#  A discretization of the Holst’s action for general relativity

Carlos E. Beltrán 111e-mail<EMAIL_ADDRESS>?? Centro
de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089,
Morelia, Michoacán, México.  José A. Zapata 222e-mail<EMAIL_ADDRESS>Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P.
58089, Morelia, Michoacán, México.

###### Abstract

We present a simplicial model for gravity written in terms of a discretized
Lorentz connection and a discretized tetrad field. Its continuum limit is
general relativity described in terms of Holst’s action. With the intention of
using it to construct spin foam modes for quantum gravity, we write two other
equivalent models written in terms of a discretized and constrained $B$ field.
The differences between our model and existing models are most likely
inessential in the sense that a quantization would lead to equivalent quantum
theories in the Wilsonian continuum limit. Nevertheless, the differences may
be advantageous in certain situations. For example, in our model there is a
pseudometric associated with each of the 4-simplices; by using it, a discrete
precursor of the causal structure could be studied.

###### Contents

  1. 1 Motivation
  2. 2 The model
    1. 2.1 Discretization
    2. 2.2 The action
    3. 2.3 The continuum limit of the action
    4. 2.4 Field equations and boundary terms
  3. 3 Formulation in terms of a constrained $B$ field
    1. 3.1 Geometricity à la EPRL
    2. 3.2 Alternative form of the geometricity conditions
    3. 3.3 Differences with other classical simplicial models related to loop quantum gravity
  4. 4 Summary and outlook

##  1 Motivation

The spin foam approach to quantum gravity [8, 9] builds a quantum theory
starting with a version of classical general relativity. Up to now, the theory
sits on a discrete structure replacing spacetime and introducing a cutoff. The
hope is that a continuum limit, in the sense of Wilsonian renormalization,
exists; if it does, the resulting theory would be compatible by design with
kinematical canonical loop quantization in the continuum [11].

It is often considered that the Holst action [6, 3] for general relativity is
the starting point for the construction of spin foam models for quantum
gravity [8]. In order to construct it, one adds an additional term to the
tetradic Palatini action

$S_{H}(e,\omega)=\dfrac{1}{2}\int_{M}\epsilon_{IJKL}e^{I}\wedge e^{J}\wedge
F^{KL}+\dfrac{1}{\gamma}\int_{M}e_{I}\wedge e_{J}\wedge F^{IJ},$ (1)

where $\gamma\in\mathbb{R}\cup\left\\{\infty\right\\}$ is called the Barbero-
Immirzi parameter, $\omega$ is a $SO(3,1)$ connection 1-form, $F$ is its
curvature, $e^{I}$ is a tetrad field (a 1-form with values on the Minkowski
space $\mathcal{M}$) and $\epsilon_{IJKL}$ is the standard skew-symmetric
Levi-Civita tensor in $\mathcal{M}$. The first term in (1) is the standard
Palatini action. The second is an additional term that does not affect the
field equations in vacuum [3].

A more detailed account is that the EPRL spin foam model for quantum gravity
(proposed by Engle, Pereira, Rovelli and Livine) [3] is constructed as a
quantization of a classical theory on a discretized spacetime, and the action
for that classical theory resembles the Holst’s action for general relativity,
but it is not a direct discretization of it: The main variable of the models,
besides a discretized connection, is a Lie algebra valued discretized 2-form,
called a $B$ field, instead of the tetrad $e$ used in the Holst action. The
argument is that in the continuum a tetrad field $e$ may be traded for an
appropriately constrained $B$ field. What is actually done is that the $B$
field together with the mentioned constraints and the connection that live in
the continuum are transformed into classical fields and constraints living on
a discretized spacetime. They are later quantized in order to obtain a spin
foam model.

The route back to a classical field theory does not precisely follow the path
suggested by retracing the path followed in the quantization procedure. This
was never a requirement; we only mention it as a curious fact. The classical
limit is studied first, and, at a second stage, the continuum/macroscopic
limit is considered: (i) The asymptotic limit of the vertex amplitude of the
model (in the sense of large quantum numbers) is governed by the action of
Regge calculus. (ii) The continuum/macroscopic limit of Regge’s action is
known to be the Einstein-Hilbert action. There are two subtleties at this
point: First, contact is made with an alternative version of Regge calculus
called “area-angle” Regge calculus [21]. Second, at the level of computer
simulations, it has been observed that the continuum limit needs a
simultaneous scaling of the Planck length and the Immirzi parameter [12].

In this article, we present a discretization of general relativity in terms of
a discretized connection and a discretized tetrad field $e$. We prove that the
macroscopic/continuum limit of the action of our model is Holst’s action for
general relativity.

We also present another model, written in terms of a discretized and
constrained $B$ field and a discretized connection, which in the nondegenerate
sector is equivalent to the first model. Two versions of the geometricity
conditions on the discretized $B$ field are considered. In the first, the main
condition is entirely analogous to the so-called “linear simplicity
constraint,” a key ingredient of the EPRL model (and of its classical prequel)
[3]. Our proof of equivalence, however, needs supplementary constraints. We
have reasons to expect that quantization of our model could be done emulating
the EPRL strategy but in the context of our curved 4-simplices. The second
version of the geometricity conditions is designed to be used in the
quantization strategy called effective spin foam models, championed by
Dittrich and collaborators [12]. We think that our model could play the role
now played by area-angle Regge calculus but with possible advantages.
Certainly, our model provides a different point of view in several of the
issues of current interest in the construction and study of models of this
type.

Only very recently have the first spin foam simulations involving more than
one simplex been performed [12, 23, 24]. Since the geometry of the current
spin foam models is based on flat 4-simplices, the curvature degrees of
freedom have barely been tested in spin foam models for quantum gravity [14].
In contrast, if a quantum version of our model is constructed, the study of a
single 4-simplex (also called spacetime atom) would necessarily include
curvature degrees of freedom.

In section 2 we present the discrete model, show that in the continuum limit
the discrete action converges to the Holst action, and deduce its field
equations. In Section 3 we write our model as a discrete constrained BF theory
in two different ways and discuss the differences between our model and other
simplicial models used in loop quantum gravity. Finally, in Section 4 we give
a summary of the main results and present avenues for future research.

##  2 The model

###  2.1 Discretization

The classical discrete model presented here uses a cellular decomposition of
the spacetime manifold $M$ introduced in Reisenberger in [4]. Start by
considering an oriented simplicial complex, denoted by $\triangle$, and its
corresponding dual cellular decomposition ${}^{*}\triangle$ as defined in [7].
Following the notation given in [4], the Greek letters $\nu,\tau,\sigma$ will
denote a 4-simplex, a 3-simplex and a 2-simplex, respectively. Furthermore,
0-simplices of a given 4-simplex will be numbered from $1$ to $5$ and
generically be denoted by the letters $P,Q,R,\dots$. The notation $\mu<\rho$
will mean that $\mu$ is a subcell or subsimplex of $\rho$. The notation
$C_{\rho}$ will denote the barycenter of $\rho$.

The discretization considered here is designed to contain elements of the
intersection between $\triangle$ and ${}^{*}\triangle$. We subdivide each
4-simplex $\nu<\triangle$ into subcells called corner cells. Given a 4-simplex
$\nu<\triangle$ and a vertex $P$ in $\nu$, we define the corner cell $c_{P}$
in $\nu$ associated with $P$ as the intersection between $\nu$ and the
4-dimensional cell in ${}^{*}\triangle$ dual to $P$. In this way, $c_{P}$ is a
hypercube with vertex set composed by the points $P$, $C_{\nu}$ and the
barycenters of all the subsimplices of $\nu$ that have $P$ as one of its
vertices. Therefore, each 4-simplex $\nu<\triangle$ is subdivided into five
corner cells, one for each vertex $P$. See Figure 1.

Now, let us consider a 2-simplex $\sigma$ and a 4-simplex $\nu<\triangle$
containing it $\sigma<\nu$. We define the wedge $s(\sigma\nu)$ associated with
$\sigma$ in $\nu$ as the intersection between the 4-simplex $\nu$ and the
2-cell in ${}^{*}\triangle$ dual to $\sigma$. The wedge $s(\sigma\nu)$ is a
plane having as vertices the barycenters of $\nu$, $\sigma$ and the
barycenters of the two 3-simplices $\tau_{1},\tau_{2}<\nu$ that share $\sigma$
(as seen in Figure 1). In four dimensions, there are 6 wedges in the boundary
of each corner cell $c_{P}$, and 10 wedges inside each 4-simplex (without
considering orientation). Inside a corner cell, a given wedge $s$ has exactly
1 dual wedge; it is the wedge that intersects $s$ only at the baricenter of
the 4-simplex. Inside a 4-simplex, for each wedge there are 3 other wedges
that intersect it at a single point.

Of special importance for our model are the lines $l(\nu\tau)$ resulting from
the intersection between $\nu$ and the 1-cell dual to the 3-simplex
$\tau<\nu$. We consider them as oriented lines starting at $C_{\nu}$ and
ending at the barycenter of $\tau$. In addition, we will denote as
$r(\tau\sigma)$ the oriented lines starting at the barycenter of $\tau<\nu$
and ending at the barycenter of the 2-simplex $\sigma<\tau$. The boundary of
an oriented wedge can be written as $\partial s=l_{1}+r_{1}-r_{2}-l_{2}$.
Oriented wedges are in one to one correspondence with ordered pairs
$(l_{1},l_{2})$.

Figure 1: A corner cell and wedges in three dimensions. Panel (a) shows
$c_{P}$, the corner cell associated with vertex $P$. Panel (b) shows $s$, the
wedge dual to $\sigma$; lines $l_{1},r_{1}$ are part of $\partial s$.

Now we describe the fields of our discrete model. Consider that there are
copies of the Minkowski space $\mathcal{M}$, the Lie group $SO(3,1)$ and the
Lie algebra $so(3,1)$ attached to the barycenter of every 4-simplex
$\nu\in\triangle$. An analogous structure is also associated to the barycenter
of each 3-simplex and each 2-simplex in $\nu$. In our conventions, capital
letters from the middle of the alphabet $I,J,\ldots$ are Minkowski space
indices, and lowercase letters from the middle of the alphabet $i,j,\ldots$
are Lie algebra indices.

A discrete tetrad field $e$ is the association of a Minkowski vector
$e_{l}^{I}$ to each oriented line $l(\nu\tau)$. For each corner cell in $\nu$,
there are four vectors that when the field is not degenerate form a basis of
the copy of Minkowski’s space attached to $C_{\nu}$. Instead of a discrete
connection 1-form $\omega$, we use a collection of parallel transport maps. We
associate an element $h_{l}\in SO(3,1)$ with each oriented line $l(\nu\tau)$,
representing the parallel transport along that curve. Analogously, we
associate an element $k_{r}\in SO(3,1)$ with each oriented line
$r(\tau\sigma)$. These are the basic fields of our model.

Notice that the tetrad field induces a (pseudo) metric in $C\nu$ associated to
each corner cell of a 4-simplex

$g_{l_{1}l_{2}}(c)=e_{l_{1}}^{I}e_{l_{2}}^{J}\eta_{IJ}\ \mbox{ for any }\
l_{1},l_{2}<c.$ (2)

In addition, the metrics of neighboring corner cells $c,c^{\prime}$ are
clearly compatible in the sense that they if two vectors contained in the
3-dimensional subspace corresponding to $c\cap c^{\prime}$ are expanded in the
basis associated to $c$ or in the basis associated $c^{\prime}$, the
calculation of their inner product in the corresponding metrics $g(c)$ or
$g(c^{\prime})$ would agree.

