#  YITP-22-23
Tensors and Algebras: An algebraic spacetime interpretation for tensor models.

Dennis<EMAIL_ADDRESS>
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

(March 2022)

The quest for a consistent theory for quantum gravity is one of the most
challenging problems in theoretical high-energy physics. An often-used
approach is to describe the gravitational degrees of freedom by the metric
tensor or related variables, and finding a way to quantise this. In the
canonical tensor model, the gravitational degrees of freedom are encoded in a
tensorial quantity $P_{abc}$, and this quantity is subsequently quantised.
This makes the quantisation much more straightforward mathematically, but the
interpretation of this tensor as a spacetime is less evident. In this work we
take a first step towards fully understanding the relationship to spacetime.
By considering $P_{abc}$ as the generator of an algebra of functions, we first
describe how we can recover the topology and the measure of a compact
Riemannian manifold. Using the tensor rank decomposition, we then generalise
this principle in order to have a well-defined notion of the topology and
geometry for a large class of tensors $P_{abc}$. We provide some examples of
the emergence of a topology and measure of both exact and perturbed Riemannian
manifolds, and of a purely algebraically-defined space called the semi-local
circle.

##  1 Introduction

One of the pinnacles of high-energy physics is the development of the standard
model of particle physics. This model uses the framework of quantum field
theory to describe the interaction of three of the four fundamental forces
between elementary particles. One of the fundamental forces, gravity, has not
yet successfully been added to this fundamental description of the universe.
Most of the calculations of the standard model rely on perturbative
renormalisation procedures, and this poses the biggest theoretical challenge
in adding gravity to this picture. This is because general relativity, the
theory that has been extremely successful in describing classical gravity [1,
2, 3, 4], is perturbatively non-renormalisable, which makes the theory lose
its predictive power at high energies [5, 6]. Experimentally it has been
proven extremely difficult to actually do measurements in the quantum regime
of gravity, since quantum effects are expected to play a role at the level of
the Planck energy, a scale far ouch of reach for direct measurements
currently.

One way to handle the perturbative renormalisation issues is to treat gravity
in a non-perturbative way. The expectation of this approach is that, though
the more straightforward perturbative renormalisation approaches do not work
for gravity, treating gravity in a non-perturbative way might solve these
issues - albeit mathematically more challenging. There are various approaches
to this, all with some levels of success. For instance, one could try to
reformulate renormalisation in a non-perturbative way and try to find a high-
energy completion of the theory this way, which is the general strategy of the
asymptotic safety programme [7, 8, 9]. Alternatively, one could consider a
canonical quantisation approach as is in loop quantum gravity, reformulating
general relativity using Ashtekar variables [10, 11]. Another non-perturbative
way of approaching quantum gravity is done by regularising the path integral
using small building blocks, which is the fundamental idea behind for instance
(causal) dynamical triangulation [12, 13, 14] and the usual tensor models [15,
16, 17].

Tensor models are an interesting approach introduced as a generalisation of
matrix models, which were successful in describing two-dimensional quantum
gravity. The general idea is that, order-$d$ tensors generate $d$-dimensional
space-times by gluing simplices together according to contractions between
tensors, however it turns out that this does not result in the emergence of
macroscopic spacetimes.222As an aside, there are also the so-called coloured
tensor models which might have a relationship to gravity through holography,
due to the emergence of so-called melonic graphs [18, 19, 20]. In the
dynamical triangulation approach, causal dynamical triangulation seemed to fix
many of these issues by introducing a notion of time by a causal requirement
on the allowed triangulations. In tensor models, however, introducing such a
restriction is unnatural. This led to the introduction of the canonical tensor
model [21, 22], which aims to describe a tensor model built from first
principles in the Hamiltonian framework with algebraic similarities to the
ADM-formalism of general relativity [23]. This comes at the expense of a
straightforward spacetime interpretation, though many connections to general
relativity have been found implying that it can be interpreted as a model for
quantum gravity [24, 25, 26]. The mathematically straightforward way of
quantising the model [27], and the interesting results from wave functions of
the model [27, 28, 29, 30, 31, 32, 33, 34, 35], make it interesting to
investigate.

Except for the canonical tensor model, all the examples above describe
approaches where the quantum theory of gravity is described by quantising
spacetime, a manifold with a (pseudo)-Riemannian metric or related variables.
Many of the difficulties when constructing this kind of quantum theory can be
traced back to the fact that this configuration space, for instance the
configuration space of all $4$-dimensional metrics modulo the diffeomorphisms,
is a very difficult configuration space to handle. The approach taken in this
work is different, and the philosophy behind it partially overlaps with
approaches in non-commutative geometry [36], namely by describing a manifold
through the algebra of functions on it.

It is a well-known fact in algebraic geometry that topological spaces,
$\mathcal{T}$, and the (real) algebra of functions on them, $C(\mathcal{T})$,
are dual to each other through the Gelfand-Naimark theorem [37]. Remarkably,
it is even possible to reconstruct the full manifold structure of a manifold
$\mathcal{M}$, including charts and atlases, purely from the algebra of real
smooth functions $C^{\infty}(\mathcal{M})$ [38]. This means that knowing only
the multiplication rules of elements of an _abstract_ algebra
$\mathcal{A}\cong C^{\infty}(\mathcal{M})$ is enough to reconstruct the
manifold. One important benefit is that, in the case of a compact Riemannian
manifold, the algebra of smooth functions is a countably-infinite dimensional
vectorspace, which as a configuration space is relatively easy to handle. This
is in contrast to for instance the configuration space of metrics over a
compact Riemannian manifold, which is overcountably infinite-dimensional.
There is one caveat here, namely that this only works for the topological
degrees of freedom. In order to reconstruct the full (pseudo-)Riemannian
manifold one needs more information, for instance through a spectral triple
approach [39, 40].

This work introduces a framework to build a theory using tensors in a
different way than the original tensor models, namely by relating them to an
algebra that is supposed to describe an algebra of functions, revisiting some
of the ideas in [41, 42, 43, 44, 45, 46, 47]. The idea is that a tensor of
order three, $P_{abc}$, describes an algebra as the structure coefficients of
the algebra for basis-elements $\\{f_{a}\\}$

$f_{a}\cdot f_{b}=\sum_{c}P_{abc}f_{c}.$

An algebra that is an algebra of functions is supposed to be commutative and
associative. While guaranteeing commutativity is straightforwardly done by
taking the tensor to be symmetric under permutations of the first two indices,
guaranteeing associativity requires more work. It is important to do this,
because in a quantum theory for gravity using the tensor $P_{abc}$ as the
fundamental variable, quantum perturbations might break the associativity. To
this end, we define the notion of the associative closure, in order to link a,
either finite- or infinite-dimensional, tensor $P_{abc}$ to an associative
algebra. This algebra might be infinite-dimensional, even if one starts with a
finite-dimensional tensor. Furthermore, part of the geometry of the Riemannian
manifold, namely the measure, is recovered by assuming the algebra to be a
Hilbert space such that $P_{abc}$ is fully symmetric. Similarly to the
spectral triple of non-commutative geometry, it will be argued that it is
possible to include (and recover) the full geometric information of the
(compact) Riemannian manifold.

This work is motivated by the canonical tensor model described above, as the
original goal was to give a potential spacetime interpretation for it. As this
model is set in the canonical (Hamiltonian) framework, the focus is on
Riemannian manifolds as they are supposed to represent spatial slices of
spacetime. It should be noted that it is not sure whether this is the most
appropriate spacetime interpretation of the canonical tensor model, and this
framework might be used for other models as well. Some potential implications
to the canonical tensor model are discussed in this work, as there seem to be
some interesting and encouraging concequences to using this interpretation.
This framework also explains some of the topological and geometric results
found before in the context of the canonical tensor model by using data
analytic methods in [48].

This work is organised as follows. In section 2 the duality between
topological spaces and the algebra of functions on them is reviewed. It is
then explained how one can construct a tensor from this algebra, and the
example of the exact circle $S^{1}$ is introduced. Section 3 introduces the
notion of an associative closure, to show how this framework can deal with
tensors that do not correspond to associative algebras directly. Section 4
then further develops this formalism, to show general ways in which an
associative closure may be found. The example of the exact circle is
revisited, and it is shown how from a five-dimensional symmetric tensor one
can reconstruct the full algebra of smooth functions with a measure on it. In
section 5 is shown how one can generate a unit, if it is not trivially present
in the algebra generated by the tensor yet. This is also where we identify a
potential way to include more information about the geometry in a tensor.
Section 6 then discusses a few examples of perturbations of the exact circle,
an example of a purely algebraically-defined space called the semi-local
circle, and the exact sphere. After this, in section 7 some implications to
the canonical tensor model are discussed and finally section 8 concludes this
work.

##  2 The duality between algebras and spaces, and the role of tensors.

In this section we briefly review the duality between smooth manifolds
$\mathcal{M}$ and the real algebra of smooth functions
$C^{\infty}(\mathcal{M})$, and develop an understanding of the role that
tensors play in describing the algebra and the measure they induce. In the end
of the section we will describe an example of the circle. While we will mainly
focus on compact Riemannian manifolds, much of this can be generalised to more
general situations. In the following section we introduce the definitions
required to understand how these algebras can be constructed from finite-
dimensional tensors. In section 4 we apply these definitions and give explicit
constructions to find the topological space and a measure corresponding to a
finite-dimensional tensor.

Let us start by considering a smooth manifold $\mathcal{M}$. The real smooth
functions on $\mathcal{M}$, denoted as $C^{\infty}(\mathcal{M})$, form an
infinite-dimensional vectorspace, equipped with a pointwise product, together
called the algebra of smooth functions on $\mathcal{M}$. In the following we
will show the duality between the space $\mathcal{M}$ and the algebra of
smooth functions on it. In particular, we will reconstruct the set of points
of the manifold and the topology. For further information on the
reconstruction of the full smooth manifold structure (including charts and
atlases) from the algebra of smooth functions we would like to refer to [38].

Consider an abstract real unital associative commutative algebra
$(\mathcal{F},\cdot)$, where $\mathcal{F}$ denotes the linear space and
$\cdot:\mathcal{F}\times\mathcal{F}\rightarrow\mathcal{F}$ the product
operation. A linear map $p:\mathcal{F}\rightarrow\mathbb{R}$ is called an
$\mathbb{R}$-algebra homomorphism if, besides the linearity conditions, it
respects the product of the algebra, i.e. $\forall
f,g\in\mathcal{F},\,p(f\cdot g)=p(f)\cdot p(g)$. The dual space of the algebra
is then defined as all $\mathbb{R}$-algebra homomorphisms of $\mathcal{F}$:

$|\mathcal{F}|:=\\{p:\mathcal{F}\rightarrow\mathbb{R}\,|\,\forall
f,g\in\mathcal{F},\,p(f\cdot g)=p(f)\cdot p(g)\\}.$ (1)

This dual space, henceforth called the algebraic dual space, is the candidate
“set of points”, the main reason being that the evaluation maps
$x\in\mathcal{M},\,p_{x}(f)\equiv f(x)$ are naturally homomorphisms. An
algebra is called geometric if the algebra $\mathcal{F}$ actually corresponds
to functions on $|\mathcal{F}|$ by identifying $f(p)=p(f)$ [38], and pointwise
defined algebras are precisely that. Note that we will often use this
identification in this work depending on the context, if we want to either
view $p$ as a point or as a homomorphism.

Figure 1: An illustration of the construction of the topology on
$|\mathcal{F}|$.

The topology on $|\mathcal{F}|$ is now defined as the weakest topology such
that the functions $f\in\mathcal{F}$ become continuous, inheriting the
continuity from $\mathbb{R}$. To make this precise, we can construct a basis
for the topology on $|\mathcal{F}|$ by taking all open subsets
$V\subset\mathbb{R}$, and using $f^{-1}(V)$ for all $f\in\mathcal{F}$ as a
basis for the topology. For an illustration of this, see figure 1. Taking an
algebra of smooth functions $C^{\infty}(\mathcal{M})$, we can thus reconstruct
$\mathcal{M}=|C^{\infty}(\mathcal{M})|$.

As mentioned before, it is possible to reconstruct the full topological
manifold structure from the algebra of smooth functions. In this work, we will
not go deeper into this, as we are mainly interested here in the
reconstruction of the topology and measure from a tensor. It is worth noting
however that we are mainly interested in algebras that correspond to smooth
algebras of compact Riemannian manifolds. This has profound implications, as a
compact Riemannian manifold carries a natural measure, locally infinitesimally
${\rm d}^{d}x\,\sqrt{\det q(x)}$ where $q$ denotes the metric tensor field,
$d$ the dimension of the Riemannian manifold, and $x$ the local coordinates.
Because of this we can define a natural inner product on the space of smooth
functions, called the $L^{2}$-inner product:

$\braket{f}{g}:=\int_{\mathcal{M}}{\rm d}^{d}x\,\sqrt{\det q(x)}f(x)g(x).$ (2)

On a compact Riemannian manifold, we arrive at the square integrable functions
$L^{2}(\mathcal{M})$ by taking the closure of $C^{\infty}(\mathcal{M})$ with
respect to the inner product above. $L^{2}(\mathcal{M})$ is a countably
infinite dimensional Hilbert space, which means that we can choose an
orthonormal Schauder basis

$\\{f_{a}\,|\,a\geq 1\\}.$ (3)

This will be important below.

In order to understand how tensors play a role in this, let us consider a
simple operator which we call the product operator

$\displaystyle\mathcal{P}:\mathcal{F}\times\mathcal{F}$
$\displaystyle\rightarrow\mathcal{F},$ (4) $\displaystyle(f,g)$
$\displaystyle\rightarrow f\cdot g.$

By construction this operator fixes the whole algebra, meaning that knowing
the decomposition of this operator in terms of any chosen basis is enough to
fix the algebra. This leads to the definition of the tensorial quantity we
have been looking for:

$P_{ab}{}^{c}:=\braket{f_{c}\,}{\,\mathcal{P}(f_{a},f_{b})}.$ (5)

Another way to view this tensor is as the structure coefficients of the
algebra, as

$f_{a}\cdot f_{b}=:\sum_{c\geq 1}P_{ab}{}^{c}f_{c}.$ (6)

Note that, as mentioned before, using this decomposition, one can recover
every possible product in the algebra. Taking $f,g\in\mathcal{F}$, with
$f=\sum_{a\geq 1}\alpha^{a}f_{a}$ and $g=\sum_{b\geq 1}\beta^{b}f_{b}$ gives

$f\cdot g=\sum_{a\geq 1}\sum_{b\geq 1}\alpha^{a}\beta^{b}f_{a}\cdot
f_{b}=\sum_{a,b,c\geq 1}\alpha^{a}\beta^{b}P_{ab}{}^{c}f_{c}.$

Equation (6) lends itself to find the dual space (1), as the elements of this
dual space are a subset of the linear dual space $\mathcal{F}^{*}$.333Note
that here we call the linear dual space the space of all (bounded) linear
functionals, while the algebraic dual space are the homomorphisms of the
algebra as defined in (1). To do this, consider a homomorphism
$p:\mathcal{F}\rightarrow\mathbb{R}$. The linear dual space is isomorphic to
$\mathcal{F}$, as we are assuming a Hilbert space structure, so we can use the
dual elements of the basis $\\{f_{a}\\}$, let us denote these elements by
$\\{\alpha^{a}\\}$ such that $\alpha^{a}(f_{b})=\delta^{a}_{b}$, as a basis of
$\mathcal{F}^{*}$. Specifically we can write $p$ as $p=\sum_{a\geq
1}p_{a}\alpha^{a}$. In terms of our original basis, the components $p_{a}$ can
now be found by evaluating $p(f_{a})=\sum_{b\geq
1}p_{b}\,\alpha^{b}(f_{a})=p_{a}$. Given $P_{ab}{}^{c}$, we can find the
algebraic dual space of the algebra by restricting $p\in\mathcal{F}^{*}$ to
the hypersurface where

$p_{a}p_{b}=\sum_{c\geq 1}P_{ab}{}^{c}p_{c},$ (7)

and we now understand that these components can be understood as evaluation
maps of the basis functions $f_{a}(p)\equiv p(f_{a})=p_{a}$.

