# Anomaly free quantum dynamics for Euclidean LQG

Madhavan Varadarajan  Raman Research Institute
Bangalore-560 080, India

###### Abstract

Classical gravitational evolution admits an elegant and compact re-expression
in terms of gauge covariant generalizations of Lie derivatives with respect to
a spatial phase space dependent $su(2)$ valued vector field called the
Electric Shift [1]. A quantum dynamics for Euclidean Loop Quantum Gravity
which ascribes a central role to the Electric Shift operator is derived in
[2]. Here we show that this quantum dynamics is non-trivially anomaly free.
Specifically, we show that on a suitable space of off shell states (a) the
(non-vanishing) commutator between a pair of Hamiltonian constraint operators
mirrors the Poisson bracket between their classical correspondents, (b) the
group of finite spatial diffeomorphisms is faithfully represented and (c) the
action of the Hamiltonian constraint operator is diffeomorphism covariant with
respect to the action of spatial diffeomorphisms.

††footnotetext: Email<EMAIL_ADDRESS>
##  1 Introduction

The constraints of classical General Relativity in its Hamiltonian formulation
admit a dramatic simplification when expressed in terms of the variables
discovered by Ashtekar [3]. These variables comprise of a self dual $su(2)$
connection $A_{a}^{i}$ and its conjugate electric field $E^{a}_{i}$ with $i\in
su(2)$ and $a$ being a tangent space index on the spatial slice $\Sigma$.
While the connection $A_{a}^{i}$ corresponds to the pull back to $\Sigma$ of
the self dual part of the spacetime spin connection, the electric field bears
the interpretation of a densitized spatial triad and hence determines the
spatial geometry. Recently [1] it was shown that not only do the gravitational
constraints simplify, but that the evolution equations also admit a simple and
geometrically intriguing form in the Ashtekar variables. Specifically the
evolution equations can be expressed in terms of certain gauge covariant
generalizations of Lie derivatives with respect to an $su(2)$ valued spatial
vector field constructed from $E^{a}_{i}$ and the lapse. This phase space
dependent object is called the Electric Shift. While the considerations of [3,
1] apply to both Lorentzian and Riemannian General Relativity, for Lorentzian
spacetime metrics the Ashtekar variables are complex and subject to certain
‘reality’ conditions which are difficult to impose directly at the quantum
level. In contrast, for Riemannian spacetime metrics (herein referred to as
‘Euclidean’) the Ashtekar variables are real and can be directly employed in
quantum theory. In [2], motivated by the structure of the classical theory
elucidated in [1], we derived an action of the quantum Hamiltonian constraint
for Euclidean Loop Quantum Gravity which accords a central role to the quantum
operator correspondent of the Electric Shift. 111For a first stab at the
problem, see [4]. Here we show that the action so derived is consistent with a
non-trivial anomaly free algebra of quantum constraints. Our interest in
Euclidean LQG in [2] and in this work stems from the fact that progress in
Lorentzian LQG is contingent on a thorough understanding of Euclidean LQG [5,
6, 7, 8].

In what follows, we shall assume familiarity with the contents of Reference
[2], which are summarized in Section 2.1 for the convenience of the reader.
Our starting point is the ‘Mixed Action’ of section 3 of Reference [2] and its
generalization in Appendix B of that work. This Mixed Action version of the
regulated Hamiltonian constraint at regulator value $\epsilon$ acts on spin
network states at their vertices and deforms the graph structure in a
neighborhood of coordinate size $\epsilon$ of each such vertex. For our
purposes it is necessary to modify these deformations slightly on scales much
smaller than $\epsilon$. As we shall see these minor modifications on the one
hand still yield valid regulated constraint operators and on the other, play a
key simplifying role in our demonstration of an anomaly free constraint
algebra. The tools and techniques we use in this work have been developed over
the last decade in a series of papers [9, 10], and most importantly, [11], in
the context of a toy model for Euclidean gravity which can be seen to arise as
a novel weak coupling limit of Euclidean gravity [12]. While familiarity with
these works would be helpful, we shall endeavour to make our presentation self
contained except for (as mentioned above) an assumed familiarity with the
contents of [2]. For readers familiar with [11], an important simplification
in this work concerns the use of Riemann Normal Coordinates as regulating
structures instead of the complicated network of coordinate patches used in
[11].

We now provide a rough summary of our considerations and results and then go
on to a detailed technical description of the layout of the paper. The intent
of the rough summary is to convey only a qualitative feel for the way in which
our calculations work and the reader should not worry if certain details are
not clear. Indeed, we urge the reader to peruse this summary a second time
after going through the technical aspects of the calculation described in
subsequent sections of this work. The mixed action of [2] of the Hamiltonian
constraint, at regulator value $\epsilon$, on a spin network is non trivial
only at vertices of the spin network which have non-zero volume. This action
at each such vertex is a sum of two types of actions, each of which deforms
the spin network graph and its labels in a small ‘$\epsilon$ size’vicinity of
the vertex. The first type closely resembles the action of a diffeomorphism on
the spin network which displaces the vertex and the structure around it by an
amount $\epsilon$. We shall refer to the deformed spin nets created by this
action as “electric diffeomorphism” type ‘children’ of the ‘parent’ spin net
being acted upon. 222The reason for the nomenclature ‘electric
diffeomorphisms’ is that this deformation is a more singular version of a
diffeomorphism and corresponds to the finite transformation at parameter
$\epsilon$ generated by the quantum counterpart of the electric shift [2]. In
contrast the second type does not result in a displaced vertex structure;
rather, it generates spin network deformations which resemble the ‘QSD’ type
deformations [5] in that an ‘extraordinary edge’ is placed in turn between
each pair of existing edges emanating from the parent vertex. Similar to the
QSD case [13], this second set of terms is responsible for the propagation of
quantum gravitational perturbations and we refer to these terms as
‘propagation’ type children. Both the electric diffeomorphism and the
propagation type actions also generate a set of kinks in each child in an
$\epsilon$ vicinity of the parental vertex being acted upon.

The off shell states we construct are linear combinations of ‘basis’ off shell
states. Each such basis state is a distributional state labelled by a complex
function $f$ on the Cauchy slice $\Sigma$ and a 3d Riemannian metric $h$ on
$\Sigma$. Such a basis off shell state can be thought of as an infinite non-
normalizable sum over spin network ‘bra states’, these bra states being of a
specific type, with the coefficients in this sum being determined by $f,h$.
The coefficient of a spin network bra summand is a product of an $h$ dependent
coefficient and an $f$ dependent one. The $f$ dependent coefficient is simply
the product of the evaluations of $f$ on each vertex of the spin network. The
$h$ dependent coefficient is a specific combination of the distances between
the kinks on the spin network graph. We are interested in the evaluation of
the (dual) action of the Hamiltonian constraint on such an off shell state in
the ‘continuum’ limit of $\epsilon\rightarrow 0$.

The (dual) action of the Hamiltonian constraint on such an off shell state has
a diffeomorphism part and a propagation part as described above. Each of these
has the effect of altering the coefficients in the sum over spin network bras.
The diffeomorphism part in the $\epsilon\rightarrow 0$ continuum limit, at
each vertex (of nonzero volume) of a bra, alters both the $f$ dependent part
as well as the $h$ dependent part of the coefficient because as
$\epsilon\rightarrow 0$ the displaced vertex as well as the kinks in the
‘children’ bras shrink to a single point corresponding to the ‘parental’
vertex. The net result is that this action replaces, in the coefficient of a
parent ‘bra’, the evaluation of $f$ at a vertex of this parent by a sum of its
derivatives along edges emanating from that vertex, multiplied by the
evaluation of the lapse at that vertex. The propagation part primarily affects
the $h$ dependent part of the coefficient through the shrinking of kinks and
the result is to reduce the coefficient of the parent bra to zero. To
summarise: the dual action of the Hamiltonian constraint on an off shell state
is the sum of an electric diffeomorphism part and a propagation part. In the
continuum limit action on an off shell state, the propagation contribution
vanishes and we are left only with the electric diffeomorphism contribution.
As a result the commutator between a pair of Hamiltonian constraint actions
acts in a manner very similar to a (sum over infinitesmal) diffeomorphisms and
matches exactly with a quantization of the Poisson bracket between two
Hamiltonian constraints, this Poisson bracket itself being, classically,
proportional to the diffeomorphism constraint.

As remarked above the electric diffeomorphism part of the action resembles a
diffeomorphism; indeed, it is not exactly a diffeomorphism but a more singular
structure which falls into the class of ‘extended’ diffeomorphisms defined in
[14]. A crucial ingredient in our calculation is an assumed volume non-
triviality of a child vertex created by this action from a parent vertex with
non-zero volume. A volume operator which satisfies this assumption is that
defined by Rovelli and Smolin [15] and we shall use their volume operator in
this work. 333A preliminary analysis of physical states in the kernel of the
constraint hints at an effective removal of ‘moduli’ parameters [16] by virtue
of the singular nature of electric diffeomorphisms alluded to here. We shall
comment further on this in our concluding section.

The layout of the paper is as follows. In section 2 we start with a review of
the form of the mixed action of [2]. Next, we specify this action in more
detail than in [2] through (a) a detailed specification of regulating
coordinate patches, (b) a precise specification of the positions of the
displaced vertex and kinks created by this action and, (c) a specification of
‘upward conical’ vertex structure at the displaced vertex. As we shall see,
these specifications are crucial (and in some sense, even sufficient) for the
working out of the calculation. The choice in (a) of Riemann Normal
Coordinates which are tied to the metric label $h$ of the off shell state
being acted upon, leads to diffeomorphism covariance of the Hamiltonian
constraint action through a mechanism very similar to that encountered in
[11]. The specification in (b) leads to the desired ‘derivative’ term from the
electric diffeormorphism part of the constraint action as well as the
trivialization of the propagation term described above in the evaluation of
the constraint action on the off shell state. The specification in (c) plays a
key role in obtaining the correct combination of terms which allow the
definition of a second constraint action and its (anomaly free) commutator.

In section 3, we construct the space of off shell states. As mentioned above
this is achieved through a specification of the set of spin network ‘bra’
summands associated with an off shell state and a specification of the
coefficient of each such summand in terms of evaluations of a vertex smooth
function $f$ and a specific combination of distances between kinks in the spin
network bra graph as computed by a Riemmanian metric $h$. If we call the set
of summands as the Bra Set $B$, we denote such an off shell state by
$\Psi_{B,f,h}$. In section 4, on this space of off shell states, we compute
the (dual) action of a single Hamiltonian constraint as well as the action of
a product of Hamiltonian constraints and, thereby, their commutator. In
section 5, we construct a quantization of the Poisson bracket between a pair
of Hamiltonian constraints whose action is a sum of infinitesmal
diffeomorphisms (as opposed to electric diffeomorphisms) so that the resulting
operator kills diffeomorphism invariant states. We evaluate the (dual) action
of this operator on the space of off shell states. We show that on this space
of off shell states, this action equals $i\hbar$ times the action of the
constraint commutator computed in section 4. This establishes a non-trivial
anomaly free representation of the commutator between a pair of Hamiltonian
constraints. In section 6, we show that the action of spatial diffeomorphisms
is anomaly free on the space of off shell states. We also demonstrate
consistency of the Hamiltonian constraint action with diffeomorphism
covariance if a certain condition of linear independence on the space of off
shell states holds. In section 7 we prove this condition of linear
independence. In section 8 we discuss an assortment of technicalities some of
which touch on the consistency of the detailed specifications (a)- (c)
described in section 2 with the interpretation of the resulting constraint
action as a valid approximant to the Hamiltonian constraint. In section 9 we
comment on the issue of regulator (in)dependence of physical states, then turn
to an account of possible (anomaly free) generalizations of the specific
anomaly free constraint action discussed hitherto together with the effect of
these generalizations on the nature of physical states and conclude with a
discussion of open problems and future directions of research.

Before we start our exposition, we introduce a useful notation to distinguish
between the notions of ‘term of leading non-trivial order in $\epsilon$’ and
‘term of order $\epsilon$:

Notation for term of order $\epsilon$: A term of $f$ of order $\epsilon$ will
be denoted as $O(\epsilon)$ by which we mean that there exist non-zero postive
numbers $M,\epsilon_{0}$ such that for all $0<\epsilon<\epsilon_{0}$, $|f|\leq
M\epsilon$.

Notation for term of leading non-trivial order in $\epsilon$: A term $f$ of
leading non-trivial order in $\epsilon$ will be denoted as ${\bf O}(\epsilon)$
by which we mean there exist non-zero positive numbers
$M_{1},M_{2},\epsilon_{0}$ such that for all $0<\epsilon<\epsilon_{0}$, we
have that $M_{1}\epsilon\leq|f|\leq M_{2}\epsilon$. Note that a term of ${\bf
O}(\epsilon)$ is also of $O(\epsilon)$ but a term of $O(\epsilon)$ may not be
of ${\bf O}(\epsilon)$.

In what follows, we choose units in which $\hbar=G=c=1$.

##  2  The action of the Hamiltonian constraint operator

Section 2.1 provides a review of essential material from Reference [2]. Its
main purpose is to set notation. Details may be found in Reference [2].
Section 2.1.1 is devoted to a review of classical aspects of Euclidean
gravity. Section 2.1.2 reviews the action of the regulated Hamiltonian
constraint operator at regulation parameter $\epsilon$ constructed in [2]. The
operator acts non-trivially only at vertices of spin network states. The
construction requires a choice of regulating coordinate patches at these
vertices of the spin network being acted upon. In [2] we constructed this
action in detail for a special class of vertices called ‘GR’ vertices without
a detailed specification of these regulating coordinates. In section 2.2 we
provide such a detailed specification. In section 2.3 we modify the action
derived in [2] at scales much smaller than $\epsilon$. As we shall see in
subsequent sections, these modifications ensure that the constraint action is
anomaly free. In section 2.4 we detail the constraint action on non-GR
vertices.

###  2.1  Review

In section 2.1.1 we review key aspects of the classical phase space
description of Euclidean gravity in terms of $SU(2)$ connections and electric
fields. In section 2.1.2 we review key aspects of the action of the quantum
Hamiltonian constraint constructed in [2].

####  2.1.1 Classical Theory

The canonically conjugate phase space variables are an $SU(2)$ connection
$A_{a}^{i}$ and a densitised electric field $E^{a}_{i}$ with
$\\{A_{a}^{i}(x),E^{b}_{j}(y)\\}=\delta^{i}_{j}\delta^{b}_{a}\delta(x,y)$
where $i,\in su(2)$ and $a,b$ are tangent space indices on the 3d Cauchy slice
$\Sigma$. The electric field defines a 3- metric $q_{ab}$ through

$E^{a}_{i}E^{ai}=qq^{ab}$ (2.1)

with $q$ being the determinant of $q_{ab}$. The phase space functions:

$\displaystyle G[\Lambda]$
$\displaystyle=\int\mathrm{d}^{3}x~{}\Lambda^{i}{\cal D}_{a}E_{i}^{a}$ (2.2)
$\displaystyle D[\vec{N}]$
$\displaystyle=\int\mathrm{d}^{3}x~{}N^{a}\left(E_{i}^{b}F_{ab}^{i}-A_{a}^{i}{\cal
D}_{b}E_{i}^{b}\right)$ (2.3) $\displaystyle H[N]$
$\displaystyle=\tfrac{1}{2}\int\mathrm{d}^{3}x~{}{N}q^{-1/3}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}F_{ab}^{k},$
(2.4)

are the Gauss law, diffeomorphism, and Hamiltonian constraints of Euclidean
gravity. The Hamiltonian constraint (2.4) is chosen to be of density weight
$\frac{4}{3}$ with the lapse $N$ carrying a density weight of $-\frac{1}{3}$.
The Poisson brackets between the constraints are:

$\displaystyle\\{G[\Lambda],G[\Lambda^{\prime}]\\}$
$\displaystyle=G[\Lambda,\Lambda^{\prime}]\;\;\;\;\;\;\\{G[\Lambda],H[N]\\}=0$
(2.5) $\displaystyle\\{D[\vec{N}],G[\Lambda]\\}$ $\displaystyle=G[{\cal
L}_{\vec{N}}\Lambda]$ (2.6) $\displaystyle\\{D[\vec{N}],D[\vec{M}]\\}$
$\displaystyle=D[\pounds_{\vec{N}}\vec{M}]$ (2.7)
$\displaystyle\\{D[\vec{N}],H[N]\\}$ $\displaystyle=H[\pounds_{\vec{N}}N]$
(2.8) $\displaystyle\\{H[N],H[M]\\}$
$\displaystyle=D[\vec{L}]+G[A\cdot\vec{L}],\qquad
L^{a}:=q^{-2/3}E_{i}^{a}E_{i}^{b}\left(M\partial_{b}N-N\partial_{b}M\right)$
(2.9)

where in the first line $G[\Lambda,\Lambda^{\prime}]$ denotes the Gauss Law
constraint smeared with the $su(2)$ Lie algebra commutator between
$\Lambda,\Lambda^{\prime}$. Note that the last Poisson bracket (between the
Hamiltonian constraints) exhibits structure functions just as in Lorentzian
gravity albeit with a different overall sign.

