# The Spacetime Picture in Quantum Gravity II

Alejandro Ascárate FaMAF-UNC (Córdoba, Argentina). Email<EMAIL_ADDRESS>
###### Abstract

As a continuation to [12], we introduce a new formalism for (part of) QG,
which we call TQG, since it’s based on the NC Tori. This allows us to obtain
numerous insights about the nature of time, like its discretization, its
regular pace at the macroscopic scale, a solution to the Problem of Time, and
a connection with the Measurement Problem and wave function collapse.

## Introduction

In the phase space of Gravity, there’s no background spacetime (both in terms
of manifold and metric), and this leads to a Hamiltonian
$\mathcal{H}_{EH}=\sum_{i}N_{(i)}\mathbf{C}^{(i)}$ given by the constraints
$\mathbf{C}^{(i)}$, which, in turn, leads to gauge invariant phase space
properties (“Dirac properties”) $F$ (i.e., the ones that commute with the
constraints, $\left\\{F,\mathbf{C}^{(i)}\right\\}=0$) that lack an adequate
time evolution (since $\frac{\partial F}{\partial
t}=\left\\{F,\mathcal{H}_{EH}\right\\}=$$\sum_{i}N_{(i)}\left\\{F,\mathbf{C}^{(i)}\right\\}=0$.)
We claim that Dirac properties are not the relevant thing to look for. For
example, one can show that the total volume of a solution $g$ is not a Dirac
property, even when it’s a diffeomorphism invariant functional [2]. But the
problem is not in the volume, it’s in the notion of Dirac property. Indeed,
the reason for the previous seemingly paradoxical issue is directly related to
the fact that the constraints in GR do not implement the full spacetime
diffeomorphism group on phase space, and this is a consequence of the fact
that the slicing of spacetime with spacelike Cauchy surfaces (which is needed
to build the phase space) is dependent on the dynamical metric [2]. Thus,
Dirac properties are phase space properties, and very tied to its structure.
The issue about diffeomorphism invariant properties in a solution, but which
are not Dirac properties, is, again, an artifact of the phase space picture,
like the “no time” problem. To avoid all of these problems, we propose to
completely dispense from the phase space picture when dealing with geometrical
properties of spacetime, and instead switch to a spacetime picture. The only
reason why we consider the Hamiltonian formulation (and, thus, the phase space
picture and Dirac properties) is because it’s needed for the process of
canonical quantization. But, once we canonically quantize a kinematical
algebra, we can build the relational spacetime algebra and start to work in
the spacetime picture and thus forget about the phase space; only if we are
trapped in the phase space picture we would need to consider Dirac properties.
When we work in basic GR in the spacetime picture and solve the field
equations in simple cases, we don’t even think about Dirac properties, we just
consider the diffeomorphism invariant properties of the solution in question,
like the spacetime volume (and, of course, also the time evolution with
respect to this solution; it doesn’t make sense to look for “time evolutions
in the phase space of GR” since there’s no spacetime in it, it’s only when we
pick a solution, a metric, that we can build the relational spacetime, as was
argued here: time evolution is the specific change with respect to a
_duration_ , the variable with respect to which one measures the change _must
have that specific physical interpretation_ ; thus, since we need anyway to
pick a solution to build the relational spacetime and the time evolution, the
need to build Dirac phase space properties in order to consider their “phase
space time evolution” completely dissolves, since by definition we must
abandon the phase space picture if we want to build a true time evolution; the
imperative of Dirac properties in QG comes from the assumption that we need,
and can, build a true time evolution in the phase space of GR, for which,
naturally, one would need to consider the phase space version of
diffeomorphism invariance, namely, Dirac properties, an a notion of relational
time not related to duration, a property of the gravitational field, but to
other fields, if that is even possible in the first place.)

Thus, the problem here will be assumed to be that of trying to find an
analogue of duration with respect to a given metric (i.e., in a spacetime
picture) but now in the quantum (gravity) realm.