The idea is to build a theory around these coefficients $P_{ab}{}^{c}$ in the
form of a tensor, where changes in the coefficients change the algebra and
thus affect the corresponding manifold. A thing to note is that, since the
algebra is commutative and the product operation on the pointwise defined
algebras discussed here is self-adjoint, the structure coefficients will be
totally symmetric:
$P_{ab}{}^{c}=P_{ca}{}^{b}=P_{bc}{}^{a}=P_{ba}{}^{c}=P_{cb}{}^{a}=P_{ac}{}^{b}$.
This means that constructing a theory that guarantees the emergence of
commutative algebras with an inner-product structure is equivalent to focusing
on totally symmetric tensors, as is done in the canonical tensor model for
instance.

Until now we have been careful to always write $P_{ab}{}^{c}$ with one upper
index and two lower, but since we have a Hilbert space structure with a
totally symmetric tensor it is not really necessary to keep this upper index
most of the time, and simply write $P_{abc}$. One can view this as “lowering
the index” with a metric $g_{ab}=\braket{f_{a}}{\,f_{b}}=\delta_{ab}$ as

$P_{abc}=g_{cd}\,P_{ab}{}^{d}.$

In this work, we will usually use the lower-index notation $P_{abc}$ as this
is also the notation used in the canonical tensor model. However, in some
cases we will still write $P_{ab}{}^{c}$ if we wish to emphasise the
relationship to the linear dual space $\mathcal{F}^{*}$.

For the correspondence between tensors and spaces mentioned above to lead to a
theory of gravity there is some more work to be done. The main factors we will
address here are as follows:

  * •

Quantum perturbations might affect the algebra such that it is not associative
anymore. Non-associative algebras can never correspond to a pointwise product,
hence there is a need to treat these algebras. In this work we try to resolve
this by looking for an associative closure of the algebra, such that a tensor
$P_{ab}{}^{c}$ actually corresponds to an associative algebra in section 3.

  * •

A theory for gravity should actually influence the metric on the Riemannian
manifold, not just the topology. We take a first step to this in section 4,
where we reproduce the measure on the manifold.

  * •

Similarly, not every algebra has a well-defined unit. We will describe how to
generate a unit in an algebra in section 5, and we will also see that this
gives us an opportunity to describe more (or all) of the geometric information
of the Riemannian manifold.

We will get back to this later, but first we turn our attention to the example
of the exact circle.

The exact circle is the example we will use to develop our understanding of
the representation of tensors, after which we will consider different algebras
later on in section 6. The exact circle is given by the 1-dimensional circle
$S^{1}$ with a flat metric on it. One choice of basis for the smooth functions
on the 1-dimensional circle is given by functions of the form444These are
exactly the eigenfunctions of the Laplace-Beltrami operator on the circle,
which will be discussed in section 5. (normalised according to the
$L^{2}(S^{1})$ inner product)

$\left\\{\frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}\sin(x),\frac{1}{\sqrt{\pi}}\cos(x),\frac{1}{\sqrt{\pi}}\sin(2x),\frac{1}{\sqrt{\pi}}\cos(2x),\frac{1}{\sqrt{\pi}}\sin(3x),\frac{1}{\sqrt{\pi}}\cos(3x),\ldots\right\\},$
(8)

with $x\in(0,2\pi]$. By evaluating the product of the basis-functions above,
we can find the structure constants by using (6). For example, from the
product

$f_{2}\cdot
f_{2}=\frac{1}{\pi}\sin(x)\cdot\sin(x)=\frac{1}{2\pi}\left(1-\cos(2x)\right)=\frac{1}{\sqrt{2\pi}}f_{1}-\frac{1}{2\sqrt{\pi}}f_{5},$
(9)

we find

$P_{22}{}^{1}=\frac{1}{\sqrt{2\pi}},\ P_{22}{}^{5}=-\frac{1}{2\sqrt{\pi}},\
P_{22}{}^{a}=0\ \text{(for }a\neq 1,5\text{).}$ (10)

Actually, we can find the entries of the tensor by considering the basic
multiplication rules

$\displaystyle\sin(n\,x)\cdot\sin(m\,x)$
$\displaystyle=\frac{1}{2}\left(\cos\left((m-n)x\right)-\cos\left((m+n)x\right)\right),$
(11) $\displaystyle\cos(n\,x)\cdot\sin(m\,x)$
$\displaystyle=\frac{1}{2}\left(\sin\left((m-n)x\right)+\sin\left((m+n)x\right)\right),$
$\displaystyle\cos(n\,x)\cdot\cos(m\,x)$
$\displaystyle=\frac{1}{2}\left(\cos\left((m-n)x\right)+\cos\left((m+n)x\right)\right).$

From this we can write out the full tensor $P_{ab}{}^{c}$. Alternatively we
can find these components by evaluating (5), for instance

$P_{22}{}^{5}=\frac{1}{\pi^{3/2}}\int_{0}^{2\pi}{\rm
d}x\,\sin(x)\sin(x)\cos(2x)=-\frac{1}{2\sqrt{\pi}}.$ (12)

The claim of this section is now that we can reconstruct the topology of the
circle, just by knowing the tensor $P_{ab}{}^{c}$. To further convince
ourselves of this, let us try to find all the homomorphisms of the first few
functions $\\{1,\sin(x),\cos(x)\\}$. By inspecting (11) we can see that all
information of the products of these functions is already present if we
restrict the tensor $P_{ab}{}^{c}$ to labels up to $a,b,c\leq N=5$. This
restriction effectively means the tensor acts on
$\left.\mathcal{F}=C^{\infty}(S^{1})\right|_{N=5}$ which is then isomorphic to
$\mathbb{R}^{5}$. In section 3 we will properly define some of the notions
described below by considering partial algebras.

Figure 2: The homomorphisms of the first three functions $f_{1},f_{2},f_{3}$
of the circle. As described in the text, we calculated $10,000$ homomorphisms
for these functions, and plotted the list $p^{i}(f_{a})$. The homomorphisms
where calculated using Mathematica by minimising
$(p_{a}p_{b}-\sum_{c=1}^{5}P_{ab}{}^{c}p_{c})^{2}$ and only keeping the ones
that evaluated to zero (excluding the trivial zero-map), then they were
ordered as described in (14) (note that the first point is arbitrary, thus
there is an arbitrary shift). This is supposed to represent a topological
circle, so slight deformations are merely an artifact of the random initial
conditions procedure for the minimizing function of Mathematica. The rigid
circle will be found once we take the measure into account in section 4.

Finding the set of points of $S^{1}$ can then be done by finding all solutions
to (7). Specifically, as we only consider the first three functions here,

$p\in\mathcal{F}^{*}\cong\mathbb{R}^{5},\,a,b\in\\{1,2,3\\}:\
p_{a}p_{b}=\sum_{c=1}^{5}P_{ab}{}^{c}p_{c}.$ (13)

All $p$ satisfying the above equation yield $|\mathcal{F}|$ as in (1). Then,
in order to introduce a topology on $|\mathcal{F}|$ we use the construction
explained below (1). To illustrate this in a figure, we calculated $10,000$
points that satisfy (13), which are labeled by $p^{i},\
i\in\\{1,\ldots,10,000\\}$. In order to visualise the topology, consider the
elementary definition of continuity,

$\lim_{\epsilon\rightarrow 0^{+}}f_{a}(p)-f_{a}(p+\epsilon)\rightarrow 0.$

Of course, when only considering a finite number of points, this will never go
to zero exactly. However, it does make sense to visualise this by arranging
the points such that the absolute value of

$f_{a}(p^{i})-f_{a}(p^{i+1})=p_{a}^{i}-p_{a}^{i+1}$

is as small as possible. To do this, start with a point, say $p^{1}$. Then,
take $p^{2}$ to be the point that minimises
$\sum_{a=1}^{5}|p_{a}^{1}-p_{a}^{2}|^{2}$. This is then done for all points
going forward, such that

$\sum_{a=1}^{5}|p_{a}^{i}-p_{a}^{i+1}|^{2}$ (14)

is minimised for every $i<10,000$. The result of this is shown in figure 2,
note that the form of these functions really resembles the functions
$\\{\frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}\sin(x),\frac{1}{\sqrt{\pi}}\cos(x)\\}$
up to deformations and a phase.

A last comment before ending this section is related to the persistent
homology technique introduced in the context of the canonical tensor model in
[48], and subsequently used in [32] for further analysis. In that analysis,
points $p^{i}$ were defined as generating vector elements of the tensor rank
decomposition of a tensor $P_{abc}=\sum_{i=1}^{R}p^{i}_{a}p^{i}_{b}p^{i}_{c}$,
where $R$ denotes the rank of the tensor, and an inner product on this space
of points was introduced for $p^{i},p^{j}\in|\mathcal{F}|$ as

$p^{i}\cdot p^{j}=\sum_{a}p_{a}^{i}p_{a}^{j}.$ (15)

Subsequently, points were argued to be “close” if their inner product was
large. From the point of view of the present work, this intuition makes total
sense to understand the topological properties of the space: it is nothing but
minimising (14). Why elements of the tensor rank decomposition may be seen as
elements of $|\mathcal{F}|$ will be discussed in section 4.

##  3 Defining the associative closure.

In the previous section we developed an understanding for the emergence of a
topology from infinite-dimensional tensors, and in the example of the exact
circle we also already got a glimpse of the emergence of a topology from just
a finite subset of such an algebra. In this section we will develop this
further, where we will define the associative closure of a (either finite- or
infinite-dimensional) tensor $P_{abc}$. After this, in section 4, we will
describe how to construct an associative closure for finite-dimensional
tensors.

First, let us explain why we want to look for an associative closure of a
tensor $P_{abc}$. Consider an $N$-dimensional tensor $P_{abc}$, and we assume
this tensor generates an algebra

$f_{a}\cdot f_{b}=\sum_{c=1}^{N}P_{abc}f_{c}.$ (16)

One could use this interpretation of the tensors and try to find an
interpretation of this algebra, but doing this would cause several issues.
First of all, we are looking at ways to understand the $P_{abc}$ as an algebra
of functions over a certain space. This algebra of functions is always
associative and commutative, so, if we hope to understand it as such, an
understanding of this tensor as an _associative_ algebra is important which is
generally not the case. Secondly, while a full theory of quantum gravity might
correspond to a certain $N\rightarrow\infty$ limit, in practice we will use
finite-dimensional tensors in for instance computer simulations. These finite-
dimensional tensors, when constructed using a certain cut-off from their
infinite-dimensional counterparts, will generate non-associative algebras when
defined as (16). Understanding how these tensors relate to the associative
algebras they are constructed from is important.

Let us explicitly show the non-associativity in the case of the five-
dimensional algebra of the exact circle of figure 2. One example of a non-
associative product is

$\displaystyle f_{2}\cdot(f_{3}\cdot f_{4})$
$\displaystyle=f_{2}\cdot\sum_{a=1}^{5}P_{34a}f_{a}=\frac{1}{2\sqrt{\pi}}f_{2}\cdot
f_{2}=\frac{1}{4\pi}(\sqrt{2}f_{1}-f_{5}),$ $\displaystyle(f_{2}\cdot
f_{3})\cdot f_{4}$ $\displaystyle=\sum_{a=1}^{5}P_{23a}f_{a}\cdot
f_{4}=\frac{1}{2\sqrt{\pi}}f_{4}\cdot f_{4}=\frac{1}{2\sqrt{2}\pi}f_{1}.$

It can be seen that this non-associativity is due to the cutoff introduced by
a finite $N$, as for $C^{\infty}(S^{1})$

$\displaystyle\sin(x)(\cos(x)\sin(2x))$
$\displaystyle=\frac{1}{2}\sin(x)(\sin(x)+\sin(3x))=\frac{1}{4}(1-\cos(4x)),$
$\displaystyle(\sin(x)\cos(x))\sin(2x)$
$\displaystyle=\frac{1}{2}\sin(2x)\sin(2x)=\frac{1}{4}(1-\cos(4x)).$

In the finite-dimensional case the $\sin(3x)$ contribution “drops out”,
because of which the product is not associative anymore.

To properly define the associative closure, we first need to introduce some
other notions first.

######  Definition 1.

Consider a vectorspace $\mathcal{F}$. A sub-vectorspace
$\mathcal{S}\subset\mathcal{F}$ with a bilinear product operator
$\mathcal{P}:\mathcal{F}\times\mathcal{F}\rightarrow\mathcal{F}$ is called a
_partial algebra_. The partial algebra is unital if
$\exists\,1\in\mathcal{S},\,\forall f\in\mathcal{S}:\mathcal{P}(1,f)=f$.
Similarly, the partial algebra is commutative if $\forall
f,g\in\mathcal{S},\,\mathcal{P}(f,g)=\mathcal{P}(g,f)$ and associative if
$\forall
f,g,h\in\mathcal{S},\,\mathcal{P}(f,\mathcal{P}(g,h))=\mathcal{P}(\mathcal{P}(f,g),h)$.

Note that for the partial algebra, for the commutativity and associativity
conditions, the result of $\mathcal{P}(f,g)$ (and the others) can be lie
outside of $\mathcal{S}$, in $\mathcal{F}$. For a finite-dimensional or
countably infinite dimensional partial algebra, the operator $\mathcal{P}$ may
be decomposed as a tensor $P_{abc}$ in a similar fashion as described in
section 2.

######  Definition 2.

The _algebraic dual-space_ of a _partial_ algebra
$\mathcal{S}\subset\mathcal{F}$ with operator $\mathcal{P}$, denoted
$|\mathcal{S}|$, are all $p\in\mathcal{F}^{*}$ that are homomorphisms of the
partial algebra in the sense that

$|\mathcal{S}|:=\left\\{p\in\mathcal{F}^{*}|\,\forall f,g\in\mathcal{S},\
p\left(\mathcal{P}(f,g)\right)=p(f)p(g)\right\\}.$ (17)

Note that while the definition is similar to (1), the crucial differences here
are that the actual product of two vectors may lie outside of the partial
algebra, and the elements $p\in|\mathcal{S}|$ are elements of
$\mathcal{F}^{*}$, but not $\mathcal{S}^{*}$, in general. In a sense, we are
looking at sub-vectorspaces with a product that is closed not in itself, but
in the whole vectorspace. The general idea is that while we want to find an
associative closure, we will not want to alter the part of the algebra that is
already associative.

Let us go back to the example of the exact circle introduced in section 2.
Here we discussed the tensor $P_{abc}$ acting on a five-dimensional
vectorspace. In the language introduced here, the algebra discussed is given
by $\mathcal{F}\cong\mathbb{R}^{5}$, and there is a partial algebra
$\mathcal{S}=\text{span}\\{f_{1},f_{2},f_{3}\\}\cong\mathbb{R}^{3}$. This
partial algebra can be verified to be unital, commutative and associative. Its
algebraic dual space, $|\mathcal{S}|\subset\mathbb{R}^{5}$, is discussed in
figure 2. The original algebra $C^{\infty}(S^{1})$ would correspond to the
associative closure we wish to define below.

In order to be able to describe more general systems than those with an
obvious candidate for a partial algebra as the exact circle, we need some more
notions. They will be introduced here, but later on in section 6 their use
will become more clear.

The _range_ of a partial algebra is defined as the sub-vectorspace
$\mathcal{K}$ of $\mathcal{F}$,
$\mathcal{S}\subset\mathcal{K}\subset\mathcal{F}$, which is reached by
evaluating products of elements of $\mathcal{S}$.