Unlike what happens in other discrete models, 4-simplices here carry
curvature. The discrete object measuring curvature in a 4-simplex $\nu$ is the
parallel transport along the boundaries of its wedges $g_{\partial
s}:=h_{l_{2}}^{-1}k_{r_{2}}^{-1}k_{r_{1}}h_{l_{1}}$.

###  2.2 The action

The action for the model can be written as a sum over 4-simplices

$S_{\triangle}(e,h,k)=\sum_{\nu<\triangle}\
\sum_{l,l^{\prime},s<\nu}\operatorname{sgn}(l,l^{\prime},s)\
e_{l}^{K}e_{l^{\prime}}^{L}\left(T_{i}\right)^{JI}P_{IJKL}\ \theta_{s}^{i},$
(3)

where a few conventions need to be clarified: In the sum over wedges each
orientation of a wedge $s$ is considered independently. The sign
$\operatorname{sgn}(l,l^{\prime},s)$ is determined trading $s$ for its
corresponding ordered pair of oriented lines
$(l^{\prime\prime},l^{\prime\prime\prime})(s)$; it is positive or negative
according to the orientation of the ordered four tuple
$(l,l^{\prime},l^{\prime\prime},l^{\prime\prime\prime})$.
$\left(T_{i}\right)_{~{}~{}J}^{I}$ is a basis for the Lie algebra, and
$\theta_{s}^{i}\left(T_{i}\right)_{~{}~{}J}^{I}$ is a discretization of the
curvature $F_{~{}~{}J}^{I}$, where
$\theta_{s}^{i}:=\text{tr}\left[T^{i}g_{\partial\bar{s}}\right]$. Finally,
$P_{IJKL}:=\epsilon_{IJKL}-\dfrac{1}{2\gamma}\epsilon_{IJMN}\epsilon_{~{}~{}~{}~{}KL}^{MN}$.

######  Definition 1 (e-field model).

The action $S_{\Delta}(e,h,k)$ defined in (3) determines the $e$-model.

There is a more explicit formula for $S_{\nu}$, the contribution to (3) from a
single 4-simplex. The lines $l<\nu$ are labeled by numbers from 1 to 5, and
written as $l_{P}$. Wedges are denoted by $s_{PQ}$. We simplify notation for
the fields writing $e_{l_{P}}=e_{P}$ and $\theta_{s_{RS}}=\theta_{RS}$. Thus,
we can write

$S_{\nu}=\sum_{T}\epsilon^{PQRST}e_{P}^{K}e_{Q}^{L}\left(T_{i}\right)^{JI}P_{IJKL}\theta_{RS}^{i}.$
(4)

Notice that the sum over $T$ corresponds to a sum over corner cells in $\nu$.

###  2.3 The continuum limit of the action

In this section, we show that the discrete action (3) converges in the
continuum limit to Holst’s action (1). This subsection follows a minor
adaptation of the procedure used by Reisenberger to study the continuum limit
of his model [4].

Let us rewrite (1) as

$S_{H}\left(e,\omega\right)=\int_{M}\Sigma_{i}(e)\wedge F^{i}(\omega),$ (5)

where
$\Sigma_{i}:=P_{IJKL}e_{\alpha}^{K}e_{\beta}^{L}\left(T_{i}\right)^{JI}dx^{\alpha}\wedge
dx^{\beta}$, and
$F^{i}:=\left(T^{i}\right)_{IJ}F_{~{}~{}~{}\mu\nu}^{IJ}dx^{\mu}\wedge
dx^{\nu}$.

######  Definition 2.

The map
$\Omega_{\triangle}:\left(\omega,e\right)\rightarrow\left(h_{l},k_{r},e_{l}\right)$
from continuum fields to simplicial fields defined on the discretization of
$M$ is given by the following expressions:

$\displaystyle h_{l}$ $\displaystyle:=Pexp\left(\int_{l}\omega\right)$ (6)
$\displaystyle k_{r}$ $\displaystyle:=Pexp\left(\int_{r}\omega\right)$ (7)
$\displaystyle e_{l}^{I}$
$\displaystyle:=\int_{l}U_{~{}~{}J}^{I}(x,C_{\nu})e^{J}(x).$ (8)

where $Pexp$ is the path-ordered exponential and
$U_{~{}~{}J}^{I}(x,C_{\nu}):=Pexp\left(\int_{C_{\nu}}^{x}\omega\right)$ is the
parallel transport of Minkowski vectors along a straight line, from $x$ to
$C_{\nu}$.

In what follows, we will work with compact spacetimes or compact regions of
spacetime. Then simplicial decompositions are finite. To prove convergence, we
use a sequence of triangulations which become finer everywhere at the same
rate. This idea is formalized by the following definition.

######  Definition 3.

Let $\left\\{\triangle_{n}\right\\}_{n=0}^{\infty}$ be a sequence of
simplicial decompositions of a compact manifold $M$, and $g_{0}$ a fixed
positive-definite metric constant on each $\nu<\triangle_{0}$. We say that
$\left\\{\triangle_{n}\right\\}_{n=0}^{\infty}$ is a uniformly refining
sequence if

  1. a)

$\triangle_{0}$ is a finite simplicial decomposition of $M$.

  2. b)

$\triangle_{n+1}$ is a finite refinement of $\triangle_{n}$, $\forall n$.

  3. c)

$\lim_{n\rightarrow\infty}r_{n}=0$, with $r_{n}$ the maximum of the
$g_{0}-$radii $r_{\nu}$ of the 4-simplices $\nu<\triangle_{n}$.

  4. d)

$r_{\nu}^{4}$ divided by the $g_{0}$ 4-volume of $\nu$ is uniformly bounded as
$n\rightarrow\infty$, for all 4-simplices $\nu<\triangle_{n}$.

We also need two preliminary lemmas. The first is an elementary calculus
result.

######  Lemma 1.

Let $x,y,x_{0},y_{0}\in\mathbb{R}$ such that

$\displaystyle\left|x-x_{0}\right|$
$\displaystyle<\text{min}\left(1,\dfrac{\epsilon}{2\left(\left|y_{0}\right|+1\right)}\right)$
$\displaystyle\left|y-y_{0}\right|$
$\displaystyle<\dfrac{\epsilon}{2\left(\left|x_{0}\right|+1\right)},$

for some $\epsilon>0$. Then $\left|xy-x_{0}y_{0}\right|<\epsilon$.

Given a k-subcell c of a 4-simplex $\nu$, let us define the characteristic
tensor of $c$ as

$t_{c}^{\alpha_{1},\dots,\alpha_{k}}:=\int_{c}dx^{\alpha_{1}}\wedge\dots\wedge
dx^{\alpha_{k}},$ (9)

where $x^{\alpha}$ are linear coordinates on $\nu$. Expression (9) is useful
for expressing the result of the following lemma.

######  Lemma 2.

Let $h$ be a continuous k-form on a compact manifold $M$,
$\left\\{\triangle_{n}\right\\}_{n=0}^{\infty}$ be a uniformly refining
sequence of simplicial decompositions of $M$, and $g_{0}$ be a positive
definite metric that is constant on each 4-simplex $\mu<\triangle_{0}$. Then
for all $\epsilon>0$ there exists $N>0$ such that for any $n>N$ and any
k-subcell $c<\nu<\triangle_{n}$ we have

$\left|\int_{c}h-h\left(C_{\nu}\right)_{\alpha_{1},\dots,\alpha_{k}}t_{c}^{\alpha_{1},\dots,\alpha_{k}}\right|<\epsilon
r_{\nu}^{k},$ (10)

where $r_{\nu}$ is the $g_{0}$ radius of $\nu$ and
$h(C_{\nu})_{\alpha_{1},\dots\alpha_{k}}$ are the components, in linear
coordinates, of $h$ evaluated in $C_{\nu}$.

The proof of Lemma 2 can be found in [4].

Now we are prepared to prove the desired convergence result.

######  Theorem 1.

Let $M$ be a four dimensional compact oriented manifold and
$\left\\{\triangle_{n}\right\\}_{n=0}^{\infty}$ a sequence of uniformly
refining simplicial decompositions of $M$. Additionally, let $(\omega,e)$ be
fields of the Holst action (with $e$ continuous and $\omega$ continuously
differentiable) and $S_{n}(e,\omega)$ the evaluation of the discrete action
(3) on the simplicial fields
$(h_{l},k_{r},e_{l})_{n}=\Omega_{\triangle_{n}}(e,\omega)$ defined on
$\triangle_{n}$ using Definition 2. Then

$\lim_{n\rightarrow\infty}S_{\triangle_{n}}(\omega,e)=S_{H}(\omega,e).$ (11)

###### Proof.

Let $g_{0}$ be a positive-definite metric that is constant on each
$\nu<\triangle_{0}$ with respect to its linear coordinates and $\epsilon>0$.
From Lemmas 1 and 2 it follows that there exists $N>0$ such that for any
$n>N$, $\nu<\triangle_{n}$ and $s,l_{1},l_{2}<\nu$

$\displaystyle\left|\theta_{s}^{i}-F_{\alpha\beta}^{i}(C_{\nu})t_{s}^{\alpha\beta}\right|$
$\displaystyle<\epsilon r_{\nu}^{2}$ (12)
$\displaystyle\left|e_{l_{1}}^{K}e_{l_{2}}^{L}\left(T_{i}\right)^{JI}P_{IJKL}-\Sigma_{i,\alpha\beta}(C_{\nu})t_{l_{1}}^{\alpha}t_{l_{2}}^{\beta}\right|$
$\displaystyle<\epsilon r_{\nu}^{2}.$ (13)

In this way, for any $n>N$

$S_{\triangle_{n}}=4\sum_{\nu<\triangle_{n}}\left[\Sigma_{i,\alpha\beta}(C_{\nu})F_{\gamma\delta}^{i}(C_{\nu})\sum_{l_{1},l_{1},s<\nu}t_{l_{1}}^{\alpha}t_{l_{2}}^{\beta}t_{s}^{\gamma\delta}\operatorname{sgn}(l_{1},l_{2},s)+\Delta
I_{n_{\nu}}\right],$ (14)

with a bounded error term $\Delta I_{n_{\nu}}$:

$\left|\Delta I_{n_{\nu}}\right|<480\epsilon
r_{\nu}^{4}\left[\epsilon+\left\|\Sigma\right\|_{C_{\nu}}+\left\|F\right\|_{C_{\nu}}\right],$
(15)

where the norm $\left\|T\right\|$ of a tensor
$T_{\alpha_{1},\dots,\alpha_{n}}^{i_{1},\dots,i_{m}}$ with both Lie algebra
and spacetime indices is

$\left\|T\right\|=\left[T_{\alpha_{1},\dots,\alpha_{n}}^{i_{1},\dots,i_{m}}T_{\beta_{1},\dots,\beta_{n}}^{j_{1},\dots,j_{m}}g_{0}^{\alpha_{1}\beta_{1}}\cdots
g_{0}^{\alpha_{n}\beta_{n}}\lambda_{i_{1}j_{1}}\cdots\lambda_{i_{m}j_{m}}\right]^{1/2},$
(16)

with $\lambda_{ij}$ the $so(3,1)$ Killing-Cartan metric.