The Electric Shift $N^{a}_{i}$ at point $p$ is defined by

$N^{a}_{i}(p)=N(p)E^{a}_{i}(p)q^{-1/3}(p)$ (2.10)

As can be checked $N^{a}_{i}(p)$ transforms like a Lie algebra valued vector
field. Segregating an ‘Electric Shift’ part in brackets, we write (2.4) as:

$H(N)=\frac{1}{2}\int\epsilon^{ijk}{(\frac{NE^{a}_{i}}{q^{\frac{1}{3}}})}F_{abk}E^{b}_{j}.$
(2.11)

On the constraint surface, the equations of motion generated by $H(N)$ can be
rewritten in a geometrically intriguing and compact form in terms of
generalizations of Lie derivatives with respect to the Electric Shift [1].
Corresponding to this, in the quantum theory [2], the action of the
Hamiltonian constraint operator on a spin network state generates deformations
of the underlying spin network graph, these deformations being associated with
the quantum Electric Shift operator. We review this action in the next
section.

####  2.1.2  Hamiltonian Constraint Operator action on GR vertices

The Electric Shift operator at point $p$ is constructed through a quantization
of a regulated version of (2.10) followed by the removal of the regulator. The
regulation employs a regulating coordinate patch $\\{x\\}$ around the point
$p$. The regulated quantum correspondent ${\hat{q}}_{\tau}^{-\frac{1}{3}}$ of
$q^{-\frac{1}{3}}$ at regulator value $\tau$ is ordered right most. Standard
regularizations of this operator using either a Thiemann like trick [5] or a
Tychonoff trick (see for e.g. [17]) result in an action which is non-trivial
only when $p$ coincides with a vertex $v$ of the gauge invariant spin network
function $S(A)$ being acted upon. This action changes the intertwiner at $v$.
The final result is

${\hat{N}}^{a}_{j}(v)S(A)=\frac{3i}{4\pi}N(x(v))\sum_{I=1}^{N}{\hat{e}}^{a}_{I}{\hat{X}}_{j\;I}S_{\lambda}(A)$
(2.12)

Here $N(x(v))$ is the evaluation of the density weight $-\frac{1}{3}$ lapse
$N$ at $v$ in the regulating coordinate patch. ${\hat{e}}^{a}_{I}$ denotes the
unit coordinate (outward pointing) $I$th edge tangent of $S$ at $v$,
${\hat{X}}_{j\;I}$ is the left invariant $SU(2)$ vector field action at $v$ on
the group element associated with the (outward pointing) $I$th edge of $S$.
$S_{\lambda}(A)$ denotes a spin network identical to $S(A)$ except for the
change of intertwiner at $v$ rendered by the operator $\lim_{\tau\rightarrow
0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v)$ with $\hat{q}_{\tau}^{-1/3}$
defined either through Thiemann like identities [5, 18] involving the Volume
operator or through the Tikhonov regularization. 444 The regularization [17]
defines the action of the inverse of a positive operator ${\hat{O}}$ on a
state $\psi$ as $\lim_{\epsilon\rightarrow
0^{+}}({\hat{O}}+\epsilon)^{-2}{\hat{O}}\psi$ where
$({\hat{O}}+\epsilon)^{-1}$ is defined through spectral analysis. The result
is an operator which annhilates the zero eigenvalue states of ${\hat{O}}$.

In the Hamiltonian constraint operator, the Electric Shift is ordered right
most. In this work we shall use the Tikhonov regularization together with the
Rovelli-Smolin Volume Operator [15]. The reason for these choices will become
clear as we go along. With these choices it follows from Footnote 4 that the
Hamiltonian constraint operator has non-trivial action only at vertices of
non-trivial volume. In section 5 of [2] we derived an action of the
Hamiltonian constraint operator ${\hat{H}}_{\epsilon}(N)$ at regulator value
$\epsilon$ on spin network vertices which had a specific vertex structure
which we call a Grot-Rovelli or ‘GR’ vertex structure. A GR vertex structure
is one for which any triple of edge tangent vectors is linearly independent.
The action at a GR vertex $v$ is:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}N(x(v))\big{(}\;\sum_{I=1}^{N}\frac{j_{I}(j_{I}+1)(S_{\lambda,I,\epsilon}-S_{\lambda})}{\epsilon}$
(2.13) $\displaystyle-$ $\displaystyle\sum_{I=1}^{N}\frac{(\sum_{J\neq
I}\sum_{K\neq I,J}S_{\lambda,(1)I,J,K,\epsilon})+(\sum_{J\neq
I}S_{\lambda,(2)I,J,\epsilon})-j_{I}(j_{I}+1)(N-1)S_{\lambda}}{\epsilon}\;\big{)}.\;\;\;\;\;\;\;\;\;\;$

Here $S_{\lambda,I,\epsilon}$ is obtained by deforming the entire vertex
structure of $S_{\lambda}$ in an $\epsilon$ vicinity of $v$ along the $I$th
edge in a specific way so that the vertex $v$ is itself displaced by an amount
$\epsilon$ to the point $v_{I,\epsilon}$. This deformation, denoted by
$\phi_{I,\epsilon}$ abruptly ‘pulls’ the remaining edges along the $I$th one
and results in a kink at each of the $J\neq I$ edges. Thus if the vertex $v$
is $N$ valent $N-1$ kinks $\\{{\tilde{v}}_{J}\\}$ are created as depicted in
Figures 1(a),1(b).

(a)

(b)

(c)

Figure 1:  Fig 1(a) shows the undeformed vertex structure at $v$.
$\phi_{I,\epsilon}$ is visualised to act on this vertex structure by deforming
it along its $I$th edge as shown in Fig 1(b) wherein the displaced vertex
$v_{I,\epsilon}$ and intersection point ${\tilde{v}}_{J}$ between the $J$th
edge and its deformed image are labelled. Fig 1(c) shows the loop
$l_{IJ,\epsilon}$ which starts from $v$, runs along the $I$th edge to
$v_{I,\epsilon}$, moves along the $J$th displaced edge to ${\tilde{v}}_{J}$
and then back to $v$ along (and in the opposite direction to) $e_{J}$.

In contrast the graph underlying the states
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ is obtained by
considering the action of $\phi_{I,\epsilon}$ on one edge at a time rather
than on the entire vertex structure. The deformed graph underlying each of
these states then consists of the original graph together with this one
deformed edge (the $K$th one in $S_{\lambda,(1)I,J,K,\epsilon}$ and the $J$th
one in $S_{\lambda,(2)I,J,\epsilon}$. As a result, instead of point
$v_{I,\epsilon}$ being a new $N$ valent vertex, it is now a kink and instead
of the $N-1$ kinks only a single kink is created (at ${\tilde{v}}_{J}$ in
$S_{\lambda,(2)I,J,\epsilon}$ and at ${\tilde{v}}_{K}$ in
$S_{\lambda,(1)I,J,K,\epsilon}$). Here by a kink we mean either a bivalent
vertex with linearly independent edge tangents or a trivalent vertex located
in the interior of one semianalytic edge from which a second semianalytice
edge emanates with edge tangent linearly independent with respect to the edge
tangent of the first edge at this vertex. The deformed states
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ are denoted in
Figures 2(a), 2(b).

(a)

(b)

Figure 2:  Figures 2(a), 2(b) depict the deformed vertex structure of
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ respectively. The
box represents the intertwiner $C_{\lambda}$ located at the position of the
undisplaced vertex $v$ of $S$ (see Figure 1(a)). Insertions of $\tau_{j}$ are
represented by circles. The touching of an object with the interwtiner box
indicates an index contraction between an index of the object and that of the
intertwiner so that whereas both the $\tau_{j}$’s have index contractions with
$C_{\lambda}$ in Figure 2(a), in Figure 2(b) only the one on the $I$th edge
has such an index contraction.

These states in general are not spin networks themselves but can be expanded
in a spin network basis and the ‘one edge at a time’ deformed graph is the
(not necessarily coarsest) graph underlying each of these spin networks. In
each of these spin networks the single deformed edge carries the same spin as
its undeformed counterpart and the only two edges which have altered spin
labels are the ones which connect $v$ to each of the two kinks. Besides the
generation of two extra kinks in each of these spin networks, it is useful for
future purposes to understand how the point $v$ presents itself in these spin
networks. More in detail, the point $v$ is a non-degenerate GR vertex in $S$.
Non-degeneracy implies that its valence is greater than 3. The question of
interest is the nature of $v$ in the spin networks generated by the
‘propagation’ part of the constraint action on $S$ at $v$. Does $v$ remain a
GR vertex of the same valence or does its valence decrease? Is it possible for
$v$ to present itself as a kink? We turn to an account of the various possible
presentations of $v$ in the spin network decompositions of
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ below.

Let the valence of $v$ in $S$ be $N$ so that $N>3$ for non-degeneracy. First
consider the spin networks in the decomposition of
$S_{\lambda,(1)I,J,K,\epsilon}$. In each of these spin networks the kink
${\tilde{v}}_{K}$ is a bivalent kink (so that the part of $e_{K}$ between $v$
and ${\tilde{v}}_{K}$ in $S$ is absent). However, the nature of the kink at
$v_{I,\epsilon}$ and the presentation of $v$ depends on details of the
decomposition as follows. Let the $I$th and $K$th edge labels be $j_{I},j_{K}$
in $S$. Then the spin label $j$ of the edge between $v$ and $v_{I,\epsilon}$
in a spin network in the decomposition must arise as one of the representation
labels in the product of the $j_{I}$ and $j_{K}$ representations. If it so
happens that $S$ is such that $j_{I}=j_{K}$, then depending on the intertwiner
at $v$ in $S$, it is possible that $j=0$ in one of the spin networks in the
decomposition, in which case $v_{I,\epsilon}$ presents itself in this spin
network as a bivalent kink and $v$ reduces its valence by 2. If $N=4$, $v$
presents itself as a bivalent kink and for $N>4$ as an $N-2$ valent GR vertex.
For the case that $j\neq 0$, $v_{I,\epsilon}$ is a trivalent kink and $v$ is
an $N-1$ valent GR vertex.

Next, consider $S_{\lambda,(2)I,J,\epsilon}$. It follows from [2], that for
every spin network in the decomposition of $S_{\lambda,(2)I,J,\epsilon}$, the
spin label of the edge between $v,{\tilde{v}}_{J}$ is equal to 1 so that the
kink at ${\tilde{v}}_{J}$ is necessarily trivalent. On the other hand the spin
label $j$ of the edge between $v,v_{I,\epsilon}$ depends on the spin labels
$j_{I}$ and $j_{J}$ of the $I$th and $J$th edges in $S$, and arises as one of
the representation labels in the product of the $j_{I}$ and $j_{J}$
representations. If a spin network with $j=0$ manifests in the decomposition,
then $v_{I,\epsilon}$ is a bivalent kink and $v$ an $N-1$ valent GR vertex
else $v_{I,\epsilon}$ presents as a trivalent kink and $v$ as an $N$ valent GR
vertex. This concludes our discussion of possible kink and vertex structures
of spin networks generated by the constraint action (2.13) on a GR vertex.

Using the nomenclature developed in section 1, we refer to the states
$S_{\lambda,I,\epsilon}$ as ‘diffeomorphism type children’ and to the states
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ as ‘propagation
type children’. Equation (2.13) may be seen to arise through a regularization
of the components of $F_{ab}^{i}$ in the constraint in terms of small loop
holonomies (depicted in Figure 1(c)) with the constraint action (2.13)
emerging as the leading order term in an expansion in small loop areas, these
areas being measured by the regulating coordinates. By independently changing
the size of these small loops for the propagation terms and the diffeomorphism
terms one can further modify the action (2.13) along the lines of Appendix C
of [2]. More in detail, an application of the methods of section 5 of [2] to
the final expression derived in Appendix C of [2] results in the modified
‘mixed’ action:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}N(x(v))\big{(}\;\sum_{I=1}^{N}\frac{j_{I}(j_{I}+1)(S_{\lambda,I,a_{I},\epsilon}-S_{\lambda})}{a_{I}\epsilon}$
(2.14) $\displaystyle-$ $\displaystyle\sum_{I=1}^{N}\frac{(\sum_{J\neq
I}\sum_{K\neq I,J}S_{\lambda,(1)I,J,K,b_{I},\epsilon})+(\sum_{J\neq
I}S_{\lambda,(2)I,J,b_{I},\epsilon})-j_{I}(j_{I}+1)(N-1)S_{\lambda}}{b_{I}\epsilon}\;\big{)}.\;\;\;\;\;\;\;\;\;\;$

Here the small loops underlying the diffeomorphism type child
$S_{\lambda,I,a_{I},\epsilon}$ have (coordinate) areas which are $a_{I}$ times
that of the corresponding small loops underlying $S_{\lambda,I,\epsilon}$ in
(2.13) and those underlying the propagation type children
$S_{\lambda,(1)I,J,b_{I},\epsilon},S_{\lambda,(2)I,J,b_{I},\epsilon}$ have
(coordinate) areas which are $b_{I}$ times that of the corresponding small
loops underlying in
$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$ in (2.13). The
construction of these small loops of prescribed area is described in section
2.3.1. In this work we shall choose $a_{I}=j_{I}(j_{I}+1)$,
$b_{I}=\frac{4}{3}a_{I}(N-1)$ so that the action of ${\hat{H}}_{\epsilon}(N)$
at a GR vertex $v$ of the spin network state $S(A)$ which will be used in this
work is:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}N(x(v))\big{(}\;\sum_{I=1}^{N}\frac{(S_{\lambda,I,\epsilon}-{\frac{1}{4}}S_{\lambda})}{\epsilon}$
(2.15) $\displaystyle-$ $\displaystyle\sum_{I=1}^{N}\frac{(\sum_{J\neq
I}\sum_{K\neq I,J}{\tilde{S}}_{\lambda,(1)I,J,K,\epsilon})+(\sum_{J\neq
I}{\tilde{S}}_{\lambda,(2)I,J,\epsilon})}{\epsilon}\;\big{)}.\;\;\;\;\;\;\;\;\;\;$

where we have set

$\displaystyle S_{\lambda,I,\epsilon}$ $\displaystyle:=$ $\displaystyle
S_{\lambda,I,a_{I}=j_{I}(j_{I}+1),\epsilon},$ (2.16)
$\displaystyle{\tilde{S}}_{\lambda,(1)I,J,K,\epsilon}$ $\displaystyle:=$
$\displaystyle\frac{3}{4(N-1)j_{I}(j_{I}+1)}S_{\lambda,(1)I,J,K,b_{I}=\frac{4}{3}(N-1)a_{I},\epsilon}\;,$
(2.17) $\displaystyle{\tilde{S}}_{\lambda,(2)I,J,\epsilon}$ $\displaystyle:=$
$\displaystyle\frac{3}{4(N-1)j_{I}(j_{I}+1)}S_{\lambda,(2)I,J,b_{I}=\frac{4}{3}(N-1)a_{I},\epsilon}\;,$
(2.18)

Beyond the factors multiplying the propagation and diffeomorphism type
children above, certain features of these deformed children will play a
central role in our considerations. These features are: (a) the location of
the displaced vertex $v_{I,\epsilon}$ in $S_{\lambda,I,\epsilon}$, (b) the
edge tangent structure at this displaced vertex in $S_{\lambda,I,\epsilon}$,
and (c) the number and placement of the kinks in
$S_{\lambda,I,\epsilon}$,$S_{\lambda,(1)I,J,K,\epsilon},S_{\lambda,(2)I,J,\epsilon}$.
While the feature (a) in this work is almost identical to that in [2], we
shall modify/further specify the features (b) and (c) relative to [2]. The
modifications will alter the structure of the deformed states on scales much
smaller then $\epsilon$ and hence (as discussed in section 8), the modified
action will continue to be seen as arising from the quantization of a
regulated approximant constraint. We discuss these
modifications/specifications in section 2.3.

###  2.2  Choice of Regulating Coordinates

Since the Hamiltonian constraint action on a spin net state is only non-
trivial at vertices of non-degenerate volume, the lapse function is evaluated
only at such vertices (2.15). Since we work with a higher than unit density
Hamiltonian constraint, the lapse is non-trivially density weighted and its
evaluation at such a spin net vertex requires a coordinate patch around that
vertex. From [2] this coordinate patch serves as a regulating coordinate patch
to define the action of the constraint operator at that vertex of the spin net
state. As remarked in section 1, we are interested in the dual action of the
constraint operator on a basis off shell states, each such basis state
$\Psi_{f,h}$ being labelled by a function $f$ and a Riemmannian metric $h$.
Since any such state resides in the algebraic dual space 555 Elements of the
algebraic dual are complex linear mappings on the finite span of spin network
states. Each such element (denoted by $\Psi$) can be thought of as a (in
general, formal) sum over spin network ‘bra’ states. The action of the complex
linear map $\Psi$ on a spin net state $|s\rangle$ is written as the
‘amplitude’ $\Psi(|s\rangle)$ to the finite span of spin network states, this
dual action is specified through the evaluation of the amplitude
$\Psi_{f,h}({\hat{H}}_{\epsilon}(N)|s\rangle)$ for every spin net $|s\rangle$.

Following [11], we tailor the choice of the regulating coordinates to the
metric label $h$ of the basis state $\Psi_{f,h}$. As detailed in [11] and
shown in section 6 of this work, such a choice ensures a transparent and
simple implementation of spatial diffeomorphism covariance. Our specific
choice of regulating coordinates at a vertex $v$ is that of Riemann Normal
Coordinates (RNCs) centered at $v$ and defined by $h$. This is in contrast to
the complicated network of coordinate patches used in [11] and simplifies the
exposition considerably with respect to that of [11].

