p_{+}+1)$
$\displaystyle\sqrt{(p+1)^{2}-1}-(p+1)\sqrt{(p_{+}+1)^{2}-1}$ $\displaystyle=$
$\displaystyle(p+1)\left[\sqrt{(p+1)^{2}-1}-\sqrt{(p_{+}+1)^{2}-1}\right]+\sqrt{(p+1)^{2}-1}$
$\displaystyle\leq$ $\displaystyle-(p+1)+\sqrt{(p+1)^{2}-1}<0.$

Hence, $f^{\prime}(p)<0$ and for $p\in\mathbb{N}$, we obtain an upper bound
for $f$,

$\sqrt{(p_{+}+1)^{2}-1}-\sqrt{(p+1)^{2}-1}\leq\sqrt{8}-\sqrt{3}<1.1=2\delta.$

Put the lower and upper bounds together, we thus have Equation (9.2).

###### Lemma 9.2

For $M$ large enough, there exists a $d\equiv d(M)$, $0.48<d<0.69$ such that
we have $\log N(M)=dM$.

Proof. First, we find an upper bound $N(M)$. Define $F:A(M)\rightarrow
A^{+}(M)$ as follows. Consider any $n$-tuple $(p_{1},\ldots,p_{n})\in A(M)$.
Since

$\sum_{u=1}^{n}\sqrt{p_{u}(p_{u}+2)}\geq\sum_{u=1}^{n}p_{u},$

thus $\sum_{u\geq 1}p_{u}=m$ for some unique $m\leq M$. Then

$F:(p_{1},\ldots,p_{n})\longmapsto(p_{1},\ldots,p_{n}+K),$

with $K$ being the smallest whole number such that $(p_{1},\ldots,p_{n}+K)\in
A^{+}(M)$. Such a $K$ exists because $m\leq M$. Clearly, $F$ is well-defined.

Suppose $F(p_{1},\ldots,p_{n})=F(q_{1},\ldots,q_{n})$. By definition of $F$,
we must have $p_{u}=q_{u}$ for $1\leq u\leq n-1$ and there exist $K$ and
$\tilde{K}$ such that $p_{n}+K=q_{n}+\tilde{K}$. If $K\neq\tilde{K}$ and hence
$p_{n}\neq q_{n}$, then the above Equation (9.2) says that $|p_{n}-q_{n}|\leq
1$. This means that at most 2 elements in $A(M)$ can be mapped to the same
element in $A^{+}(M)$.

Hence $N(M)\leq 2N_{+}(M)$. From the proof in [52], we see that
$N_{+}(M)=(C/2)2^{M}$ for some constant $C$. Therefore, we have $N(M)\leq
2N_{+}(M)=C2^{M}$.

Now we find a lower bound for $N(M)$. Define $G:A^{-}(M)\rightarrow A(M)$ as
follows. Consider any $n$-tuple $(p_{1},\ldots,p_{n})\in A^{-}(M)$. Since

$\sum_{u=1}^{n}\sqrt{p_{u}(p_{u}+2)}=\sum_{u=1}^{n}\sqrt{(p_{u}+1)^{2}-1}\leq\sum_{u=1}^{n}(p_{u}+1)=M,$

we have $(p_{1},\ldots,p_{n})\in A(m)$ for some $m\leq M$.

Then

$G:(p_{1},\ldots,p_{n})\longmapsto(p_{1},\ldots,p_{n}+L),$

with $L$ being the smallest whole number such that $(p_{1},\ldots,p_{n}+L)\in
A(M)$. Such a $L$ exists because $m\leq M$ and it is well-defined.

Suppose $(p_{1},\ldots,p_{n}),(q_{1},\ldots,q_{n})\in A^{-}(M)$ such that
$G(p_{1},\ldots,p_{n})=G(q_{1},\ldots,q_{n})$. By definition of $G$, we must
have $p_{u}=q_{u}$ for $1\leq u\leq n-1$. Since
$\sum_{u=1}^{n}(p_{u}+1)=\sum_{u=1}^{n}(q_{u}+1)=M$, we see that
$p_{n}=q_{n}$, hence $G$ is an injective map.