Since the fields $\Sigma$ and $F$ are continuous on $M$, which is assumed to
be compact, they are bounded. We can then write

$\left|\Delta I_{n_{\nu}}\right|<\epsilon r_{\nu}^{4}k,$ (17)

with $k$ a finite positive constant. Condition d) in Definition 3 says that
the ratio $r_{\nu}^{4}/V_{\nu}$ (where $V_{\nu}$ is the $g_{0}$ volume of
$\nu$) is uniformly bounded. In other words, $r_{\nu}^{4}<RV_{\nu}$ for some
$R>0$. We then have that $\left|\Delta I_{n_{\nu}}\right|<\epsilon RkV_{\nu}$,
and the sum of errors is bounded by

$\left|\sum_{\nu<\triangle_{n}}\Delta I_{n_{\nu}}\right|<\epsilon RV_{M}k,$
(18)

where $V_{M}$ is the $g_{0}$ volume of $M$.

On the other hand, it is possible to show that

$\sum_{PQRST}t_{l_{P}}^{\alpha}t_{l_{Q}}^{\beta}t_{s_{RS}}^{\gamma\delta}\epsilon^{PQRST}=\dfrac{1}{4}t_{\nu}^{\alpha\beta\gamma\delta}.$
(19)

By using all the results given above, we obtain

$\lim_{n\rightarrow\infty}S_{\triangle_{n}}=\lim_{n\rightarrow\infty}\left(\left[\sum_{\nu<\triangle_{n}}\Sigma_{i,\alpha\beta}(C_{\nu})F_{\gamma\delta}^{i}(C_{\nu})t_{\nu}^{\alpha\beta\gamma\delta}\right]+\Delta
I_{n}\right)=\int_{M}\Sigma_{i}\wedge F^{i}.$ (20)

∎

###  2.4 Field equations and boundary terms

Before starting the calculations, we will further simplify the expression for
the action (3) and point out a feature of our discrete fields that is not
present in closely related discrete models for gravity like the classical EPRL
model [3].

Let us introduce $\hat{J}_{s}$, a function of the tetrad field $e_{l}$, and
$J^{s}$, a function of $\hat{J}_{s}$.

$\hat{J}_{s_{PQ}~{}i}:=e_{l_{P}}^{K}e_{l_{Q}}^{L}\left(T_{i}\right)^{JI}P_{IJKL}.$
(21) $J^{s}_{i}:=\sum_{s^{\prime}<\nu}\operatorname{sgn}(s^{\prime},s)\
\hat{J}_{s^{\prime}~{}i}.$ (22)

We can also write $J^{PQ}_{i}:=\sum_{T}\epsilon^{PQRST}\hat{J}_{{RS}~{}i}$.

By using (22), the action (3) can be written as

$S_{\triangle}(e,h,k)=\sum_{\nu<\triangle}\left[\sum_{s<\nu}J^{s}_{i}\theta_{s}^{i}\right].$
(23)

We then discover an identity of our discrete model and something that looks
like a symmetry of the model but is not so. They follow from the fact that the
function taking the $\hat{J}_{s}$ variables to the $J^{s}$ is a linear
transformation from $so(3,1)^{10}$ to itself which is not 1 to 1. The image of
the transformation is the solution space of the relations

$\sum_{s>l_{P}}J^{s}_{i}=0,$ (24)

one such relation for each $l_{P}>\nu$. That is, if we consider the previous
relations as a function of the $\hat{J}_{s}$ variables they become identities;
they could be called “closure identities.” The calculation is simple:
$\sum_{s>l_{P}}J^{s}_{i}=\sum_{TQ}\epsilon^{PQRST}\hat{J}_{{RS}~{}i}=0$. On
the other hand, the kernel of the transformation is given by elements of the
form $K_{PQ~{}i}=\left(\delta_{P}^{I}-\delta_{Q}^{I}\right)k_{I~{}i}$, with
$k_{I~{}i}\in so(3,1)$.

Our basic fields are not the $\hat{J}$ variables but the tetrad field $e$, but
let us consider (23) as a function of the discrete connection and $\hat{J}$.
In that case, the transformation
$\hat{J}_{s~{}i}\rightarrow\hat{J}^{\prime}_{s~{}i}=\hat{J}_{s~{}i}+K_{s~{}i}$
would be a gauge symmetry. For our model, however, the mentioned
transformation is not a symmetry because it generically takes $\hat{J}$
variables that can be calculated from a tetrad $e$ following (21) to variables
that cannot be expressed in the same way.333If we were considering a
discretization of BF theory with $\hat{J}$ as a basic field, the mentioned
transformation would be a gauge symmetry of the model.

Now we begin calculating the field equations. They are obtained by taking the
variation of (3) or (23) with respect to the fields $h_{l}$, $k_{r}$ and
$e_{l}$, while maintaining the variables $k_{r}$ fixed at the boundary of the
region.

The variation of (23) with respect to $h_{l}$ can be calculated by
parametrizing the variations of this variable in the form $h_{l}+\delta
h_{l}=h_{l}\text{exp}(\lambda\alpha_{l}^{i}T_{i})$, where
$\lambda\in\mathbb{R}$ and $\alpha_{l}^{i}T_{i}\in so(3,1)$. For any $s$ with
$l<s$, this induces a variation on $\theta_{s}^{i}$ given by

$\delta\theta_{s}^{i}:=\dfrac{d\theta_{s}^{i}}{d\lambda}\bigg{\rvert}_{\lambda=0}=\alpha_{l}^{j}\text{tr}\left[T_{j}T^{i}g_{\partial
s}\right].$ (25)

Therefore, the variation of (23) with respect to $h_{l}$ is given by

$\delta
S_{\triangle}:=\dfrac{dS_{\triangle}}{d\lambda}\bigg{\rvert}_{\lambda=0}=\alpha_{l}^{j}\sum_{s>l}w^{s}_{j},$
(26)

with

$\displaystyle w^{s}_{j}=$ $\displaystyle J^{s}_{i}\
\text{tr}\left[T_{j}T^{i}g_{\partial s}\right]$ $\displaystyle=$
$\displaystyle\left(\sum_{l_{1},l_{2}<\nu}\operatorname{sgn}(l_{1},l_{2},s)e_{l_{1}}^{K}e_{l_{2}}^{L}\right)\left(T_{i}\right)^{JI}P_{IJKL}\text{tr}\left[T_{j}T^{i}g_{\partial
s}\right].$ (27)

Thus, the extremality condition with respect to variations in $h_{l}$ is

$\sum_{s>l}w^{s}_{j}=0.$ (28)

Notice that when $g_{\partial s}=\mbox{id}$, due to (24), this “Gauss law”
reduces to an identity. Once the tetrad field is given, the “Gauss law”
demands that the discretized connection be frame-compatible.

The variation of the action with respect to $k_{r}$ can be calculated by
writing $k_{r}+\delta k_{r}=\text{exp}(\lambda\beta_{r}^{j}T_{j})k_{r}$. Since
$g_{\partial s}=h_{l_{2}}^{-1}k_{r_{2}}^{-1}k_{r_{1}}h_{l_{1}}$, then the
variation of $\theta_{s}^{i}$ is given by

$\delta\theta_{s}^{i}=\beta_{r_{1}}^{j}\text{tr}\left[T^{i}h_{l_{2}}^{-1}k_{r_{2}}^{-1}T_{j}k_{r_{1}}h_{l_{1}}\right],$

and the variation of (23) is

$\delta
S_{\triangle}=\alpha_{r_{1}}^{j}u_{\sigma\nu_{1}~{}j}+\alpha_{r_{1}}^{j}u_{\sigma\nu_{2}~{}j},$
(29)

where $\nu_{1}$ and $\nu_{2}$ are the two 4-simplices that share the line
$r_{1}$, and
$u_{\sigma\nu~{}j}:=k_{r_{1}}h_{l_{1}}w^{s(\sigma\nu)}_{j}h_{l_{1}}^{-1}k_{r_{1}}^{-1}$.

Extremization of $S_{\triangle}$ with respect to $k_{r_{1}}$ implies

$u_{\sigma\nu_{1}}=u_{\sigma\nu_{2}}.$ (30)

Notice that $u_{\sigma\nu}$ is obtained by the parallel transport of
$w^{s(\sigma\nu)}$ from $C_{\nu}$ to $C_{\sigma}$ following the lines
$l_{1},r_{1}$ in a positive direction. Field equation (30) implies that the
variables $u_{\sigma\nu,j}$ are independent of the 4-simplices $\nu$ and are
then uniquely associated with the 2-simplices $\sigma$ in $\triangle$.

Finally, extremization with respect to the field $e_{l_{P}}^{K}$ yields

$\sum_{T}\epsilon^{PQRST}e_{Q}^{L}\left(T_{i}\right)^{JI}P_{IJKL}\theta_{RS}^{i}=0.$
(31)

Equations (28), (30) and (31) are the field equations that define the dynamics
of our model.

If the spacetime region considered has a boundary, the contributions to the
action from 4-simplices meeting the boundary may need to be modified for the
purpose of arriving to the correct bulk field equations without any equations
conflicting with the boundary conditions. Clearly, if the gauge field $k_{r}$
is kept fixed at the boundary, no modification is needed. On the other hand,
if its momentum, the variable $u_{\sigma}$, is kept fixed at the boundary, we
need to avoid conflict between field equation (30) and the boundary condition.
Consider that $r_{1}<s$ is part of the boundary; then in the action we should
replace
$\theta_{s}^{i}:=\text{tr}\left[T^{i}h_{l_{2}}^{-1}k_{r_{2}}^{-1}k_{r_{1}}h_{l_{1}}\right]$
with
$\tilde{\theta}_{s}^{i}:=\text{tr}\left[T^{i}h_{l_{2}}^{-1}k_{r_{2}}^{-1}\text{id}h_{l_{1}}\right]$.
In [4], the analogous modification to the action at boundaries was proposed.

##  3 Formulation in terms of a constrained $B$ field

The discrete $B$ field assigns an element of the copy of $\mathfrak{so}(3,1)$
associated with $C_{\nu}$ to every wedge $s<\nu$. We write it as
$B_{s}=B_{si}T^{i}\in\mathfrak{so}(3,1)$. Such a field may be considered the
result of integrating the $B$ field in the continuum over $s$, in a way that
is compatible with gauge transformations following a prescription similar to
(6).

The degrees of freedom inside each spacetime atom (4-simplex) can be neatly
divided into bulk degrees of freedom and boundary degrees of freedom. The $B$
field and the $h$ field live in the bulk, and the only basic field living at
the boundary is the boundary connection $k$.

Our starting point will be the action

$S_{\Delta}(B,h,k)=\sum_{\nu<\Delta}\sum_{s,s^{\prime}<\nu}\text{sgn}(s,s^{\prime})\left(B_{si}-\dfrac{1}{\gamma}{\ast}B_{si}\right)\theta^{i}_{s^{\prime}}$
(32)

together with a set of constraints on the $B$ field that makes it a discrete
model for gravity.

The idea is that if $B_{PQ}=\ast\ e_{P}\wedge e_{Q}$, then the action (32)
reduces to (3)444Notice that
$\hat{J}_{s,i}=B_{s,i}-\dfrac{1}{\gamma}{\ast}B_{s,i}$, but in this section
$B$ is a basic field and not just a convenient variable.. The contribution to
the action from a 4-simplex can also be written as a sum over corner cells

$S_{\nu}=\sum_{T}\epsilon^{PQRST}\left(B_{PQi}-\dfrac{1}{\gamma}{\ast}B_{PQi}\right)\theta^{i}_{RS}.$

###  3.1 Geometricity à la EPRL

We will introduce a series of geometricity/simplicity conditions, formally
stated below as 4, 5 and 6, which by construction will ensure that
$B_{PQ}=\ast\ e_{P}\wedge e_{Q}$.