Two key properties of RNCs are as follows. First, the coordinate edge tangent
vector ${\hat{e}}^{a}_{I}$ along the $I$th edge at the vertex $v$, when
evaluated with respect to the RNCs at $v$ defined by the metric $h$, is
exactly the edge tangent vector with unit norm with respect to $h$ at $v$ i.e.

$h_{ab}{\hat{e}}^{a}_{I}{\hat{e}}^{b}_{I}|_{v}=1.$ (2.19)

This follows directly from the definition of RNCs. Next let a point
$p_{\epsilon}$ be located at a geodesic distance $\epsilon$ away from the
point $p$ for small enough $\epsilon$ that $p_{\epsilon}$ is in a convex
normal neighborhood of $p$. Note that from the definition of the RNCs at $p$
this distance is the same as the coordinate distance of $p_{\epsilon}$ from
$p$. Let the RNCs centered at $p$ be $\\{x\\}$ and at $p_{\epsilon}$ be
$\\{x\\}_{\epsilon}$. Let the Jacobian of the former with respect to the
latter be $J$ so that

$J^{i}\;_{j}=\frac{\partial x^{i}}{\partial x^{j}_{\epsilon}}$ (2.20)

Then at $p_{\epsilon}$ we have that

$\det{J}{|_{p_{\epsilon}}}=1+O(\epsilon^{2})$ (2.21)

where $\det{J}$ is the determinant of $J$. For a proof, see Appendix A.

###  2.3  Modified Action on GR vertices

From [2], the constraint action (2.15) at a spin net vertex depends on the
specification of the action of the ‘electric diffeomorphisms’ defined by the
quantum shift at that vertex, this action being non-trivial in a small
vicinity of the vertex. The specification of the action of these electric
diffeomorphisms turns out to be simple and straightforward for GR vertices
(recall that these are vertices at which any triple of edge tangents are
linearly independent). Similar to the QSD case [5], we would like the
resultant deformation of the vertex structure to be such that the deformed
edges and their undeformed counterparts have simple and intuitively reasonable
intersection structure. Whereas this is straightforward to achieve for GR
vertex structures, for Non-GR (NGR) vertices, it is quite an intricate affair
to define a deformation and the consequent routing of the deformed edges so
that no undesired intersections occur. In [2] we focussed on the GR case. Here
we find it necessary to modify the deformations of GR vertices specified in
[2] on scales much smaller than the regulating parameter scale $\epsilon$ for
reasons described in section 1. Since the modifications are small scale they
do not alter the validity of the derivation of the resulting action when
viewed through the lense of approximants to leading order in $\epsilon$. More
in detail, recall that the small loop holonomies discussed in section 2.1.2
play a key role in the action of the constraint and each of them provide a
leading order contribution proportional to their area, this area being of
order $\epsilon^{2}$. The modifications described in this section can be
thought of as very small scale deformations of these small loops. Since (as
can be seen explicitly in Appendices B, D) these deformations can be made
arbitrarily small, they change the small loop area only by terms of higher
order than $\epsilon^{2}$. This is why they are called ‘small scale’ and this
is why they do not change the validity of the derivation of the resulting
constraint action.

In section 2.3.1 we describe the deformations of a GR vertex $v$ as
constructed in [2]. Our description will be slightly qualitative in the
interest of pedagogy. A technically complete construction suitable for our
purposes (with a slightly altered placement of $v_{I,\epsilon}$ in accordance
with the latter part of section 2.3.1) may be found in Appendix B.1. In
sections 2.3.2 and 2.3.3 we detail the small scale modifications of these
deformations. We relegate attendant technicalities to Appendices B.2, B.3 and
D.

####  2.3.1  Deformation of a GR Vertex in Reference [2]

From section 2.2 the regulating coordinates are RNCs centered at $v$ and
associated with the metric $h$. From the discussion in section 2.2 and as
depicted in Figure 1(b), the deformation associated with the $I$th edge
tangent contribution to the Electric shift displaces the original vertex $v$
along the edge $e_{I}$ to $v_{I,\epsilon}$. In [2] $v_{I,\epsilon}$ is located
a coordinate distance $\epsilon$ along the $I$th edge from the vertex $v$ i.e.
the length of the part of the $I$th edge between the points $v$ and
$v_{I,\epsilon}$ as measured by the regulating coordinates is $\epsilon$.

The $J$th deformed edge meets its undeformed counterpart $e_{J}$ at the kink
${\tilde{v}}_{J}$ and joins this kink to $v_{I,\epsilon}$. In the tangent
space $T_{v}$ at $v$ consider the $N-1$ planes $P_{IJ},J\neq I$ formed by the
$I$th and the $J$th edge tangents. Since $v$ is GR, these planes are distinct.
It follows that for small enough $\epsilon$, we can confine the $J$th deformed
edge to within a small vicinity of the $IJ$ coordinate plane and in this way,
other than the intersection structure at ${\tilde{v}}_{J},v_{I,\epsilon}$
described above, avoid any further intersections between the undeformed and
deformed edges as well between the deformed edges themselves. It also follows
that the vertex $v_{I,\epsilon}$ remains GR.

The small loops $l_{IJ,\epsilon}$ alluded to in section 2.2 and depicted in
Figure 1(c) run from $v$ to $v_{I,\epsilon}$ along $e_{I}$, from
$v_{I,\epsilon}$ to ${\tilde{v}}_{J}$ along the $J$th deformed edge and then
back from ${\tilde{v}}_{J}$ to $v$ along the undeformed $J$th edge. As
discussed in section 2.2, the areas of these loops are adjusted so as to
obtain the expression (2.15). From the discussion in [2] these loops have
areas of ${\bf O}(\epsilon^{2})$ so that ${\tilde{v}}_{J}$ are located at
distances of ${\bf O}(\epsilon)$ from $v$. The desired loop areas may then be
obtained by adjusting the locations of the kinks ${\tilde{v}}_{J}$. This
concludes our brief (and slightly more precise) description of the deformation
of the graph structure around a GR vertex as defined in [2].

In what follows we shall change the placement of $v_{I,\epsilon}$ by an amount
$O(\epsilon^{2})$. This, by itself, does not change the small loop areas to
${\bf O}(\epsilon^{2})$. We shall also ‘smoothen’ all but 3 of the kinks and,
further, change the placement of these remaining kinks. These modifications in
the kink structure and placement are implemented in such a way that the loop
areas are unchanged to ${\bf O}(\epsilon^{2})$. As a consequence the resulting
constraint action retains its validity as a leading order in $\epsilon$
approximant. Moreover the modifications can be seen to be on scales much
smaller than $\epsilon$.

####  2.3.2  Modified placement of $v_{I,\epsilon}$ and modified choice and
placement of kinks

We modify the placement of $v_{I,\epsilon}$ as follows. Recall that in [2]
$v_{I,\epsilon}$ is placed such that the coordinate length of the $I$th edge
between $v$ and $v_{I,\epsilon}$ is $\epsilon$. Since this edge is not
necessarily a coordinate straight line this edge length can differ from the
coordinate distance between the two points $v_{I,\epsilon}$ and $v$. However,
since the edge is $C^{r},r>>1$, this difference is of $O(\epsilon^{2})$. We
modify the placement of $v_{I,\epsilon}$ by a coordinate distance
$O(\epsilon^{2})$ so that the coordinate distance between the new position of
$v_{I,\epsilon}$ and $v$ is $\epsilon$. Thus, $v_{I,\epsilon}$ is now placed
on $e_{I}$ in such a way that the coordinate length of the coordinate straight
line connecting $v$ to $v_{I,\epsilon}$ is $\epsilon$. Since the coordinates
are RNCs centered at $v$, this coordinate straight line distance is also the
geodesic distance between $v$ and $v_{I,\epsilon}$.

While the modifed placement of $v_{I,\epsilon}$ is implemented for the
‘diffeomorphism’ and the ‘propagation’ type deformations, we shall modify the
‘diffeomorphism’ deformations in a slightly different manner from the
‘propagation’ deformations with regard to their ${\tilde{v}}_{J}$ kink
structure. Accordingly first consider the ‘diffemorphism’ deformation along
the edge $e_{I}$ depicted in Fig 1(b) with $N-1$ kinks
$\\{{\tilde{v}}_{J},\;J\neq I\\}$. As depicted in Fig 3, it is possible to
alter the deformed edge emanating from the kink ${\tilde{v}}_{J}$ in an
arbitrary small vicinity of ${\tilde{v}}_{J}$ so as to make it meet its
undeformed counterpart ‘smoothly’. At a technical level, by ‘smoothly’ we mean
in a manner which preserves the first $r$ derivatives. In this regard, recall
that the edges are $C^{r}$ semianalytic. This smoothening can then be achieved
by suitable ‘polynomial’ fits discussed in, for example, [18]. For a detailed
account see Appendix B of this paper.

Figure 3: The kink ${\tilde{v}}_{L}$ between the pair of edges on the left is
smoothened to give the single semianalytic edge on the right

In this manner we smoothen all but 3 of the kinks. Let the remaining kinks be
${\tilde{v}}_{\hat{J}_{i}},i=1,2,3$ As depicted in Figure 5, we can then move
each of these kinks along $e_{J_{i}}$ as close to $v$ as we desire with an
arbitrarily small area change of the loop $l_{IJ_{i},\epsilon}$. In the
modified loop, the deformed edge emanates from the new kink position and runs
‘almost’ along $e_{J_{i}}$ till it reaches the vicinity of the old position of
the kink, and, subsequently moves towards $v_{I,\epsilon}$ as before. In
detail this can be achieved in one of two ways: either by polynomial $C^{r}$
joins or by the action of appropriate semianalytic diffeomorphisms of compact
support (see Appendices B and D).

Figure 4:  Figure 5:  The first figure depicts the movement of the kink
${\tilde{v}}_{J}$ towards $v$ through the procedure outlined in Appendices B
and D. The second figure shows the modification in the small loop
$l_{IJ,\epsilon}$ required to achieve this movement. The lower strand of the
modified loop on the right curves in the vicinity of the original position of
the kink and connects to the displaced kink. The closer this strand is to the
undeformed edge $e_{J}$, the smaller is the change in area of the modified
loop relative to the original one on the left.

We shall place these 3 kinks as follows. Let $\epsilon$ be small enough that
the 3 kinks are in a single Convex Normal Neighborhood (with respect to the
metric $h$) of $v$. We shall place the 3 kinks, such that two of them are at a
distance of ${\bf O}(\epsilon^{q}),q\geq 2$ from $v$ with the third at a
distance of ${\bf O}(\epsilon^{p}),p>>q$, in such a way that the 3 interkink
geodesic distances (with respect to $h$) are unequal, with the ratio of the
smallest interkink distance to the largest being equal to $1/2$ to order
$\epsilon^{q}$ i.e.

$\frac{d_{min}}{d_{max}}=\frac{1}{2}\;+\;O(\epsilon^{q}),\;\;\;\;\;d_{min}:=\min_{i\neq
j}d({{\tilde{v}}_{J_{i}},{\tilde{v}}_{J_{j}}})\;\;\;\;d_{max}:=\max_{i,j}d({{\tilde{v}}_{J_{i}},{\tilde{v}}_{J_{j}}})$
(2.22)

with

$d({{\tilde{v}}_{J_{i}},{\tilde{v}}_{J_{j}}})\sim{\bf O}(\epsilon^{q\geq
2})\;\;\;,i,j\in\\{1,2,3\\}$ (2.23)

where $d({p_{1},p_{2}})$ denotes the geodesic distance between the points
$p_{1},p_{2}$ (see Appendix C for a demonstration that such a placement
exists). Recall that the state obtained by an electric diffeomorphism of
$S_{\lambda}$ was denoted by $S_{\lambda,I,\epsilon}$ in (2.16),section 2.1.
We shall denote its modification (with only 3 kinks placed as above and with
the modified placement of $v_{I,\epsilon}$ described above) by

$S_{\lambda,I,{\vec{\hat{J}}},\epsilon}$ (2.24)

where ${\vec{\hat{J}}}:=({\hat{J}}_{1},{\hat{J}}_{2},{\hat{J}}_{3})$ denotes
the 3 chosen edges on which the kinks are placed. In addition, the notation
will also be assumed to encode the placement of each of the kinks as follows.
The kink closest to $v$ is denoted by ${\tilde{v}}_{\hat{J}_{3}}$ Of the two
remaining kinks, the kink closer to ${v}$ is chosen to be
${\tilde{v}}_{\hat{J}_{2}}$ (this corresponds to $v_{1}$ in Appendix C) The
third (remaining) kink, corresponding to $v_{2}$ in Appendix C is then at
${\tilde{v}}_{\hat{J}_{1}}$.

Next, consider the ‘propagation’ deformations associated with the $I$th
component of the electric shift as depicted in Fig 2. Each such deformation
creates a pair of kinks, one at $v_{I,\epsilon}$ and the other at
$\tilde{v_{J}}$. While the location of $v_{I,\epsilon}$ has been described
above and is the same as for the diffeomorphism deformation, we move the kink
$\tilde{v_{J}}$ to a coordinate distance of ${\bf O}(\epsilon^{q}),q\geq 2$
from $v$, where we remind the reader that these coordinates are RNC’s centered
at $v$. As for the ‘diffeomorphism’ type deformation this can be done with an
arbitarily small change in the coordinate area of the loop $l_{IJ,\epsilon}$.
Since the metric is $C^{r-1},r>>1$, it is straightforward to conclude that the
geodesic distance of this kink from $v$ is of the same order as its coordinate
distance from $v$. Since $v_{I,\epsilon}$ is at a geodesic distance $\epsilon$
from $v$, it then follows that for the propagation deformation with a kink at
${\tilde{v}}_{J}$ and for small enough $\epsilon$ that

$d({\tilde{v}}_{J},v_{I,\epsilon})=\epsilon+O(\epsilon^{q\geq 2}).$ (2.25)

From here on the states (2.17), (2.18) will be assumed to have kink placements
as detailed in (2.25).

####  2.3.3  Modified edge tangent structure at displaced vertex

From section 2.3.1 and Appendix B.1, the ‘diffeomorphism’ type deformation of
a GR vertex $v$ along the edge $e_{I}$ preserves the GR property so that
$v_{I,\epsilon}$ is also GR. We shall modify this deformation in an
arbitrarily small vicinity of $v_{I,\epsilon}$ in a manner which preserves the
GR property as well as the location of $v_{I,\epsilon}$, and endows the edge
tangent configuration with a stiff ‘upward’ conical structure around the edge
$e_{I}$. By a ‘stiff upward cone’ we mean that the component of the $J$th edge
tangent at $v_{I,\epsilon}$ orthogonal to ${\hat{e}}^{a}_{I}$ is much smaller
than the component along $e_{I}$. The small orthogonal contribution to each of
the $N-1$ edge tangents is required to be such that the vertex
$v_{I,\epsilon}$ remains GR. More precisely, we require that for small enough
$\epsilon$:

${\hat{e}^{\prime}}^{a}_{J}|_{v_{I,\epsilon}}={\hat{e}}^{a}_{I}|_{v_{I,\epsilon}}+O(\epsilon^{2})v^{a}_{J}.$
(2.26)

where $O(\epsilon^{2})v^{a}_{J}$ is non-vanishing for sufficiently small
$\epsilon,\;\epsilon>0$, $v^{a}_{J}$ is of metric norm ${\bf O}(1)$ and the
set of vectors $\\{{\hat{e}}^{a}_{I}|_{v_{I,\epsilon}},v^{a}_{J},J\neq I\\}$
is such that no triple is linearly dependent. We depict the desired
modification of the edge tangent structure in Fig 6.

Figure 6: The edge tangent configuration at the vertex $v_{I,\epsilon}$ on the
left is modified to the upward pointing configuration shown on the right.

As shown in Appendices B, D this modification can be implemented both by using
exclusively polynomial $C^{r}$ joins as well as through the use of such joins
in conjunction with semianalytic diffeomorphisms of compact support.

####  2.3.4  Modified constraint action on GR vertices

The ‘upward conical’ modification of edge tangent structure described in the
previous section is applied to the vicinity of the vertex $v_{I,\epsilon}$ of
the 3-kink modified state $S_{\lambda,I,{\vec{\hat{J}}}\epsilon}$ (2.24). To
summarise: The modification of the ‘diffeomorphism child’ $S_{I,\epsilon}$ is
three fold: the first modification is the $O(\epsilon^{2})$ adjustment of
section 2.3.2 to the placement of $v_{I,\epsilon}$, the second is to the kink
structure as described in section 2.3.2 and the third modification is the
upward conical modification described above. The modification of each of the
‘propagation’ children is two fold
${\tilde{S}}_{\lambda,(1)I,J,K,\epsilon},{\tilde{S}}_{\lambda,(2)I,J,\epsilon}$:
first, we have the modified placement of the kink $v_{I,\epsilon}$, identical
to that for the diffeomorphism child and second, the kinks (at
${\tilde{v}}_{K}$ , respectively ${\tilde{v}}_{J}$) are placed as described in
section 2.3.2. We shall abuse notation slightly and continue to denote the
resulting ‘vertex placement, upward conical, 3-kink’ modification of
$S_{\lambda,I,\epsilon}$ by $S_{\lambda,I,{\vec{\hat{J}}},\epsilon}$ and also
continue to denote the ‘kink placement’ modifications of the propagation
children by
${\tilde{S}}_{\lambda,(1)I,J,K,\epsilon},{\tilde{S}}_{\lambda,(2)I,J,\epsilon}$.