$\mathcal{K}^{(\mathcal{S})}:=\\{\mathcal{P}(f,g)|\,f,g\in\mathcal{S}\\}.$

A _system_ of partial algebras is a set of partial algebras
$\\{\mathcal{S}_{i}|\,i\in\mathcal{I}\\}$ with ranges
$\\{\mathcal{K}_{i}|\,i\in\mathcal{I}\\}$, where $\mathcal{I}$ is some index
set, such that for every pair $(\mathcal{S}_{i},\mathcal{S}_{j})$ their dual
spaces have a nontrivial intersection, i.e. $|S_{i}|\cap|S_{j}|\neq\emptyset$.
The range of the system is defined as the union of the ranges of the partial
algebras

$\mathcal{K}^{(\\{\mathcal{S}_{i}\\})}:=\bigcup_{\\{i\in\mathcal{I}\\}}\mathcal{K}^{(\mathcal{S}_{i})}.$

Note that this is the union and not the sum of vector spaces. The system
$\\{\mathcal{S}_{i}\\}$ is called _maximal_ if there is no system, say
$\\{\mathcal{T}_{j}\\}$, with a larger range, i.e.
$\mathcal{K}^{(\\{\mathcal{S}_{i}\\})}\subset\mathcal{K}^{(\\{\mathcal{T}_{j}\\})}$.
This system is said to be _covering_ $\mathcal{F}$ if its range spans the
whole algebra, i.e.
$\cup_{i\in\mathcal{I}}\mathcal{K}^{(\\{\mathcal{S}_{i}\\})}=\mathcal{F}$. The
_algebraic dual-space_ of a system of partial algebras $\\{\mathcal{S}_{i}\\}$
is the intersection of all of the dual spaces, i.e.

$|\\{\mathcal{S}_{i}\\}|:=\bigcap_{i}|\mathcal{S}_{i}|.$

In the example of the exact circle, we have several unital commutative
associative partial algebras. Some examples are
$\mathcal{S}_{0}=\operatorname{span}\\{1\\}$,
$\mathcal{S}_{1}=\operatorname{span}\\{1,\sin(x)\\}$,
$\mathcal{S}_{2}=\operatorname{span}\\{1,\cos(x)\\}$,
$\mathcal{S}_{3}=\operatorname{span}\\{1,\sin(x),\cos(x)\\}$. Their ranges are
$\mathcal{K}_{0}=\operatorname{span}\\{1\\}$,
$\mathcal{K}_{1}=\operatorname{span}\\{1,\sin(x),\cos(2x)\\}$,
$\mathcal{K}_{2}=\operatorname{span}\\{1,\cos(x),\cos(2x)\\}$ and
$\mathcal{K}_{3}=\operatorname{span}\\{1,\sin(x),\cos(x),\sin(2x),\cos(2x)\\}$
respectively. An example of a non-trivial system would be the system of
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ though they are not maximal nor
covering. However, the system of only $\mathcal{S}_{3}$ is a covering system.
Another equivalent covering system would be all
$\mathcal{S}_{\alpha}=\\{1,\sin(\alpha)\sin(x)+\cos(\alpha)\cos(x)\\}$ for
$\alpha\in[0,2\pi)$. Note that every partial algebra in this system is two-
dimensional, but taking all of them together will still cover all of
$\mathcal{F}$ since
$(\sin(\alpha)\sin(x)+\cos(\alpha)\cos(x))(\sin(\alpha)\sin(x)+\cos(\alpha)\cos(x))=\frac{\sin(\alpha)^{2}}{2}(1-\cos(2x))+\sin(\alpha)\cos(\alpha)\sin(2x)+\frac{\cos(\alpha)^{2}}{2}(1+\cos(2x))$.

The reason why we are introducing all this terminology with respect to partial
algebras is twofold. Firstly, we wish to properly define a way to find
potential “signs of associativity”, since a pointwise algebra as discussed in
section 2 should be associative. These should be a good starting point of
constructing an associative closure. Secondly, we would like these signs of
associativity to actually still be present in the full associative closure. If
we for instance consider the covering partial algebra in figure 2, we would
like these functions to look the same in the final associative closure and not
look wildly different.

We now turn to the more general situation where we have a (finite-dimensional
or countably infinite-dimensional) Hilbert space $\mathcal{F}$ with a basis
$\mathcal{B}_{\mathcal{F}}=\\{f_{a}|\,a\geq 1\\}$ and a symmetric tensor
$P_{abc}$ acting on it. This tensor will certainly span an algebra by the rule
(6), however this algebra might not be unital nor associative. Here we will
assume that this algebra spanned is already unital, i.e. it contains a unit
$\mathcal{F}\ni 1\equiv\alpha_{a}f_{a}$ (using the Einstein summation
convention555In this work, whenever repeated indices are used the Einstein-
summation convention should be assumed, unless specified otherwise.) such that

$\displaystyle\forall f_{b}\in\mathcal{B}_{\mathcal{F}}$
$\displaystyle,\,1\cdot f_{b}=\alpha_{a}P_{abc}f_{c}=f_{b},$ (18)
$\displaystyle\Rightarrow\alpha_{a}P_{abc}=\delta_{bc}.$

A tensor $P_{abc}$ that generates a unital algebra as above is called a
_unital_ tensor. In section 5 we will describe a procedure to construct such a
unit in many cases, but for now we assume that the tensor $P_{abc}$ already
contains it. Note that the unit in the tensor representing the exact circle is
given by $1=\sqrt{2\pi}f_{1}$.

We now define an associative extension of a tensor $P_{abc}$ as follows

######  Definition 3.

An associative extension of a tensor $P_{abc}$ acting on a Hilbert space
$\mathcal{F}$ with basis $\mathcal{B}_{\mathcal{F}}$ is an algebra
$(\mathcal{A},\cdot)$, consisting of a Hilbert space $\mathcal{A}$, which is
an extension of $\mathcal{F}\subset\mathcal{A}$, and a product operation
$\cdot:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ satisfying:

  1. 1.

The algebra is unital, associative and commutative.

  2. 2.

The product operation $\cdot$ reduces to $P_{abc}$ on $\mathcal{F}$ in the
sense that

$\forall
f_{a},f_{b},f_{c}\in\mathcal{B}_{\mathcal{F}}\subset\mathcal{A}:\,P_{abc}=\braket{f_{c}}{\,f_{a}\cdot
f_{b}},$ (19)

  3. 3.

Every element of the algebraic dual space $p\in|\mathcal{A}|$ projected to
$\mathcal{F}^{*}$, i.e. $p|_{\mathcal{F}^{*}}$, is contained in the dual space
of some maximal system of partial algebras
$|\\{\mathcal{S}_{i}\\}|\subset\mathcal{F}^{*}$. Furthermore this projection
is injective.

Let us first explain the meaning of the requirements that are present in this
definition. The first two should be clear, we want an associative algebra that
is on $\mathcal{F}$ given by the tensor $P_{abc}$, as is expected from
pointwise defined algebras. The third condition above is introduced for two
reasons. The first reason is physical, as we do not want low-energy functions
to suddenly look differently once we are probing higher energies. Take for
instance the unit function, we do not want to consider extensions of the
algebra $\mathcal{F}$ where the unit of $\mathcal{F}$ does not correspond to
the pointwise unit, as an associative extension might have a wildly different-
looking dual space. Secondly, and this reason mainly refers to the injectivity
requirement of the restriction, we do not want to consider extensions that are
not connected to the current algebra through the product (for instance direct
sums).

The associative extension is a first step towards the notion of the
associative closure, and actually the associative closure we will define below
is an associative extension. Consider for instance the algebra of smooth
functions over a circle, from this definition it certainly is an associative
extension of the five-dimensional algebra we considered around (13) as will be
shown below. But the notion of associative extension does not have everything
we want yet. For one, it is generally not unique for a tensor $P_{abc}$.
Secondly, in the case of a finite-dimensional tensor that corresponds to a
smooth manifold, there is an infinite-dimensional extension which we are
looking for, but there are many finite-dimensional ones too. Thus we need a
way to construct the infinite-dimensional one we are looking for (for instance
the algebra of all smooth functions over a circle), if it exists.

Let us elaborate the example of the exact circle a bit more. The algebra we
are after, $C^{\infty}(S^{1})$, equipped with the $L^{2}(S^{1})$ inner product
is an algebraic extension. We can explicitly check this by going through the
three requirements mentioned above. The first two points are relatively
straightforward. Since the algebra is defined as a pointwise product, it is
necessarily commutative and associative. Furthermore, the unit is a smooth
function and thus included in this algebra making it a unital algebra. By the
definition of the tensor in (5), the algebra reduces to $P_{abc}$ when
restricted to $\mathcal{F}\cong\mathbb{R}^{5}$. However, there are many other
possible extensions of $P_{abc}$. As will be shown in section 4, using the
tensor rank decomposition it is possible to find a seven-dimensional algebra
with seven points in its algebraic dual space that is an associative extension
of the five-dimensional $P_{abc}$ from the example. In fact, there are many of
these associative extensions possible, and in a heuristic manner we can see
the associative closure we want as a kind of ”union” of those. It is however
not useful to look at the union of the algebras themselves or their dual
spaces, since the algebraic extensions are hardly comparable since they have
different dimensions. What is comparable though are the extensions and their
dual spaces when projected to $\mathcal{F}$ and $\mathcal{F}^{*}$
respectively. The restriction of an associative extension $\mathcal{A}$ to
$\mathcal{F}$ just yields $\mathcal{A}|_{\mathcal{F}}=\mathcal{F}$ as it is an
extension, thus we are left with considering the projection of the algebraic
dual space $|\mathcal{A}||_{\mathcal{F}^{*}}$.

From this, we are led to the following definition.

######  Definition 4.

A _potential homomorphism_ of a tensor $P_{abc}$ acting on $\mathcal{F}$ is a
homomorphism of an associative extension $(\mathcal{A},\cdot)$ projected to
$\mathcal{F}^{*}$. The _space of potential homomorphisms_ of $\mathcal{F}$,
denoted $|\mathcal{F}|^{(P)}$, is the collection of all homomorphisms of all
associative extensions projected to $\mathcal{F}^{*}$.

The key point of this definition is to allow us to not only consider the
homomorphisms of $\mathcal{F}$ under $P$, but consider all homomorphisms of
some bigger algebra that could generate $P$. We can think of this as the
actual space of points that we are considering. This thought then finally
leads us to the definition of the associative closure.

######  Definition 5.

An associative closure of $\mathcal{F}$ is an associative extension
$\mathcal{A}$ such that the restriction of its algebraic dual space to the
linear dual space of $\mathcal{F}$ is exactly the space of potential
homomorphisms of $\mathcal{F}$, i.e.
$\left.|\mathcal{A}|\right|_{\mathcal{F}^{*}}=|\mathcal{F}|^{(P)}$.

######  Corollary 1.

The algebraic dual space of an algebraic closure is isomorphic to the space of
potential homomorphisms, since the projection map is both injective and
surjective.

Note that while the associative closure is not necessarily unique, this is
physically not really a problem. We assume here that what we can actually
measure are the functions included in $\mathcal{F}$, and the physical space is
the space of potential homomorphisms. The associative closure is in that sense
mainly a mathematical tool to be sure that there exists a topology on the
space of potential homomorphisms. In the sections below we will explicitly
construct an associative closure for finite-dimensional tensors, but it might
be interesting to find out if the existence holds more generally. Furthermore,
though this has not been investigated further, it might be the case that the
different associative closures are related by some well-defined
transformation, for instance a diffeomorphism.

As an example, let us show that the $C^{\infty}(S^{1})$ algebra is an
associative closure of the example of the exact circle.

######  Proposition 1.

$C^{\infty}(S^{1})$ is an associative closure of the five-dimensional tensor
$P_{abc}$ discussed in the end of section 2.

###### Proof.

As has already been argued before, $C^{\infty}(S^{1})$ is an associative
extension of $P_{abc}$. What we would like to show now is that every potential
homomorphism of $P_{abc}$ is an element of
$|C^{\infty}(S^{1})||_{\mathcal{F}^{*}}$.

In this case we can use the knowledge of the partial algebra
$\mathcal{S}\cong\mathbb{R}^{3}$ mentioned above to our advantage. Firstly,
the dual space of any covering system of partial algebra has to lie in the
dual space of $\mathcal{S}$ since other systems are either equivalent, like
$\mathcal{S}_{\alpha}$ mentioned above, or more restrictive. Thus, from the
definition of any associative extension $\mathcal{A}$, the restriction of
$|\mathcal{A}|$ to $\mathcal{F}^{*}$ has to lie in $|\mathcal{S}|$, i.e.
$|\mathcal{A}||_{\mathcal{F}^{*}}\subset|\mathcal{S}|$. In other words, every
potential homomorphism must lie in $|\mathcal{S}|$. This means that, if
$|C^{\infty}(S^{1})||_{\mathcal{F}^{*}}=|\mathcal{S}|$, then necessarily every
potential homomorphism lies in $|C^{\infty}(S^{1})||_{\mathcal{F}^{*}}$ and
thus $C^{\infty}(S^{1})$ is an associative closure.

To do this, let us take an element $p\in|\mathcal{S}|$. From (13), this is any
$p\in\mathbb{R}^{5}$ such that $\forall a,b\leq 3$

$p_{a}p_{b}=\sum_{c=1}^{5}P_{abc}p_{c}.$

We can convince ourselves that for $p_{1},p_{2},p_{3}$ this already fixes the
components as evaluation maps of $f_{1}\sim
1,f_{2}\sim\sin(x),f_{3}\sim\cos(x)$. This is because the evaluation maps of
these functions are precisely all the homomorphisms
$p:C^{\infty}(S^{1})\rightarrow\mathbb{R}$, and since all the information of
their products is contained within the five-dimensional tensor $P_{abc}$, the
solutions to the above equation are nothing more but the restriction of these
homomorphisms to $\mathbb{R}^{3}$, so
$|C^{\infty}(S^{1})||_{\mathbb{R}^{3}}=|\mathcal{S}||_{\mathbb{R}^{3}}$. We
now need to fix the last two elements and show that these correspond to
restrictions of the full $C^{\infty}(S^{1})$ homomorphisms.

However, since the $C^{\infty}(S^{1})$ algebra is defined pointwise, this is
necessarily the case. For instance, let us take the case of $f_{5}$. From (9)

$f_{5}=\sqrt{2}f_{1}-2\sqrt{\pi}f_{2}\cdot f_{2},$

and taking the same homomorphism $p\in\mathbb{R}^{5}$

$f_{5}(p)=p_{5}=\sqrt{2}p_{1}-2\sqrt{\pi}p_{2}p_{2}=\sqrt{2}f_{1}(p)-2\sqrt{\pi}f_{2}(p)f_{2}(p),$

we get a pointwise definition of the function $f_{5}\sim\cos(2x)$. A similar
statement holds for $f_{4}\sim\sin(2x)$. Since every homomorphism of
$C^{\infty}(S^{1})$ necessarily respects this, we see that actually

$|C^{\infty}(S^{1})||_{\mathcal{F}^{*}}=|\mathcal{S}|.$

∎

A final note for this section is about the general structure of these
definitions. While the algebra $C^{\infty}(S^{1})$ is of course very nice in
the sense that it exactly corresponds to a smooth manifold, these definitions
can both reconstruct these nicer spaces or less “nice” spaces. The latter we
will call fuzzy spaces, since as we will see later in section 6, their
behaviour can vary from the usual continuous topological spaces.

##  4 Constructing the associative closure using a measure.

In this section we continue to understand the associative closure. We will
first describe the appearance of a measure for a broad class of tensors, and
subsequently describe a way to construct the associative closure for finite-
dimensional tensors using the tensor rank decomposition. Along the way we will
again refer to the example of the exact circle, where we reconstruct the full
infinite-dimensional algebra and the measure on the dual space from the five-
dimensional tensor constructed in section 2. Note that in this section we
still assume the tensor $P_{abc}$ to be unital as explained in section 3, but
not associative.

Generally, the associative closure of $P_{abc}$ acting on $\mathcal{F}$ may be
constructed in two steps: 1) Generate the space of potential homomorphisms
$|\mathcal{F}|^{(P)}$, 2) construct the associative closure by treating the
potential homomorphisms as evaluation maps of functions and taking pointwise
products between them to generate new functions. The result of this will then
be a new algebra that is pointwise defined, and thus associative by
construction, with an algebraic dual space isomorphic to
$|\mathcal{F}|^{(P)}$. In order to simplify the notation a bit, in this
section we will assume $P_{abc}$ to be a finite-dimensional tensor. We will
first introduce the notion of measure generated tensors, after which we will
show that these tensors have a natural associative closure. After this we will
show how to actually construct the space of potential homomorphisms and find
such a measure, where we will also state a curious conjecture: every unital
tensor is measure-generated. Along the way, we will further develop the
example of the exact circle mentioned in section 2.