######  Definition 4 (Condition G1($\nu$)).

The $B$ field in spacetime atom $\nu$ is such that for any internal edge
$l<\nu$ there is a vector $\vec{n}_{l}\in{\mathbb{R}}^{3,1}-\\{\vec{0}\\}$
such that for any wedge $s$ with $l<s<\nu$ we have

$n_{lI}B_{s}^{IJ}=0.$

This constraint is analogous to the so-called “linear simplicity” constraint
[15], which is at the core of the construction of the EPRL model [3]. An
important difference is that at each atom our constraint involves bulk degrees
of freedom only.

Once we have a $B$ field satisfying Condition 4, let us fix a choice of set of
vectors $\\{n_{l}\\}_{l<\nu}$ participating in 4; it will be clear later that
our considerations are independent of this choice. Condition 4 holds if and
only if the $B$ field inside $\nu$ can be written as $B_{PQ}=\lambda_{PQ}\ast
n_{P}\wedge n_{Q}$ for a set of scalars $\\{\lambda_{PQ}=\lambda_{QP}\\}$.

If we had a “tetrad” written as $e_{P}=\lambda_{P}n_{P}$ for some coefficients
$\lambda_{P}$, then the relation between $B$ and $e$ induces the system of
equations

$\lambda_{PQ}=\lambda_{P}\lambda_{Q}.$ (33)

In this system of equations, we consider $\lambda_{PQ}$ as given data and
$\lambda_{P}$ as unknown parameters that will determine the “tetrad.” For
being able to determine $\lambda_{P}$, we add a second set of conditions on
$B$.

Recall that two wedges $s,s^{\prime}<c$ contained in the same corner cell are
transversal if and only if $\text{sgn}(s,s^{\prime})\neq 0$.

######  Definition 5 (Condition G2($c$)).

For any two pairs of transversal wedges $(s,s^{\prime})$,
$(s^{\prime\prime},s^{\prime\prime\prime})$ contained in the same corner cell
$c$ the following equation holds

$\text{sgn}(s,s^{\prime})\epsilon_{IJKL}B_{s}^{IJ}\wedge
B_{s^{\prime}}^{KL}=\text{sgn}(s^{\prime\prime},s^{\prime\prime\prime})\epsilon_{IJKL}B_{s^{\prime\prime}}^{IJ}\wedge
B_{s^{\prime\prime\prime}}^{KL}.$

This condition states that the 4-volume of the corner cell $c$ may be computed
as $V(c):=4^{4}\text{sgn}(s,s^{\prime})\epsilon_{IJKL}B_{s}^{IJ}\wedge
B_{s^{\prime}}^{KL}$ for any pair of transversal wedges $(s,s^{\prime})$
contained in $c$.555 The 4-volume of the corner cell $c<\nu$ may be calculated
by dividing the corner cell into $4!$ “flags” (4-simplices of the baricentric
subdivision of $\nu$). The volume of any flag in corner cell $c_{5}$ is
$4^{3}/6$ times the volume of the parallelepiped determined by
$l_{1},l_{2},l_{3},l_{4}$. The field is considered to be nondegenerate at $c$
if $V(c)\neq 0$. Conditions of this type have also previously appeared in
discrete gravity models [16, 4]. In addition, a similar condition is
considered as part of the classical EPRL model [3]. In the quantum EPRL model,
however, the analogous condition is automatically satisfied. For a discussion,
see [8].

For the sake of concreteness, let us examine the conditions for corner cell
$c_{5}$ and study the case of a nondegenerate field satisfying Conditions G1
and G2. Condition G1 implies that the normal vectors $n_{1},n_{2},n_{3},n_{4}$
are defined. Non degeneracy is equivalent to
$\epsilon_{IJKL}n_{1}^{I}n_{2}^{J}n_{3}^{K}n_{4}^{L}\neq 0$, and Condition G2
may be stated as

$\epsilon^{PQRS5}\neq 0\ \ \ \mbox{ implies }\ \ \
\lambda_{PQ}\lambda_{RS}=\lambda_{12}\lambda_{34}\doteq\Lambda_{5}.$ (34)

A set of independent coefficients generating the rest (in the nondegenerate
case) is $\lambda_{(12)}$, $\lambda_{(13)}$, $\lambda_{(14)}$, $\Lambda_{5}$.
The system of equations (33), (34) is easily solvable. We define
$\lambda_{1}^{2}\doteq\frac{\lambda_{13}\lambda_{14}}{\lambda_{34}}=\frac{\lambda_{12}\lambda_{13}\lambda_{14}}{\Lambda_{5}}$.
In general, we can write
$\lambda_{P}^{2}\doteq\frac{\lambda_{PP+1}\lambda_{PP+2}\lambda_{PP+3}}{\Lambda_{5}}$
using appropriate cyclic conventions. It is clear that if
$P,Q\in\\{1,2,3,4\\}$ are different then
$\lambda_{P}^{2}\lambda_{Q}^{2}=(\lambda_{PQ})^{2}$. Thus, if all the
coefficients $\lambda_{P}^{2}$ are positive, we have found real solutions to
the system of equations that let us determine $e$ in terms of $B$ for the
variables associated with the corner cell $c_{5}$.

An example showing that the positivity requirements are not trivial is
constructed staring with a $B$ that is of the form $B_{PQ}=\ast e_{P}\wedge
e_{Q}$ and defining $B^{\prime}=-B$. This new field satisfies Condition 4
(with the same vectors $n_{P}$) and also satisfies Condition 5. On the other
hand, since $\lambda^{\prime}_{PQ}=-\lambda_{PQ}$ the abovementioned
positivity requirements are not met.

Associated with cell $c_{5}$, we have positivity requirements for
$\lambda_{1}^{2}$, $\lambda_{2}^{2}$, $\lambda_{3}^{2}$ and $\lambda_{4}^{2}$,
but since $P\neq Q$ implies
$\lambda_{P}^{2}\lambda_{Q}^{2}=(\lambda_{PQ})^{2}$, either all the
coefficients $\lambda_{P}^{2}$ are positive or all of them are negative.

Below we will state a condition, written directly in terms of the $B$ field,
which is equivalent to the above positivity conditions. Since
$\lambda_{1}^{2}$, when the $B$ field is geometric, it has the interpretation
of being the proportionality factor between $n_{1}\cdot n_{1}$ and $e_{1}\cdot
e_{1}$, it is not hard to guess that the appropriate expression is reminiscent
of the $(1,1)$ component of Urbantke’s metric [22]. A straightforward
calculation leads to

$\frac{1}{12V(c_{5})}f_{ijk}\epsilon^{ABCD5}B^{i}_{1A}B^{j}_{BC}B^{k}_{D1}=\lambda_{1}^{2}\
(\eta_{IJ}n_{1}^{I}n_{1}^{J}).$

Since the left-hand side of this expression is independent of the choice of
vectors $n_{P}$, two comments follow: First, the positivity conditions are
independent of the choice of vectors $n_{P}$. Let us explain. Assume that
conditions 4 and 5 hold. A different choice of the vectors
$\tilde{n}_{P}=k_{P}n_{P}$ for some scalars $k_{P}\neq 0$ (which is all the
freedom of choice, once we are given a nondegenerate $B$ field satisfying
Condition 4) leads to a right hand side of the form $\tilde{\lambda}_{1}^{2}\
(\eta_{IJ}\tilde{n}_{1}^{I}\tilde{n}_{1}^{J})=\tilde{\lambda}_{1}^{2}\
k_{1}^{2}\ (\eta_{IJ}n_{1}^{I}n_{1}^{J})$. Thus, $\tilde{\lambda}_{1}^{2}>0$
if and only if $\lambda_{1}^{2}>0$. Second, since the coefficients
$\lambda_{P}^{2}$ are either all positive or all negative, the two available
options lead to bilinear forms of different signatures. This observation leads
to our last geometricity condition.

######  Definition 6 (Condition G3($c$)).

At each corner cell the bilinear form defined in terms of the $B$ field by

$g_{QR}(c_{P})=\frac{1}{12V(c_{P})}f_{ijk}\epsilon^{ABCDP}B^{i}_{QA}B^{j}_{BC}B^{k}_{DR}$
(35)

has the same signature as Minkowski’s metric $\eta$.

We have constructed a system of equations with an explicit solution
corresponding to a four tuple of vectors $e_{l}$ for each of the five corner
cells determined by the the restriction of the discrete $B$ field to that
corner cell. If the solutions from different corner cells were compatible we
would have a way to find a discrete tetrad field $e$ in terms of a discrete
$B$ field. Consider the solutions found for $\lambda_{1}^{2}$ associated with
corner cells $c_{5}$ and $c_{4}$. For corner cell $c_{5}$ we wrote,
$\lambda_{1}^{2}(c_{5})=\frac{\lambda_{12}\lambda_{13}\lambda_{14}}{\Lambda_{5}}$.
Following the same procedure for $c_{4}$ we obtain
$\lambda_{1}^{2}(c_{4})=\frac{\lambda_{12}\lambda_{13}\lambda_{15}}{\Lambda_{4}}$
(with $\Lambda_{4}=\lambda_{12}\lambda_{35}$). A simple calculation shows that
$\frac{\lambda_{1}^{2}(c_{5})}{\lambda_{1}^{2}(c_{4})}=\frac{\lambda_{35}\lambda_{14}}{\lambda_{34}\lambda_{15}}=\frac{\Lambda_{2}}{\Lambda_{2}}=1$,
which proves compatibility.

We have arrived constructively to the main result of this subsection.

######  Theorem 2.

A nondegenerate $B$ field may be written in terms of a tetrad $e$, which is
unique up to sign, as $B_{PQ}=\ast e_{P}\wedge e_{Q}$ if and only if
conditions 4, 5 and 6 hold.

It is not difficult to see that under the conditions of the above theorem the
pseudometrics (2) and (35) agree. Then if $c$ and $c^{\prime}$ are neighboring
corner cells in a 4-simplex, the corresponding metrics are compatible as
stated in Subsection 2.1.

Now we will define another discrete model.

######  Definition 7 (B-field model 1).

The action $S_{\Delta}(B,h,k)$ defined in (32), together with Conditions 4 and
5 restricting the $B$ field, determines $B$-model 1.

A discrete tetrad field $e$ is nondegenerate at a corner cell $c$ if and only
if $\\{e_{P}\\}_{l_{P}<c}$ is a basis for the copy of Minkowski’s space
sitting at $c$. It is clear that if a tetrad field and a $B$ field are related
by $B_{PQ}=\ast e_{P}\wedge e_{Q}$ then $B$ is nondegenerate if and only if
$e$ is nondegenerate.

######  Corollary 1.

$B$-model 1, in its nondegenerate sector and complemented with Condition 6, is
equivalent to the $e$-model in its nondegenerate sector.

###  3.2 Alternative form of the geometricity conditions

First, a set of quadratic geometricity conditions GG1 (strengthening condition
G2) is imposed at each corner cell.