## Background

We review and summarize below some of the basic notions from [12] that will be
relevant in the present paper, which is a continuation. We follow here the
numeration for the definitions and propositions from that reference, where a
more detailed discussion can be found.

$\mathbf{Definition\;1.1}$: The (kinematical) phase space of GR is defined as

$X=\left\\{\left[h_{ab},\pi^{ab}\right]\diagup h_{ab},\pi^{ab}\in
C^{\infty}(\varSigma)\,(\mathrm{as\,fields}\,\mathrm{on}\,\varSigma)\right\\},$

where $h_{ab}$ and $\pi^{ab}$ are, respectively, a smooth riemannian metric on
a spacelike Cauchy hypersurface $\Sigma$ in a compact and boundaryless
spacetime $M$, foliated by $\Sigma$ as usual, and the conjugate momentum
tensor density$.\,\blacksquare$

$\mathbf{Definition\;1.2}$: The subset $\mathcal{F}\subset C(X)$ consists of
the phase space functionals of the form

$F_{f}\left(\left[h_{ab},\pi^{ab}\right]\right)\doteq\int_{\varSigma}f\boldsymbol{\epsilon}(h_{ab}),\,\forall\left[h_{ab},\pi^{ab}\right]\in
X,$

where $f\in C^{\infty}(\varSigma)$ and $\boldsymbol{\epsilon}(h_{ab})$ is the
volume element of $h_{ab}$, i.e.
$\boldsymbol{\epsilon}(h_{ab})=\sqrt{h}\,\mathrm{d}^{3}x$
($h=\mathrm{det}\,(h_{ab})$)$.\,\blacksquare$

$\mathbf{Proposition\;1.1}$: The assignment $f\,\longmapsto F_{f}$ is
injective.$\,\square$

$\mathbf{Definition\;1.3}$: We now define a mapping $R:\mathcal{F}\times
X\,\longrightarrow C^{\infty}(\varSigma)\times Obj(\mathsf{Hil})$ (where
$Obj(\mathsf{Hil})$ is the collection of objects in the category
$\mathsf{Hil}$ of Hilbert spaces) by

$(F_{f},\left[h_{ab},\pi^{ab}\right])\,\longmapsto
R(F_{f},\left[h_{ab},\pi^{ab}\right])\doteq(f,L^{2}(\mathcal{S},\boldsymbol{\epsilon}(h_{ab})),$

(where $\mathcal{S}$ is the $C^{\infty}(\varSigma)-$module of smooth spinor
fields in $\varSigma$, which from now on we assume allows a spin
structure)$.\,\blacksquare$

$\mathbf{Definition\;1.4}$: For _fixed_ $\left[h_{ab},\pi^{ab}\right]\in X$
and _variable_ $F_{f}\in\mathcal{F}$, we denote the first component of
$R(F_{f},\left[h_{ab},\pi^{ab}\right])$ as $R_{h}(F_{f}).\,\blacksquare$

$\mathbf{Definition\;1.5}$: We make $\mathcal{F}$ into an (unital) algebra
$(\mathcal{F},$$\cdot_{sp.})$, where the product $\cdot_{sp.}$ in
$\mathcal{F}$ is defined as

$\left(F_{f_{1}}\cdot_{sp}F_{f_{2}}\right)\left(\left[h_{ab},\pi^{ab}\right]\right)\doteq
F_{f_{1}f_{2}}\left(\left[h_{ab},\pi^{ab}\right]\right),\,\forall\left[h_{ab},\pi^{ab}\right]\in
X$