###  4.1 Measure generated tensors

In this section we introduce the notion of measure generated tensors. This is
a broad class of tensors, that roughly speaking can be interpreted as
constructed using a set of points with a measure on them. The precise
definition is as follows.666For some of the notions we use from measure
theory, we would like to refer to appendix A.

######  Definition 6.

A tensor $P_{abc}$ acting on $\mathcal{F}$ is said to be _measure-generated_
if there is an associative closure $\mathcal{A}$ equipped with a measure $\mu$
on $|\mathcal{A}|$ such that

$\forall f,g\in\mathcal{A}:\,\braket{f}{\,g}=\int_{|\mathcal{A}|}{\rm
d}\mu(p)f(p)g(p).$ (20)

Note that $|\mathcal{A}|$ comes with a natural topology as explained in figure
1, thus we can construct the Borel $\sigma$-algebra on this space, on which
the measure is defined (see appendix A).

######  Proposition 2.

For any $f,g\in\mathcal{F}$

$\braket{f}{\,g}=\int_{|\mathcal{A}|}{\rm
d}\mu(p)f(p)g(p)=\int_{|\mathcal{F}|^{(P)}}{\rm d}\mu(p)f(p)g(p),$

###### Proof.

$|\mathcal{F}|^{(P)}$ is the projection of homomorphisms $p\in|\mathcal{A}|$
to $\mathcal{F}^{*}$ by the definition of the associative closure, so for
$f\in\mathcal{F},p(f)=p^{*}(f)$, where $p^{*}$ denotes the projection of $p$
to $\mathcal{F}^{*}$. Since we are only considering functions in
$\mathcal{F}$, and the spaces are isomorphic, we only have to consider
$|\mathcal{F}|^{(P)}$. ∎

######  Corollary 2.

In particular, for any two basis elements
$f_{a},f_{b}\in\mathcal{B}_{\mathcal{F}}$

$\delta_{ab}=\int_{|\mathcal{F}|^{(P)}}{\rm d}\mu(p)f_{a}(p)f_{b}(p).$ (21)

######  Proposition 3.

A measure-generated unital tensor is given by

$P_{abc}=\int_{|\mathcal{F}|^{(P)}}{\rm d}\mu(p)f_{a}(p)f_{b}(p)f_{c}(p).$
(22)

###### Proof.

Take $\mathcal{A}$ to be an associative closure of $P_{abc}$. For an element
of its algebraic dual space $p\in|\mathcal{A}|$ we have

$p(f_{a})p(f_{b})=p(f_{a}\cdot f_{b}).$

Furthermore, for any product of the basis function
$f_{a},f_{b}\in\mathcal{B}_{\mathcal{F}}$

$f_{a}\cdot f_{b}=P_{abc}f_{c}+g,$

where $P_{abc}f_{c}\in\mathcal{F}$ and $g\in\mathcal{A}\setminus\mathcal{F}$.
This comes from the definition of an associative extension, such that
$P_{abc}=\braket{f_{c}}{\,f_{a}\cdot f_{b}}$, so $\braket{f}{\,g}=0$ for all
$f\in\mathcal{F}$. In particular, for any $p\in|\mathcal{A}|$

$p(f_{a})p(f_{b})=p(f_{a}\cdot f_{b})=p(P_{abc}f_{c})+p(g).$

Since $p(f)=p^{*}(f)$ for $f\in\mathcal{F}$, where $p^{*}$ denotes the
projection of $p$ to $\mathcal{F}^{*}$, we find

$\int_{|\mathcal{F}|^{(P)}}{\rm
d}\mu(p)p(f_{a})p(f_{b})p(f_{c})=\int_{|\mathcal{A}|}{\rm
d}\mu(p)p(f_{a})p(f_{b})p(f_{c})=\int_{|\mathcal{A}|}{\rm
d}\mu(p)p(P_{abd}f_{d})p(f_{c})+\int_{|\mathcal{A}|}{\rm
d}\mu(p)p(g)p(f_{c}).$

On the right-hand-side we have the inner product as defined in (20), so we get

$\int_{|\mathcal{F}|^{(P)}}{\rm
d}\mu(p)p(f_{a})p(f_{b})p(f_{c})=P_{abd}\braket{f_{d}}{\,f_{c}}+\braket{g}{\,f_{c}}=P_{abc},$

where the last inner product is zero since $g\not\in\mathcal{F}$. ∎

In the example of the exact circle, it is clear that the above definition is
satisfied. First, the inner product on the smooth functions of the exact
circle is given by (2), this means that we can interpret the measure above as
the canonical Riemannian measure. The equation (22) follows from (5).

Let us develop some more intuition for these notions before continuing. Let us
first assume that the basis functions are given by simple functions777See
appendix A.

$f_{a}=\sum_{i=1}^{R}p^{i}_{a}\mathbf{1}_{A_{i}},$ (23)

where $R$ is the amount of disjoint regions $A_{i}\subset|\mathcal{F}|^{(P)}$,
$p^{i}_{a}$ denotes the value of the function $f_{a}$ in region $A_{i}$, and
$\mathbf{1}_{A_{i}}$ is the region’s indicator function. Given a measure on
$|\mathcal{F}|^{(P)}$ we then find

$\int_{|\mathcal{F}|^{(P)}}{\rm
d}\mu(p)f_{a}=\sum_{i=1}^{R}p^{i}_{a}\mu(A_{i})\equiv\sum_{i=1}^{R}p^{i}_{a}\beta_{i}.$

Here $\beta_{i}\equiv\mu(A_{i})>0$. Note that we assume that all of the basis
functions have the same decomposition in terms of the indicator functions.
This means that if we evaluate a product we get a similar decomposition

$f_{a}\cdot f_{b}=\sum_{i=1}^{R}p^{i}_{a}p^{i}_{b}\mathbf{1}_{A_{i}},$

and, by using the definition above in (20),

$\braket{f_{a}}{\,f_{b}}=\int_{|\mathcal{F}|^{(P)}}{\rm
d}\mu(p)f_{a}(p)f_{b}(p)=\sum_{i=1}^{R}\mu(A_{i})p^{i}_{a}p^{i}_{b}\equiv\sum_{i=1}^{R}\beta_{i}p^{i}_{a}p^{i}_{b}.$
(24)

Moreover, by using (22) we find

$P_{abc}=\sum_{i=1}^{R}\beta_{i}p^{i}_{a}p^{i}_{b}p^{i}_{c}.$ (25)

This expression may be recognised as a tensor rank decomposition of the tensor
$P_{abc}$, and it is a first sign of why the tensor rank decomposition is so
useful and has been so successful when applied to the canonical tensor model
in the past. Usually the basis functions of an algebra of functions over a
space are not given by simple functions, but a finite-dimensional part of the
algebra may still be represented by them. In section 4.3 we will see that the
tensor rank decomposition actually generates a set of $p^{i}_{a}$ and
$\beta_{i}$ which represents the algebra in a pointwise manner, and actually
corresponds to an associative extension. We will argue that taking all
possible tensor rank decompositions (under certain restrictions) will
correspond to an associative closure.

Going back to the example of the exact circle, we took $R=1.000$ points of the
10.000 points calculated in figure 2 at random. We then used Mathematica to
find a solution for (25). In the case of the circle, the $\beta_{i}$ can be
interpreted as the length of each line-segment since line-segments generate
the measurable sets of the circle and $\beta_{i}=\mu(A_{i})$ is their length.
This implies that we can approximate the functions $f_{a}$ as simple functions

$f_{a}(x)=\begin{cases}p_{a}^{1}\ ,\ 0\leq x<\beta_{1},\\\ p_{a}^{2}\ ,\
\beta_{1}\leq x<\beta_{1}+\beta_{2},\\\ \ldots\\\ p_{a}^{R}\ ,\
\sum_{i=1}^{R-1}\beta_{i}\leq x<\sum_{i=1}^{R}\beta_{i}.\\\ \end{cases}$ (26)

The result of this procedure may be found in figure 3. Note that the
circumference of the circle (the range of the plots) exactly matches $2\pi$,
and the functions look exactly like the functions we started out with in the
construction of the tensor in (8). It should be noted that this is merely an
approximation of the functions, but in the limit they will exactly represent
the original functions $\\{1,\sin(x),\cos(x)\\}$.

Figure 3: A plot of the first three basis functions of the circle, as given by
(26). As mentioned in the text, 1.000 points are used. Note that the
deformations of figure 2 disappear due to the inclusion of the information of
the measure.

While the solution in figure 3 looks exactly like the functions we are after,
it is not a definition of the full measure yet by the $\beta_{i}$. To
reconstruct the full measure, one needs to take all possible finite subsets of
$|\mathcal{F}|^{(P)}$, and take all possible solutions to (24) (or
equivalently (25)). The collection of all these different solutions will then
correspond to the measure over different regions. In terms of measure theory,
this would allow us to then integrate any function using Lebesgue integration
since we know the measure-value of the indicator functions of any region.

Note that in practice, it usually suffices to take only one set of points and
find a solution like in figure 3, similarly to the fact that for many purposes
taking a finite Riemann sum gives a good approximation of many integrals. In
section 4.2 we will show that the 1.000 points evaluated here can very
accurately generate new basis elements, at least up to $N=11$.

The notion of a measure-generated tensor is important here, and one might be
worried that this limits the scope of this section considerably. However, in
section 4.3 we will argue that many - if not all - unital tensors actually
have this property.

###  4.2 Construction of an associative closure

In this section we will show how to construct an associative closure from a
measure-generated tensor. We will assume that one already constructed the
space of potential homomorphisms. In section 4.3 we will show how one could do
this.

Let us take two basis functions $f_{a},f_{b}\in\mathcal{B}_{\mathcal{F}}$, and
consider a potential homomorphism $p\in|\mathcal{F}|^{(P)}$ and treat it as a
point, i.e. evaluation map. The pointwise product of these functions can then
be decomposed as

$f_{a}(p)f_{b}(p)=P_{abc}f_{c}(p)+g(p)=\braket{f_{c}}{\,f_{a}\cdot
f_{b}}f_{c}(p)+g(p).$ (27)

Here $g(p)$ corresponds to the difference between the pointwise product to the
product induced by $P_{abc}$. If $g(p)=0$ it means that the tensor already
describes a pointwise algebra for $f_{a},f_{b}$,888As an aside, this also
implies that there is an associative commutative partial algebra spanned by
$f_{a},f_{b}$. and we try another combination of basis functions such that
$g(p)\neq 0$ for them. If there is no such combination, the algebra described
by $P_{abc}$ is already a pointwise (associative) algebra and we are done.
From here on we assume that $g(p)\neq 0$.

The general idea is now that we see $g(p)$ as a new function that is not in
our algebra yet. By construction, this $g$ is a map

$g:|\mathcal{F}|^{(P)}\rightarrow\mathbb{R}$

In a sense we “close” the algebra by adding $g$ as a new basis element. Note
that by construction it is true that $\forall
f_{a}\in\mathcal{B}_{\mathcal{F}}$

$\int_{|\mathcal{F}|^{(P)}}{\rm d}\mu(p)f_{a}(p)g(p)=0.$

This means that actually we can extend the inner product to $g$ as well. Say
the dimension of $\mathcal{F}$ is $N$, then we can now add a new basis element

$f_{N+1}(p)=\frac{g(p)}{\|g\|},$

where $\|g\|=\sqrt{\braket{g}{\,g}}$ is the norm induced by the inner product.
$f_{N+1}$ together with $\mathcal{B}_{\mathcal{F}}$ forms an orthonormal basis
with the same inner product

$\delta_{ab}=\braket{f_{a}}{\,f_{b}}=\int_{|\mathcal{F}|^{(P)}}{\rm
d}\mu(p)f_{a}(p)f_{b}(p),$ (28)

where $a$ now runs from $1$ to $N+1$. Let us now denote this vectorspace by
$F^{(N+1)}$, and introduce a new tensor
$P_{abc}^{(N+1)}=\braket{f_{c}}{\,f_{a}\cdot f_{b}}$ for all $a,b,c\leq N+1$
with the inner product of (28).

Next we pick new $f_{a},f_{b}\in\mathcal{F}^{(N+1)}$ and restart this process
starting with (27), where we treat $\mathcal{F}^{(N+1)}$ as the vectorspace
with tensor $P_{abc}^{(N+1)}$. When we keep iterating, we will either
eventually reach a point where for every $f_{a},f_{b}$ the product is exactly
described by $P_{abc}$, i.e. $f_{a}(p)f_{b}(p)=P_{abc}f_{c}(p)$, or we
continue finding new functions. In the former, at that point we have a proper
definition of the associative closure, which will be finite-dimensional. In
the latter case, we define the associative closure as the inductive limit of
this process, producing a countably infinite-dimensional algebra.

Let us now explicitly check that the algebra constructed like this is indeed
an associative closure. For this we first need to confirm that it is an
associative extension. First note that for every element in
$p^{*}\in|\mathcal{F}|^{(P)}$ we constructed an element in the algebraic dual
space of $\mathcal{A}$, $p\in|\mathcal{A}|,p(f_{a})=f_{a}(p^{*})$, since we
basically constructed a new algebra where the original potential homomorphisms
become proper homomorphisms. Referring to definition 3, it is clear that the
first two points are satisfied by construction. Furthermore, since the every
element of the dual space $p\in|\mathcal{A}|$ projected to $\mathcal{F}^{*}$
must be an element of some associative extension by the definition of
potential homomorphisms, it necessarily lies in $|\mathcal{S}|$ for a maximal
system of partial algebras $\mathcal{S}_{i}\subset\mathcal{F}$. Since every
element $p\in|\mathcal{A}|$ is constructed from a
$p^{*}\in|\mathcal{F}|^{(P)}$, this is necessarily injective. Similarily it
follows that it is an associative closure, since for every $p\in|\mathcal{A}|$
it must hold that $p^{*}\equiv p|_{\mathcal{F}^{*}}\in|\mathcal{F}|^{(P)}$ by
definition of the space of potential homomorphisms, and we constructed a
$p\in|\mathcal{A}|$ from every $p^{*}\in|\mathcal{F}|^{(P)}$, we have an
isomorphism and thus $|\mathcal{A}||_{\mathcal{F}^{*}}=|\mathcal{F}|^{(P)}$.

Figure 4: A plot of the 10th and 11th basis functions of the exact circle
generated as described by the procedure in section 4.2 and using the simple
function inner product of (24) with the same $1.000$ points as before in
figure 3.

Let us now go back to the exact circle. As we already know that the
associative closure $\mathcal{A}\cong C^{\infty}(S^{1})$ is infinite-
dimensional, this procedure would never terminate. We used this construction
together with the simple function approximation introduced in section 4.1 to
calculate the inner product in order to extend the algebra
$\mathcal{F}\cong\mathbb{R}^{5}$ to an extension
$\mathcal{A}\cong\mathbb{R}^{11}$. The result for $f_{10}(x)\sim\sin(5x)$ and
$f_{11}(x)\sim\cos(5x)$ (up to some phase due to the random choice of the
first point) is given in figure 4. Note that technically the construction with
a finite amount of points is an associative extension and not a closure, but
using more and more points would create a more and more accurate
representation of the functions up to very high dimensions and in the limit
the associative closure.

###  4.3 Generating potential homomorphisms

In this section we show how to generate the potential homomorphisms more
generally. In the previous sections we always assumed that we knew the
potential homomorphisms of an algebra already, and in the case of the exact
circle we “got lucky” since we could simply take the partial algebra
homomorphisms as explained in (13) and properly defined in definition 2. In
this section we will use an important tool that has been extensively used in
the context of the canonical tensor model [48], the tensor rank decomposition.
We will end this section by again examining the example of the exact circle.

Let us consider a tensor rank decomposition for some (not necessarily the
lowest) $R$ of the unital tensor $P_{abc}$999Note that we assume a tensor rank
decomposition to be any decomposition of $P_{abc}$ such that (29) is satisfied
as in [49]. Oftentimes it is defined instead to be the decomposition for the
lowest $R$ possible, which we call a _minimal_ tensor rank decomposition.