######  Definition 8 (Condition GG1($c$)).

For all $s,s^{\prime}<c$

$\epsilon_{IJKL}B_{s}^{IJ}\wedge
B_{s^{\prime}}^{KL}-\frac{1}{12}\text{sgn}(s,s^{\prime})\sum_{s^{\prime\prime},s^{\prime\prime\prime}<c}\text{sgn}(s^{\prime\prime},s^{\prime\prime\prime})\epsilon_{IJKL}B_{s^{\prime\prime}}^{IJ}\wedge
B_{s^{\prime\prime\prime}}^{KL}=0.$

A wellknown result of De Pietri and Freidel [16] (see also [7]), translated to
our Lorentzian case, says that the space of solutions of GG1 is divided into
sectors: In the sector of interest, we have (i) $B_{PQ}=\ast e_{P}\wedge
e_{Q}$; other sectors of solutions are composed of $B$ fields of the form (ii)
$B_{PQ}=e_{P}\wedge e_{Q}$, (iii) $B_{PQ}=-e_{P}\wedge e_{Q}$ and (iv)
$B_{PQ}=-\ast e_{P}\wedge e_{Q}$. We will impose constraints GG2 and GG3,
which discard the unwanted sectors.

Clearly, condition G2 from the previous subsection is included in the set of
constraints GG1. Thus, the 4-volume of a corner cell may be calculated as
$V(c):=4^{4}\text{sgn}(s,s^{\prime})\epsilon_{IJKL}B_{s}^{IJ}\wedge
B_{s^{\prime}}^{KL}$ for any pair of transversal wedges $(s,s^{\prime})$
contained in $c$.

######  Definition 9 (Condition GG2($c$)).

The $B$ field in corner cell $c$ is such that $V(c)>0$.

######  Definition 10 (Condition GG3($c$)).

At each corner cell, the bilinear form defined in terms of the $B$ field by

$g_{QR}(c_{P})=\frac{1}{12V(c_{P})}f_{ijk}\epsilon^{ABCDP}B^{i}_{QA}B^{j}_{BC}B^{k}_{DR}$

has the same signature as Minkowski’s metric $\eta$.

Now we can state the main result of this subsection.

######  Theorem 3.

A nondegenerate $B$ field may be written in terms of a tetrad $e$, which is
unique up to sign, as $B_{PQ}=\ast e_{P}\wedge e_{Q}$ if and only if
conditions 8, 9 and 10 hold.

###### Proof.

First, a simple calculation shows that a field of the form $B_{PQ}=\ast
e_{P}\wedge e_{Q}$ satisfies conditions GG1, GG2 and GG3. Let us see that the
set of conditions are sufficient.

The mentioned result of [16] states that the set of solutions of GG1 is
composed by sectors (i) to (iv) listed above. Since (i) satisfies condition
GG2 and $V(c)[\ast B]=-V(c)[B]$, we see that condition GG2 discards sectors
(iii) and (ii). On the other hand, since condition GG3 is cubic and it is
satisfied by sector (i), it rules sector (iv) out. ∎

Now we introduce yet another discrete model.

######  Definition 11 (B-field model 2).

The action $S_{\Delta}(B,h,k)$ defined in (32), together with Condition 8
restricting the $B$ field, determines $B$-model 2.

######  Corollary 2.

$B$-model 2, complemented with Conditions 9 and 10, is equivalent to the
$e$-model in its nondegenerate sector.

###  3.3 Differences with other classical simplicial models related to loop
quantum gravity

Differences among models obtained from a field theory by a cutoff are to be
expected, but in a quantum continuum limit inessential differences are washed
out by universality. From this point of view, the differences that we will be
talking about in this section are most likely inessential. Among equivalent
models, the possible advantages of one of them over another have two possible
origins. Either one model leads to a certain insight that is not as clear in
the other model, or one model leads to more efficient simulations (in general
or when a particular observable is studied).

The geometric description of our model differs from that of other classical
models related to spin foam quantum gravity.

The most direct geometric description of our model stems from our discrete
tetrad field, and the geometry of the classical prequel to the EPRL model
could be thought of as a Regge geometry [17]. In a quantum theory, however,
properties of the space of histories play a central role, and the properties
of the space of classical solutions are relevant only in the appropriate
regime. In this respect, in our $e$-model the geometry of histories follows
directly from the tetrad field, and for the EPR model, the relevant geometry
is that of twisted geometry [13]; below we compare these geometries. We are
not ready to present a geometric picture for our $B$-models for all histories.

If the geometricity conditions are sharply imposed, the geometric picture
would agree with the tetradic geometry and may be described using the
pseudometric (35). This metric would provide a tool for describing aspects of
the geometry of 4-simplices, such as a causal structure, in terms of the $B$
field. But if we somehow adapted the EPRL quantization method to be applied to
our model, the geometricity conditions would be enforced only softly, and the
geometric picture would include a certain twist in the matching of neighboring
corner cells within the same 4-simplex.

It is evident that the contribution to the action from a 4-simplex in our
model is more complicated than the terms of the action in the classical EPRL
model. Considering that our 4-simplices include curvature (in a complete set
of directions for the discrete geometry), what we should compare, however, is
the complexity of the action of one of our 4-simplices with the action
associated with the star of a vertex in the triangulation of a Regge
discretization. From this perspective, the action of our model does not seem
to be overly complicated; the difference seems to be mostly how degrees of
freedom are organized with respect to the simplicial decomposition. The
following paragraph discusses an issue that initially looks like a difference
in the geometric content of our model with respect to the classical EPRL
model. A careful look shows that the difference is also related to the
different organization of the degrees of freedom with respect to the
simplicial decomposition.

In Reisenberger’s discretization, a spin foam model may be constructed after
integrating over the bulk degrees of freedom of spacetime atoms. It is then
natural to ask how does our model look like when described in terms of
boundary data. In our classical model, the set of boundary data is composed of
the boundary connection and its momenta $u_{\sigma}$“$=\frac{\delta S}{\delta
k_{r}}=-\frac{\delta S}{\delta k_{r^{\prime}}}$” (where $\partial
s(\sigma\nu)=l+r-r^{\prime}-l^{\prime}$). The formula in terms of the tetrad
field is

$u_{\sigma\nu~{}j}:=k_{r_{1}}h_{l_{1}}\left(\sum_{l_{1},l_{2}<\nu}\operatorname{sgn}(l_{1},l_{2},s)e_{l_{1}}^{K}e_{l_{2}}^{L}\left(T_{i}\right)^{JI}P_{IJKL}\text{tr}\left[T_{j}T^{i}g_{\partial
s}\right]\right)h_{l_{1}}^{-1}k_{r_{1}}^{-1}.$

Notice that the momentum variable associated with a wedge $s$ is an $so(3,1)$
element which is not simple in the generic case: it cannot be written as a
wedge product of two Minkowski vectors. This looks like a clear difference
with the classical prequel of the EPRL model and its twisted geometry [13]. At
the level of corner cells boundaries, however, simplicity is present. Consider
a corner cell $c<\nu$ and a wedge $s(\sigma\nu)$ intersecting it. Let us
define $u_{\sigma\nu}|_{c}$ as the restriction of the sum given above where
$\sum_{l_{1},l_{2}<\nu}$ is replaced by $\sum_{l_{1},l_{2}<c}$. In the
resulting sum, there are only two nonvanishing terms corresponding to the
wedge product of two Minkowski vectors. In our B-models, the $B$ field is
shared by neighboring corner cells within the same 4-simplex, and this gluing
of neighboring elements seems to admit a twist similar to that of twisted
geometries. Thus, this apparent difference seems to also be related to the
different arrangement of the degrees of freedom within the simplicial complex.

On the other hand, in the EPRL model, neighboring 4-simplices share the $B$
field, and in our model they share the gauge field $k$. In our model metric
degrees of freedom of neighboring 4-simplices are related only on-shell. When
solving the geometricity conditions, one finds two sectors related by
$B\mapsto B^{\prime}=-B$. For the EPRL model, these are two global sectors,
but for our model this could mean two sectors per 4-simplex. At the quantum
level, this could mean that finding a quantized version of our discrete
Condition G3 may be essential, even when in the EPRL there is no condition of
this type.

Our $e$-model, makes sense also for a general tetrad field $e$ (possibly
degenerate). One may argue that its continuum limit is an extension of general
relativity. In the same way, our $B$-model 1 makes sense for a general $B$
field. Its continuum limit would not need an analog of the discrete Condition
G3: continuity would ensure that the signature of the field would not
fluctuate wildly from point to point. This heuristic argument tells us that at
the level of a discrete quantum model, omitting Condition G3 may be a decision
that drastically changes the model. On the other hand, our condition G3 can
only be imposed after restricting to nondegenerate fields, and in quantum
gravity at the most fundamental level, it does not seem reasonable to consider
only nondegenerate fields. We are thinking of an scenario that considers
coarsegrained fields; in this scenario, asking for non degeneracy could be
reasonable. Similar comments would apply for our $B$-model 2. Currently,
however, these heuristic arguments are only speculation.

##  4 Summary and outlook

We presented a discrete model for gravity, the e-field model in Section 2, and
proved that its action converges in the continuum limit to Holst’s action for
general relativity.

In our model, there is a copy of Minkowski’s space associated with each
spacetime atom. When the gravitational field ($e$ or $B$) is not degenerate,
one can use it to provide a relative reference frame in that copy of
Minkowski’s space. (For a expression of the spacetime metric written in terms
of the tetrad field $e$ see Subsection 2.1, and for a expression written in
terms of the $B$ field see Subsection 3.1.) The role played by the connection
field is to provide a communication system between those reference frames.

Striving to provide a new point of view for spin foam models for quantum
gravity, we gave two versions of our model, which trade the discrete tetrad
field $e$ for a discrete $B$ field (B-field models 1 and 2 in Section 3). With
the first version, the plan is to follow an adapted version of the EPRL
quantization [3]. This will be done first at the level of each corner cell of
a 4-simplex, and then the pieces will be glued and projected to solve a
quantum version of our G3 condition for constructing a variant of the Conrady-
Hnybida extension [18] of the EPRL vertex. We expect that the intuitive
geometric picture for the resulting quantum theory will be a twisted geometry
where the 2-surfaces that are glued with a twist are internal to each
4-simplex (the 2-surfaces at the boundary of the intersection between two
neighboring corner cells). The second version was constructed with the
intention of convincing researchers that it can be used to construct an
effective spin foam model along the lines proposed by Dittrich et al in [12].
In our view, their use of area-angle Regge calculus holds them back from
achieving even more efficient simulations. A classical model formulated over a
Reisenberger discretization would be readily compatible with boundary spin
network states and would have superior gluing properties. One option is to use
the model presented in the article. Another bonus is that, as mentioned above,
the model also provides a new perspective on the causal structure of
histories. A third quantization avenue would be a direct path integral
quantization of our tetradic model by considering a 4-simplex $\nu$ (or
spacetime atom) with a fixed connection on its boundary (the $k_{r}$ for
$r<\partial\nu$) and integrating out the bulk degrees of freedom, which are
the bulk connection and the tetrad (the $h_{l}$ and the $e_{l}$ for $l<\nu$)
using an appropriate measure. It is natural to ask how do these three
quantization avenues compare to each other. This type of study was proposed in
[19], but since our model does have an actual tetrad field, it may provide an
interesting new perspective.