Therefore, $N_{-}(M)\leq N(M)$. From the proof in [52], we see that for some
constants $D$ and $E$,

$N_{-}(M)=Da_{+}^{M}+Ea_{-}^{M},$

whereby $a_{\pm}$ are the roots to the equation $x^{2}=x+1$. We let
$a_{+}=(1+\sqrt{5})/2$ be the larger root.

When $M$ is large, $a_{+}^{M}$ will dominate, so we will have the lower bound
$N(M)\geq Da_{+}^{M}$. Putting all the bounds together, we will have

$Da_{+}^{M}\leq N(M)\leq C2^{M},$

for large enough $M$. Take $\log$, we will have

$\log D+M\log a_{+}\leq\log N(M)\leq\log C+M\log 2.$

Thus, for large enough $M$, there exists a $\log[(1+\sqrt{5})/2]<d(M)<\log 2$
such that $\log N(M)=d(M)M$.

Given $M\in\mathbb{N}$, we note that

$[A_{0}-\bar{\delta}\Lambda,A_{0}+\bar{\delta}\Lambda]=\beta\Lambda[M-\delta,M+\delta],$

for some

$A_{0}=\beta\Lambda M,\quad\bar{\delta}:=\beta\delta.$

Let the number of quantum states whose real eigenvalues lie inside
$[A_{0}-\bar{\delta}\Lambda,A_{0}+\bar{\delta}\Lambda]$ be denoted by
$\tilde{N}(A_{0})$. From Lemma 9.2, we will have

$\log\tilde{N}(A_{0})=\log
N(M)=\frac{4d(M)}{\beta\sqrt{\pi}}\frac{c^{3}}{\hbar G}A_{0}.$

Thus entropy

$S=k\log\tilde{N}(A_{0})=\left(\frac{4d(M)}{\beta\sqrt{\pi}}\right)\frac{kc^{3}}{\hbar
G}A_{0}.$

###### Remark 9.3

We refer the reader to Equation (47) in [6], whereby the entropy is given by

$S=\frac{\mathbf{c}}{\gamma}\frac{k}{4\hbar G}A,$

and $\mathbf{c}\sim 1/4\pi$. There is a constant $\gamma$, for which the
meaning is unclear. If one compares this expression with our expression, then
one may try to identify $\gamma$ with $\beta$. But in our expression, $\beta$
is the piercing number, which has no association with $\gamma$.

A simple calculation will give us

$1.08<\frac{4d(M)}{\sqrt{\pi}}<1.56.$

Now, $1.56/6\approx 0.26$ and $1.08/5\approx 0.216$. Thus by choosing an
appropriate piercing number $\beta$, the proportionality constant is somewhere
between

$0.216<\frac{4d}{\beta\sqrt{\pi}}<0.26.$

So, we showed that entropy of a black hole is proportional to its area. The
constant obtained by Bekenstein was 0.25, but the constant we obtained lies
between 0.216 and 0.26. The authors in [54] gave a partial explanation for
this constant, but remarked that the full significance of this proportionality
constant is still not well-understood. Alternatively, we could have made some
choice of $q$ to obtain the exact proportionality constant, as done in [8].

We would like to mention the work done by the authors in [8], whereby they
obtained the correct constant 0.25 in the Bekenstein-Hawking formula. They did
this by introducing the Barbero-Immirzi $\gamma$ to the area operator and
defined it accordingly to obtain the right constant. Note that in obtaining a
lower bound for the counting of states, they only consider states whereby each
$j_{u}=1/2$.

##  10 Final Remarks

We would like to conclude this article with the following remarks.

###  10.1 Gravitons

Gravitons are predicted to exist in QFT. However, using spin networks,
gravitons play little or no role. See [3]. But this is not the case in loop
representation of quantum gravity.

We have already explained that the matter hyperlink should represent