$P_{abc}=\sum_{i=1}^{R}\phi^{i}_{a}\phi^{i}_{b}\phi^{i}_{c}.$ (29)

Since the tensor algebra is unital, we know for the unit $1=\alpha^{a}f_{a}$,

$\delta_{bc}=\alpha^{a}P_{abc}=\sum_{i=1}^{R}(\alpha^{a}\phi^{i}_{a})\phi^{i}_{b}\phi^{i}_{c}\equiv\sum_{i=1}^{R}\gamma_{i}\phi^{i}_{b}\phi^{i}_{c}.$
(30)

We call the tensor rank decomposition _positive_ if $\gamma_{i}>0$ for all
$i$. If we now define $\beta_{i}=(\gamma_{i})^{1/3}>0$ and
$p^{i}_{a}\equiv(\beta_{i})^{-1/3}\phi^{i}_{a}$, we see that (29) and (30)
exactly reduce to

$\displaystyle P_{abc}$
$\displaystyle=\sum_{i=1}^{R}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i},$ (31)
$\displaystyle\delta_{ab}$
$\displaystyle=\sum_{i=1}^{R}\beta_{i}p_{a}^{i}p_{b}^{i},$

in a similar fashion to (25) and (24) with $\beta_{i}>0$.

The similarity between (31) and (25) suggests that the tensor rank
decomposition has a relationship with the simple function approach using
potential homomorphisms of section 4.1. The idea in the following is to treat
the $p_{a}^{i}$ above as potential homomorphisms. If the
$p^{i}\in\mathcal{F}^{*}$ indeed correspond to potential homomorphisms
$p^{i}\in|\mathcal{F}|^{(P)}\subset\mathcal{F}^{*}$ we could keep finding
tensor rank decompositions in order to generate more and more points.
Furthermore, if we consider the interpretation introduced in section 4.1, we
already get the information of the measure for free through the $\beta_{i}$.
Then, using the procedure described in 4.2 we can reconstruct the whole
associative closure.

######  Definition 7.

A (not necessarily minimal) positive tensor rank decomposition
$P_{abc}=\sum_{i=1}^{R}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i}$ is called a
_pointwise decomposition_ if every $p_{a}^{i}\in|\mathcal{F}|^{(P)}$.

By this definition, the decomposition in (25) is a pointwise decomposition.
Thus we immediately can conclude that

######  Corollary 3.

Every measure-generated tensor has a pointwise decomposition.

Naturally, this bring up several questions. Firstly, whether or not every
tensor have a positive tensor rank decomposition. There is no conclusive proof
of this fact yet, but some numerical calculations for $N=3,4,5$ imply that it
seems to be true. The calculation was done using Mathematica and generating
1.000 random unital symmetric tensors $P_{abc}$ and trying to obtain a
positive tensor rank decomposition. The tensors were ensured to be unital by
fixing $P_{1ab}=P_{a1b}=P_{ab1}=\delta_{ab}$. In all cases a good
decomposition with an error
$(\sum_{i=1}^{R}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i}-P_{abc})^{2}<10^{-20}$
could be found. This leads to the formulation of the conjecture:

######  Conjecture 1.

Every unital symmetric real tensor has a positive tensor rank decomposition.

Though the numerical evidence suggests this to be true, it still remains to be
proven.

Secondly, if we have such a positive tensor rank decomposition, will it be a
pointwise decomposition? The answer to this question in general is no. Note
that in the above requirement that $\gamma^{i}>0$ is similar to requiring that
the unit of $P_{abc}$ becomes a pointwise unit. It does not necessarily mean
that all other functions also suddenly have proper pointwise representations.
This seems to make it necessary to check every time if indeed all of the
elements of the tensor rank decomposition $p^{i}$ correspond to potential
homomorphisms. However, the example of the exact circle that is discussed
below suggests that it seems to be the case for the _minimal_ positive tensor
rank decomposition. This leads to a second conjecture:

######  Conjecture 2.

For a unital tensor $P_{abc}$ that admits a covering system of partial
algebras, a minimal positive tensor rank decomposition is a pointwise
decomposition.

It is at the moment not known how to prove this statement, and whether it is
applicable even more generally. However, if the tensor does not admit a
covering system of partial algebras, it seems necessary to actually verify if
the $p^{i}$ are potential homomorphisms. In practice this is done by checking
if they are in the dual space of a maximal system of partial algebras. This is
also a spot where the algebraic approach and the pure tensor rank
decomposition approach might not always agree, which might also have to do
with the tensors in the quantum- and geometric phases of the canonical tensor
model [33]. In the future it might prove beneficial to either restrict the
tensor rank decomposition approach further, or broaden the definition of the
associative extension.

Once one finds the collection $p^{i}$ as described above, it is then possible
to construct an associative extension, and with all possible $p^{i}$ a
closure, of the tensor $P_{abc}$ by using the method described in section 4.2.

Using what is written above, we can then interpret the positive tensor rank
decomposition as a simple functions representation of $f\in\mathcal{F}$, as
introduced in (23). As was already pointed out in [33], if the tensor
$P_{abc}$ has a symmetry such that for a Lie-group transformation $G_{ab}$

$G_{aa^{\prime}}G_{bb^{\prime}}G_{cc^{\prime}}P_{a^{\prime}b^{\prime}c^{\prime}}=P_{abc},$

there is a continuous degeneracy of the tensor rank decomposition, since for
every $p^{i}_{a}$ belonging to a tensor rank decomposition,
$G_{aa^{\prime}}p^{i}_{a^{\prime}}$ belongs to an element of another tensor
rank decomposition

$G_{aa^{\prime}}G_{bb^{\prime}}G_{cc^{\prime}}P_{a^{\prime}b^{\prime}c^{\prime}}=\sum_{i=1}^{R}\beta_{i}(G_{aa^{\prime}}p_{a^{\prime}}^{i})(G_{bb^{\prime}}p_{b^{\prime}}^{i})(G_{cc^{\prime}}p_{c^{\prime}}^{i})=P_{abc}.$

This is the reason that we can only say that a tensor rank decomposition
corresponds to an associative extension, since not all potential homomorphisms
are generally included. Once we consider all pointwise decompositions, we will
get the whole space of potential homomorphisms $|\mathcal{F}|^{(P)}$, which we
can then use to construct the full associative closure as in section 4.2. In
[30, 29] it was already argued that the canonical tensor model seems to prefer
symmetric states, with the current discussion this implies that we can expect
the emergence of (almost) continuous spacetimes.

Figure 5: A plot of the potential homomorphisms generated by the tensor rank
decomposition of the second basis function of the exact circle, using the
representation in (26). As mentioned in the text, we used from left to right
1, 5 and 150 tensor rank decompositions, leading to 7, 35 and 1050 points
respectively. Note the agreement between the rightmost figure and figure 3
(except for a phase difference due to the random choice of the first point).

Before finishing this section we will examine the exact circle once more. The
tensor described in (9) has rank $R=7$ which can be verified by finding
solutions to
$0=(\sum_{i=1}^{R}\phi_{a}^{i}\phi_{b}^{i}\phi_{c}^{i}-P_{abc})^{2}$. In the
following, we denote by $M$ the amount of tensor rank decompositions
generated. We have found $M=150$ minimal tensor rank decompositions using
Mathematica and treated the points as defined in (31). As expected from the
discussion, these points $\\{p^{i}\\}$ appear to be potential homomorphisms
since they satisfy

$s_{1}^{a}s_{2}^{b}p_{a}^{i}p_{b}^{i}=s_{1}^{a}s_{2}^{b}P_{abc}p_{c}^{i}$ (32)

for every $s_{1},s_{2}\in\mathcal{S}_{3}$, the covering partial algebra as
defined in section 3. Moreover; we seem to find all possible homomorphisms.
After generating $M$ minimal tensor rank decompositions, the $I$th one denoted
by
$\sum_{i=1}^{R}\beta_{i}^{(I)}{p_{a}^{(I)}}^{i}{p_{b}^{(I)}}^{i}{p_{c}^{(I)}}^{i}$,
we sum them together as

$P_{abc}=\frac{1}{M}\left(\sum_{i=1}^{R}\beta_{i}^{(1)}{p_{a}^{(1)}}^{i}{p_{b}^{(1)}}^{i}{p_{c}^{(1)}}^{i}+\ldots+\sum_{i_{M}=1}^{R}\beta_{i}^{(M)}{p_{a}^{(M)}}^{i}{p_{b}^{(M)}}^{i}{p_{c}^{(M)}}^{i}\right)=\frac{1}{M}\sum_{I=1}^{M}\sum_{i=1}^{R}\beta^{(I)}_{i}{p_{a}^{(I)}}^{i}{p_{b}^{(I)}}^{i}{p_{c}^{(I)}}^{i}.$
(33)

Using the same simple function representation as in (26), we plotted the
values of $f_{2}(x)$ using the same ordering as discussed at (14). We did this
for three cases: using only $M=1$ tensor rank decomposition, using $M=5$
tensor rank decompositions and using $M=150$ tensor rank decompositions.

We can now view the sum in (33) as one big tensor rank decomposition,
consisting of $M*R$ points. For the measure-values $\beta_{i}$ we need to make
sure that the sum $\sum_{i=1}^{M*R}\beta_{i}=\sum_{i=1}^{R}\beta^{(I)}_{i}\
\forall_{I\leq M}$, as otherwise we would get a tensor rank decomposition for
$M*P_{abc}$. In the above, we used the rule, for $I\leq M,$ and $j\leq R$

$\beta_{(I-1)*R+j}=\frac{1}{M}\beta^{(I)}_{j}.$ (34)

There is a certain ambiguity in doing so. Making a different choice for any

$\beta_{(I-1)*R+j}=A_{I}\beta^{(I)}_{j},$

such that $A_{I}>0,\ \sum_{I=1}^{M}A_{I}=1$, and technically this would
correspond to a different associative closure. It is expected that such a
“transformation” corresponds to performing a diffeomorphism on the manifold,
since it can be interpreted as a deformation. Here we chose simply (34) and
the result of this exercise may be found in figure 5.

Figure 6: A plot of the elements generated by a non-minimal tensor rank
decomposition, showing how they deviate from the results in figure 5. In this
image, $10$ tensor rank decompositions of rank $8$ are combined into the plot,
using the same procedure as before. Some points still look similar to figure
5, but there are clear discrepancies from what we are looking for.

We end this section with a remark about the requirement of the tensor rank
decomposition to be _minimal_. It might not seem clear from the discussion
above why this should be the case, but the empirical evidence points towards
this. Let us consider the example of the exact circle cited above again. If we
attempt to find $R=8$ tensor rank decompositions, instead of the minimal $R=7$
above, we see that the resulting tensor rank decomposition does not consist of
potential homomorphisms. This is shown visually in figure 6, but can also be
verified by verifying (32). For the case of $R=7$, all of the elements
$p^{i}_{a}$ in the tensor rank decomposition exactly satisfied (32), but for
$R=8$ there always seem to be two elements $p_{a}^{i}$ that deviate a lot (as
in,
$|s_{1}^{a}s_{2}^{b}p_{a}^{i}p_{b}^{i}-s_{1}^{a}s_{2}^{b}P_{abc}p_{c}^{i}|\sim
O(1)$) and the other six deviate a little (where the deviation was $\sim
O(0.1)$), explaining why there seem to be some random points in figure 6 along
with points that look very similar to figure 5. It should be noted that the
$f_{1}$ function, proportional to the unit function, does always yield the
correct result.

##  5 Defining a unital algebra and introducing geometry.

Up until now we have always assumed that the tensors we used already have a
unit, in the sense that there is a $\mathbf{1}=\alpha_{a}f_{a}\in\mathcal{F}$
such that

$\alpha_{a}P_{abc}=\delta_{bc}.$ (35)

For a general symmetric tensor $P_{abc}$ there is no such element
$\mathbf{1}$. In this section we introduce a way to generate a new tensor
$\tilde{P}_{abc}$ which does have a unit, from a large class of tensors
$P_{abc}$. While this might seem like an ad hoc construction, it will be
pointed out that this might have useful physical implications since this gives
us the opportunity to describe the full geometric information of a Riemannian
manifold in the tensor $P_{abc}$.

First let us note that the unit in (35) was derived from the requirement that
$\forall f=\beta_{a}f_{a}\in\mathcal{F}$

$\mathbf{1}\cdot f=\alpha_{a}\beta_{b}P_{abc}f_{c}=\beta_{c}f_{c}=f.$

This notion might be generalised to the requirement that the unit is its own
unit

$\mathbf{1}\cdot\mathbf{1}=\alpha_{a}\alpha_{b}P_{abc}f_{c}=\alpha_{c}f_{c}=\mathbf{1},$

which implies

$\alpha_{a}\alpha_{b}P_{abc}=\alpha_{c}.$ (36)

This might be recognised as the Eigen-problem of a tensor $P_{abc}$. For a
real tensor, there is always at least one real solution to this [50]. If there
are several real solutions, one has to make a choice. We will get back to this
later in the section.

After finding an $\alpha_{a}$ such that (36) holds, we consider the matrix

$M_{bc}=\alpha_{a}P_{abc}.$ (37)

If $\alpha_{a}$ would correspond to a true unit, this matrix would simply
reduce to $\delta_{bc}$ as in (35). If this is not the case, we can
diagonalise the matrix, such that in the new basis (without Einstein-
summation)

$M_{ab}=w_{a}\delta_{ab}.$ (38)

We are restricting ourselves to tensors that have a solution to (36) such that
all $w_{a}>0$. In the new basis, we can then redefine the tensor (without
using Einstein-summation)

$\tilde{P}_{abc}=\frac{1}{\sqrt{w_{a}w_{b}w_{c}}}P_{abc},$ (39)

which means we get a new tensor $\tilde{P}_{abc}$ that has a unit given by
$\mathbf{1}=\sum_{a}\tilde{\alpha}_{a}f_{a}=\sum_{a}\sqrt{w_{a}}\alpha_{a}f_{a}$.
This can be seen by (without using Einstein-summation)

$\sum_{a}\alpha_{a}\sqrt{w_{a}}\tilde{P}_{abc}=\sum_{a}\alpha_{a}P_{abc}\frac{1}{\sqrt{w_{b}w_{c}}}=\frac{w_{b}\delta_{bc}}{\sqrt{w_{b}w_{c}}}=\delta_{bc}.$

In order to understand under which conditions the matrix $M_{ab}$ is positive
definite, consider the functional

$f(\alpha):=\alpha_{a}\alpha_{b}\alpha_{c}P_{abc},$ (40)

and suppose we are interested in finding the local extrema of this functional
under the condition that $|\alpha|^{2}=\alpha_{a}\alpha_{a}=1$. This can be
done by introducing a Lagrange-multiplier $k$,

$g(\alpha):=\alpha_{a}\alpha_{b}\alpha_{c}P_{abc}+k(1-|\alpha|^{2}).$

Finding the local extrema can then be done by taking the derivative with
respect to $\alpha_{a}$

$\partial_{\alpha_{c}}g(\alpha)=3\alpha_{a}\alpha_{b}P_{abc}-2k\alpha_{c}=0.$

Note that if we define $\alpha_{a}^{\prime}=\frac{2k}{3}\alpha_{a}$, this
exactly reproduces the Eigen-problem of (36):

$\alpha_{a}^{\prime}\alpha_{b}^{\prime}P_{abc}=\alpha_{c}^{\prime}.$

Thus, by the definition of the matrix $M_{ab}$, the extremal value
$\alpha_{a}$ is an eigenvector of $M_{ab}$

$\alpha_{a}M_{ab}=\alpha_{a}\alpha_{c}^{\prime}P_{abc}=\frac{2k}{3}\alpha_{a}\alpha_{c}P_{abc}=\left(\frac{2k}{3}\right)^{2}\alpha_{b}.$

It can be seen that the eigenvalues are positive for real solutions. This also
implies

$\alpha_{a}\alpha_{b}\alpha_{c}P_{abc}=\frac{2k}{3}.$

This sets the first eigenvalue to $\left(\frac{2k}{3}\right)^{2}$, for a
condition for the other eigenvalues, let us consider a second order
perturbation of (40) denoted by $\epsilon\delta_{a}$, with $\delta_{a}$ a
unit-size vector. It follows from the restriction $|\alpha|^{2}=1$ that
$\alpha_{a}\delta_{a}=0$. The first-order contribution is zero since we are
considering the extremal value. The second-order contribution is given by

$3\alpha_{a}\delta_{b}\delta_{c}P_{abc}=\frac{9}{2k}M_{bc}\delta_{b}\delta_{c},$

this means that the requirement that $M_{ab}$ is positive-definite coincides
with the functional (40) either having a local minimum with a positive value
($k>0$), or a local maximum with a negative value ($k<0$). Here, we will
choose the positive value as the canonical choice.