The histories in our model have a clear combinatorial geometry which differs
from that of other models; see Subsection 3.3. We plan to use this geometric
picture to investigate the corresponding notion of causal structure. We would
like to compare a kinematical/geometric notion of causal structure with a
dynamical one, which will emerge from the study of perturbation propagation in
the linearization of our model. The motivation behind such a study is its use
in spin foam models, since experience in another approach to quantum gravity
[20] suggests that it may be crucial to only sum over histories that are
causally nondegenerate in a certain sense.

We mentioned our plan to study perturbation propagation in the linearization
of the model. This will also provide more elements in the classical side to
compare with a graviton propagator that could be calculated in a quantized
model. Recall that the atoms of our model are curved. Thus, even if a study of
the graviton propagator included only one atom of our model, it would be
testing the quantum effects of curvature. This is to be contrasted with using
a single 4-simplex in quantum Regge calculus, where curvature would not play
any role [14]. We clarify that even when we attach some significance to this
calculation, a physically relevant calculation would be reached only after a
continuum limit (in the Wilsonian sense) is considered. Intuitively,
stability/convergence should be reached after gluing a large number of atoms.
In the search for an effective vertex, one certainly needs to include all
types of degrees of freedom. Could an atom with curvature degrees of freedom
and without the simplicity restriction at the boundary, like the one used in
our model, be used to compute an effective vertex? We will investigate it.

## Acknowledgements

We thank M. Reisenberger for fruitful discussions. Carlos E. Beltrán wishes to
thank Reisenberger for his help in reproducing some of the calculations
presented in [4].

## References

  * [1] _Macroscopic observables from the comparison of local reference systems_ ,
C. Meneses and J. A. Zapata, Class. Quant. Grav. 36 (2019) 23, 235011.

  * [2] _The action of $SU(N)$ lattice gauge theory in terms of the heat kernel on the group manifold_,
P. Menotti and E. Onofri, Nucl. Phys. B 190 (1981) 288-300.

  * [3] _LQG vertex with finite Immirzi parameter_ ,
J. Engle, E. Livine, R. Pereira and C. Rovelli, Nucl. Phys. B 799 (2008)
136–149. arXiv:0711.0146 [gr-qc]

  * [4] _A left-handed simplicial action for euclidean general relativity_ ,
M. P. Reisenberger, Class. Quantum Grav. 14, (1997) 1753–1770. arXiv:gr-
qc/9609002 [gr-qc]

  * [5] _A lattice worldsheet sum for 4-d Euclidean general relativity_ ,
M. P. Reisenberger, arXiv, e-print, (1997). [arXiv:gr-qc/9711052].

  * [6] _Barbero’s Hamiltonian derived from a generalized Hilbert- Palatini action_ ,
S. Holst, Phys. Rev. D. 53, (1996) 5966–5969.

  * [7] _Modern Canonical Quantum General Relativity_ ,
T. Thiemann, Cambridge Monographs on Mathematical Physics, (Cambridge
University Press, Cambridge; New York, 2007).

  * [8] _The Spin-Foam Approach to Quantum Gravity_ ,
A. Perez, Living Rev. Relativity, 16, (2013), 3.

  * [9] _Covariant Loop Quantum Gravity_.
C. Rovelli and F. Vidotto, (Cambridge University Press, Cambridge; United
Kingdom, 2015).

  * [10] _Multisymplectic effective General Boundary Field Theory_.
M. Arjang and J. A. Zapata, (Class. Quant. Grav. 31, 095013 (2014)).
arXiv:1312.3220 [math-ph]

  * [11] _Loop quantization as a continuum limit_.
E. Manrique, R. Oeckl, A. Weber and J. A. Zapata, (Class. Quant. Grav. 23
(2006), 3393-3404) arXiv:hep-th/0511222 [hep-th]

  * [12] _Effective Spin Foam Models for Four-Dimensional Quantum Gravity_.
S. K. Asante, B. Dittrich and H. M. Haggard, (Phys. Rev. Lett. 125 (2020)
no.23, 231301) arXiv:2004.07013 [gr-qc]
_Effective spin foam models for Lorentzian quantum gravity_.
S. K. Asante, B. Dittrich and J. Padua-Arguelles, (Class. Quant. Grav. 38
(2021) no.19, 195002) arXiv:2104.00485 [gr-qc]
_Discrete gravity dynamics from effective spin foams_.
S. K. Asante, B. Dittrich and H. M. Haggard, Class. Quant. Grav. 38 (2021)
no.14, 145023 [arXiv:2011.14468 [gr-qc]].

  * [13] _Twisted geometries: A geometric parametrisation of SU(2) phase space_.
L. Freidel and S. Speziale, Phys. Rev. D 82 (2010), 084040 arXiv:1001.2748
[gr-qc]

  * [14] _The Spinfoam Framework for Quantum Gravity_.
E. R. Livine, arXiv:1101.5061 [gr-qc]

  * [15] _A new spin foam model for 4D gravity_.
L. Freidel and K. Krasnov, (Class. Quantum Grav., 25, 125018 (2008), 125018).
arXiv:0708.1595 [gr-qc]

  * [16] _so(4) Plebanski Action and Relativistic Spin Foam Model_.
R. De Pietri and L. Freidel, (Class. Quantum Grav., 16, 2187–2196 (1999),
2187-2196)

  * [17] _General relativity without coordinates_.
T. Regge, Nuovo Cim. 19 (1961), 558-571

  * [18] _A spin foam model for general Lorentzian 4-geometries_.
F. Conrady and J. Hnybida, (Class. Quant. Grav. 27 (2010), 185011)
arXiv:1002.1959 [gr-qc]

  * [19] _Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation_.
J. Engle, M. Han and T. Thiemann, (Class. Quant. Grav. 27 (2010), 245014)
arXiv:0911.3433 [gr-qc]

  * [20] _Reconstructing the universe_.
J. Ambjorn, J. Jurkiewicz and R. Loll, (Phys. Rev. D 72 (2005), 064014)
arXiv:hep-th/0505154 [hep-th]

  * [21] _Area-angle variables for general relativity_.
B. Dittrich and S. Speziale, (New J. Phys. 10 (2008), 083006) arXiv:0802.0864
[gr-qc]

  * [22] _On integrability properties of SU(2) Yang-Mill fields. I. Infinitesimal Part_.
H. Urbantke, (J. Math. Phys. 25 (1984) no.7, 2321-2324)

  * [23] _Numerical analysis of spin foam dynamics and the flatness problem_.
P. Donà, F. Gozzini and G. Sarno, Phys. Rev. D 102, no.10, 106003 (2020)
[arXiv:2004.12911 [gr-qc]].
_Hypercuboidal renormalization in spin foam quantum gravity_.
B. Bahr and S. Steinhaus, Phys. Rev. D 95, no.12, 126006 (2017)
[arXiv:1701.02311 [gr-qc]].

  * [24] _Spinfoam on a Lefschetz thimble: Markov chain Monte Carlo computation of a Lorentzian spinfoam propagator_. M. Han, Z. Huang, H. Liu, D. Qu and Y. Wan, Phys. Rev. D 103, no.8, 084026 (2021) [arXiv:2012.11515 [gr-qc]].
_Spin-foams as semi-classical vertices: gluing constraints and a hybrid
algorithm_. S. K. Asante, J. D. Simão and S. Steinhaus, [arXiv:2206.13540 [gr-
qc]].
_Finiteness of spinfoam vertex amplitude with timelike polyhedra and the
regularization of full amplitude_.
M. Han, W. Kaminski and H. Liu, Phys. Rev. D 105, no.8, 084034 (2022)
[arXiv:2110.01091 [gr-qc]].


# Threshold Saturation on Channels with Memory via Spatial Coupling

Shrinivas Kudekar1 and Kenta Kasai2

1 New Mexico Consortium and Center for Non-linear Studies, Los Alamos National
Laboratory, NM, USA
Email<EMAIL_ADDRESS>2 Dept. of Communications and Integrated Systems,
Tokyo Institute of Technology, 152-8550 Tokyo, Japan.
Email<EMAIL_ADDRESS>
###### Abstract

We consider spatially coupled code ensembles. A particular instance are
convolutional LDPC ensembles. It was recently shown that, for transmission
over the memoryless binary erasure channel, this coupling increases the belief
propagation threshold of the ensemble to the maximum a-posteriori threshold of
the underlying component ensemble. This paved the way for a new class of
capacity achieving low-density parity check codes. It was also shown
empirically that the same threshold saturation occurs when we consider
transmission over general binary input memoryless channels.

In this work, we report on empirical evidence which suggests that the same
phenomenon also occurs when transmission takes place over a class of channels
with memory. This is confirmed both by simulations as well as by computing
EXIT curves.

##  I Introduction

It has long been known that convolutional LDPC (or spatially coupled)
ensembles, introduced by Felström and Zigangirov [1], have excellent
thresholds when transmitting over general binary-input memoryless symmetric-
output (BMS) channels. The fundamental reason underlying this good performance
was recently discussed in detail in [2] for the case when transmission takes
place over the binary erasure channel (BEC).

In particular, it was shown in [2] that the BP threshold of the spatially
coupled ensemble (see the last paragraph of this section for a definition) is
essentially equal to the MAP threshold of the underlying component ensemble.
It was also shown that for long chains the MAP performance of the chain cannot
be substantially larger than the MAP threshold of the component ensemble. In
this sense, the BP threshold of the chain is increased to its maximal possible
value. This is the reason why they call this phenomena threshold saturation
via spatial coupling. In a recent paper [3], Lentmaier and Fettweis
independently formulated the same statement as conjecture. They attribute the
observation of the equality of the two thresholds to G. Liva. The phenomena of
threshold saturation seems not to be restricted to the BEC. It was also shown
recently in [4] that the same phenomena manifests itself when we consider
transmission over more general BMS channels.

The principle which underlies the good performance of spatially coupled
ensembles is very broad. It has been shown to apply to many other problems in
communications, and more generally computer science. To mention just a few,
the threshold saturation effect (dynamical threshold of the system being equal
to the static or condensation threshold) of coupled graphical models has
recently been shown to occur for compressed sensing [5], and a variety of
graphical models in statistical physics and computer science like the so-
called $K$-SAT problem, random graph coloring, or the Curie-Weiss model [6].
Other communication scenarios where the spatially coupled codes have found
immediate application is to achieve the whole rate-equivocation region of the
BEC wiretap channel [7].

It is tempting to conjecture that the same phenomenon occurs for transmission
over general channels with memory. We provide some empirical evidence that
this is indeed the case. In particular, we compute EXIT curves for
transmission over a class of channels with memory known as the Dicode Erasure
Channel (DEC). We show that these curves behave in an identical fashion to the
ones when transmission takes place over the memoryless BEC. We also compute
fixed points (FPs) of the spatial configuration and we demonstrate again
empirically that these FPs have properties identical to the ones in the BEC
case.

For a review on the literature on convolutional LDPC ensembles we refer the
reader to [2] and the references therein. As discussed in [2], there are many
basic variants of coupled ensembles. For the sake of convenience of the
reader, we quickly review the ensemble $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$.
This is the ensemble we use throughout the paper as it is the simplest to
analyze.