The modified version of the constraint action is then:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}N(x(v))\big{(}\;\sum_{I=1}^{N}\frac{(S_{\lambda,I,{\vec{\hat{J}}}\epsilon}-{\frac{1}{4}}S_{\lambda})}{\epsilon}$
(2.27) $\displaystyle-$ $\displaystyle\sum_{I=1}^{N}\frac{(\sum_{J\neq
I}\sum_{K\neq I,J}{\tilde{S}}_{\lambda,(1)I,J,K,\epsilon})+(\sum_{J\neq
I}{\tilde{S}}_{\lambda,(2)I,J,\epsilon})}{\epsilon}\;\big{)}.\;\;\;\;\;\;\;\;\;\;$

In the above expression, the diffeomorphism type deformation depends on an
arbitrary choice of kink triples specified through ${\vec{\hat{J}}}$. Since
there is no natural choice for ${\vec{\hat{J}}}$ we sum over all such choices.
For fixed $I$ there are ${}^{N-1}\\!P_{3}$ choices of ${\vec{\hat{J}}}$ for a
GR vertex of valence $N$. Each such choice yields a legitimate constraint
action (2.27). Hence we sum over such actions and divide by the factor
${}^{N-1}\\!P_{3}$ to obtain our final expression for the constraint action on
a GR vertex $v$:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}N(x(v))\big{(}\;\sum_{I=1}^{N}\frac{([\frac{1}{{}^{N-1}\\!P_{3}}\sum_{\vec{\hat{J}}}S_{\lambda,I,{\vec{\hat{J}}}\epsilon}]-{\frac{1}{4}}S_{\lambda})}{\epsilon}$
(2.28) $\displaystyle-$ $\displaystyle\sum_{I=1}^{N}\frac{(\sum_{J\neq
I}\sum_{K\neq I,J}{\tilde{S}}_{\lambda,(1)I,J,K,\epsilon})+(\sum_{J\neq
I}{\tilde{S}}_{\lambda,(2)I,J,\epsilon})}{\epsilon}\;\big{)}.\;\;\;\;\;\;\;\;\;\;$

###  2.4  Form of assumed action on non-GR vertices

We assume the following form for the constraint action on an $N$ valent non GR
vertex $v$:

${\hat{H}}_{\epsilon}(N)S(A)=\frac{N(x(v))}{\epsilon}\big{(}\;(\sum_{I=1}^{N}A_{I}S_{\lambda,I,\epsilon})+BS_{\lambda}+(\sum_{I=1}^{N}\sum_{J\neq
I}\sum_{\alpha_{IJ}}P_{\alpha_{IJ}}S_{\alpha_{IJ},\epsilon})\;\;\big{)}$
(2.29)

Here $S_{\lambda,I,\epsilon}$ is a diffeomorphism type deformation of
$S_{\lambda}$ which displaces the vertex $v$ along $e_{I}$ to $v_{I,\epsilon}$
by a coordinate distance of ${\bf O}(\epsilon)$. We shall assume that this
deformation preserves the NGR nature of the vertex so that $v_{I,\epsilon}$ is
also an NGR vertex. In addition we make a further assumption which we state
using certain nomenclature defined in section 3.1 and which we justify through
a construction described in Appendices D.2 and B.3. We assume that, similar to
the GR case, three of the deformed edges of $S_{\lambda,I,\epsilon}$ meet
their undeformed counterparts at kinks Such a deformation may be obtained by
the action on $S_{\lambda}$ of a semianalytic diffeomorphism of the type
constructed in section D.2 to yield a state $S_{diff}$ followed by the
introduction of the desired kinks. Since the diffeomorphism is identity
outside a small neighborhood $U$ of the vertex, we may place kinks $k$ at the
chosen edges in $S_{diff}$ at the intersection of these edges with the
boundary of this neighborhood through the following procedure. Choose a small
neighborhood of $k$ such that no other edge than the edge of interest
intersects this neighborhood and fix a semianalytic coordinate patch thereon .
Remove part of the edge connecting $k$ to a point $p$ in this neighborhood and
join $k$ to $p$ by a a pair of coordinate straight lines one from $k$ to a
suitable point $q$ in this neighborhood and then from $q$ to $p$ so that these
lines do not intersect the edge of interest except at $k,p$ and such that
$k,q,p$ are kinks. Finally, smoothen the kinks at $q,p$ using the techniques
of Appendix B.3 thereby leaving a kink at $k$. This concludes our account of
the state $S_{\lambda,I,\epsilon}$.

$S_{\lambda}$ is, as defined for the GR case, a spin net identical to $S$
except for a different intertwiner at the NGR vertex $v$. The subscript
${\alpha}_{IJ}$ denotes a ‘propagation’ type deformation of the vertex
structure at $v$ in an $\epsilon$ size vicinity of $v$. By ‘propagation’ type
we mean that, similar to the GR case of propagation type deformations, the
deformation structure is as follows. As in the GR case, the deformations which
contribute to the constraint action are expected to arise from the
nontriviality of the curvature components
$F_{ab}^{i}{\hat{e}}^{a}_{I}{\hat{e}}^{b}_{J}$. Hence the $IJ$ label ranges
over pairs of edges with linearly independent tangents at $v$. Consider the
points $v_{I,\epsilon}$ along the $I$th edge $e_{I}$ of $S$ and
${\tilde{v}}_{J}$ on the edge $e_{J}$ of $S$ with ${\tilde{v}}_{J}$ at a
coordinate distance of ${\bf O}(\epsilon^{q\geq 2})$ from $v$. 666Similar to
the GR case, we assume that this kink positioning can be implemented without
affecting the interpretation of the constraint action at parameter $\epsilon$
as that of a leading order approximant to the constraint. Join these 2 points
with a semianalytic edge $e^{\prime}_{J}$ which does not intersect the graph
$\gamma$ underlying $S$ except at these two points and whose edge tangents at
${\tilde{v}}_{J},v_{I,\epsilon}$ are linearly independent with respect to the
edge tangents along $e_{J},e_{I}$ at ${\tilde{v}}_{J},v_{I,\epsilon}$
respectively. 777This can be achieved, for example, through the procedure
detailed in (ii) section 8.  Then the graph underlying each of the deformed
states $S_{\alpha_{IJ},\epsilon}$ is

$\gamma_{IJ}=\gamma\cup e^{\prime}_{J}$ (2.30)

Using the notation $e(p,q)$ to denote the semianalytic edge containing $p$ to
$q$, consider the edges
$e(v,v_{I,\epsilon}),e(v_{I,\epsilon},{\tilde{v}}_{J}),e({\tilde{v}}_{J},v)\subset\gamma_{IJ}$.
These edges combine to form the loop $l_{IJ,\epsilon}$. The colorings of
$\gamma_{IJ}-l_{IJ,\epsilon}\subset\gamma$ in each $S_{\alpha_{IJ},\epsilon}$
are the same as for their counterparts in $S$. While the coloring of
$e(v_{I,\epsilon},{\tilde{v}}_{J})\equiv e^{\prime}_{J}$ in
$S_{\alpha_{IJ},\epsilon}$ is the same as that of $e_{J}$ in $S$, the
colorings of the remaning edges of $l_{IJ,\epsilon}\subset\gamma_{IJ}$ are
different from those in $S$ and can include the trivial $j=0$ coloring,
888Since any edge with $j=0$ coloring is absent, it is possible that the
removal of such an edge from the set of edges at $v$ in $S$ can convert $v$
from an NGR vertex in $S$ to a GR vertex in $S_{\alpha_{IJ},\epsilon}$. in
which case $\gamma_{IJ}$ is not the coarsest graph underling
$S_{\alpha_{IJ},\epsilon}$. The points $v_{I,\epsilon},{\tilde{v}_{J}}$ are
kinks in $S_{\alpha_{IJ},\epsilon}$ (for a precise definition of kinks see
section 3.1) and depending on the edge colorings of $l_{IJ,\epsilon}$ have
uniquely defined intertwiners by virtue of their bivalent or trivalent nature.
Finally, the intertwiner at $v$ in $S_{\alpha_{IJ},\epsilon}$ differs from
that in $S$. Thus the label ${\alpha_{IJ},\epsilon}$ denotes a deformation
which introduces the pair of kinks $v_{I,\epsilon},{\tilde{v}}_{J}$ together
with the colored edge $e(v_{I,\epsilon},{\tilde{v}}_{J})$ and encodes the
changes in colorings on the edges $e(v,v_{I,\epsilon}),e({\tilde{v}}_{J},v)$
and the change in the intertwiner at $v$ relative to $S$. If
$e(v,v_{I,\epsilon})$ is colored with $j=0$, $v_{I,\epsilon}$ is a bivalent
kink. If $e(v_{I,\epsilon},{\tilde{v}}_{J})$ is colored with $j=0$,
${\tilde{v}}_{J}$ is a bivalent kink. Clearly the presence of each such
bivalent kink reduces the valence of $v$ relative to its valence in $S$ by 1.

Since the vertex $v$ is nondegenerate in $S$ it has a minimum valence of 4.
Since it is NGR in $S$, it is straightforward to see that reduction of its
valence by 1 or 2 can lead to its presentation in $S_{\alpha_{IJ},\epsilon}$
as any of the following: a GR or NGR vertex, a $C^{m<r}$ bivalent or trivalent
kink, a $C^{r}$ trivalent kink 999See section 3.1 for the definition of a
$C^{m}$ kink. or an interior point on a single semianalytic edge. Note that if
both $v_{I,\epsilon}$ and ${\tilde{v}}_{J}$ are trivalent there is no
reduction in valence and $v$ continues to be an NGR vertex with unchanged edge
tangent structure relative to its presentation in $S$.

It follows from the positioning of $v_{I,\epsilon},{\tilde{v}}_{J}$ described
above that the coordinate distance between ${\tilde{v}}_{J}$ and
$v_{I,\epsilon}$ is of ${\bf O}(\epsilon)$. It also follows straightforwardly
that the geodesic distances as measured by the metric $h$ are:

$\displaystyle d(v_{I,\epsilon},v)$ $\displaystyle=$ $\displaystyle{\bf
O}(\epsilon)$ (2.31) $\displaystyle d({\tilde{v}}_{J},v)$ $\displaystyle=$
$\displaystyle{\bf O}(\epsilon^{q\geq 2})$ (2.32) $\displaystyle
d({\tilde{v}}_{J},v_{I,\epsilon})$ $\displaystyle=$ $\displaystyle{\bf
O}(\epsilon)$ (2.33)

Note that, for sufficiently small $\epsilon$, as $\epsilon$ is decreased, the
deformed spin network $S_{\alpha_{IJ},\epsilon}$ has the same
$\epsilon$-independent vertex intertwiner at $v$ as well as the same
$\epsilon$ independent abstract colored graph structure (see section 3.1 for a
definition of this abstract structure). This $\epsilon$ independent abstract
structure is encoded by the index $\alpha_{IJ}$. The extra label $\epsilon$ in
$\alpha_{IJ},\epsilon$ indicates that the deformation is confined in an
$\epsilon$ vicinity of $v$ with the placements of the kinks at
$v_{I,\epsilon},{\tilde{v}}_{J}$ as detailed above. 101010More precisely
$\alpha_{IJ}$ is a label for the embeddable abstract spin network whose
embedding into $S_{\alpha_{IJ},\epsilon}$ is specified by
${\alpha_{IJ},\epsilon}$ (see section 3.1 for the definition of an embeddable
abstract spin network).

$A_{I},B,P_{\alpha_{IJ}}$ are (not necessarily non-vanishing) complex
coefficients. While it is not necessary for our demonstration of anomaly free
constraint action to specify the deformations and coefficients in (2.29)
beyond our discussion above, we shall present one possible concrete and
complete such specification and justify its validity in section 8.

Note: We note here that the deformed spin network states (in the spin network
decomposition of the deformed states) which arise in the constraint action
(2.15) on GR vertices are also of the type $S_{\alpha_{IJ},\epsilon}$ so that
(2.15) is of the type (2.29). We reiterate this observation in the first part
of section 4.3.

###  2.5  Constraint action on any spin net

Equations (2.15) and (2.29) imply that the constraint action on any spin net
$|s\rangle$ is:

$\displaystyle{\hat{H}}_{\epsilon}(N)S(A)$ $\displaystyle:=$
$\displaystyle\frac{3}{8\pi}\sum_{v\in
V_{GR}(s)}N(x(v))\big{(}\;\sum_{I_{v}=1}^{N_{v}}\frac{([\frac{1}{{}^{N_{v}-1}\\!P_{3}}\sum_{\vec{\hat{J}}_{v}}S_{\lambda_{v},I_{v},{\vec{\hat{J}}}_{v},\epsilon}]-{\frac{1}{4}}S_{\lambda_{v}})}{\epsilon}$
(2.34) $\displaystyle-$
$\displaystyle\sum_{I_{v}=1}^{N_{v}}\frac{(\sum_{J_{v}\neq
I_{v}}\sum_{K_{v}\neq
I_{v},J_{v}}{\tilde{S}}_{\lambda_{v},(1)I_{v},J_{v},K_{v},\epsilon})+(\sum_{J_{v}\neq
I_{v}}{\tilde{S}}_{\lambda_{v},(2)I_{v},J_{v},\epsilon})}{\epsilon}\;\big{)}\;\;\;\;\;\;\;\;\;\;$
$\displaystyle+$ $\displaystyle\sum_{v\in
V_{NGR}(s)}\frac{N(x(v))}{\epsilon}\big{(}\;(\sum_{I_{v}=1}^{N_{v}}A_{vI_{v}}S_{\lambda_{v},I_{v},\epsilon})+B_{v}S_{\lambda_{v}}+(\sum_{I_{v}=1}^{N_{v}}\sum_{J_{v}\neq
I_{v}}\sum_{\alpha_{vI_{v}J_{v}}}P_{\alpha_{vI_{v}J_{v}}}S_{\alpha_{vI_{v}J_{v}},\epsilon})\big{)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

Here $V_{GR}(s)$ is the set of volume non-degenerate GR vertices of $s$ and
$V_{NGR}(s)$ is the set of volume non-degenerate NGR vertices of $s$. The $v$
subscripts are used in obvious notation to denote dependence on the vertex
$v$.

It is useful to denote the contribution from the GR vertices in the first two
lines as ${\hat{H}}_{\epsilon}(N)S(A)|_{V_{GR}(s)}$ and that from the NGR
vertices as ${\hat{H}}_{\epsilon}(N)S(A)|_{V_{NGR}(s)}$ so that

${\hat{H}}_{\epsilon}(N)S(A)={\hat{H}}_{\epsilon}(N)S(A)|_{V_{GR}(s)}+{\hat{H}}_{\epsilon}(N)S(A)|_{V_{NGR}(s)}\;.$
(2.35)

For future purposes it is important to emphasise certain features of the
deformed states generated by the constraint action (2.34). These features
follow directly from the discussion and analysis in sections 2.1.2, 2.3 and
2.4 :
(i) In the first line of (2.34) the GR vertex $v$ in $S$ is replaced by one at
$v_{I,\epsilon}$ in $S_{\lambda_{v},I_{v},\vec{\hat{J}}_{v},\epsilon}$.
(ii) In the second line of (2.34), in the state
${\tilde{S}}_{\lambda_{v},(2)I_{v},J_{v},\epsilon}$ can be decomposed into
spin networks. In each such spin network, there is a trivalent kink on the
$J_{v}$th edge and either a trivalent or a bivalent kink at
$v_{I_{v},\epsilon}$ in which case the $v$ remains GR and has either the same
valence as in $S$ or this valence reduces by 1. The state
${\tilde{S}}_{\lambda_{v},(1)I_{v},J_{v},K_{v},\epsilon}$ can also be
decomposed into spin networks. In each of these spin networks there are a pair
of kinks, a bivalent one along the $K_{v}$th edge emanating from $v$ and a
bivalent or trivalent one at $v_{I,\epsilon}$. The point $v$ is either a
bivalent kink or a GR vertex of valence 3 or more.
(iii) In the 3rd line of (2.34), the NGR vertex $v$ in $S$ is replaced by an
NGR vertex at $v_{I,\epsilon}$ in $S_{\lambda_{v},I_{v},\epsilon}$.
(iv) In the 3rd line of (2.34), the deformations $\alpha_{vI_{v}J_{v}}$ create
a pair of kinks one along the $J_{v}$th edge and another at $v_{I,\epsilon}$.
Depending on the properties of $\alpha_{vI_{v}J_{v}}$, either or both of these
kinks could be bivalent or trivalent. Consequently the valence of $v$ in
$S_{\alpha_{vI_{v}J_{v}},\epsilon}$ could reduce by 0 or 1 or 2 relative to
its valence in $S$. Depending on the original vertex structure at $v$ in $S$,
this could result in $v$ presenting itsef in
$S_{\alpha_{vI_{v}J_{v}},\epsilon}$ as a GR vertex of valence 3 or more, an
NGR vertex of valence 3 or more, a trivalent kink or a bivalent kink or an
interior point of a single semianalytic edge.