Let us remark on how to choose the $\alpha_{a}$ if there are several solutions
to the above, such that the matrix $M_{ab}$ is positive definite. Since we are
looking for “almost unit” functions, it makes sense to look at the minimal
solution for

$\sum_{a=1}^{N}(w_{a}-1)^{2}.$

This would then produce a candidate unit which is as close to the real unit as
possible.

A potentially worrisome fact is that we are considering only tensors that have
solutions such that $w_{a}>0$, in other words such that $M_{ab}$ is positive-
definite. However, this might not be such a big issue due to two main reasons.
Firstly, in [30, 29] it has been shown that the canonical tensor model wave
functions tend to have peaks around symmetric configurations. This suggests
that while quantum fluctuations from a symmetric spacetime might make
$P_{abc}$ non-unital or non-associative, the deviation from this symmetric
tensor is not expected to be that large, and thus the $w_{a}$ above will not
deviate that much from the unit of the symmetric tensor. Secondly, if a tensor
$P_{abc}$ does have negative eigenvalues, this might be a sign that this
tensor does not correspond to a Riemannian manifold. Instead, it is expected
that the tensor might describe a _pseudo_ -Riemannian manifold. As our current
aim is to first develop an understanding of algebras corresponding to compact
Riemannian manifolds, it is reasonable to leave these cases for future study.

The values $w_{a}$ have an interesting interpretation, that might be useful
for model-building. Considering a compact Riemannian manifold
$(\mathcal{M},g)$, we can equip the Hilbert space as introduced in section 2
with a compact self-adjoint operator
$\mathcal{O}:L^{2}(\mathcal{M})\rightarrow L^{2}(\mathcal{M})$. According to
the spectral theorem [51], there exists an orthonormal basis of
$L^{2}(\mathcal{M})$ consisting of eigenvectors of $\mathcal{O}$. Let us now
consider the basis $\\{f_{a}\\}$ we introduced in section 2, and assume that
this basis consists of eigenvectors of $\mathcal{O}$. The eigenvalues
$\lambda_{a}$, have the property $\lim_{a\rightarrow\infty}\lambda_{a}=0$.
Instead of $\tilde{P}_{abc}=\braket{f_{c}}{\,f_{a}\cdot f_{b}}$, we could
instead consider the tensor

$P_{abc}:=\braket{\mathcal{O}f_{c}}{\,\mathcal{O}(f_{a})\cdot\mathcal{O}(f_{b})}=\lambda_{a}\lambda_{b}\lambda_{c}\tilde{P}_{abc}.$
(41)

Comparing this to (39), this gives the $w_{a}$ an interpretation of the
eigenvalues of a compact self-adjoint operator, according to the relationship

$w_{a}=\lambda_{a}^{2}$ (42)

Here we will assume the eigenvalues of such an operator to be positive, such
that the values $w_{a}$ can directly be interpreted without any loss of
information.

In order to construct such a compact operator, let us consider a different
operator called the Laplace-Beltrami operator. This is a prominent operator
that has an intrinsic connection to the geometry of Riemannian manifolds, see
appendix B for more information on the Laplace-Beltrami operator. This
operator $\Delta:C^{\infty}(\mathcal{M})\rightarrow C^{\infty}(\mathcal{M})$
is known to generate an orthonormal basis of $L^{2}(\mathcal{M})$ as described
above, though it is not compact. Its eigenvalues are negative, so often we
consider $-\Delta$ instead. The eigenvalues are denoted by $-\mu_{a}$ and
given by

$\displaystyle-\Delta f_{a}$ $\displaystyle=\mu_{a}f_{a},$ $\displaystyle 0$
$\displaystyle=\mu_{1}<\mu_{2}\leq\mu_{3}\leq\mu_{4}\leq\ldots,$

with its limit $\lim_{a\rightarrow\infty}\mu_{a}=\infty$. The intrinsic
connection between this operator and the geometry (encoded by the metric) of a
Riemannian manifold may already be seen in its representation in local
coordinates

$\frac{1}{\sqrt{\det g}}\partial_{j}(g^{ij}\sqrt{\det g}\partial_{i}f),$ (43)

and in fact it is possible to actually determine the metric from the
information of the Laplacian acting on the functions $f\in
C^{\infty}(\mathcal{M})$ [52]. The Laplacian has been used in studies of the
geometry in quantum gravity before, for instance in the case of causal
dynamical triangulation [53, 54] and the canonical tensor model [48].

Combining the information of the last two paragraphs, we find an interesting
new interpretation of the $w_{a}$, and we have the opportunity to construct an
operator $\mathcal{O}$ that not only serves to generate an orthogonal basis
but also includes important geometric information. An example of such an
operator would be

$\mathcal{O}=\operatorname{e}^{\Delta}.$ (44)

A similar operator has actually been used in the canonical tensor model before
to generate the tensor similarly to (41), with the reasoning that this would
smoothen the cutoff when considering an $N$-dimensional tensor instead of an
infinite-dimensional one [48]. In this work we find that the cutoff might
actually not be that much of a problem, as we can reconstruct the topology and
infinite-dimensional algebra using the associative closure. However, adding
this extra operator actually adds geometric information which might in part
explain the success of the approach in [48] to extract topological and
geometric data using the tensor rank decomposition.

It still has to be seen which operator would be good to use in a certain
context. It may be expected that for a certain model, for instance the
canonical tensor model, one needs to specify which operator is considered in
order to make the complete link to gravity.

##  6 Examples: Exact and fuzzy spaces.

In this section we will discuss several examples of the formalism introduced
in this work. The exact circle has been developed throughout this paper as a
concrete example of how a tensor can correspond to a Riemannian manifold. We
have shown that the topology, measure and the full algebra of functions can be
reconstructed from the tensor constructed in section 2. It is remarkable that
we can reconstruct this whole structure from a five-dimensional symmetric
tensor with just $35$ entries. However, it is important to show that this
formalism can handle a wide range of spaces. For this we look at three main
areas. We analyse the behaviour of perturbations of the exact circle in
section 6.1, then we analyse an inherently fuzzy space namely the semi-local
circle in section 6.2, and lastly we will consider the exact sphere in section
6.3 to show that the framework works just as well in higher dimensions.

In this section, for every example a benchmark will be cited. This is the
maximal error value $\Delta$, below which a tensor rank decomposition will be
accepted

$(P_{abc}-\sum_{i=1}^{R}\phi_{a}^{i}\phi_{b}^{i}\phi_{c}^{i})^{2}<\Delta.$
(45)

Often, this value can be taken to be extremely small ($\Delta\sim
O(10^{-30})$), especially if the tensor has a certain continuous symmetry such
that there are many minimal tensor rank decompositions, but in some cases
where the minimal tensor rank decomposition is harder to find we have to
increase this value. As a general rule, we will always require it to be at
most $\Delta<10^{-6}$.

###  6.1 Perturbations around the exact circle

Looking at perturbations around a given tensor is a useful first step to
understanding the behaviour of space when altering the algebra of functions.
Furthermore, given the strong peaks of the quantum wave function of the
canonical tensor model around symmetric configurations [30], it seems that we
should expect small perturbations to occur around tensors representing smooth
spaces.

As we have a relatively good understanding of the exact circle by now, let us
consider perturbations around this tensor. The vectorspace
$\mathcal{F}\cong\mathbb{R}^{5}$ is kept the same. We will denote the original
tensor, as described in section 2, as $P_{abc}$ and any perturbed tensor by
$\tilde{P}_{abc}$

$\tilde{P}_{abc}=P_{abc}+Q_{abc},$

where $Q_{abc}$ is the symmetric tensor characterising the perturbation. All
tensors described here act on the same vectorspace $\mathcal{F}$, not on
extensions of it or such. Generally, $\tilde{P}_{abc}$ will not be unital
anymore, so we will have to generate a new unit according to the procedure in
section 5. We will consider several different kinds of perturbations:

  * •

Using the potential homomorphisms found using the tensor rank decomposition in
section 4.3, we perturb the $\beta_{i}$ measure factors.

  * •

Using the potential homomorphisms found using the tensor rank decomposition in
section 4.3, we perturb one of the points $p^{i}$ themselves.

  * •

We add a random “high-energy” perturbation, meaning a perturbation of
$Q_{abc}$ where $Q_{abc}\neq 0$ if $a,b,c\geq 4$.

What we will see is that in the case of perturbing the tensor rank
decomposition itself, it is reasonably well-behaved and the perturbation
mainly results in a deformation of the manifold. When altering or adding
points, it depends on the size of the perturbation. Small perturbations still
remain smooth, whereas bigger perturbations or random high-energy
perturbations lead to a breaking of the smooth structure and only a finite set
of points remain.

It should be noted that for perturbations it was sometimes more difficult to
find an exact minimal tensor rank decomposition. In principle, one could use a
higher-rank decomposition as well, but then one has to filter out the
decompositions that do not correspond to potential homomorphisms as explained
in section 4.3. In this work we take an approximate minimal tensor rank
decomposition, where we are satisfied with an error
$(P_{abc}-\sum_{i=1}^{R}\beta_{i}p^{i}_{a}p^{i}_{b}p^{i}_{c})^{2}<\Delta=10^{-6}$.
Since the main goal here is to demonstrate how these perturbations affect the
shape of the functions, finding a method to find higher-rank tensor rank
decompositions that correspond to potential homomorphisms only is left for
later study. Furthermore, this section is not meant as a systematic analysis
of perturbations using this framework but mainly a proof of concept to show
the potential of types of spaces that can be analysed within this framework.
There are a lot of interesting things to find out about them, but this is out
of the scope of the present paper.

Case 1: Perturbing the measure. Perturbing the measure was done as follows.
First the tensor $P_{abc}^{(u)}$ was defined as the five-dimensional tensor in
section 2. Then, an extra damping factor was introduced using the operator
$\mathcal{O}f_{a}=\operatorname{e}^{\Delta/25}f_{a}=\operatorname{e}^{-n_{a}^{2}/25}f_{a}$,
with $n_{a}=\\{0,1,1,4,4\\}$, in order to demonstrate the reconstruction of
the unital tensor $P_{abc}^{(u)}$ as described in section 5,101010Note that
the factor $25$ is arbitrary, and any number could have been taken.

$P_{abc}=\operatorname{e}^{-n_{a}^{2}/25-n_{b}^{2}/25-n_{c}^{2}/25}P_{abc}^{(u)}.$
(46)

The unital tensor $P_{abc}^{(u)}$ was then reconstructed using the procedure
in section 5. This means that the solutions to

$\alpha_{a}\alpha_{b}P_{abc}=\alpha_{c}$

were found. This was done using Mathematica and finding solutions to
$\sum_{c=1}^{5}(\alpha_{a}\alpha_{b}P_{abc}-\alpha_{c})^{2}=0$ by using
minimisation. Then, the matrix $M_{ab}$ was found (without Einstein-summation)

$M_{ab}=\sum_{c=1}^{5}\alpha_{c}P_{abc}=w_{a}\delta_{ab},$

where $w_{a}$ are exactly given by

$w_{a}=\\{1,{\rm e}^{-2/25},{\rm e}^{-2/25},{\rm e}^{-8/25},{\rm
e}^{-8/25}\\}.$

Note that these exactly correspond to the square of the eigenvalues, as also
expected from (42). From this tensor111111Note that this tensor now includes
some information about the metric. Here we will not go deeper into this, but
it would be interesting to examine what part of the metric we can reconstruct
from this more closely in a future study. we now reconstruct the unital tensor

$P_{abc}^{(u)}=\frac{1}{\sqrt{w_{a}w_{b}w_{c}}}P_{abc}.$

Using the tensor rank decomposition, as described in section 4.3, $1400$
points $p^{i}$ with their measure-factor $\beta_{i}$ were generated. This
gives a decomposition of

$P_{abc}^{(u)}=\sum_{i=1}^{1400}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i}.$

This leads to an equivalent picture as in figure 5. In this case, we want to
perturb the measure. The perturbation tensor is defined as

$Q_{abc}=\epsilon\sum_{i}\beta_{i}^{\prime}p_{a}^{i}p_{b}^{i}p_{c}^{i},$

where $\beta_{i}^{\prime}$ characterises which points $p^{i}$ will be
perturbed. We also added a factor of $\epsilon>0$ to alter the general size of
the perturbation. This means that, for example, $\beta_{i}^{\prime}=0$
corresponds to no perturbation, or for example $\beta_{1}^{\prime}=1$
corresponds to a perturbation of the first point. The measure factors
$\beta_{i}$ were altered in two ways. The first approach was to change the
measure factor for only the first point, $\beta_{1}$, such that

$\beta_{i}^{\prime}=\delta_{i1}.$ (47)

The second approach was to alter all of the factors $\beta_{i}$, using a
Gaussian distribution

$\beta_{i}^{\prime}=\operatorname{e}^{-(i-700)^{2}/36}.$ (48)

The factor $36$ was taken such that there would be a reasonable range, such
that not just a few points would be affected. The full new, perturbed, tensor
is then given by

$\tilde{P}_{abc}=P_{abc}^{(u)}+\epsilon
Q_{abc}=\sum_{i=1}^{1400}(\beta_{i}+\epsilon\beta_{i}^{\prime})p_{a}^{i}p_{b}^{i}p_{c}^{i},$

hence the claim that we are perturbing the measure.

Figure 7: A plot of the functions $f_{2}(x)$ and $f_{3}(x)$ of the circle with
a perturbed measure, where the function values are defined as in (26). The top
two images represent the case of (47) using $\epsilon=0.4$ and
$\Delta=7.6\cdot 10^{-31}$. The bottom two images represent the case of (48)
using $\epsilon=0.5$ and $\Delta=2.8\cdot 10^{-30}$. In both cases, 200 tensor
rank decompositions of rank 7 were used, yielding 1400 points.

After altering the $\beta_{i}$’s, a new unit has to be found according to the
procedure in section 5, since generally the new tensor $\tilde{P}_{abc}$ is
not unital anymore. Then, the tensor rank decomposition method of section 4.3
was applied to find the potential homomorphisms, and they were ordered in the
same way as for the exact circle in (14). The tensor rank used remained
unchanged at $R=7$, and $1400$ points were generated this way. The results of
this are displayed in figure 7.

Three things are interesting here. Firstly, the measure is clearly disturbed
which can be seen by the deformation of the functions and the size of the
circle (in both cases it became smaller), but the general properties of the
circle are still present. This is encouraging, since it means that we can
describe interesting spaces that do not exactly correspond to the algebra of a
circle. Secondly, the reason why we can find this continuous amount of
potential homomorphisms, is because the associative extension was defined in
terms of any maximal partial algebra. In the case of the exact circle, there
was a partial algebra $\mathcal{S}_{3}$ that already contained all possible
potential homomorphisms, but very few potential homomorphisms in figure 7
actually correspond to homomorphisms of the same partial algebra. Since we
allow them to come from different maximal partial algebras we get this rich
structure. It should be noted that all of these points in the associative
closure will become proper homomorphisms. Lastly, due to the defining of a new
unit, there is a sort of mixing that happens. Changing $\beta_{1}$ only does
not simply make that part of the dual space bigger, but deforms the region
around it too. Note that the perturbations here are by no means small, so the
strong deformations are to be expected.