###  I-A $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ Ensemble [2]

We assume that the variable nodes are at sections $[-L,L]$, $L\in\mathbb{N}$.
At each section there are $M$ variable nodes, $M\in\mathbb{N}$. Conceptually
we think of the check nodes to be located at all integer positions from
$[-\infty,\infty]$. Only some of these positions actually interact with the
variable nodes. At each position there are
$\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}M$ check nodes. It remains to describe
how the connections are chosen. We assume that each of the $d_{\mathrm{l}}$
connections of a variable node at position $i$ is uniformly and independently
chosen from the range $[i,\dots,i+w-1]$, where $w$ is a “smoothing” parameter.
In the same way, we assume that each of the $d_{\mathrm{r}}$ connections of a
check node at position $i$ is independently chosen from the range
$[i-w+1,\dots,i]$.

A discussion on the above ensemble and a proof of the following lemma can be
found in [2].

######  Lemma 1 (Design Rate)

The design rate of the ensemble $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$, with
$w\leq 2L$, is given by

$\displaystyle R(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$
$\displaystyle=(1-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}})-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}\frac{w+1-2\sum_{i=0}^{w}\bigl{(}\frac{i}{w}\bigr{)}^{d_{\mathrm{r}}}}{2L+1}.$

In the next section we provide the channel model and the joint iterative
decoder. We also present the density evolution analysis of the joint iterative
decoder when we consider $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC
ensembles. In the section on main results, we demonstrate the threshold
saturation phenomena by using spatially coupled codes.

##  II Channels with Memory: The Dicode Erasure Channel

The particular class of channel with memory that we consider is the Dicode
Erasure Channel (DEC). The DEC is a binary-input channel defined as follows.
The output of a binary-input linear filter $(1-D)$ ($D$ is the delay element)
is erased with probability $\epsilon$ and transmitted perfectly with
probability $1-\epsilon$. For this channel we will be interested in the
symmetric information rate (SIR), i.e., the capacity assuming i.i.d Bern(1/2)
signalling. In this case, the Shannon threshold for a given rate $r$ is given
by $\frac{1-r}{4}+\frac{1}{4}\sqrt{(1-r)^{2}+8(1-r)}$. The details on the
definition of the channel and the analytical formula for the SIR can be found
in the thesis of Pfister [8] and in [9].

###  II-A Joint Iterative Decoder, Density Evolution and the Extended BP Fixed
Points

We use the joint iterative decoder (JIT) of Pfister and Siegel [9]. More
precisely, we consider a turbo equalization system, which performs one channel
iteration (BCJR step) for each iteration over the LDPC code. As a result, in
every iteration, first the channel detector uses the extrinsic information
provided by the LDPC code to compute its extrinsic erasure fraction. This is
then fed to the LDPC decoder which then again computes the usual variable node
and check node erasure messages.

The simplicity of the DEC gives an analytical formula for the erasure fraction
of the message which is passed from the channel detector to the LDPC code (see
[9] for a derivation). This is given by

$f(x)=\frac{4\epsilon^{2}}{(2-x(1-\epsilon))^{2}},$

where $x$ represents the fraction of erasures entering the channel detector
from the LDPC code. $f(.)$ represents the extrinsic erasure information
provided by the channel detector.

To summarize: the density evolution111See [9] for a rigorous justification of
the density evolution analysis. (DE) equation for the case of
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble is given by

$\displaystyle
x=f((1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}})(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}.$

Note that the term inside the brackets in $f(.)$ represents the probability
that a variable node is in erasure as given by the LDPC code. Also it is not
hard to see that $f(x)\leq 1$ for any $x$.

###### Example 2

Consider JIT decoding of the DEC with $(5,15)$-regular LDPC ensemble. The
design rate of this code is $2/3$. Using the SIR formula
($=1-2\epsilon^{2}/(1+\epsilon)$) from [9] we get that the Shannon threshold
at rate=2/3 is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. Figure
1 shows the performance of the JIT decoder. We see that the threshold is given
by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx 0.363471$,
which is far away from the capacity. Throughout the paper we will use
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ to denote the threshold of the JIT
decoder when we use $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble
and transmit over the DEC.

(50,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(36,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$

$(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$

Figure 1:  The BP curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15)$-regular
ensemble and transmission over the DEC. The threshold of the JIT decoder is
given by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx
0.363471$.

#### The EXIT curve

The EXIT curve222To be very precise, we should call the curves we plot as
EXIT-like curves. The reason being that we do not provide any operation
interpretation of these curves, like the Area theorem [10] in this work. The
curves serve only to illustrate the capacity achieving nature of coupled-
codes. plots all the fixed-points of the DE equation. The curve is given by
the parametric curve
$\\{(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}},\epsilon(x)\\}$. We obtain
$\epsilon(x)$ by solving for $\epsilon$ in the DE equation.

As an example, we plot the EXIT curve for various
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles as shown in Figure 2.
The JIT threshold is got by dropping a vertical line from the leftmost point
on any given curve. We note that for every $\epsilon>\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$, there are exactly 3
fixed-points. One of them being the trivial 0 fixed-point. This “C” shape of
the EXIT curve is also what we observe when we transmit through a memoryless
BEC using $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble. Also we
remark that as the degrees increase, keeping the design rate fixed, the JIT
threshold keeps on decreasing. This is also the case for transmission over
memoryless BEC. In fact, for memoryless BEC case, the BP threshold goes to
zero as we increase the degrees.

(0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-14,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$

$(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$

$(3,9)$

$(5,15)$

$(7,21)$

$(10,30)$

$(30,90)$

Figure 2:  The EXIT curve for regular LDPC ensembles with
$(d_{\mathrm{l}},d_{\mathrm{r}})$ given by $(3,9)$, $(5,15)$, $(7,21)$,
$(10,30)$, $(30,90)$, and transmission over the DEC. We observe that the JIT
threshold moves to the left and eventually will go to zero as degrees go to
infinity.

We can also show the same result for the DEC. More precisely, we have

######  Lemma 3 (JIT Threshold Goes to Zero)

For any $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble we have

$\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\leq\sqrt{\frac{1}{\sqrt{d_{\mathrm{r}}-1}(1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}})}}.$

###### Proof:

We claim that the necessary condition for the JIT decoder to succeed is given
by

$\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x,$

for all $x\in(0,1]$. Indeed, suppose on the contrary that there exists a
$c\in(0,1]$ such that the above inequality is violated. Thus we have
$\epsilon^{2}(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$ Since
$f(x)\geq\epsilon^{2}$ for all $x\in[0,1]$ we get

$f(c)(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$

This implies that there exists a FP of DE for the DEC for some value in
$[c,1]$. It is not hard to see that this implies the JIT decoder will get
stuck at this FP, resulting in unsuccessful decoding.

Thus we must have that for all $x\in(0,1]$

$\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x.$

For the choice of $x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ we get the statement
of the lemma. To see this computation first write $(1-x)^{d_{\mathrm{r}}-1}$
as $e^{(d_{\mathrm{r}}-1)\log(1-x)}$. Then use $\log(1-x)\leq-x$ and
$x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ to get $(1-x)^{d_{\mathrm{r}}-1}\leq
e^{-\sqrt{d_{\mathrm{r}}-1}}$. After this use

$\displaystyle(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1}$
$\displaystyle=1-(1-(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1})$
$\displaystyle\geq 1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}},$

to complete the argument. ∎

As a consequence of Lemma 3 we get that, with the ratio
$d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed,
$\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})=0$.

##  III Main Results

In this section we show, empirically, that spatially coupled-codes achieve the
Shannon capacity of the DEC. We recall that we are consider SIR which is give
by the formula SIR$=1-2\epsilon^{2}/(1+\epsilon)$. For the sake of exposition,
we demonstrate our results only for rate equals $2/3$. The Shannon threshold
for this rate is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. For
other rates similar results can be observed. From the preceding section we see
that standard $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles do not
saturate the JIT threshold (to the Shannon threshold).

We begin by writing down the DE equation for the coupled-codes.

###  III-A Density Evolution

Consider the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Recall that there
are $2L+1$ sections of variable nodes. Each section has $M$ variable nodes. We
transmit variable nodes sectionwise over the DEC. More precisely, the variable
nodes in section $-L$ are transmitted first, followed by variable nodes in
section $-L+1$ and so on so forth till we finally transmit all the variable
node in section $L$. As a consequence we have a channel detected factor graph
sitting on top of each section of the coupled-code.

To perform the DE analysis, we already take the limit $M\to\infty$. As a
result of this limit, one can ignore the boundary effects of the channel
detector and treat the channel detectors as disconnected333Another way to
think about this is to imagine that we transmit a known sequence of bits of
length equal to the memory of the channel after we transmit all the variable
nodes in each section. Since the channel memory is finite, this induces a rate
loss going to zero as $M\to\infty$. Now the known sequence is the initial
state for each of the channel detectors and hence we can consider them
disconnected..

Let $x_{i}$, $i\in\mathbb{Z}$, denote the average erasure probability which is
emitted by variable nodes at position $i$. For $i\not\in[-L,L]$ we set
$x_{i}=0$. For $i\in[-L,L]$ the DE is given by

$\displaystyle x_{i}$
$\displaystyle=\epsilon_{i}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1},$
(1)

where $\epsilon_{i}$ is given by

$\displaystyle\epsilon_{i}=f\Big{(}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}}\Big{)},$
(2)

where recall that $f(\cdot)$ is the channel extrinsic transfer function. We
will use the notation $\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ to denote the threshold of the JIT
decoder when we use the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble for
transmission. As a shorthand we use $g(x_{i-w+1},\dots,x_{i+w-1})$ to denote
$\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1}.$

######  Definition 4 (FPs of Density Evolution)

Consider DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Let
$\underline{x}=(x_{-L},\dots,{x}_{L})$. We call $\underline{x}$ the
constellation. We say that $\underline{x}$ forms a FP of DE with channel
$\epsilon$ if $\underline{x}$ fulfills (1) for $i\in[-L,L]$. As a shorthand we
then say that $(\epsilon,\underline{x})$ is a FP. We say that
$(\epsilon,\underline{x})$ is a non-trivial FP if $\underline{x}$ is not
identically equal to $0\,\forall\,i$. Again, for $i\notin[-L,L]$, $x_{i}=0$. ∎

######  Definition 5 (Forward DE and Admissible Schedules)

Consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble.
More precisely, pick a channel $\epsilon$. Initialize
$\underline{x}^{(0)}=(1,\dots,1)$. Let $\underline{x}^{(\ell)}$ be the result
of $\ell$ rounds of DE. More precisely, $\underline{x}^{(\ell+1)}$ is
generated from $\underline{x}^{(\ell)}$ by applying the DE equation (1) to
each section $i\in[-L,L]$,

$\displaystyle x_{i}^{(\ell+1)}$
$\displaystyle=\epsilon_{i}g(x_{i-w+1}^{(\ell)},\dots,x_{i+w-1}^{(\ell)}).$

We call this the parallel schedule. The important difference with the
memoryless BEC case is that the channel $\epsilon_{i}$ is not fixed for the
DEC and decreases with increasing iterations according to (2).

More generally, consider a schedule in which in each step $\ell$ an arbitrary
subset of the sections is updated, constrained only by the fact that every
section is updated in infinitely many steps. We call such a schedule
admissible. Again, we call $\underline{x}^{(\ell)}$ the resulting sequence of
constellations. ∎

One can show that if we perform forward DE under any admissible schedule, then
the constellation $\underline{x}^{(\ell)}$ converges to a FP of DE and this FP
is independent of schedule. This statement can be proved similar to the one in
[2].