$=\int_{\varSigma}f_{1}f_{2}\boldsymbol{\epsilon}(h_{ab}).\,\blacksquare$

$\mathbf{Corollary\;1.1}$: The map $R_{h}$ is a _bijection_ beetwen
$\mathcal{F}$ and $C^{\infty}(\varSigma)$ (by Proposition 1.1), and is a
faithfull algebra representation of $\mathcal{F}$
into111$\mathcal{B}(\mathcal{H})$ is the algebra of all the _bounded_
operators that act on the Hilbert space $\mathcal{H}$, with the operator
composition $\circ$ as algebraic product.
$\mathcal{B}\left(L^{2}(\mathcal{S},\boldsymbol{\epsilon}(h_{ab}))\right)$ by
multiplication operators, i.e. $R_{h}(F_{f})(\psi)(x)\doteq
f(x)\psi(x),\,\forall\psi\in
L^{2}(\mathcal{S},\boldsymbol{\epsilon}(h_{ab}))$$.\,\square$

We call $R_{h}$ the “ _relational representation_ ”, since _it gives a way of
obtaining the algebra of physical $3-$space purely from Field properties_
(represented here by the phase space functionals). _Switching to a relational
and algebraic frame of mind, we can take this representation as the actual way
of defining how to build space from the phase space algebra of the
gravitational field._

$\mathbf{Proposition\;1.2}$: If one starts with a subset
$\mathcal{A}_{\varSigma}$ (with a commutative product $\cdot_{sp}$) of the
phase space algebra $\mathcal{A}_{X}$ (i.e.
$\mathcal{A}_{\varSigma}\subset\mathcal{A}_{X}$, but only as a set and _not_
as a subalgebra222This convention will be maintained whenever the symbol
$\subset$ is used, unless some other sense is explicitly stated.), which
correspond, respectively, to $\mathcal{F}$ and $C(X)$ but _seen as abstract
algebras_ , then there exist a _purely algebraic_ (i.e. which _doesn’t_ make
use of the commutative manifold structure of space for its definition)
faithful representation
$\left(\widetilde{R}_{h}\left[\mathcal{A}_{\varSigma}\right],\mathcal{H}_{\widetilde{R}_{h}}\right)$
of $\mathcal{A}_{\varSigma}$ and isomorphisms
$\pi_{X}:\mathcal{A}_{\varSigma}\,\longrightarrow\mathcal{F}$ and
$\pi_{\varSigma}:\widetilde{R}_{h}\left[\mathcal{A}_{\varSigma}\right]\,\longrightarrow
C^{\infty}(\varSigma)$, such that the following diagram commutes:

$\textstyle{\mathcal{A}_{\varSigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{R}_{h}}$$\scriptstyle{\pi_{X}}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{h}}$$\textstyle{\widetilde{R}_{h}\left[\mathcal{A}_{\varSigma}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\varSigma}}$$\textstyle{C^{\infty}\left(\varSigma\right)}$

i.e. such that

$R_{h}=\pi_{\varSigma}\circ\widetilde{R}_{h}\circ\pi_{X}^{-1}.\,\square$

Note that,_in the general, non-commutative case, only the left hand side of
the previous diagram survives_. In this way, _all the geometrical information
of the space is now contained in the spectral triple_.