Case 2: Perturbing a point. The perturbation of a point was done by shifting
one point of a tensor rank decomposition. This was done using the unperturbed
tensor rank decomposition
$P_{abc}=\sum_{i=1}^{7}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i}$ with 7 points as
discussed above. The first point $p^{1}$, which should be noted is arbitrarily
picked, was taken and shifted with a vector $q$

$p^{1}_{a}\rightarrow p^{1}_{a}+\epsilon q_{a},$

where again $\epsilon>0$ is introduced to set the size of the perturbation.
This yields a new tensor with a tensor rank decomposition

$\tilde{P}_{abc}=\beta_{1}(p^{1}+\epsilon q)_{a}(p^{1}+\epsilon
q)_{b}(p^{1}+\epsilon
q)_{c}+\sum_{i=2}^{7}\beta_{i}p_{a}^{i}p_{b}^{i}p_{c}^{i}.$

In terms of the perturbation tensor $Q_{abc}$, this corresponds to

$Q_{abc}=\epsilon\beta_{1}(p^{1}_{a}p^{1}_{b}q_{c}+p^{1}_{a}q_{b}p^{1}_{c}+q_{a}p^{1}_{b}p^{1}_{c}+\epsilon
p^{1}_{a}q_{b}q_{c}+\epsilon q_{a}p^{1}_{b}q_{c}+\epsilon
q_{a}q_{b}p^{1}_{c}+\epsilon^{2}q_{a}q_{b}q_{c}).$ (49)

A similar procedure as in the previous case was then followed, taking a random
normalised vector

$q=(0.,0.714853,-0.514041,-0.247008,0.40464),$ (50)

generated by Mathematica. Note that in order to not interfere too much with
the unit, the first component of the vector was taken to be $0$. Then, the
algebra was redefined as in section 5 and tensor rank decompositions were
generated just like above. The result of this can be found in figure 8.

Figure 8: A plot of the functions $f_{2}(x)$ of the circle with a shifted
point as described in (49), where the function values are defined as in (26).
On the left using $\epsilon=0.1,\Delta=8.1\cdot 10^{-31}$, in the middle
$\epsilon=0.25,\Delta=5.6\cdot 10^{-30}$ and on the right
$\epsilon=0.75,\Delta=2.3\cdot 10^{-28}$. In all cases, 200 tensor rank
decompositions of rank 7 were used, yielding 1400 points with the random
perturbation vector of (50), and the displayed functions are in the original
basis in order to make direct comparison to the smooth functions on a circle
possible. Note that in the plot on the right, many of the points overlap.

Interestingly, for small values of $\epsilon$, the functions do not change
that much and the shape of the functions is just deformed, similarly to
changing the measure as described above. However, when the perturbation
becomes large, $\epsilon\sim O(1)$, the smooth structure breaks down.

Case 3: Random high-energy perturbation. Adding a random perturbation is a
very uncontrolled way of adding a perturbation, but nonetheless it is
interesting because a priori it is not known what kind of perturbations one
should expect from a quantum theory. The perturbation is now characterised by
the tensor

$Q_{abc}=\begin{cases}\epsilon RandomReal(-1,1)&,\ \text{if }a,b,c\geq 4,\\\
0&,\ \text{otherwise}.\end{cases}$ (51)

Here $RandomReal$ denotes a function giving a random number in the range
$[-1,1]$ according to a uniform distribution. For $\epsilon$, a value of
$\epsilon=0.01$ was chosen. We can characterise this perturbation by four
elements; $Q_{444}$, $Q_{445}$, $Q_{455}$, and $Q_{555}$. In the current
example, we used specifically (randomly generated)

$\displaystyle Q_{444}$ $\displaystyle=-0.00616405,$ (52) $\displaystyle
Q_{445}$ $\displaystyle=0.000446477,$ $\displaystyle Q_{455}$
$\displaystyle=-0.00686687,$ $\displaystyle Q_{555}$
$\displaystyle=-0.00208394.$

Then, the same procedure was repeated as before. Note firstly that the tensor
$Q_{abc}$ does not alter the unit of the tensor, since $Q_{1bc}=0$, so finding
a new unit is not necessary. Furthermore, in this case it appears easier to
find a higher rank-decomposition and required its to lie in the dual space of
the partial algebra generated by $\mathcal{S}_{3}=\\{f_{1},f_{2},f_{3}\\}$,
since this is still a covering partial algebra.121212Note that in a future
study it would be interesting to include other partial algebras into this
picture too, as they might not all be equivalent anymore due to this
perturbation. Adding the perturbation will break some of the symmetric
properties, so it is expected that this tensor will not correspond to a smooth
space anymore.

Figure 9: A plot of the functions $f_{1}(x),f_{2}(x)$ and $f_{3}(x)$ of the
circle with a random high-energy parturbation as described in (51), where the
function values are defined as in (26), using the perturbation tensor as (52),
and generating 87 rank-8 tensor rank decompositions yielding 696 points
(though many of them overlap).

The result of this procedure is given in figure 9. It can be seen that
apparently, for this perturbation, the resulting tensor has $8$ potential
homomorphisms. It is expected that for higher energy perturbations (meaning
perturbations for $a,b,c>M$ for some large $M$) there would be many more
potential homomorphisms but still a possible breaking of the smooth structure.
This suggests that, if the quantum theory for gravity admits any random
perturbations, spacetime on a trans-Planckian level is actually discrete,
though it could also be the case that symmetric configurations with smooth
emerging spaces might be preferred as suggested in the canonical tensor model
[29, 30].

A last note here to the general structure of the definitions in section 3. The
definitions of the associative extension and associative closure attempt to
make sure that as much as possible of the partial algebras is retained. It
might, in the future, turn out that it is more beneficial to relax the
requirements slightly to allow more points. At present, it is not clear what
would be a better way to define it, so in this picture these kind of
perturbations likely lead to some fundamental discreteness of space.

###  6.2 The semi-local circle

The semi-local circle is an example of a tensor that is expected to reproduce
a fuzzy space. It is defined using a constant $0<\alpha<1$ and $a,b,c\leq N$
for some positive integer $N$ as

$P^{\alpha}_{abc}=\begin{cases}1,\ \text{if }a=b=c,\\\ \alpha,\ \text{if
}a=b=c\pm 1\parallel a=b\pm 1=c\parallel a\pm 1=b=c,\\\ 0,\
\text{otherwise,}\end{cases}$ (53)

where we use circular boundary conditions, i.e. $N+1=1$ and $1-1=N$. It has a
discrete version of a translation symmetry, as it is symmetric under
$(a,b,c)\rightarrow(a+1,b+1,c+1)$. We consider a set of local functions as
having the property that (without Einstein-summation)

$f_{a}\cdot f_{b}=\delta_{ab}f_{a},$

which would be the case for

$P_{abc}=\begin{cases}1,\ \text{if }a=b=c,\\\ 0,\
\text{otherwise}.\end{cases}$

Together with the cyclic property it possesses this explains the terminology
“semi-local circle”.

Figure 10: A plot of the functions $f_{2}(x)$ of the semi-local circle with
$\alpha=0.1$ as defined in (53), where the function values are defined as in
(26), generated using various ranks of tensor rank decompositions. From top-
left to bottom-right, they are $R=7,9,10,11$. The amount of points generated
are, respectively, $70,90,100,99$. Note that the first point is always
selected at random, so there is no meaning behind the phase-difference.

This tensor gives an interesting example because of its discrete nature and
easy way to construct. Besides, since it is not constructed using a smooth
background like the examples in section 6.1, it is an example of an
intrinsically algebraic space.

The procedure followed to analyse this tensor is the same as before. First one
generates a unit and redefines the tensor as in section 5, then one uses the
tensor rank decomposition of section 4.3 to generate potential homomorphisms.
This was done for $N=7$. As mentioned before, we always target a
$\Delta<10^{-6}$, however in this case we also evaluated decompositions with a
lower rank, as there is some interesting behaviour going on. The results for
$f_{2}(x)$ for $R=7,9,10,11$ are shown in figure 10.

Interestingly, though the $R=7$ case is not very precise with a
$\Delta=10^{-2}$, but it shows a discretised space of $7$ points, similar to
what one would naively expect from the tensor described above. However, since
the decomposition does not seem to be correct, one should increase $R$. For
$R=9$ the decomposition is still not great with $\Delta=4\cdot 10^{-3}$, but
some actual fuzzy behaviour can be observed which only increases with $R=10$.
Finally, with $R=11$ proper decompositions can be found with
$\Delta=10^{-25}$. Interestingly we see that it starts to look more like a
circle, but with some fuzzy points included and the regions that were present
in the smaller $R$ cases dominate with large steps in between. It seems to be
a very interesting example that would be insightful to investigate further,
for instance by taking various different $N$ and checking if all points indeed
are included in the dual space of some system of partial algebras. In the
current check this has not been performed rigorously, since one needs to find
a candidate maximal system of algebras first which is more difficult than in
the case of the circle or the sphere. This would be interesting to take a look
at in a future study.

###  6.3 The exact sphere

The exact sphere is an example of a 2-dimensional Riemannian manifold. This
example is taken to demonstrate the ability of this framework to handle
higher-dimensional manifolds as well. The tensor representing the exact sphere
is found using the procedure in section 2. First, one chooses a basis for the
real smooth functions $C^{\infty}(S^{2})$. One such basis is given by the real
spherical harmonics, $Y_{l}^{m}(\theta,\phi)$, which are eigenfunctions of the
Laplace-Beltrami operator. The ordering of eigenfunctions is determined by
$a\in\\{1,2,3,\ldots\\}\rightarrow(l,m)\in\\{(0,0),(1,-1),(1,0),(1,1),(2,-2),\ldots\\}$.
The tensor $P_{abc}$ is then generated by

$P_{abc}=\int_{S^{2}}{\rm d}\Omega\,Y_{a}(\Omega)Y_{b}(\Omega)Y_{c}(\Omega),$

where $\Omega=(\theta,\phi)$ and ${\rm d}\Omega=\sin(\theta){\rm d}\theta{\rm
d}\phi$.

In this example, the smallest non-trivial tensor was used corresponding to the
case where we cut the tensor off at $N=9$, such that the products for
$l=\\{0,1\\}$ are all fully included. Determining the potential homomorphisms
can be done in two ways, either by evaluating a similar equation as in (13)

$p\in\mathbb{R}^{9},\,a,b\in\\{1,\ldots,4\\}\,:\,p_{a}\,p_{b}=\sum_{c=1}^{9}P_{abc}p_{c},$

or by using the tensor rank decomposition. Here, the former approach was used
to construct $1200$ points.

A more difficult thing is how to illustrate the topology of the space. In
section 2 the proper mathematical way of defining the topology was described
in figure 1. Even though this definition is unambiguous, in practice it is
more difficult to visualise this in a two-dimensional case. Here a simple way
was considered, since this is only meant to be a proof of concept, but in the
future it would be good to have a more rigid approach.

One way that can be used is by defining a topological distance between the
points as

$d(p^{i},p^{j})=\sqrt{\sum_{a=1}^{N}|p_{a}^{i}-p_{a}^{j}|^{2}},$

similar to the approach for the circle. Taking as a reference point $p^{1}$,
we can then choose two points close to $p^{1}$, let’s say $p^{q}$ and $p^{r}$.
Then we define the direction $p^{q}$ as the $x$-axis, and the length of
$p^{q}$ is simply $d(p^{1},p^{q})$. The $y$-axis is then found by solving
$d(p^{1},p^{r})=\sqrt{(p^{r}_{x})^{2}+(p^{r}_{y})^{2}}$ and
$d(p^{q},p^{r})=\sqrt{(p^{q}-p^{r}_{x})^{2}+(p^{r}_{y})^{2}}$. Now we have
defined an $x$\- and $y$-axis and we can define a position of every point
$p^{i}$ in the $(x,y)$-plane by solving
$d(p^{q},p^{i})=\sqrt{(p^{q}-p^{i}_{x})^{2}+(p^{i}_{y})^{2}}$ and
$d(p^{r},p^{i})=\sqrt{(p^{r}_{x}-p^{i}_{x})^{2}+(p^{r}_{y}-p^{i}_{y})^{2}}$.
Note that this is a very crude way of defining part of the topology, as we use
a local coordinate system and naively extrapolate this to the whole space. In
figure 11 the result of this is shown for two of the eigenfunctions, however
it should be noted that using this way of defining coordinates, not all
eigenfunctions generally look this nice as for some eigenfunctions the “front”
and “back” values of the sphere are mixed beyond a small local patch.

Figure 11: Density plots of the functions $f_{3}(x,y)$ on the left, supposed
to represent $Y^{1}_{0}$, and $f_{7}(x,y)$ on the right, supposed to represent
$Y^{2}_{0}$, using the definition of the coordinates as defined in the text.
Though the way of defining the coordinates in the text is not perfect, a clear
resemblance to the spherical harmonics may be observed. 1200 points were used
to create these density plots.

The points in figure 11 were constructed using the dual space of the partial
algebra, an approach already presented in section 2 in the example of the
exact circle. No information on the measure or geometry has been taken into
account, as this is merely meant as a proof of concept to describe higher-
dimensional spaces using this algebraic interpretation. However, it could have
also been done using the tensor rank decomposition. A sanity check was
performed, using $\Delta=5.\cdot 10^{-12}$, it is possible to find rank $R=18$
tensor rank decompositions. The measure will also be regained when doing this,
showing that the volume of the sphere is indeed $\sum_{i=1}^{R}\beta_{i}=4\pi$
(up to 6 decimal places). More research has to be done here, to find a good
way to visualise these properties properly and show exactly what the measure
looks like.

##  7 Implications for the canonical tensor model.

The main goal of this work was to lay a foundation for the understanding of
the emergence of spacetime in the canonical tensor model (ctm). While it is
not fully clear if this is the best and only spacetime interpretation of the
ctm, at the very least this seems like an interesting approach to build a
tensor-model that can describe (fuzzy) spaces, in a very different way than
the original tensor models [15, 16, 17]. In this section we will highlight
some of the benefits of this interpretation for such a model, by interpreting
important previous results of both the links of the ctm to gravity and the
quantum version of the model. For a brief introduction to the ctm, we would
like to refer to appendix C.

The ctm is a tensor model where the fundamental configuration is given by a
real symmetric tensor of degree three, $P_{abc}$. It is built in the canonical
framework in order to introduce time to a tensor model. This means in the
present interpretation that the ctm corresponds to an algebra of functions
over a (fuzzy) space, and both the topology through section 3 and the geometry
through section 5 are allowed to fluctuate. Evaluating the time-evolution
would then correspond to adding new spatial slices, in a similar sense to the
ADM-formalism [55].

One important immediate result from the formalism presented here, is that in
order to analyse countably infinite-dimensional function spaces it is possible
to analyse the finite-dimensional counterpart and assume the associative
closure. This makes analysis of a finite-dimensional tensor useful in the
first place, as in [42] it was already pointed out that finite-dimensional
tensors could only directly correspond to completely local associative
algebras. With the formalism developed here, we have shown that we could for
instance take only the $5$-dimensional tensor written in section 2 and
reconstruct the full topology and measure on the circle. Besides, while to
keep the mathematical formulation more simple we often assumed a finite-
dimensional tensor, but the formalism seems to be able to be extended to
countably infinite-dimensional tensors. This means that whether or not the
final theory of quantum gravity is actually infinite-dimensional or finite-
dimensional, the formalism seems to be able to handle both cases as long as
the space is assumed to be compact.

In prior work, the authors already found that one can extract a remarkable
amount of topological and geometric information from tensors [48, 33], using
tools from data analysis. Especially the tensor rank decomposition and
persistent homology turned out to be useful. In the present work we discover
why these tools actually work so well. Firstly, the tensor rank decomposition
indeed generates points of the dual space of an algebra according to section
4.3, which is the reason why using the tensor rank decomposition worked so
well. Furthermore, if we choose a discrete number of these points we argued in
(14) that two points are to be considered close in a topological sense if
their inner product in $\mathbb{R}^{N}$ is large. This is exactly what was
used in [48] in order to use persistent homology to extract topological data.
In these papers, a damping function was introduced corresponding to (44). The
original idea was to use this damping in order to make the cutoff more smooth
and it was found that the results for extracting the topology and geometry
became better. In the current paper we find that the inclusion of these
damping functions actually corresponds to including explicit geometric
information in the tensor. It appears crucial to reconstruct the full
Riemannian manifold.