##  3  Off Shell Basis States $\Psi_{B,f,h}$

The off shell states are finite linear combinations of basis states. Each
basis state is labelled by a $C^{r-1}$ 3d Riemannian metric $h$ and a $C^{r}$
function $f$ on the semianalytic $C^{r}$ Cauchy slice $\Sigma$. Permissible
metric labels are elements of the set ${\cal H}_{h_{0}}$ of distinct
diffeomorphic images of the metric $h_{0}$ where $h_{0}$ is a metric with no
conformal isometries. An off shell basis state is an element of the algebraic
dual to the finite span of spin net states and can be thought of as a (in
general) non-normalizable formal sum over spin network bra states (see
Footnote 5) . The set of these bra state summands $B$ is referred to as the
Bra Set and the Bra Set serves as an additional label for basis off shell
states. Thus $\Psi_{B,f,h}$ denotes an off shell basis state labelled by a
$C^{r}$ function $f$, a metric $h$ with $h\in{\cal H}_{h_{0}}$ and a Bra Set
$B$. The state is then specified by the coefficients of each bra in the Bra
Set, each such coefficient being determined by $f,h$ and the bra summand being
multiplied by that coefficient. The coefficient of a bra $\langle s|\in B$ is
then simply the evaluation of the algebraic dual off shell basis state on the
ket counterpart of $\langle s|$ i.e. the coefficient of $\langle s|$ is
$\Psi_{B,f,h}(|s\rangle)$. While the $f$ dependence of the coefficient is just
the product of the evaluations of $f$ on every GR vertex of $\langle s|$, the
$h$ dependence is more complicated and depends on the structure of the set of
kinks in the graph underlying the spin net label $s$.

The layout of this section is as follows. Subsection 3.1 introduces
nomenclature and definitions which play a key role in the articulation of the
subsequent sections. In subsection 3.2 we specify the properties of
permissible Bra Set labels. In subsection 3.3 we isolate a particular $h$
dependent structural feature of the set of kinks on a graph which is of use in
specifying the $h$ dependence of the coefficent. In subsection 3.4 we specify
$\Psi_{B,f,h}$ through a specification of the coefficients
$\Psi_{B,f,h}(|s\rangle)$ for every $\langle s|\in B$. In what follows we
shall often refer to $\Psi_{B,f,h}(|s\rangle)$ as the amplitude of the off
shell state on $|s\rangle$.

###  3.1  Preliminaries

In order to specify the off shell state it is necessary to define the
following:

  1. 1.

Bivalent Kink: A Bivalent $C^{m}$ kink (with $0\leq m<r$) is an intersection
$k$ between two unoriented semianalytic edges 111111A semianalytic edge is
defined [25] to be a 1d semianalytic manifold with 2 point boundary.
$e_{i},i=1,2$ such that $k$ is a common endpoint for these edges and which has
the following property. Orient one of the edges to be incoming and the other
outgoing at $k$. Fix a semianalytic $C^{r}$ coordinate patch around $k$. Then
for the case $m>0$
(a) there exists a parameterization of each edge consistent with its
orientation in which the first $m$ derivatives at $k$ satisfy
$\frac{d^{m}}{dt_{1}}e_{1}^{\mu}=\frac{d^{m}}{dt_{2}^{m}}e_{2}^{\mu}$ and in
which at $k$ the $(m+1)$th derivatives are unequal i.e.
$\frac{d^{m+1}}{dt_{1}^{m+1}}e_{1}^{\mu}\neq\frac{d^{m+1}}{dt_{2}^{m+1}}e_{2}^{\mu}$,
and,
(b) there is no choice of parameterizations in which the first $m+1$
derivatives are equal at $k$. For $m=0$ a bivalent $C^{0}$ kink is one at
which the edge tangents at $k$ are linearly independent. It is straightforward
to see that these definitions are independent of the choice of coordinates and
the choice of which edge is taken to be incoming. Consequently the notion of a
$C^{m<r}$ bivalent kink is diffeomorphism invariant with respect to $C^{r}$
semianalytic diffeomorphisms.

  2. 2.

Trivalent Kink: A trivalent kink $k$ is the intersection of an unoriented
semianalytic edge $e$ located in its interior, with a distinct unoriented
semianalytic edge $e^{\prime}$ with an end point at $k$. Trivalent kinks can
be of type $C^{m}$, $0\leq m\leq r$. These kink types are defined as follows:

2.1 Choose an orientation for $e,e^{\prime}$. If the edge tangents of $e$ and
$e^{\prime}$ are linearly independent at $k$ the kink is defined to be a
$C^{0}$ kink. Clearly this notion is invariant under the action of $C^{r}$
semianalytic diffeomorphisms.

2.2 The point $k$ is defined to be a trivalent $C^{m}$ kink for $0<m<r$ if
there exists an orientation assignment for $e$ with $e^{\prime}$ assigned an
outgoing orientation such that the incoming part of $e$ at $k$ together with
$e^{\prime}$ (which has been chosen to be outgoing at $k$) form a $C^{m}$
bivalent kink. Clearly the notion of such a $C^{m}$ kink is invariant under
the action of $C^{r}$ semianalytic diffeomorphisms.

2.3 The point $k$ is defined to be a trivalent $C^{r}$ kink if
(i) $e^{\prime}$ is assigned an outgoing orientation, and
(ii) there exists an orientation assignment and parameterization $t$
consistent with this assignment for $e$ together a paramaterization
$t^{\prime}$ of $e^{\prime}$ consistent with its assigned outgoing orientation
at $k$ such that the first $r$ derivatives, $\frac{d^{m}}{dt^{\prime
m}}e^{\prime\mu},m=1,..,r$ along the outgoing part of $e^{\prime}$ agree with
their counterparts, $\frac{d^{m}}{dt^{m}}e^{\mu},m=1,..,r$ along the incoming
part (as well as, by virtue of $e$ being a single edge, the outgoing part) of
$e$ at $k$. A $C^{r}$ kink $k$ can also be seen as the intersection of 3 edges
$e_{1},e_{2},e_{3}$ in which two of the edges (say $e_{2},e_{3}$) merge into
the remaining edge $e_{1}$ in a $C^{r}$ manner (i.e. with $e_{2},e_{3}$
incoming and $e_{1}$ outgoing there exist parameterizations
$t_{1},t_{2},t_{3}$ of the three edges consistent with their orientations such
that $\frac{d^{m}}{dt_{i}^{m}}e_{i}^{\mu},i=1,2,3$ agree with each other for
each $m\in\\{1,..,r\\}$).
It can be checked that this definition is coordinate independent so that the
notion of a $C^{r}$ trivalent kink is invariant under the action of $C^{r}$
semianalytic diffeomorphisms.

  3. 3.

Vertex: An $N$ valent vertex $v$ for $N>3$ is the intersection of $N$ distinct
semianalytic edges each with a single endpoint at $v$ (thus a loop through $v$
would contribute 2 distinct edges). A 3 valent vertex is the intersection of 3
distinct semianalytic edges which is not a trivalent kink. Note that the
trivalent vertex can be NGR (i.e. have linearly dependent edge tangents)
without being a kink.

  4. 4.

Embeddable Abstract Spin Network: We develop the notion of an embeddable
abstract spin network as an abstract labelled structure with the labels
specifying the nature of realizations (i.e. embeddings) of this structure as
an embedded spin network.

Our starting point is the abstract spin network defined by Baez [19]. We
review that definition here, for details see section 2 of Reference [19]. An
abstract (oriented) graph is specified by its edges, vertices and connectivity
i.e. which edges meet at which vertices. More in detail an abstract graph
$\gamma$ comprises of a finite set $E$ of edges, a finite set $V$ of vertices,
and functions $s:E\rightarrow V$, $t:E\rightarrow V$ where the vertex $s(e)$
is called the source of the edge $e$, and the vertex $t(e)$ the target of the
$e$. An edge is outgoing at its source and incoming at its target. An abstract
spin network consists of an abstract graph $\gamma$, a labelling of each its
edges $e_{I}$ by spin quantum numbers $j_{I}$ (i.e. by representations of
$SU(2)$) and an assignation of invariant tensors called intertwiners ${\cal
I}_{v}$ , one for every vertex $v$, which map the product of incoming edge
representations to that of outgoing edge representations. Note that there is a
slight degeneracy in the set of labels as follows. Consider an abstract spin
network state specified by $\gamma,\\{j_{I},\\},\\{{\cal I}_{v}\\}$. Change
$\gamma$ to $\gamma^{\prime}$ by flipping some or all of the orientations of
the edges of $\gamma$. Retain the same spin labels for the edges in
$\gamma^{\prime}$ as for their counterparts in $\gamma$. Then from standard
$SU(2)$ representation theory their exists a unique (at most upto rescaling by
a constant $c_{v}$ such that $\prod_{v}c_{v}=1$) choice of vertex intertwiners
$\\{{\cal I}_{v}^{\prime}\\}$ such that the same spin network state is
specified by the labels $\gamma^{\prime},\\{j_{I},\\},\\{{\cal
I}^{\prime}_{v}\\}$. In what follows, we shall append the qualifier ‘abstract’
to any element of the abstract graph whenever there is a possibility of
confusion between abstract and embedded structures. In particular for abstract
vertices as defined above, we do not distinguish between vertices and kinks,
whereas we do distinguish between embedded images of such vertices as kinks
and vertices as defined above.

We now embellish the definition of an abstract spin network by augmenting the
intertwiner specification with additional finitely specified information
(called a ‘decoration’) at each of its vertices. To this end, given an
abstract graph we define its decorated counter part as follows.

First consider the case of bivalent vertices. We shall always work with a
representative of the abstract spin network with orientation such that at any
bivalent kink there is one incoming and one outgoing edge (it is
straightforward to see that this is always possible by suitable orientation
flips and intertwiner choices) with the intertwiner simply being the identity
matrix. We tailor our definitions below in such a way that the only abstract
vertices which are permitted to be bivalent are those which are embeddable as
$C^{m<r}$ kinks. Accordingly, for a bivalent vertex the decoration consists of
a single integer valued between $0$ and $r-1$, the idea being that permissible
embeddings would embed the two edges at this vertex (more, precisely, their
unoriented counterparts) in such a way that the decorated vertex is embedded
as a bivalent $C^{m<r}$ kink.

Next consider trivalent vertices. For each trivalent vertex, we specify an
intertwiner (note that for this 3 valent case the intertwiner choice is
limited to rescaling of the standard Clebsch-Gordon coefficient appropriate to
the edge colorings and orientations at such a vertex). The decoration of a
trivalent vertex consists of the following labels. Let the 3 unoriented edges
intersecting at the abstract vertex be $e_{I_{1}},e_{I_{2}},e_{I_{3}}$. The
first label is an integer $-2\leq m_{0}\leq r$. If $m_{0}=-2$, the vertex is
to be embedded as an NGR vertex, if $m_{0}=-1$ the vertex is to be embedded as
a GR vertex. If $m_{0}\neq-1,-2$, the abstract vertex is to be embedded as a
$C^{m_{0}}$ kink. To complete the specification of this embedding, we append
an additional label comprising of one of the edge labels $e_{I_{i}}$. If
$m_{0}<r$ the abstract vertex is to be embedded as a trivalent $C^{m_{0}<r}$
kink such that 2.1 (for $m_{0}=0$) or 2.2 (for $r>m_{0}>0$) above is satisfied
with $e_{I_{i}}=e^{\prime}$ and $\cup_{j\neq i}e_{I_{j}}=e$ embedded as a
single $C^{r}$ edge (see 2.1, 2.2 above). If $m_{0}=r$ the vertex is to be
embedded as a $C^{r}$ trivalent kink, with 2.3 satisfied with
$e_{I_{i}}=e_{1}$ and the two remaining edges merging in a $C^{r}$ manner with
this ${I_{i}}$th edge (see 2.3 above).

If the abstract vertex has valence greater than 3, we specify an intertwiner.
In contrast to the trivalent case, there is a finite (and greater than one)
dimensional vector space of such intertwiners. The decoration label consists
of a single integer $n$, $n\in\\{0,1\\}$. If $n=0$ the vertex is to be
embedded as a GR vertex, else as an NGR vertex.

A decorated abstract spin network consists of a decorated graph, a labelling
of its edges by spin quantum numbers (i.e. by representations of $SU(2)$) and
an assignation of intertwiners, one for every vertex of valence greater than
2, which maps the product of incoming edge representations to that of outgoing
edge representations (modulo the degeneracy in this labelling through
orientation and intertwiner change as discussed above).

We refer to the (usual) spin networks defined on graphs in $\Sigma$ as
embedded spin networks. Consider a representative of an embedded spin network
defined on the coarsest graph underlying it (in which case the only remaining
choice of representative is its edge orientation and the associated
intertwiners, this choice being the exact counterpart of the one discussed
above for abstract spin networks). Such a representative, through its label
set, naturally defines an associated decorated abstract spin network. A
decorated abstract spin network ${\bf S}$ is defined to be an embeddable
abstract spin network iff there exists an embedded spin network $S$ with
associated decorated abstract spin network ${\bf S}$. In such a case, $S$ will
be called an embedding of ${\bf S}$ and ${\bf S}$ will be said to embed into
$S$. Thus an embeddable abstract spin network is a decorated abstract spin
network which admits at least one embedding into $\Sigma$ as an embedded spin
network. Clearly, while an embeddable abstract spin network will have
infinitely many distinct embeddings, two distinct (i.e. non-isomorphic
121212An isomorphism ${\iota}$ between two embeddable abstract spin networks
${\bf S_{1}},{\bf S_{2}}$ is a bijection from the vertex set $V_{1}$ of ${\bf
S_{1}}$ to the vertex set $V_{2}$ of ${\bf S_{2}}$ in such a way that (a)
$v_{2}={\iota(v_{1}})$ has the same decorations as $v_{1}$, (b) for every
incoming (outgoing) colored edge at $v_{1}$ there is a corresponding
identically colored and oriented edge at $v_{2}$ and vice versa, and (c)
$v_{2}$ and $v_{1}$ have identical intertwiners. and inequivalent in the sense
of the orientation-intertwiner label degeneracy described above) embeddable
abstract spin networks cannot embed into the same embedded spin network.

  5. 5.

Asymmetric Colored Graph: Consider the unoriented colored graph
$\alpha_{c}({\bf S})$ of a decorated abstract spin network ${\bf S}$ obtained
by dropping the orientation, intertwiner and decoration information from ${\bf
S}$. Thus $\alpha_{c}({\bf S})$ is specified by (a) the vertex set $V(S)$, (b)
edge set $E$, (c) a map $b$ from $E$ to $V\times V$ such that $b(e\in
E)=\\{x,y\\}=\\{s(e),t(e)\\}$ with $x,y$ called the endpoints of $e$ and (d) a
labelling of elements $e$ of $E$ by spins $j(e)$. Let the number of edges with
end points $\\{x,y\\}$ be $\mu(x,y)$.

A colored graph automorphism of $\alpha_{c}$ is a bijection of $\alpha_{c}$
which is defined through a permutation $\sigma$ on $V({\bf S})$ such that for
each edge $e$ with end points $\\{x,y\\}$ there exists an edge $e_{\sigma}$
with end points $b(e_{\sigma})=\\{\sigma(x),\sigma(y)\\}$ with
$j(e_{\sigma})=j(e)$ and $\mu(\sigma(x),\sigma(y))=\mu(x,y)$. An asymmetric
colored graph is a colored graph which has no automorphisms apart from the
identity.

  6. 6.

Asymmetric Spin Network Basis: The Volume operator acts at vertices of spin
networks as a self adjoint operator on vertex intertwiners [15, 20]. For fixed
vertex valence and edge colorings and edge orientations, the intertwiner space
is finite dimensional and the self adjoint operator preserves each such finite
dimensional space and thereby restricts to a self adjoint operator on this
finite dimensional vector space of intertwiners. The eigen states of the
operator action in this finite dimensional space constitute an orthonormal
basis of intertwiners for non-degenerate eigen values and for degenerate eigen
values some orthonormal basis spanning the eigen space can be chosen and fixed
once and for all. By an orthonormal basis we mean the following. Consider an
$N$ valent vertex at which there are $M$ outgoing edges with spins
$j_{1},..,j_{M}$ and $N-M$ incoming edges with spins $j_{M+1},..j_{N}$ so that
an intertwiner $C$ at this vertex has the index structure
$C^{A_{1}..A_{M}}{}_{A_{M+1}...A_{N}}$ with $A_{I}$ being a spin $j_{I}$
representation space label. Then an orthornormal basis $\\{C_{(\mu)}\\}$ of
intertwiners for this configuration of oriented colored edges satisfies:

$\sum_{A_{I},I=1,..N}C^{A_{1}..A_{M}}_{(\mu)}{}_{A_{M+1}...A_{N}}(C^{A_{1}..A_{M}}_{(\nu)}{}_{A_{M+1}...A_{N}})^{*}=(\prod_{I=1}^{N}\sqrt{2j_{I}+1})\;\;\delta_{\mu,\nu}$
(3.1)

We call this basis of intertwiners as a volume intertwiner basis. It is
straightforward to check that the normalization condition (3.1) for $\mu=\nu$
ensures that the embeddings of abstract spin networks with intertwiners so
normalised are of unit norm with respect to the Ashtekar Lewandowski inner
product on the kinematic Hilbert space of LQG.