###  III-B Forward DE – Simulation Results

We consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble.
More precisely, we fix an $\epsilon$ and initialize all $x_{i}$ for
$i\in[-L,L]$ to 1. Then we run the DE given by (1) till we reach a fixed-
point. We fix $L=250$. For $d_{\mathrm{l}}=3$ and $d_{\mathrm{r}}=9$, we have
that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(3,9,300,3)\approx
0.49815$. If we increase the degrees we get $\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(5,15,300,5)\approx 0.49995$, $\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(7,21,300,7)\approx 0.499989$ and
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(9,27,300,9)\approx 0.499996$.
We observe that for increasing the degrees the threshold approaches the
Shannon threshold of $0.5$.

###  III-C The EXIT Curve for Coupled Ensembles

We now come to the key point of the paper, the computation of the EXIT curve.
Before we do this, we define the entropy of a constellation
$\underline{x}=(x_{-L},\dots,x_{L})$ as

$\displaystyle\chi=\frac{1}{2L+1}\sum_{i=-L}^{L}x_{i}.$

To plot the EXIT curve we first fix $\chi\in[0,1]$ and then run DE such that
the resulting FP constellation has entropy equal to $\chi$. This is the
reverse DE procedure as described in [11]. We remark that $f(x)$ is an
increasing function of $\epsilon$, hence in the reverse DE procedure one can
easily find an appropriate $\epsilon$ by the bisection method.

Figure 3 shows the plot of the EXIT curve for the $(5,15,L,5)$ ensemble with
$L=2,4,8,16,32,64,128,256,512$. We see that the curves look very similar to
the curves when transmitting over a BMS channel. For very small values of $L$,
the curves are far to the right due to significant rate loss that is incurred
at the boundary. As $L$ increases the rate loss diminishes and the JIT
threshold is very close to the Shannon threshold. This picture strongly
suggests that the same threshold saturation effect ($\epsilon^{\text{\tiny
JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\approx\epsilon^{\text{\tiny
MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$) also occurs for
the DEC as it was shown analytically in [2].

$L\\!=\\!2$

$L\\!=\\!4$

$L\\!=\\!8$

$L\\!=\\!16$

(0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-16,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$

EXIT

Figure 3:  The EXIT curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15,L,5)$
ensemble and transmission over the DEC for $L=2,4,8,16,32,64,128,256,512$. The
curves keep moving to the left as $L$ increases similar to the curves when
transmitting over BMS. The “vertical” drop in the EXIT curves occurs at
$\approx 0.5$ for $L\geq 32$. Also shown in light gray is the BP exit curve
for the uncoupled $(5,15)$-regular ensemble.

###  III-D Shape of Fixed Point of Density Evolution

We plot the constellation representing the unstable FP of DE. This FP cannot
be reached via forward DE and is obtained via reverse DE procedure. We recall
that this FP played a key role in proving the threshold saturation phenomena
when transmitting over the BEC. Let us describe the (empirically observed)
crucial properties of this constellation.

  * (i)

The constellation is symmetric around $i=0$ and is unimodal. The constellation
has $\epsilon\approx 0.49995$.

  * (ii)

Let $x_{\text{s}}(\epsilon)$ denote a stable FP of DE. The value in the flat
part in the middle is $\approx 0.4434$ which is very close to the stable FP of
DE for the underlying uncoupled $(5,15)$-regular ensemble.

  * (iii)

The transition from close to zero to close to $x_{\text{s}}(\epsilon)$ is very
quick.

(6,0)(14.4,0)[b]$\text{-}16$,$\text{-}14$,$\text{-}12$,$\text{-}10$,$\text{-}8$,$\text{-}6$,$\text{-}4$,$\text{-}2$,0,2,4,6,8,10,12,14,16
Figure 4:  The constellation representing FP of DE for $(5,15,33,5)$ ensemble
and entropy fixed to $\chi=0.2$. This is an unstable FP constellation. The
constellation is very similar to any unstable FP constellation when
transmitting over memoryless BEC. The constellation is unimodal. There is a
long tail of zeros followed by a sharp transition and then a long flat part
with values close to $x_{\text{s}}(\epsilon)$. The constellation has
$\epsilon\approx 0.49995$.

##  IV A Possible Proof Approach

Till now we gave empirical evidence of the threshold saturation phenomena when
transmitting over the DEC using coupled-codes. Before we proceed to give the
proof idea for the threshold saturation, we first show that coupling indeed
helps. More precisely we have the following lemma,

######  Lemma 6 (Spatial Coupling Helps)

For $d_{\mathrm{l}},d_{\mathrm{r}}\to\infty$ with the ratio
$d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed, we have

$\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}.$

###### Proof:

Since $\epsilon_{i}$ is an increasing function of $x_{i-w+1},\dots,x_{i+w-1}$,
we have $\epsilon_{i}\leq
f(1)\leq\frac{4\epsilon^{2}}{(1+\epsilon)^{2}}\leq\epsilon.$ Combining this
with the DE equation for the coupled-codes, we get

$\displaystyle x_{i}\leq\epsilon g(x_{i-w+1},\dots,x_{i+w-1}),$

for all $i\in[-L,L]$. But we know from Theorem 10 in [2] that
$\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny
BP}}_{\text{BEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\to\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$.
Thus for $\epsilon<\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$ the right-hand-side
of the above inequality goes to zero. Hence the lemma. ∎

As an example, consider the $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble
with $d_{\mathrm{l}}/d_{\mathrm{r}}=1/3$ (rate equal to $2/3$) . For
$L\to\infty$, the rate of the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ goes to
$2/3$. From Lemma 3 we have that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\to 0$ and from Lemma 6 we have that
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}=\frac{1}{3}$.
Thus spatial coupling indeed boosts the JIT threshold. However the empirical
evidence suggests that the boost is all the way up to the Shannon threshold
(which is $0.5$ in this case). Since there is ample similarity between the DEC
and the BEC, the guideline for a proof is similar to when we are transmitting
over the BEC.
(i) Existence of FP: A key ingredient in proving the result for the BEC was to
show the existence of a special FP of DE $(\underline{x},\epsilon^{*})$. In
principle, the BEC proof should extend. The only difference is that instead of
a constant channel $\epsilon$, we have a channel value which depends on the FP
constellation itself. However, since the functions involved are rational, this
should not be a big hurdle.
(ii) Shape of the constellation and the transition length: The next task is to
show that the FP guaranteed by the above theorem has the properties as given
in Section III-D. Proving this would first involve showing that the underlying
regular ensemble has a “C” shaped EXIT curve. Intuitively, this means that the
FP constellation (of the coupled-code) can only hover around the stable FPs of
DE (of the underlying regular ensemble), implying that it has either a large
tail of zeros or a large flat part with values close to
$x_{\text{s}}(\epsilon^{*})$.
(iii) Construction of the EXIT curve and the Area Theorem: Another key part of
the BEC proof was to construct a family of FPs (not necessarily stable FPs)
using the special FP guaranteed by the Existence theorem. The EXIT curve plus
the fast transition would allow us to show that this special FP must have an
associated channel parameter, $\epsilon^{*}$, very close to the Shannon
threshold (for large degrees.)444For finite degrees, $\epsilon^{*}$ should be
very close to the MAP threshold of the
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble. One should be able to
prove this by formulating an appropriate Area theorem (see Section 3.20 in
[10]).
Operational interpretation: The proof would be completed by providing an
operation meaning to the EXIT curve. Loosely speaking, the EXIT constructed
above would have a vertical drop at
$\epsilon\approx\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$ (cf.
Figure 3). This would help to show that for any
$\epsilon<\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$, the JIT
decoder will go to the trivial FP.

##  V Conclusions

In this paper we show that empirically coupled-codes saturate the JIT
threshold on the DEC. For the channel extrinsic transfer function we consider
the case when there is no precoding. We list below some comments and open
questions.

  * (i)

An obvious future direction is to complete the proof of threshold saturation.
The guidelines provided above serve as a starting point. Following this route,
in principle, it should be possible to prove the capacity achieving nature of
these codes on the DEC.

  * (ii)

Another interesting question is that whether the threshold saturation
phenomena can be shown to be true for all channel extrinsic transfer functions
$f(.)$ which are non-decreasing both in $\epsilon$ and $x$ (threshold
saturation holds when $f(.)$ represents precoding).

  * (iii)

A proof of the threshold saturation phenomena should also pave the way for the
justification of the Maxwell construction to determine $\epsilon^{\text{\tiny
MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ for the DEC.

  * (iv)

Recently, it was observed that coupled MacKay-Neal (MN) codes with bounded
degree exhibit the BP threshold very close to the Shannon threshold over the
BEC [12]. It is interesting to see if the coupled MN codes have the JIT
threshold close to the SIR over the DEC.

##  VI Acknowledgments

SK acknowledges support of NMC via the NSF collaborative grant CCF-0829945 on
“Harnessing Statistical Physics for Computing and Communications.” SK would
also like to thank Rüdiger Urbanke, Misha Chertkov and Henry Pfister for their
encouragement.

## References

  * [1] A. J. Felström and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” _IEEE Trans. Inform. Theory_ , vol. 45, no. 5, pp. 2181–2190, Sept. 1999.
  * [2] S. Kudekar, T. Richardson, and R. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC,” 2010, e-print: http://arxiv.org/abs/1001.1826.
  * [3] M. Lentmaier and G. P. Fettweis, “On the thresholds of generalized LDPC convolutional codes based on protographs,” in _Proc. of the IEEE Int. Symposium on Inform. Theory_ , Austing, TX, USA, June 2010, pp. 709–713.
  * [4] S. Kudekar, C. Méasson, T. Richardson, and R. Urbanke, “Threshold saturation on BMS channels via spatial coupling,” Apr. 2010, e-print: http://arxiv.org/abs/1004.3742.
  * [5] S. Kudekar and H. D. Pfister, “The effect of spatial coupling on compressive sensing,” in _Proc. of the Allerton Conf. on Commun., Control, and Computing_ , Monticello, IL, USA, 2010.
  * [6] S. H. Hassani, N. Macris, and R. Urbanke, “Coupled graphical models and their thresholds,” in _Proc. of the IEEE Inform. Theory Workshop_ , Dublin, Ireland, Sept. 2010.
  * [7] V. Rathi, R. Urbanke, M. Andersson, and M. Skoglund, “Rate-equivocation optimally spatially coupled LDPC codes for the BEC wiretap channel,” 2010, e-print: http://arxiv.org/abs/1010.1669.
  * [8] H. D. Pfister, “On the capacity of finite state channels and the analysis of convolutional accumulate-$m$ codes,” Ph.D. dissertation, UCSD, San Diego, CA, USA, 2003.
  * [9] H. D. Pfister and P. H. Siegel, “Joint iterative decoding of LDPC codes for channels with memory and erasure noise,” _IEEE J. Sel. Area. Commun._ , vol. 26, no. 2, pp. 320–337, Feb. 2008.
  * [10] T. Richardson and R. Urbanke, _Modern Coding Theory_. Cambridge University Press, 2008.
  * [11] C. Méasson, A. Montanari, T. Richardson, and R. Urbanke, “The generalized area theorem and some of its consequences,” _IEEE Trans. Inform. Theory_ , vol. 55, no. 11, pp. 4793–4821, Nov. 2009.
  * [12] K. Kasai and K. Sakaniwa, “Spatially-coupled bounded-density capacity-achieving codes,” in _Proc. Symp. on Inf. Theory and its Applications_ , Dec. 2010, pp. 1–6, (in Japanese).