In [24], one of the first explicit connections between the ctm and General
Relativity was made. Here the authors investigated the $N=1$ case, and found
that when rewriting the Hamiltonian to an action, the action exactly
corresponded to the minisuperspace action of the Friedmann-Robertson-Walker
(frw) universe. This result fits the current framework very well, as here the
$N=1$ case would mean that we are only considering the constant function over
a space. In other words, the $N=1$ case is the case of a completely isotropic
universe, and the fact that the ctm leads to the frw universe is an important
check for the consistency of the ctm and specifically when interpreted using
the framework developed in this work.

In terms of the quantum ctm, there has been a lot of work on the finding and
analysing of wave functions [28, 29, 30, 31, 32, 33, 34, 35]. Here it was
shown that the ctm seems to prefer symmetric tensors, since there is evidence
for strong peaks of these in a prominent wave function [29, 30]. In the
current context this would imply that the ctm, as already speculated, seems to
have good indications for the emergence of macroscopic spaces. For instance a
smooth manifold with a lot of symmetry seems to be preferred over a very odd-
behaved fuzzy space.

There are also two main papers considering a formal continuum limit of the ctm
[25, 26]. The formalism developed in the present work is not directly
applicable to these cases, since these cases consider an emergent space
$\mathbb{R}^{d}$ which is not compact and thus the smooth functions
$C^{\infty}(\mathcal{M})$ do not have a Schauder basis which is countable.
However, the insights from these papers might be important in the future to
extend the formalism to non-compact spaces.

##  8 Summary and future prospects.

Finding a consistent theory of quantum gravity starts with a seemingly simple
question: “How do we describe gravity?”. In this work we introduce a new way
to describe a gravitational theory in terms of tensors. While tensor models,
such as the canonical tensor model (ctm) [21], already existed, it is
important to fully understand how one can translate their results to a
spacetime picture as we expect from a theory of quantum gravity. The benefit
of the framework introduced is that it is possible to describe all topological
and geometric information of a compact Riemannian manifold in an object that
only has countably infinite-dimensional degrees of freedom. Besides, it makes
sense of finite-dimensional tensors too.

In this work we introduced the concept of an associative closure in order to
deal with tensors that do not generate associative algebras. This helps to
make the approach more stable, as was seen in for instance section 6.1 where
the exact circle was perturbed. Furthermore, this made sure that even finite-
dimensional tensors can correspond to smooth manifolds, making finite-
dimensional calculations worthwhile. As it turned out, the symmetry of a
tensor in the first two indices corresponds to generating a commutative
algebra, but requiring the tensor to be fully symmetric and supposing a
Hilbert-space structure gives us the opportunity to define a measure. It was
then argued that it is possible to introduce geometric information by means of
the Laplacian or a similar operator into this formalism.

In principle, it is thus possible to have a countably infinite-dimensional
symmetric order-three tensor $P_{abc}$ describing the full geometry of a
compact Riemannian manifold. If one were only interested in the topology and
measure, even a finite-dimensional tensor would suffice. This means that this
would present a potential interpretation for models like the ctm, since except
for a formal continuum limit this model generally works with finite-
dimensional tensors. In section 7 several implications for the ctm were
discussed, and it was found that it seems consistent with several results.
Especially since the ctm seems to prefer symmetric tensors, this
interpretation would yield strong evidence for the emergence of macroscopic
spacetimes.

There is a lot of interesting areas with research opportunities with this kind
of framework. They are discussed below in three categories: The framework
itself, the connection to the ctm, and the opportunity to build tensor models.

Firstly, the framework itself still has several places where it can be
refined. For instance in section 4.3 two conjectures were phrased, which would
be interesting to see if there is a way to prove these right or wrong, and
under what conditions. Furthermore, there are still several questions in
relation to section 5. Though for an infinite-dimensional tensor we can
reconstruct the full geometry if we know the deviation of the candidate for a
unit as generated by the tensor $P_{abc}$ with the actual unit, for a finite-
dimensional tensor this interpretation is less clear. One should either extend
the operator to the full associative closure somehow, or only define the
metric using functions up to the dimension of the tensor $N$ which will only
give an approximate metric (which could be interpreted as fuzziness). Another
question is about the case where the $w_{a}$ as in (39) are not strictly
positive. It is expected that we can still make sense of this as a manifold,
possibly as a pseudo-Riemannian manifold. Besides, we demonstrated in this
paper how to reconstruct the topology and measure of a space, but it would be
good to demonstrate that one could explicitly reconstruct the metric. A last
remark about the framework itself goes to the example spaces that have been
considered. While the analysis has been done thoroughly, the main goal was to
provide “proof of concepts”, but it would be very interesting to analyse these
examples further and understand more of the properties of these emergent
(fuzzy) spaces.

A second interesting research area relates to the ctm. We already discussed
some of the implications for this model in section 7, but more has yet to be
done. For instance, several extensions [56, 57] and use cases [58, 59, 60] of
the ctm have been identified in the past and it would be interesting to
revisit them and see if there is any interesting interpretation of those.
Furthermore, this paper is mainly looking at the kinematical question “what is
the interpretation of the object are we dealing with?”, but it would be very
interesting to see if we can use the dynamics of the ctm to see how the
emergent spaces would behave through time. Related to the last remark, it
would be interesting to see if the exact spaces introduced here, or exact
spaces with a modified unit giving them an interesting geometric
interpretation according to section 5, are actually included in the tensors
highlighted by the ctm wave-function [29, 30]. This would be strong evidence
for the quantum-ctm to reproduce macroscopic spaces.

A last research area to consider is the construction of other models using
this framework. While the ctm seems to be a good candidate to apply this
framework to, it is not sure if that is actually the case. It might be that
this framework is best-used for a different kind of model. From section 5 we
know that one could attempt to construct a model with any operator that
satisfies certain requirements. It would be interesting if we could derive a
Hamiltonian for instance, starting with a certain operator and seeing how
these equations should be to yield a theory similar to general relativity. And
in the case of the ctm: Which operator should we use to relate it to general
relativity? Lastly, how can we describe other matter- and force-fields in this
description. While they can be described as a manifold (fibre bundles) and are
thus subject to the same kind of algebras as all other manifolds, they are
expected to behave very differently. It might even be possible to give some
meaning to non-commutative cases in this context, since from a tensor-model
perspective this simply means dropping the symmetry-requirement on the tensor.

The connection between tensors, algebras and their dual topological spaces
seems to open up a different way of interpreting gravity. The duality between
geometry and algebra has been an active field of mathematics for a while, and
combined with tensor models, it seems to provide an opportunity to describe
spacetime in a way that would lead to a mathematically more straightforward
formulation of quantum gravity.

##  9 Acknowledgements

The author would like to thank N. Sasakura for all the fruitful discussions,
advice and encouragement that made this work possible.

##  Appendix A Measure theory.

Throughout this work, some notions from measure theory are used and discussed.
This section serves to fix some of the definitions. For a comprehensive
introduction to the topic, we would like to refer to the literature, for
instance [61].

The general idea behind measure theory is to define a general notion of
“volume of a subset”, and use this to define a way of integration that is more
generally applicable than Riemann integrals. The volume of a subset is
basically a function which takes a subset and generates a positive real
number, called a _measure_. However, it is generally not possible to
consistently define a measure that works on any collection of subsets, so one
first needs to define what subsets one considers. This is done by means of a
$\sigma$-algebra.

######  Definition 8.

Let $X$ be an arbitrary set. A collection $\mathcal{A}$ of subsets is called a
$\sigma$-algebra if

  * •

$X\in\mathcal{A}$,

  * •

for each set $A\in\mathcal{A}$, $A^{c}\in\mathcal{A}$,

  * •

for each infinite sequence $\\{A_{i}\\}$ with $A_{i}\in\mathcal{A}$, the set
$\bigcup_{i}A_{i}\in\mathcal{A}$,

  * •

for each infinite sequence $\\{A_{i}\\}$ with $A_{i}\in\mathcal{A}$, the set
$\bigcap_{i}A_{i}\in\mathcal{A}$.

A $\sigma$-algebra $\mathcal{A}$ is basically a family of subsets of a set $X$
on which we can consistently define a measure. For this reason, a subset
$A\in\mathcal{A}$ is called a _measurable set_. An important $\sigma$-algebra
that is mentioned in the text is the _Borel $\sigma$-algebra_ of a topological
space $\mathcal{T}$, denoted $\mathcal{B}(\mathcal{T})$. It is generated by
the open sets in a topological space, in the sense that one takes all open
sets, and takes infinite intersections of these to add them to the Borel
$\sigma$-algebra. This means that there is a canonical way to define a
$\sigma$-algebra from a topological space.

We are now set to define a measure.

######  Definition 9.

A _measure_ is a function from a $\sigma$-algebra $\mathcal{A}$ to the
positive real numbers

$\mu:\mathcal{A}\rightarrow\mathbb{R}_{+}\cup\\{+\infty\\},$

that is countably additive

$\mu(\cup_{i}A_{i})=\sum_{i}\mu(A_{i}),$

for each infinite sequence $\\{A_{i}\\}$ of disjoint measurable sets.

The triplet $(X,\mathcal{A},\mu)$ of a set, a $\sigma$-algebra and a measure
is often called a _measure space_ , and only $(X,\mathcal{A})$ is a
_measurable space_.

One of the most important and useful achievements of measure theory is the
_Lebesgue-integral_. For this, let us first define the notion of a measurable
function, of which several similar and equivalent definitions exist, which are
the functions we can actually define an integral for.

######  Definition 10.

For $(X,\mathcal{A})$ a measurable space and $A\in\mathcal{A}$, a measurable
function with respect to $\mathcal{A}$ $f$ is a function

$f:A\rightarrow\mathbb{R},$

such that for every real number $t\in\mathbb{R}$, the set $\\{x\in
A:f(x)<t\\}\in\mathcal{A}$.

Two important notions used for the Lebesgue integral, and used in section 4,
are indicator functions and simple functions. An _indicator function_
$\mathbf{1}_{A}:X\rightarrow\mathbb{R}$ is a function that has the value $1$
on the measurable set $A$, and $0$ elsewhere, and may be seen as a
generalisation of a step function. We define the integral of an indicator
function as the volume of the set $A$, i.e.

$\int_{X}\mathbf{1}_{A}{\rm d}\mu(x):=\mu(A),$

which makes sense since $A$ is measurable. A _simple function_
$f:X\rightarrow\mathbb{R}$ consists of a finite sum of these indicator
functions for disjoints measurable sets $A_{i}$ and real numbers $\alpha_{i}$

$f(x)=\sum_{i=1}^{n}\alpha_{i}\mathbf{1}_{A_{i}}.$

Note that since $A_{i}$ are all measurable and disjoint, we can readily define
an integral over these simple functions

$\int_{X}f(x){\rm d}\mu(x):=\sum_{i=1}^{n}\alpha_{i}\mu(A_{i}).$

Let us denote the collection of all simple functions on $(X,\mathcal{A})$ as
$\mathcal{S}$. The Lebesgue-integral for a positive valued function
$f_{+}(x)\geq 0$ is now defined as

$\int_{X}f_{+}(x){\rm d}\mu(x):=\sup\left\\{\int_{X}g(x){\rm
d}\mu(x)\,|\,g\in\mathcal{S},\ g(x)<f_{+}(x)\forall x\in X\right\\}.$

This can be seen as approximating a function $f_{+}$ by simple functions. For
a general measurable function $f$ we then split the function up into two
positive functions $f=f_{+}-f_{-},f_{+}\geq 0,f_{-}\geq 0$. Finally, the
Lebesgue integral for any measurable function is defined as

$\int_{X}f(x){\rm d}\mu(x):=\int_{X}f_{+}(x){\rm d}\mu(x)-\int_{X}f_{-}(x){\rm
d}\mu(x).$

The Lebesgue integral is a more general notion of integration than the Riemann
integral, but importantly, if the Riemann integral of a function $f(x)$
exists, the Lebesgue integral also exists and they are equal.

##  Appendix B The Laplace-Beltrami operator.

This section serves as an introduction to the Laplace-Beltrami operator, which
generalises the Laplacian from Euclidean space $\mathbb{R}^{n}$ to Riemannian
manifolds $(M,g)$. More information on this operator may be found in
literature, for instance [62]. Since we are mainly interested in smooth
functions in this work, we will always consider the function spaces of smooth
functions $C^{\infty}(\mathbb{R}^{n})$ and $C^{\infty}(\mathcal{M})$
respectively.

The Laplacian on Euclidean space $\mathbb{R}^{n}$ is an operator

$\Delta:C^{\infty}(\mathbb{R}^{n})\rightarrow C^{\infty}(\mathbb{R}^{n}),$

and is given by

$\Delta:=\partial_{1}^{2}+\ldots+\partial_{n}^{2},$

and it shows up in various contexts throughout physics. If one wishes to
generalise this operator to a Riemannian manifold $(\mathcal{M},g)$, one needs
to define it in a coordinate-free way. This is achieved as

$\Delta:=\text{div}\circ\nabla.$

Here $\nabla$ denotes the gradient of a function, and div the divergence. This
definition now works for general Riemannian manfolds. Locally, it is given by

$\Delta f=\frac{1}{\sqrt{\det g}}\partial_{j}(g^{ij}\sqrt{\det
g}\partial_{i}f).$

On an n-dimensional Riemannian manifold, one can define a measure by the
integration form

${\rm d}^{n}x\sqrt{\det g}.$

Using this integration form, one can define an inner product on
$C^{\infty}(\mathcal{M})$,

$\braket{f}{\,g}:=\int_{\mathcal{M}}{\rm d}^{n}x\sqrt{\det g}f(x)g(x).$

Taking the closure of $C^{\infty}(\mathcal{M})$ with respect to this inner
product leads to the Hilbert space of square integrable functions
$L^{2}(\mathcal{M})$. Using this inner product, the Laplace-Beltrami operator
is actually self-adjoint for compactly supported functions [52]

$\braket{\Delta f}{\,g}=\braket{f}{\,\Delta g}.$

A final remark goes to the link to geometry [62]. Not only is the Laplace-
Beltrami operator determined by the metric, but this works the other way
around too. Knowing how the Laplace-Beltrami operator acts on functions
actually fixes the metric completely.

##  Appendix C The canonical tensor model.

This section gives a brief review of the canonical tensor model (ctm). The ctm
is a tensor model for gravity in the canonical (Hamiltonian) formalism, and it
is the main motivation to introduce the formalism of this paper. The tensors
are fully symmetric tensors of rank three, denoted by $Q_{abc}$. In the case
of an $N$-dimensional underlying vectorspace, this means that the
configuration space is $\mathbb{R}^{\mathcal{N}}$, where
$\mathcal{N}=\frac{1}{6}N(N+1)(N+2)$ is the amount of independent entries in
the tensor.

Since the model is constructed in the canonical formalism, we construct a
phase space which is isomorphic to $\mathbb{R}^{2\mathcal{N}}$, where the
canonically conjugate pair is denoted by $(Q_{abc},P_{abc})$. On this phase
space, the Poisson bracket is given by

$\\{Q_{abc},P_{abc}\\}=\sum_{\sigma}\delta_{a\sigma_{d}}\delta_{b\sigma_{e}}\delta_{c\sigma_{f}},$

and all other brackets vanish, where $\sigma$ denote the permutations of
$\\{d,e,f\\}$.

Similarly to the adm-formalism [55], the Hamiltonian of the theory consists of
two constraints. One analogous to the spatial diffeomorphism constraint of the
adm-formalism, the generator of $SO(N)$ transformations,

$\mathcal{J}_{ab}=\frac{1}{4}\left(Q_{acd}P_{bcd}-Q_{bcd}P_{acd}\right).$

The other constraint, anolagous to the Hamiltonian constraint in the adm-
formalism, is given by

$\mathcal{H}_{a}=\frac{1}{2}\left(P_{abc}P_{bde}Q_{cde}-\lambda
Q_{abb}\right),$

where $\lambda$ is a real constant. This Hamiltonian has been shown to be
unique under some physically reasonable assumptions [23].

Just like the adm-formalism, the constraints span an algebra given below,