Our aim is to construct a well defined basis for embedded spin network states.
For future purposes we require the basis set to map into itself under the
action of diffeomorphisms and electric diffeomorphisms. Even just for
diffeomorphisms this does not seem possible for arbitrary states because of
the possible existence of graph symmetries [21] which can map a spin network
on a graph with such symmetries to a distinct spin network on the same graph.
However for sufficiently asymmetric spin networks it turns out that we can
construct the desired basis as follows.

Consider a colored unoriented graph $\gamma_{c}$ underlying some embeddable
abstract spin network. Restrict attention to $\gamma_{c}$ with the following
properties:
(a) $\gamma_{c}$ is an asymmetric colored graph.
(b) If $v$ is a bivalent vertex of $\gamma_{c}$ then $v$ is connected to
itself by a single edge i.e. the only bivalent vertices in $\gamma_{c}$ are
the single vertices in loops with each loop corresponding to a single
connected component of $\gamma_{c}$.
(c) $\gamma_{c}$ has no loops which start and end at vertices with valence
greater than 2 i.e. there is no edge with identical endpoints such that this
endpoint has valence greater than 2.

Note from (b) that $\gamma_{c}$ consists of the disjoint union of one or more
connected components which are not single loops with a set of zero or one or
more single loops 131313 Reference [25] defines a semianalytic edge as one
with distinct endpoints; this necessitates a slightly involved definition of
embedding of a loop. We refrain from commenting on this further as we will be
interested in loops with multiple kinks, for which there is no tension between
abstract and embedded notions. i.e. $\gamma_{c}=\gamma_{0c}\cup L$ where
$\gamma_{0c}$ has no bivalent vertices and $L$ is the disjoint union of a set
of loops or $L$ is empty. If $L$ is not empty, the asymmetry of $\gamma_{c}$
implies the edge in each distinct loop must be of a distinct color. In order
to accomodate the action of electric diffeomorphisms, we are interested in
defining a basis for the space of embeddings of all possible abstract
decorated spin networks living on the (by definition) undecorated graph
$\gamma_{0c}$ as well as on any graph $\gamma{\prime}_{0c}$ obtained from
$\gamma_{0c}$ by the introduction of bivalent vertices in the ‘interior’ of
its edges (we explain what we mean by this below). Since electric
diffeomorphisms act non-trivially only on non-degenerate vertices, we shall
treat the the ‘loop’ sector seperately. Accordingly we focus first on spin
network labels associated with $\gamma_{0c}$ and subsequently discuss the spin
net labels associated with the loop sector.

We choose an orientation of the edges of $\gamma_{0c}$ and fix this choice
once and for all. As discussed in 4, intertwiners at trivalent vertices are
then fixed by $SU(2)$ representation theory upto a constant rescaling. The
normalization (3.1) fixes this constant upto phase and we fix this phase once
and for all by demanding that the intertwiners be real (see for e.g [22]).
With this choice fixed, we omit explicit intertwiner specificaton for 3 valent
vertices, it being understood implicitly that we use the above fixed choice.
At every $N>3$ valent vertex $v$ in the vertex set of $\gamma_{0c}$ we fix an
orthonormal volume intertwiner basis $\\{C_{v(\mu_{v})}\\}$ with appropriate
index structure and choose the intertwiner at $v$ to be one of the volume
intertwiner basis intertwiners.

Next, introduce bivalent vertices in the ‘interior’ of each edge i.e. in an
edge $e$ with color $j(e)$ and end points $v,\bar{v}$ introduce bivalent
vertices $k_{1},k_{2},..,k_{n},n\geq 1$ such that $k_{j}$ is connected to
$k_{j-1},k_{j+1}$ by an edge with color $j(e)$ with
$k_{0}:=v,k_{n+1}:=\bar{v}$. Let the orientation of the edges be consistent
with that of $e$ and fix the intertwiners in accordance with 4 above. It is
straightforward to check that the ‘identity’ intertwiner so fixed satisfies
(3.1). Decorate the vertices of the resulting spin nets and for simplicity
restrict the decorations of the bivalent vertices to correspond to their
embedding as $C^{0}$ bivalent kinks. Note that the above introduction of
bivalent vertices changes the colored graph $\gamma_{0c}$. It is
straightforward to see that the new colored graph $\gamma^{\prime}_{0c}$ so
obtained is also asymmetric and hence satisfies (a), (c) above. 141414If not
there exists an automorphism $\sigma$ which sends the sequence of vertices
$v,k_{1},..k_{n},\bar{v}\neq v$ connected to each other by single edges to
another such sequence. By virtue of (b),(c) above this sequence must start at
$\sigma(v)\in\gamma_{0c}$ and end at
$\sigma(\bar{v})\neq\sigma(v)\in\gamma_{0c}$ with identical connectivity so
that this sequence must arise through the introduction of bivalent kinks on
some edge $\sigma(e)$. Restricting the automorphism to the vertex set of
$\gamma_{0c}$ (so that no bivalent kinks are involved) yields an automorphism
of $\gamma_{0c}$ which must be the identity so that
$\sigma(v)=v,\sigma(\bar{v})=\bar{v},\sigma(e)=e$, which in turn implies that
the sequence $\\{\sigma(k_{i})i=0,..,n\\}$ is associated with the edge $e$.
This implies that $\sigma(k_{1})=k_{1},\sigma(k_{n})=k_{n}$, which implies
$\sigma(k_{2})=k_{2},\sigma(k_{n-1})=k_{n-1}$ and so on, so that the
automorphism $\sigma$ is necessarily the identity, and the new graph is
necessarily asymmetric. As implied by Footnote 14 and discussed further below
every $N\geq 3$ valent vertex of $\gamma^{\prime}_{0c}$ can be identified
uniquely with a vertex in $\gamma_{0c}$ and as we shall see below, this
identification allows us to choose the same orthornormal volume intertwiner
basis at the $N>3$ valent vertices so identified. Varying over all possible
basis intertwiners consistent with this choice, and all possible decorations,
we obtain a set of abstract decorated spin network states living on the
colored graph $\gamma_{0c}$ and colored graphs obtained from it through the
introduction of bivalent $C^{0}$ kinks in the manner detailed above. Let the
subset of embeddable elements of this set be ${\bf E}_{\gamma_{0c}}$. 151515It
seems straightforward to argue that all members of the set are embeddable but
this is not necessary for our purposes. Since the embellishment of an abstract
spin network with decorations does not, by definition, change its underlying
colored graph, the graphs underlying the set ${\bf E}_{\gamma_{0c}}$ of
abstract decorated spin networks also satisfy (a),(c).

We briefly digress here to explain the role of property (c). Let $\gamma_{0c}$
be endowed with an orientation as above. This orientation naturally induces an
orientation on the graphs $\gamma^{\prime}_{0c}$ in the manner described
above. Since the only automorphism of $\gamma_{0c},\gamma^{\prime}_{0c}$ is
the identity and since, by virtue of (c), every edge has distinct vertices,
the identity automorphism is orientation preserving. In contrast if property
(c) did not hold, the identity automorphism of any loop could still be
consistent with orientation reversal of its embedding and this could change
the embedded spin network function while leaving its underlying embedded
colored graph untouched. This ends the digression and we return to the
consideration of the set ${\bf E}_{\gamma_{0c}}$ of decorated abstract spin
networks on $\gamma_{0c},\gamma^{\prime}_{0c}$ and turn to a discussion of
their embeddings.

Accordingly, let the set of embedded spin networks obtained as all possible
embeddings of elements of ${\bf E}_{\gamma_{0c}}$ be ${E}_{\gamma_{0c}}$.
Consider ${\bf s}\in{\bf E}_{\gamma_{0c}}$ which happens to have colored graph
$\gamma_{0c}$. Let ${\bf s}$ embed to $s\in{E}_{\gamma_{0c}}$ via the
embedding map $F:{\bf s}\rightarrow s$ and call the colored graph underlying
$s$ as $\gamma_{0c}(s)$. It follows that $s$ is the unique embedding of ${\bf
s}$ on $\gamma_{0c}(s)$ i.e. there exists no
${\bar{s}}\in{E}_{\gamma_{0c}},{\bar{s}}\neq s$ such that
$\gamma_{0c}({\bar{s}})=\gamma_{0c}(s)$. To see this, proceed as follows. Let
there exist ${\bar{s}}={\bar{F}}({\bf s})$ for some embedding ${\bar{F}}\neq
F$ with $\gamma_{0c}({\bar{s}})=\gamma_{0c}(s)$ and ${\bar{s}}\neq s$. The
absence of nontrivial automorphisms of $\gamma_{0c}$ implies that
${\bar{F}}^{-1}F\gamma_{0c}=\gamma_{0c}$ so that if $V$ is the vertex set and
$E$ the set of edges of $\gamma_{0c}$, we have that $F(v)={\bar{F}}(v)\forall
v\in V,F(e)={\bar{F}}(e)\forall e\in E$. By virtue of our digression above
$F(e),{\bar{F}}(e)$ have the same orientations. Since $F,{\bar{F}}$ yield the
same embedded oriented colored graph and map the intertwiner on every vertex
$v$ of ${\bf s}$ to $F(v)={\bar{F}}(v)$, it must be the case that
$s={\bar{s}}$. Similar comments apply to embeddable abstract spin nets with
colored graphs of type $\gamma^{\prime}_{0c}$ by virtue of their asymmetry.
Moreover, since any $\gamma^{\prime}_{0c}$ is obtained by the introduction of
bivalent vertices in the ‘interior’ of edges of $\gamma_{0c}$, there is a
natural 1-1 mapping of those vertices of $\gamma^{\prime}_{0c}$ which are of
valence greater than 2 to the vertex set of $\gamma_{0c}$ so that we may label
these vertices of $\gamma^{\prime}_{0c}$ with elements of the vertex set $V$
of $\gamma_{0c}$. As indicated in Footnote 14, with this labeling of such
vertices there is a unique association of edge connected sequences of bivalent
vertices between end points $v,\bar{v}$ in $\gamma^{\prime}_{0c}$ with an edge
with these end points in $\gamma_{0c}$. It follows that at any such vertex
$v$, one can uniquely identify incoming and outgoing edges on the colored
graph $\gamma^{\prime}_{0c}$ with the chosen fixed orientation above with
their counterparts in $\gamma_{0c}$ with the chosen fixed orientation. If the
embeddable abstract spin networks ${\bf s},{\bf s}^{\prime}$ with colored
graphs $\gamma_{0c},\gamma^{\prime}_{0c}$ both have intertwiner $C_{v}$ at
vertex $v$, we shall say that ${\bf s},{\bf s}^{\prime}$ have the same
intertwiner at $v$ and it is in this sense that our choice of orthornormal
volume intertwiner basis is the same for
$v\in\gamma_{0c},\gamma^{\prime}_{0c}$. It follows that these identifications
are also induced onto their embedded images. In particular if these spin
networks are embedded through embeddings $F,F^{\prime}$, we say that $F({\bf
s}),F({\bf s}^{\prime})$ have the same intertwiners at $F(v),F^{\prime}(v)$.

Let us now consider the loop component $L$ of $\gamma_{c}$ (when non-empty)
and color each loop $l_{j}$ with spin $j$ so that $l_{1}$ is colored with
$j=1$, $l_{2}$ with $j=2$ and so on. Let us denote $L$ by $L_{m}$ when $L$ has
$m$ such loops with $L_{0}$ being the empty set. We restrict attention to the
case in which the bivalent vertex $v$ is to be decorated with a $C^{0}$ kink
decoration. Next we introduce a pair of bivalent vertices on each $l_{j}$,
decorate one of them as a $C^{1}$ kink and the second as a $C^{2}$ kink. We
call $l_{j}$ embellished with these two kinks as $l^{(3)}_{j}$. We call the
collection $\\{l^{(3)}_{j}\\}$ so obtained as $L^{(3)}_{m}$. Clearly
$L^{(3)}_{m}$ has no decoration preserving colored graph automorphisms except
the identity. We fix an orientation of each $l^{(3)}_{j}$ once and for all
corresponding to a traversal from the $C^{0}$ kink $v$ to the $C^{1}$ kink and
thence to the $C^{2}$ kink and finally back to $v$. Next consider any
embedding $F$ of $l^{(3)}_{j}$ into $F(l^{(3)}_{j})$. Note that
$F(l^{(3)}_{j})$ is a continuous oriented loop with orientation acquired from
the fixed orientation chosen for $l^{(3)}_{j}$. With $F(l^{(3)}_{j})$
associate the spin network function $\text{Tr}h_{F(l^{(3)}_{j})}(F(v),F(v))$
where $h_{F(l{(3)}_{j})}$ is the holonomy, in the spin $j$ representation,
around the loop $F(l^{(3)}_{j})$ from its starting point $F(v)$ to its end
point which is also $F(v)$ with Tr denoting the trace in the spin $j$
representation. Since $l^{(3)}_{j}$ has no nontrivial decoration preserving
colored graph automorphisms, similar to $\gamma_{0c}$ it admits a unique
embedding to any of its embedded images. Finally, with the embedding
$F(L^{(3)}_{m})$ associate the spin network function
$\prod_{j=1}^{m}\text{Tr}h_{F(l^{(3)}_{j})}(F(v),F(v))$.

Let ${\bf E}_{L^{(3)}_{m}},m>0$ be the set containing $L^{(3)}_{m}$ and let
the set of embedded spin networks obtained as above through holonomy traces
from this single element of ${\bf E}_{L_{m}}$ be $E_{L_{m}}$. Define ${\bf
E}_{\gamma_{m,c}}:={\bf E}_{\gamma_{0c}}\otimes{\bf E}_{L^{(3)}_{m}}$ so that
${\bf E}_{\gamma_{c}}$ is the set of ordered pairs $({\bf s}\in{\bf
E}_{\gamma_{0c}},{\bf l}\in{\bf E}_{L^{(3)}_{m}})$ with ${\bf l}=L^{(3)}_{m}$.
Such an ordered pair naturally defines an abstract spin network on the union
of some decoration $d$ of $\gamma_{0c}$ (resp. $\gamma^{\prime}_{0c}$) with
$L^{(3)}_{m}$. Let us call the $\gamma_{0c},\gamma^{\prime}_{0c}$ so decorated
as $\gamma_{0c,d},\gamma^{\prime}_{0c,d}$ 161616Note that the bivalent
vertices on $\gamma^{\prime}_{0c}$ are restricted to be decorated so as to
embed to $C^{0}$ kinks. Hence the only freedom left is the decoration of the
vertices of $\gamma^{\prime}_{0c}$ which are also vertices of $\gamma_{0c}$
and the notation indicates that these vertices are decorated exactly as in
$\gamma_{0c}$. and define the decorated colored graphs
$\gamma_{m,c,d}=\gamma_{0c,d}\cup L^{(3)}_{m}$,
$\gamma^{\prime}_{m,c,d}=\gamma^{\prime}_{0c,d}\cup L^{(3)}_{m}$, Given such
an element of ${\bf E}_{\gamma_{m,c}}$ and an embedding $F$ of
$\gamma_{m,c,d}$ (resp. $\gamma^{\prime}_{m,c,d}$), we define the embedding of
this element $({\bf s},{\bf l})$ by $F$ as the embedded spin network function
obtained as a product of the embedded spin network function $s=F({\bf s})$
with the embedded spin network function defined by $F({\bf l})$. The set of
all distinct embedded spin network functions obtained for all possible
embeddings of elements of ${\bf E}_{\gamma_{m,c}}$ is denoted as
$E_{\gamma_{m,c}}$ where we have set ${\bf E}_{\gamma_{0,c}}={\bf
E}_{\gamma_{0c}}$. Finally, note that embedded spin networks whose underlying
embedded colored graphs do not coincide in $\Sigma$, are orthogonal.

From the discussion above it is straightforward to see that the following
statements hold:

(i) The elements of ${E}_{\gamma_{m,c}}$ provide a well defined basis of
states on spin networks living on all possible embeddings of (a) decorations
$\gamma_{0c,d}$ of the undecorated colored graph $\gamma_{0c}$ together with
$L^{(3)}_{m}$ as well as of (b) any decorated graph $\gamma{\prime}_{0c,d}$
obtained from $\gamma_{0c,d}$ by the introduction of bivalent ‘$C^{0}$ kink’
vertices in the ‘interior’ of its edges together with $L^{(3)}_{m}$.

(ii) Since the image of any embedding by a diffeomorphism is also an
embedding, the set ${E}_{\gamma_{m,c}}$ is stable under the action of
diffeomorphisms. Moreover vertices, oriented edges and interwtiners are
naturally identifed between any of these images. In particular we may
identify, in obvious notation $F(v),F(e),F({\cal I}_{v})\in F({\bf s})$ with
$F_{\phi}(v),F_{\phi}(e),F_{\phi}({\cal I}_{v})\in F_{\phi}({\bf s})$ for
$v,e\in\gamma_{0c}$ where the embedding $F_{\phi}$ is defined as $\phi\circ F$
for some diffeomorphism $\phi$.