$\mathbf{Definition\;2.1}$: Consider the phase space of GR,
$(X,C(X),\left\\{\cdot,\cdot\right\\}_{GR})$, and a background independent
Poisson sub-algebra $\mathcal{A}_{GR}$ of phase space functionals there. The
basic ${}^{*}-$algebra, $\widehat{\mathcal{A}}_{QG}$, in QG will be the
algebra $S_{\mathcal{A}_{GR}}$ freely generated (with field $\mathbb{C}$) by
the classical algebra elements333Which is just the complex vector space
generated by the basis $\left\\{e_{S}\right\\}$, where $S$ runs over all the
possible ordered, finite sequences of elements from $\mathcal{A}_{GR}$ (e.g.
$S=$$(a_{1},a_{2},...,a_{k}),\,k>0$) and the algebra product is given at the
basis level by $e_{S}\star e_{T}\doteq e_{(S,T)}$. and with the relations
imposed by the ideals $\mathcal{I}_{*LP}$ (where $*$ is the involution, $L$ is
from linear and $P$ from $\mathrm{i}$ multiplied by the Poisson brackets. This
process simply imposes the familiar Dirac commutation relations of canonical
quantization to $\widehat{\mathcal{A}}_{QG}$, that is:
$\left[\widehat{a},\widehat{a}^{\prime}\right]_{\widehat{\mathcal{A}}_{QG}}\doteq$$\widehat{a}\star_{\widehat{\mathcal{A}}_{QG}}\widehat{a}^{\prime}-\widehat{a}^{\prime}\star_{\widehat{\mathcal{A}}_{QG}}\widehat{a}=$$\mathrm{i}\widehat{\left\\{a,a^{\prime}\right\\}}_{GR}$,
where $\star_{\widehat{\mathcal{A}}_{QG}}$ is the quantum probability
algebraic product of $\widehat{\mathcal{A}}_{QG}$). That is,
$\widehat{\mathcal{A}}_{QG}$ is the quotient:

$\widehat{\mathcal{A}}_{QG}\doteq\frac{S_{\mathcal{A}_{GR}}}{\mathcal{I}_{*LP}}.\,\blacksquare$

The next task is to identify a _subalgebra_
$\widehat{\mathcal{A}}_{SP.}\subset\widehat{\mathcal{A}}_{QG}$_that plays a
role equivalent to the one of $\mathcal{A}_{\varSigma}$ in the classical case
of Proposition 1.2._

$\mathbf{Definition\;2.2}$: Consider the subset $\\{f_{S}\\}$ of all elements
$f_{S}$ in $\mathcal{F}$ with $S\doteq supp\,f_{S}\subseteq\varSigma$, then we
can see $\mathcal{F}$ as the result of the application of an _indexation
relation_
$\widetilde{I}_{\mathcal{F}}:I\,\longrightarrow\mathcal{F},\,S\,\longmapsto\widetilde{I}_{\mathcal{F}}(S)\doteq\\{f_{S}\\}$,
where $I$ is the collection of all the supports $S$. We now _replace_ $I$ by a
subset $I_{Dis}\subset I$ which can be _at most countably infinite_ , and
compute the Poisson brackets in
$\mathcal{F}_{Dis}\doteq\widetilde{I}_{\mathcal{F}}[I_{Dis}]$: the subset
$I_{Dis}$ should be selected in a way that makes
$(\mathcal{F}_{Dis},\left\\{\cdot,\cdot\right\\}_{GR})$ a Poisson subalgebra
and, in particular, one whose structure constants _don’t_ depend on the
differentiable manifold details of $\Sigma$ nor on the details of the
functions $f_{S}$ varying over it, _but only on $I_{Dis}$, as an indexing
set._ With this set up, we choose $\mathcal{A}_{GR}$ such that
$\mathcal{F}_{Dis}\subset\mathcal{A}_{GR}$ (so that
$\widehat{\mathcal{F}}_{Dis}\subset\widehat{\mathcal{A}}_{QG}$) and define:

$\widehat{\mathcal{A}}_{SP.}\doteq\widehat{\mathcal{F}}_{Dis}.\,\blacksquare$

$\mathbf{Definition\;2.3}$: Assuming one has the family of possible
representations
$\left(R_{QG}\left[\widehat{\mathcal{A}}_{QG}\right],\mathcal{H}_{R_{QG}}\right)$
of $\widehat{\mathcal{A}}_{QG}$, we define as

$\widehat{\mathcal{A}}_{sp.}\doteq
R_{QG}\left[\widehat{\mathcal{A}}_{SP.}\right]$

the _non-commutative_ algebra of quantum physical $3-$space that “relationally
arises” from the quantum gravitational field (this algebra will be the quantum
analogue of $\widetilde{R}_{h}\left[\mathcal{A}_{\varSigma}\right]$, and
$R_{QG}$ that of $\widetilde{R}_{h}$, in the left hand side of the diagram of
Proposition 1.2)$.\,\blacksquare$

$\mathbf{Definition\;2.6}$:  The analogue of a given classical metric $h_{ab}$
will be given by a $3-$(spectral) dimensional _real first order spectral
triple_

$\mathcal{T}_{D_{QG}}^{R_{QG}}\doteq\left(R_{QG}\left[\widehat{\mathcal{A}}_{SP.}\right],\mathcal{H}_{R_{QG}},D_{QG}\right).\,\blacksquare$

$\mathbf{Remark\;2.2}$: In particular, the _Dirac-like operators_ $D_{QG}$
could be used to calculate _non-commutative_ space intervals (i.e. distances)
and _non-commutative_ volume integrals, which, given the interpretations we
made, _are the genuinely quantum distances and volumes_ , since they are based
on the _quantized space algebra $.\,\blacksquare$_

### 2.3. TQG Relational Time

Regarding time, we have the following situation. If we recall the definition
of the variables used in our relational construction of the space picture
(Definition 1.2), the natural generalization needed to include time, and in
this way to get a _spacetime_ picture, would be

$F_{f}\left(\left[h,\pi\right]\right)\doteq\int_{M}f(t,\overrightarrow{x})\,\sqrt{-g}\,\mathrm{d}^{4}x.$

Thus, we can see that if we want to get time into the picture, we need to
consider test functions $f$ whose support is spacetime rather than just space.
Nevertheless, there are two problems with the previous generalization, namely:
i) how do we get an identification
$g_{ab}\,\longleftrightarrow\left[h,\pi\right]$ between smooth spacetime
metrics $g_{ab}$ which are solutions and phase space points
$\left[h_{ab},\pi^{ab}\right]$?; ii) even if we solve that, is the resulting
assignment $f\,\longmapsto F_{f}$ injective? Regarding point i), it’s well
known that the Gravitational Field Equations have a well-posed initial value
formulation, that is, an initial value $\left[h_{ab},\pi^{ab}\right]\in X$
which satisfies the initial value constraints, and a gauge fixing of the lapse
$N$ and shift $N^{a}$ in $\varSigma$ (this if we use the so called “wave
gauge” [11, 1]) determines a unique smooth and globally hyperbolic solution
$g_{ab}\,\underset{N,N^{a}}{\longleftrightarrow}\left[h_{ab},\pi^{ab}\right]$.
For point ii), we have the following proposition.

$\mathbf{Proposition\;2.4}$: For spacetime functions $f$ with support in
$[0,\epsilon)\times\varSigma$, the assignment $f\,\longmapsto F_{f}$ is
injective in the limit $\epsilon\,\longrightarrow 0$.

_Proof_ : from now on, we take, once and for all, $N=1$ and $N^{a}=0$ (both,
in $\varSigma$): thus, we will work with the identification
$g_{ab}\,\underset{1,0}{\longleftrightarrow}\left[h_{ab},\pi^{ab}\right]$.
Now, at the infinitesimal level, the infinitesimal arc of the graph of a real
variable function $h(t)\equiv\mathrm{det}\,(h_{ab})$ at some point (say,
$t=0$) can be approximated by the tangent line at that point. To describe the
latter, we need the value $h(0)$ and its derivative $\dot{h}(0)$. In the wave
gauge, $\sqrt{-g}=N\sqrt{h}$ in $\varSigma$, i.e., at $t=0$, and, with our
choice $N(0)=1$, we get $\sqrt{-g}(0)=\sqrt{h}(0).$ For $t\neq 0$,

$g\equiv\mathrm{det}\,(g_{ab})=-g_{00}h+\sum_{i}g_{0i}d_{i},$

where $d_{i}$ are the remaining determinants. In this way,

$\dot{g}=-\dot{g}_{00}h-g_{00}\dot{h}+\sum_{i}\dot{g}_{0i}d_{i}+\sum_{i}g_{0i}\dot{d}_{i}.$