(iii) Let $S\in{E}_{\gamma_{m,c}}$ have a non-degenerate vertex $v$ (so that
$v$ lies in the non-loop component of ${S}$). Let
$S_{\lambda_{v},I_{v},\epsilon}$ be an electric diffeomorphism deformation of
$S_{\lambda_{v}}$ at $v$. Then $S_{\lambda_{v}}$ and
$S_{\lambda_{v},I_{v},\epsilon}$ have the same intertwiner at
$v,v_{I_{v},\epsilon}$. Moreoever, since the intertwiners are volume
eigenstates, it follows that modulo the same factor,
$S_{\lambda_{v}},S_{\lambda_{v},I_{v},\epsilon}$ are elements of
${E}_{\gamma_{m,c}}$. Accordingly, the abstract correspondent of
$S_{\lambda_{v},I_{v},\epsilon}$ (upto an overall factor) is in ${\bf
E}_{\gamma_{m,c}}$ so that any electric diffeomorphism child of $S$, in the
sense of 7 below is in ${E}_{\gamma_{m,c}}$.

Finally, any possible parent (in the sense of 8 below) of
$S_{\lambda_{v},I_{v},\epsilon}$ must be an element of ${E}_{\gamma_{m,c}}$.
To see this, consider a state ${\bar{S}}$ which when acted upon by the
constraint yields an electric diffeomorphism child $C$ with abstract colored
graph isomorphic to that underlying $S_{\lambda_{v},I_{v},\epsilon}$. Since
these graphs are asymmetric this isomorphism is unique and we may identify
these graphs using this isomorphism. Note that the embedded colored graph
underlying ${\bar{S}}$ is obtained by the removal of a triplet of kinks from
the non-loop part of the colored graph underlying $C$ so that the abstract
colored graph obtained from removing the corresponding set of bivalent
vertices from the abstract colored graph which embeds to the colored graph
underlying $C,S_{\lambda_{v},I_{v},\epsilon}$ must embed to the embedded
colored graph underlying ${\bar{S}}$ which implies that
${\bar{S}}\in{E}_{\gamma_{m,c}}$.

Thus, given a state in $S$ all its children by electric diffeomorphism in the
sense of 7, and all its possible parents by electric diffeomorphism in the
sense of 8, are in ${E}_{\gamma_{m,c}}$. We refer to this property as the
stability of the set ${E}_{\gamma_{m,c}}$ under the action of electric
diffeomorphisms.

By considering all distinct (i.e. non-isomorphic) choices of $\gamma_{0c}$,
all choices of decorations $d$ thereof and all values of $m\geq 0$, we obtain
a basis on the space of all embeddings of embeddable abstract spin networks
living on all corresponding choices of
$\gamma_{m,c,d},\gamma^{\prime}_{m,c,d}$ This basis is stable under the action
of diffeormorphisms and electric diffeomorphisms. We call this basis as the
Asymmetric Spin Network Basis. An embeddable abstract spin network which
embeds into any element of this basis will be called an abstract asymmetric
basis spin network

  7. 7.

Electric diffeomorphism Child of an Asymmetric Spin Network Basis element: Let
$S$ be an asymmetric spin network basis element and let its abstract
asymmetric counterpart be ${\bf S}$. The action of the electric diffeomorphism
part of the constraint (2.34) at sufficiently small $\epsilon$ on $S$ yields,
besides $S_{\lambda}$ states $S_{\lambda_{v},I_{v},{\hat{J}_{v}},\epsilon}$
which are related to $S$ through the action of electric diffeomorphisms at
nondegenerate vertices of $S$. Since each of these children have a decorated
abstract colored graph structure which is obtained by the embellishment of
some of the edges of $S$ with bivalent $C^{0}$ kinks, the discussion in 6
above implies that each of these children (up to overall factors) are also
asymmetric spin network basis elements. Clearly for all sufficiently small
$\epsilon$ the states $S_{\lambda_{v},I_{v},{\vec{J}}_{v},\epsilon}$ for fixed
$I_{v},{\vec{J}}_{v}$ (upto overall factors) are embeddings of the same
abstract asymmetric spin network ${\bf C}(I_{v},{\vec{J}}_{v})$. We shall
refer to any embedding of any ${\bf C}(I_{v},{\vec{J}}_{v})$ as a child of $S$
(note that ${\bf C}(I_{v},{\vec{J}}_{v})$ for distinct $(I_{v},{\vec{J}}_{v})$
need not be distinct).

  8. 8.

Possible parent by electric diffeomorphism of an Asymmetric Spin Network Basis
element: Let $S$ be an asymmetric spin network basis element with abstract
asymmetric counterpart ${\bf S}$. $P$ is called a possible parent of $S$ if
all embeddings of ${\bf S}$ are children of $P$ in the sense of 7 above. Since
the abstract colored graph structure of $P$ is obtained by removal of a triple
of bivalent $C^{0}$ kinks from the non loop component of ${\bf S}$, the
discussion in 6 above shows that $P$ is also an asymmetric spin network basis
element.

  9. 9.

Useful Notation: For the purposes of the next section and section 7 it is
useful to define the following notation which builds on that introduced in 6
above. Accordingly, within the set ${\bf E}_{\gamma_{m,c}}$ defined above
consider those elements which correspond to a fix decoration $d$ of the
vertices of $\gamma_{0c}$. Call this subset of ${\bf E}_{\gamma_{m,c}}$ as
${\bf E}_{\gamma_{m,c,d}}$. Let the set of embeddings of elements of this set
be ${E}_{\gamma_{m,c,d}}$.

Define the union of the sets ${\bf E}_{\gamma_{m,c}}$ for all finite $m$ to be
${{\mathbfcal E}}_{\gamma_{0c}}$ and the set of embeddings of elements of this
union to be ${{\cal E}}_{\gamma_{0c}}$ so that ${{\mathbfcal
E}}_{\gamma_{0c}}=\cup_{m=0}^{\infty}{\bf E}_{\gamma_{m,c}}$, ${\cal
E}_{\gamma_{0c}}=\cup_{m=0}^{\infty}{E}_{\gamma_{m,c}}$, where we recall from
6 above that we have set $\gamma_{(0)c}\equiv\gamma_{0c}$.

Similarly define ${{\mathbfcal E}}_{\gamma_{0c,d}},{{\cal E}}_{\gamma_{0c,d}}$
as the fixed decoration (with decoration $d$ of $\gamma_{0c}$) elements of
${{\mathbfcal E}}_{\gamma_{0c}},{{\cal E}}_{\gamma_{0c}}$ so that
${{\mathbfcal E}}_{\gamma_{0c,d}}=\cup_{m=0}^{\infty}{\bf
E}_{\gamma_{m,c,d}}$, ${\cal
E}_{\gamma_{0c,d}}=\cup_{m=0}^{\infty}{E}_{\gamma_{m,c,d}}$.

###  3.2  The Bra Set $B$

Recall that the Bra Set label $B$ of an off shell state contains all the bra
summands which when summed over with appropriate coefficients yield a formal
sum which represents the off shell state $\Psi_{B,f,h}$. The dual action of
the constraint ${\hat{H}}_{\epsilon}(N)$ on this off shell state is defined
through its ‘amplitude’, $\Psi_{B,f,h}({\hat{H}}_{\epsilon}(N)S)$, for any
spin net state $S$. The $\epsilon\rightarrow 0$ limit of these amplitudes
defines the continuum limit constraint action. Clearly only those spin net
states generated by the constraint action ${\hat{H}}_{\epsilon}(N)S$ which
have non-trivial inner product with one or more elements of $B$ contribute to
the amplitude for $S$. The spin net states which do contribute in this way
will be said to have overlap in $B$. In order to have adequate control on the
calculations of these amplitudes it suffices, that for any $S$ with
nondegenerate vertices, either all the spin nets generated by the electric
diffeomorphism part of the constraint action on nondegenerate vertices of $S$,
together with $S$ have overlap in $B$ or that none of them do (in which case
the amplitudes will be seen to vanish in the continuum limit). 171717In a
previous version of this work, similar to [11] we attempted to impose a
similar condition by also including propagation type children; however it
turns out the weaker condition here suffices and can be imposed consistently.
Further, it is reasonable to anticipate that this feature of overlap or lack
thereof should persist in an $\epsilon$-independent way for sufficiently small
$\epsilon$ so that the continuum limit amplitudes exist. The properties (i)-
(iii) below of permissible Bra Sets are motivated by the discussion above. In
their articulation, we use the definitions of the previous section. In
particular we distinguish between vertices and kinks.

We require that the Bra Set label $B$ of an off shell basis state have the
following properties:

(i) Each element $S$ of $B$ is an asymmetric spin network basis element whose
vertices are all GR and which has at least one such vertex. If $S\in B$ then
we can replace the volume intertwiner basis element at each vertex of valence
greater than 3 by a distinct such basis element to get a state $S^{\prime}$.
(Recall that the volume intertwiner basis is associated with abstract
asymmetric basis spin networks which embed to their (embedded) asymmetric spin
network basis counterparts). We require that any $S^{\prime}$ obtained by such
a replacement is also in $B$.

(ii) If $S\in B$ and ${\bf S}$ is the embeddable abstract spin net which
embeds as $S$, then $B$ contains all embeddings of ${\bf S}$.

(iii) Let $C$ be an electric diffeomorphism type child of $S\in B$ as defined
in 7 (Clearly, all vertices of $C$ are GR and $C$ has at least one such
vertex). Then $C\in B$. If $P$ is a possible parent of $S\in B$ with parentage
via electric diffeomorphism as defined in 8 then $P\in B$ (Clearly, all
vertices of $P$ are GR and $P$ has at least one such vertex).

The asymmetry requirement in (i) in conjunction with the role of abstract
structures in (ii) together with closure of $B$ with respect to appropriate
parent-child relations in (iii) ensures that $B$ is stable under the action of
diffeomorphisms and electric diffeomorphisms as discussed in section 3.1. The
role of property (i) is also to ensure sufficent overlap in $B$ given the fact
that there is a finite dimensional space of intertwiners. Property (ii)
ensures $\epsilon$-independent overlap as well as $h$-independent overlap (see
the discussion in section 4.4); the former is necessary for the continuum
limit constraint action to exist and the latter is an important ingredient of
our proof of diffeomorphism covariance of the constraint action in section 6.
Property (iii) ensures that $B$ is closed under suitable ‘parent-child’
relations which in turn ensures the kind of overlap required in the discussion
in the first paragraph of this section. The reason we insist here on ‘at least
one GR vertex’ in (i) is for the ensuing simplicity of expression for the off-
shell amplitudes in section 3.4.

Our demonstration of anomaly freedom of hamiltonian constraint commutators in
section 4 and 5 uses only properties (i)-(iii) without any further
specification of Bra Sets. The considerations of section 6 which demonstrate
diffeomorphism covariance require a certain property of linear independence of
off -shell states to hold. This property is proved in section 7. The proof of
linear independence for off shell states with a single Bra Set label but
different metric and ‘vertex smooth’ function labels requires Bra Sets to
satisfy an additional property (iv)(a). The proof for the general case of
different Bra Set labels requires a further property (iv)(b). It seems
plausible that the properties (iv)(a,b) are implied by property (iii) above.
However a putative proof is beyond the scope of this work. Instead in section
7 we explicitly construct a rich family of Bra Sets which satisfy properties
(i)-(iv).

More in detail: It is straightforward to see that properties (i)-(iii) hold if
we choose as our Bra Set, the set ${E}_{\gamma_{m,c,d}}$ (see 9 of section 3.1
above) with $\gamma_{m,c,d}$ chosen so that all vertices of states in this set
are GR and there is at least one such vertex. Properties (i),(ii) hold by
virtue of this choice and due to the realization of elements of
${E}_{\gamma_{m,c,d}}$ as embeddings of abstract structures and (iii) holds by
virtue of the procedure of adding bivalent kinks to the interior of edges of
$\gamma_{0c}$ as outlined in 6 of the previous section. Clearly identical
argumentation applies to the set ${\cal E}_{\gamma_{0c,d}}$ with
${\gamma_{0c,d}}$ chosen so that all vertices of states in this set are GR and
there is at least one such vertex. Thus ${\cal E}_{\gamma_{0c,d}}$ for any
such ${\gamma_{0c,d}}$ also satisfies (i)-(iii) As we shall see in section
7.7, a family of Bra Sets for which we are able to conclusively demonstrate
satisfaction of properties (i)-(iii) and (iv), is provided by the family of
sets ${\cal E}_{\gamma_{0c,d}}$ (one for each distinct isomorphism class of
$\gamma_{0,c}$ with appropriate $d$).

NOTE: In the next and subsequent sections, we shall refer to a certain
‘distance’ function between two points also by $d$. However from the context
it will be amply clear whether the symbol $d$ refers to this distance or to
the decorations of graph defined above and there will be no scope for
confusion.

###  3.3  Kink Sets

Let $S$ be a spin net. The set of bivalent and trivalent kinks in the graph
underlying $S$ is called the Kink Set ${\bf K}$ of $S$. We are interested in
segregating elements of ${\bf K}$ into subsets of ‘closest kinks’ with respect
to the metric $h$.

To this end, we first define the function $d(a_{1},a_{2})$ between any two
distinct points $a_{1},a_{2}\in\Sigma$ as follows: If there exists a unique
geodesic (with respect to the metric $h$) with length $l,l<1$ which joins
$a_{1}$ to $a_{2}$ then we define $d=l$ else we set $d=1$. We shall refer to
$d$ as a ‘distance’ function. Next, compute the distances $d(k,k^{\prime})$
between all pairs of elements $k\neq k^{\prime}$ of ${\bf K}$ and follow the
following algorithm to segregate elements of ${\bf K}$ into subsets.

Look for a set of 3 kinks such that their interkink distances are smaller than
those between the remaining kinks and also smaller than the distance between
any remaining kink and any member of this triplet. If such a set exists call
it $K^{(1)}_{3}$. If no such set exists, look for a pair of closest kinks so
that the distance between them is smaller than the distances between the
remaining kinks and also smaller than the distance between any remaining kink
and any member of this pair. If such a set exists, call it $K^{(1)}_{2}$ and
stop the procedure so that there are only two subsets, namely $K^{(1)}_{2}$
and its complement $K^{\prime}$. If no such set exists there is only one
relevant subset ${\bf K}$.

If $K^{(1)}_{3}$ exists, remove $K^{(1)}_{3}$ from ${\bf K}$ and repeat the
above procedure for the resulting set of kinks. Clearly, the procedure then
comes to a halt after some number $n\geq 1$ of iterations with the final
‘closest’ kink set being either a 2 kink set or a 3 kink set. Denote by
$K^{(i)}_{3}$ the 3- kink set obtained at the $i$th iteration and by
$K^{(n)}_{2}$ the 2-kink set (if it exists) obtained at the (necessarily)
$n$th iteration. It follows that we have a segregation of ${\bf K}$ into
either of

Case 0: ${\bf K}$ (here the procedure ends necessarily at the first step $n=1$
with no ‘closest’ 2 kink or 3 kink sets) or

Case 1: $K^{(i)}_{3},i=1,\ldots n-1$, $K^{(n)}_{2}$, ${\bf
K}-(\cup_{i=1}^{n-1}K^{(i)}_{3}\cup K^{(n)}_{2})$ or

Case 2: $K^{(i)}_{3},i=1,\ldots n$, ${\bf K}-(\cup_{i=1}^{n}K^{(i)}_{3})$.

Above, for Case 1, it is understood that if $n=1$ the procedure yields a
segregation of ${\bf K}$ into a single 2 point kink set and its complement.

For future purposes it is useful to define the minimum and maximum interkink
distances in any 3 kink set $K^{(i)}_{3}$ as

$d^{(i)}_{\max}=\max_{k,k^{\prime}\in
K^{(i)}_{3}}d(k,k^{\prime}),\;\;\;\;d^{(i)}_{\min}=\min_{k\neq k^{\prime}\in
K^{(i)}_{3}}d(k,k^{\prime})$ (3.2)

Here, if all 3 interkink distances are equal we define the maximum and minimum
distance both to be equal to this interkink distance.

###  3.4  The coefficients

Consider an off shell state $\Psi_{B,f,h}$. The coefficient of the bra
$\langle s|$ (also referred to as the off shell amplitude for $s$) in the
formal sum over bras associated with $\Psi_{B,f,h}$ is
$\Psi_{B,f,h}(|s\rangle)$. Consider the set of spin network states $B_{\perp}$
such that every element of $B_{\perp}$ is orthogonal to every element of $B$.
Then since $\Psi_{B,f,h}$ is a sum over states in $B$, the amplitude of this
off shell state on any element of $B_{\perp}$ vanishes. and, since
$\Psi_{B,f,h}$ is a linear map on the finite span of spin nets, we may specify
$\Psi_{B,f,h}$ through its amplitudes on elements of $B$ (each of which are
asymmetric spin network basis states, see 6, section 3.1 and (i), section
3.2). In section 3.4.1 we specify the amplitude for any element of $B$. In
section 3.4.2 we derive the amplitude for any spin network state based on its
specification in section 3.4.1. In section 3.4.3 we discuss a useful property
of the amplitude of section 3.4.2.