But now, at $t=0,$ $g_{00}=-1$, $g_{0i}=0,$ and then $d_{i}=0$. Also,
$\dot{h}=-2hK$ (where $K=-\frac{1}{2}h^{ab}\dot{h}_{ab}$; note that
$\pi=2\sqrt{h}K$), while it can be determined from the wave gauge conditions
that $\dot{g}_{00}=-K$ (recall that the wave gauge condition on the
coordinates is

$\sum_{\mu,\nu}g^{\mu\nu}\varGamma_{\mu\nu}^{\alpha}=0,\,\forall\alpha,$

which means that the coordinates satisfy the wave equation; in our variables,
at $t=0$, we get from it:

$\dot{g}_{00}+Nh^{ab}K_{ab}=0.)$

Thus, we get $\dot{g}(0)=-h(0)K(0)$. What all this means is that the tangent
line to the graph of $\sqrt{-g}$ at $t=0$ only depends on
$\left[h_{ab},\pi^{ab}\right]\in X$. Of course, due to the second order
character of the Gravitational Field Equations, the initial data
$\left[h_{ab},\pi^{ab}\right]\in X$ is arbitrary, which means that the tangent
line can be arbitrary. Note that this is not trivial and we got it thanks to
the wave gauge condition: if this condition implied, instead, that
$\dot{g}_{00}=-2K$, then we would get $\dot{g}(0)=0$, which means that this
derivative cannot be arbitrary, even if $\left[h_{ab},\pi^{ab}\right]\in X$
is. Of course, it’s actually this derivative the most important thing that we
need here in order to probe beyond $\varSigma$ in the time
direction$.\,\square$

$\mathbf{Example\;2.3}$: Let’s analyse first these type of spacetime variables
in the case of standard QFT on a fixed curved spacetime ($M,g_{ab})$ [1, 5].
Consider the usual scalar field $\varphi$ with a linear wave equation

$g_{ab}\nabla^{a}\nabla^{b}\varphi=0.$

Since this equation has a well-posed initial value formulation, one can
identify the phase space with the space $\mathrm{Sol}.$ of smooth solutions.
Then, for a spacetime test function $f\in C_{0}^{\infty}(M)$, one defines the
following variable on that phase space:

$\underline{\varphi}(f)(\varphi)\doteq\int_{M}\varphi
f\boldsymbol{\epsilon}(g).$

The Poisson brackets for these variables are then given by

$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}=-\Omega(Ef,Ef^{\prime})$

$=-\int_{M}fEf^{\prime}\boldsymbol{\epsilon}(g)=\int_{M}f^{\prime}Ef\boldsymbol{\epsilon}(g),$

where $\Omega$ is the symplectic form on $\mathrm{Sol}.$ and the linear map
$E:C_{0}^{\infty}(M)\,\longrightarrow\mathrm{Sol}.,$$\,f\,\longmapsto Ef$ is
the advanced minus the retarded solution of the wave equation with source
$f\in C_{0}^{\infty}(M)$. Now, consider the case in which the supports of $f$
and $f^{\prime}$ are _causally disconnected_ , i.e., if $f(x)f^{\prime}(y)\neq
0\,\Longrightarrow y\notin J^{+}(x)\bigcup J^{-}(x)$ (where $J^{+/-}$ are the
causal future and past, respectively.) But we also have that:

$\int_{M}f^{\prime}Ef\boldsymbol{\epsilon}(g)=\int_{\mathscr{R}_{1}}f^{\prime}(Af)\boldsymbol{\epsilon}(g)-\int_{\mathscr{R}_{2}}f^{\prime}(Rf)\boldsymbol{\epsilon}(g),$

where $\mathscr{R}_{1}=supp\left(f^{\prime}\right)\bigcap
J^{-}\left[supp\left(f\right)\right]$ and
$\mathscr{R}_{2}=supp\left(f^{\prime}\right)\bigcap
J^{+}\left[supp\left(f\right)\right]$, since $A(f)$ and $R(f)$ are,
respectively, the advanced and retarded solutions. Thus, if the supports of
$f$ and $f^{\prime}$ are causally disconnected, then, because of well-
posedness of the field equation, clearly we have that
$\mathscr{R}_{1}=\mathscr{R}_{2}=\textrm{\O}$ and in this way we get