####  3.4.1  Amplitude for states in $B$ and perpendicular to $B$

The amplitude $\Psi_{B,f,h}(|s\rangle)$ for any asymmetric spin network (ASN)
basis state $|s\rangle$ is:

$\displaystyle\Psi_{B,f,h}(|s\rangle)$ $\displaystyle=$ $\displaystyle
0\;\;\;\;\;{\rm if}\;\;\langle s|\in B_{\perp}$ (3.3) $\displaystyle=$
$\displaystyle(\prod_{v\in V(s)}f(v))g_{s,h}\;\;\;\;\;{\rm if}\;\;\langle
s|\in B$ (3.4)

where we recall again that any state in $B$ is necessarily an asymmetric spin
network basis state. Here $V(s)$ is the set of vertices of $s\in B$, each such
vertex being GR by virtue of property (i), section 3.2. $f$ is a $C^{r}$
function on $\Sigma$ so that the off shell amplitude depends on the product of
evaluations of $f$ at each vertex of $s$. $g_{s,h}$ is a (positive) function
$g({\bf K},h)$ of the Kink Set ${\bf K}$ of $s$ segregated with respect to the
metric $h$ which depends on the network of interkink distances defined in
terms of $h$ between elements of ${\bf K}$ endowed with the $h$ dependent
segregation structure defined in section 3.2.

Let the Kink Set segregation procedure of section 3.2 terminate in $n$ steps,
$n\geq 1$ yielding the (exhaustive) segregation structures defined in Cases 0
to 2 of that section. For Case 0, ${\bf K}$ does not admit any non-trivial
segregation and we set

$g({\bf K},h)=1$ (3.5)

For Case 1, we define $g$ to be:

$\displaystyle g({\bf K},h)$ $\displaystyle:=$
$\displaystyle\left(\prod_{i=1}^{n-1}\left(\frac{d^{(i)}_{\min}}{d^{(i)}_{\max}}\right)^{2}\right)\left(d(k^{(n)}_{1},k^{(n)}_{2})\right)^{2}\;\;{\rm
if}\;n>1$ (3.6) $\displaystyle:=$
$\displaystyle\big{(}d(k^{(1)}_{1},k^{(1)}_{2})\big{)}^{2}\;\;{\rm if}\;n=1$
(3.7)

Finally, for Case 2, we define $g({\bf K},h)$ to be:

$g({\bf
K},h):=\prod_{i=1}^{n}\left(\frac{d^{(i)}_{\min}}{d^{(i)}_{\max}}\right)^{2}$
(3.8)

Thus, we set $g_{s,h}=g({\bf K},h)$ with $g({\bf K},h)$ defined through
equations (3.5)-(3.8).

####  3.4.2  Amplitude for arbitrary spin network state

Consider any expansion of a spin network state $|s\rangle$ in terms of other
spin network states. Clearly, the only spin networks which contribute non-
trivially to this expansion are those for which all spin net labels agree with
those of $s$ except, possibly, the vertex intertwiner labels. Hence all these
spin networks live on the same embedded colored graph as $s$ (by embedded
colored graph we mean the graph as a subset of $\Sigma$ with specified edge
colorings). This immediately implies that any spin network state $|s\rangle$
may either be expanded in terms of a finite number of asymmetric spin network
basis states (see 6, section 3.1) or is orthogonal to every such basis state.
In the former case we have that:

$|s\rangle=\sum_{i=1}^{p}c^{s}_{i}|s_{i}\rangle.$ (3.9)

where $|s_{i}\rangle$ are asymmetric spin network basis states. Recall from
section 3.2 that a spin net $s$ will be said to have overlap in $B$ if $s$ has
non-vanishing inner product with at least one element of $B$, where for
convenience we have ommitted to represent the spin nets involved in bra-ket
notation. Let $s$ has overlap in $B$. Hence it shares all its labels except
perhaps intertwiners with some element of $B$. The discussion above in
cojunction with property (i), section 3.2, implies that (a) this element of
$B$ is an element of the asymmetric spin network basis and (b) all states
living on the same embedded colored graph as $s$ with all possible choices of
volume basis intertwiner labels at vertices are in $B$. It follows that if $s$
has overlap in $B$, it admits an expansion (3.9) in which every $s_{i}$ is in
$B$. Conversely, if $s$ does not have overlap in $B$, then either $s$ admits
an expansion (3.9) in which no $s_{i}$ with $c^{s}_{i}\neq 0$ is in $B$ or $s$
is orthogonal to every asymmetric spin net basis element It then follows from
(3.4) that:

$\displaystyle\Psi_{B,f,h}(|s\rangle)$ $\displaystyle=$ $\displaystyle
0\;\;\;\;\;{\rm if}\;\;|s\rangle\;\;{\rm has}\;{\rm no}\;{\rm overlap}\;{\rm
in}\;\;B$ (3.10) $\displaystyle=$ $\displaystyle(\prod_{v\in
V(s)}f(v))g_{s,h}\gamma_{s}\;\;\;\;\;{\rm if}\;\;|s\rangle\;\;{\rm has}\;{\rm
overlap}\;{\rm in}\;\;B$ (3.11)

where, referring to the expansion (3.9),

$\gamma_{s}:=\sum_{i=1}^{p}{c^{s}_{i}}\;\;\;\;s_{i}\in B,\;i=1,..,p$ (3.13)

####  3.4.3  Useful properties of $\gamma_{s}$

Let $v$ be a vertex of the spin network $s$. Consider a neighborhood $R_{v}$
of $v$ which is small enough that it contains no vertex of $s$ other than $v$.
Denote the Rovelli-Smolin (RS) Volume operator associated with this region by
${\hat{V}}_{R_{v}}$. From 7., section 3.1, independent of the choice of
$R_{v}$, ${\hat{V}}_{R_{v}}$ acts as a self adjoint operator on the
intertwiner at $v$. Denote this ‘RS vertex volume operator’ by
${\hat{V}}_{v}$. Clearly, every asymmetric spin network basis element with a
vertex at $v$ is an eigen state of ${\hat{V}_{v}}$.

Next, consider the action of the inverse determinant metric operator
$\lim_{\tau\rightarrow 0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v)$ on
$|s\rangle$. As discussed in Reference [2] and reiterated in section 2.1.2,
this operator also acts as a linear operator on the space of intertwiners at
$v$ in $s$. Let us call this linear operator as ${\hat{\lambda}}_{v}$. Recall
that in this work the operator $\lim_{\tau\rightarrow
0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v)$ is defined via a Tychonoff
regulation. It then follows from the construction of this operator in [2] that
upto a factor ${\hat{\lambda}}_{v}$ is defined via Footnote 4 with
${\hat{O}}:=({\hat{V}}_{v})^{\frac{2}{3}}$ so that eigenstates of
${\hat{V}}_{v}$ with eigen values $\nu_{v}\neq 0$ are eigen states of
${\hat{\lambda}}_{v}$ with upto a numerical factor) eigen values
$(\nu_{v})^{-\frac{2}{3}}$ and so that the zero eigen value eigen states of
${\hat{V}}_{v}$ are annihilated by ${\hat{\lambda}}_{v}$. We shall refer to
${\hat{\lambda}}_{v}$ as the RS inverse volume operator.

Note that for a spin net $s$ with vertex $v$, just as the action of
${\hat{V}}_{R_{v}}$ on $s$ yields an intertwiner change through the action of
${\hat{V}}_{v}$, the discussion above indicates that the action of
$\lim_{\tau\rightarrow 0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v)$ on $s$
also yields an intertwiner change through the action of ${\hat{\lambda}}_{v}$.

Next, let $|s\rangle$ be a spin network state with basis expansion (3.9). Let
$s$ have a GR vertex at $v$. Consider an electric diffeomorphism type
deformation, without first acting with ${\hat{\lambda}}_{v}$, of $|s\rangle$
along its $I_{v}$th edge at $v$ which yields the state
$|s_{I,\epsilon}\rangle$. The vertex $v$ and its vertex structure is displaced
to $v_{I,\epsilon}$ as detailed in sections 2.1.2, 2.3.1 \- 2.3.3. Recall
that:
(i) $s_{I,\epsilon}$ differs from $s$ only in an $\epsilon$ size neighborhood
of $v$.
(ii) In this neighborhood of $v$, the vertex $v$ in $s$ is replaced by the
vertex $v_{I,\epsilon}$ in $s_{I,\epsilon}$. The displaced vertex
$v_{I,\epsilon}$ in $s_{I,\epsilon}$ has the same intertwiner as that of $v$
in $s$ and the edges emanating from $v_{I,\epsilon}$ in $s_{I,\epsilon}$ are
the deformed counterparts of those from $v$ in $s$ with the same edge
colorings.
(iii) Bivalent kinks are created in this $\epsilon$-neighborhood of $v$.
(iv) While the edge tangent structure at $v_{I,\epsilon}$ in $s_{I,\epsilon}$
is, in general, not diffeomorphic to that at $v$ in $s$, the RS Volume
operator is insensitive to this difference so that as operators on
intertwiners, the RS vertex volume operator at $v_{I,\epsilon}$ in
$|s_{I,\epsilon}\rangle$ and that at $v$ in $|s\rangle$ are identical. Similar
conclusions hold for the RS vertex inverse volume operator as well for vertex
operators composed of products of powers of the RS vertex volume and inverse
volume operators.

It follows from (i)-(iii) above together with the detailed construction of the
ASN basis (see (iii) of 6, section 3.1) that $|s_{I,\epsilon}\rangle$ has the
basis expansion:

$|s_{I,\epsilon}\rangle=\sum_{i=1}^{p}c^{s}_{i}|(s_{i})_{I,\epsilon}\rangle$
(3.14)

where $|(s_{i})_{I,\epsilon}\rangle$ is the electric diffeomorphism type
deformation (i.e. without first acting with ${\hat{\lambda}}$) of
$|s_{i}\rangle$ and its coefficient $c^{s}_{i}$ in the expansion (3.14) is
identical to the coefficient of $|s_{i}\rangle$ in the expansion (3.9). An
immediate consequence of this is that:

Next let $v$ be a non-degenerate vertex of $s$. From the discussion above it
follows that the action of $\lim_{\tau\rightarrow
0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v)$ on $|s\rangle$ results in a
change of intertwiner at $v$ through the action of ${\hat{\lambda}}_{v}$. As
in the rest of this paper, we denote the resulting state by
$|s_{\lambda_{v}}\rangle$. In this notation,
$|(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle$ is obtained from the
action of $\lim_{\tau\rightarrow
0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}(v_{I,\epsilon})$ on
$|s_{I,\epsilon}\rangle$ with the intertwiner change in
$|(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle$ effected through the
action of ${\hat{\lambda}}_{v_{I,\epsilon}}$ on the intertwiner at
$v_{I,\epsilon}$ in $|s_{I,\epsilon}\rangle$. From (i)-(iv) above the new
intertwiner in $|s_{\lambda_{v}}\rangle$ at $v$ and in
$|(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle$ at $v_{I,\epsilon}$ are
identical. It follows that the state obtained through an electric
diffeomorphism type deformation at $v$ of $|s_{\lambda_{v}}\rangle$ is the
same as $|(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle$. This
relationship is expressed in our notation as:

$|s_{\lambda_{v},I,\epsilon}\rangle=|(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle.$
(3.15)

Similar argumentation may be applied to each $s_{i}$ in the expansion (3.9).
Since $s_{i}$ is an eigen vector of ${\hat{\lambda}}_{v}$ with some eigen
value $\lambda_{i,v}$, it then follows that $s_{I,\epsilon}$ is an eigen
vector of ${\hat{\lambda}}_{v_{I,\epsilon}}$ with same eigen value.
181818Here, as above, we have suppressed the ‘$v$’ subscripts on the edge
index $I$, as well as the additional specification of edge triples ${\vec{J}}$
to avoid notational clutter. Accordingly we have:

$\displaystyle{\hat{\lambda}}_{v}|s_{i}\rangle$ $\displaystyle=$
$\displaystyle\lambda_{i,v}|s_{i}\rangle$ (3.16)
$\displaystyle|((s_{i})_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}\rangle$
$\displaystyle:=$
$\displaystyle{\hat{\lambda}}_{v_{I,\epsilon}}|(s_{i})_{I,\epsilon}\rangle=\lambda_{i,v}|(s_{i})_{I,\epsilon}\rangle$
(3.17)

From (3.15) and (3.17) we have that:

$|(s_{i})_{\lambda_{v},I,\epsilon}\rangle=\lambda_{i,v}|(s_{i})_{I,\epsilon}\rangle$
(3.18)

From (3.9) and (3.16) we have that:

$|s_{\lambda_{v}}\rangle=\sum_{i}c_{i}^{s}\lambda_{i,v}|s_{i}\rangle$ (3.19)

Replacing $s$ by $s_{\lambda_{v}}$ in (3.14) yields:

$\displaystyle|s_{\lambda_{v},I,\epsilon}\rangle$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}^{s}\lambda_{i,v}|(s_{i})_{I,\epsilon}\rangle$
(3.20) $\displaystyle=$
$\displaystyle\sum_{i}c_{i}^{s}|(s_{i})_{\lambda_{v},I,\epsilon}\rangle$
(3.21)

where we have used (3.18) in the last line above. Note that for $v$ non-
degenerate in $s_{i}$, $(s_{i})_{\lambda_{v},I,\epsilon}$ is, upto a factor of
$\lambda_{i,v}$ an electric diffeomorphism child of $s_{i}$. From (iii),
section 3.2, it follows that if $s_{i}\in B$ and $v$ is non-degenerate in
$s_{i}$, then $(s_{i})_{I,\epsilon}$ is in $B$. Hence
$s_{\lambda_{v},I,\epsilon}$ has overlap in $B$ if $s$ has overlap in $B$ and
$v$ is non-degenerate in $s$. From (3.20), (3.19), (3.15) it follows that for
$s$ with overlap in $B$:

$\gamma_{(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}}}}=\gamma_{s_{\lambda_{v},I,\epsilon}}=\gamma_{s_{\lambda_{v}}}=\sum_{i}c_{i}^{s}\lambda_{i,v}\;.$
(3.22)

Repeating these arguments for an application of the inverse determinant metric
operator twice at $v$ in $|s\rangle$ (instead of once as above) and denoting
the result by $|s_{\lambda_{v},\lambda_{v}}\rangle$, we have in obvious
notation that:

$\gamma_{(s_{I,\epsilon})_{\lambda_{v_{I,\epsilon}},\lambda_{v_{I,\epsilon}}}}=\gamma_{s_{\lambda_{v},\lambda_{v}}}\;.$
(3.23)

Finally, consider the action of the operator $\lim_{\tau\rightarrow
0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}({\bar{v}})$, with ${\bar{v}}\neq
v$, on $|s_{\lambda_{v}}\rangle$ with ${\bar{v}},v$ nondegenerate. Since this
operator changes the intertwiner exclusively at ${\bar{v}}$ it commutes with
the action of an electric diffeomorphism at $v$ which is supported away from
${\bar{v}}$. It follows that:

$|(s_{\lambda_{v},I,\epsilon})_{\lambda_{\bar{v}}}\rangle=|((s_{\lambda_{v}})_{\lambda_{\bar{v}}})_{I_{v},\epsilon}\rangle,\;\;\;{\bar{v}}\neq
v$ (3.24)

where the notation signifies that the state on the left hand side state is
obtained by an electric diffeomorphism type deformation of
$|s_{\lambda_{v}}\rangle$ along its $I_{v}$th edge followed by the action of
$\lim_{\tau\rightarrow 0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}({\bar{v}})$
and that the state on the right hand side is obtained by the action of
$\lim_{\tau\rightarrow 0}\tau^{-2}{\hat{q}}_{\tau}^{-\frac{1}{3}}({\bar{v}})$
on $|s_{\lambda_{v}}\rangle$ to yield $|(s_{\lambda_{v}})_{\bar{v}}\rangle$
followed by an electric diffeomorphism type deformation along the $I_{v}$th
edge at $v$ of $|(s_{\lambda_{v}})_{\bar{v}}\rangle$. If $s$ has overlap in
$B$ then similar arguments as above imply that the states in (3.24) have
overlap in $B$. Next, an application of ${\hat{\lambda}}_{\bar{v}}$ on (3.19),
(3.20) yields:

$\displaystyle{\hat{\lambda}}_{\bar{v}}|s_{\lambda_{v}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i}c_{i}^{s}\lambda_{i,\bar{v}}\lambda_{i,v}|s_{i}\rangle$
(3.25)
$\displaystyle{\hat{\lambda}}_{\bar{v}}|s_{\lambda_{v},I,\epsilon}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i}c_{i}^{s}\lambda_{i,\bar{v}}\lambda_{i,v}|(s_{i})_{I,\epsilon}\rangle$
(3.26)

It then follows from (3.24), (3.25), (3.26) that:

$\gamma_{(s_{\lambda_{v},I,\epsilon})_{\lambda_{\bar{v}}}}=\gamma_{((s_{\lambda_{v}})_{\lambda_{\bar{v}}})_{I_{v},\epsilon}}=\gamma_{(s_{\lambda_{v}})_{\lambda_{\bar{v}}}}=\sum_{i}c_{i}^{s}\lambda_{i,\bar{v}}\lambda_{i,v},\;\;\;{\bar{v}}\neq
v.$ (3.27)

Finally, given $s$ with expansion (3.9) and its image $s_{\phi}$ by a
diffeomorphism $\phi$, we have that:

$|s_{\phi}\rangle={\hat{U}}(\phi)|s\rangle=\sum_{i}c^{s}_{i}|(s_{i})_{\phi}\rangle$
(3.28)

where we have defined $|(s_{i})_{\phi}\rangle={\hat{U}}(\phi)|s_{i}\rangle$.
From (ii), 6 of section 3.1 we have that $\\{(s_{i})_{\phi}\\}$ are also