$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}=-\Omega(Ef,Ef^{\prime})=0$

for this case. On the other hand, if the supports are _causally connected_ ,
we get

$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}=-\Omega(Ef,Ef^{\prime})\neq
0.\,\blacksquare$

With the above in mind, let’s now go back to GR. In this theory, the field
equations do not have the simple form of the previous linear wave equations,
but comprise (in the so-called “wave gauge”), instead, what’s known as a
_quasi-linear_ system. For a scalar field with a quasi-linear field equation,
the latter takes the form

$g_{ab}(x;\varphi;\nabla^{c}\varphi)\nabla^{a}\nabla^{b}\varphi=F(x;\varphi;\nabla^{c}\varphi),$

where $g_{ab}$ is a smooth lorentzian metric and $F$ a smooth function. These
type of equations still have a well-posed initial value formulation, but they
are quite different to the standard, ordinary wave equation. Indeed, the
lorentzian metric that defines the character of the principal symbol of the
differential operator now depends on the field variable. Of course, this makes
the field equations non-linear. But, since the principal symbol of the
operator is what determines how causal influences on this field propagate, it
also means that the very causal structure is tied now to the variation of the
field variable. Unfortunately, this fact makes all of the previous analysis of
QFT in fixed curved spacetimes to be inapplicable here. Indeed, even to this
date, the general form of the Poisson bracket for functionals like $F_{f}$ is
_not_ known [7]444Nevertheless, it’s shown in this reference that we still
get, for the quasi-linear case,
$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}=0$ if
the supports are causally disconnected and, presumably,
$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}\neq
0$ if the supports are causally connected. Thus, in the quantization below,
all stages in which only this property of the brackets is used can be
considered also applicable to the general case, while the ones in which we use
the explicit form
$\left\\{\underline{\varphi}(f),\underline{\varphi}(f^{\prime})\right\\}=-\Omega(Ef,Ef^{\prime})$,
_from the linear case_ , are approximative.. Thus, a complete and rigorous
quantization of time by following the recipe of our appraoch is definitely out
of reach for now. Thus, it’s at this point when we start to make _several_
approximations, which, we warn, may or may not be ultimately valid. We will
judge that by the reasonability of their nature and of the results that follow
from assuming them.

For simplicity, from now on we pretend a metric solution $g_{ab}$ is just a
scalar field $\varphi$. These details will not matter here since the present
analysis is only approximate and merely structural.

$\mathbf{Definition\;2.14}$: Consider a compact curve555Actually, we take
$\gamma$ as a narrow timelike/spacelike spacetime cylinder centered at the
original curve, in order to avoid a cluttering of delta functions on the
integrals. (timelike or spacelike) segment $\gamma$ in $M$. Then, on the phase
space $X$ of GR, we define the following variables:

$\underline{\varphi}(f_{\gamma})(\varphi)\doteq\int_{M}\varphi
f_{\gamma}\boldsymbol{\epsilon}(\varphi),\,supp\,(f_{\gamma})=\gamma,\,f_{\gamma}\in
C_{0}^{\infty}(M).\,\blacksquare$

$\mathbf{Proposition\;2.5}$: The previous variables in Definition 2.14 _can_
indeed be used to obtain $C^{\infty}(\gamma)$ via $R_{\varphi}$ (in the limit
$\epsilon\,\longrightarrow 0$.)

_Proof_ : as in Proposition 2.1 [12], we define the following product

$(\underline{\varphi}(f_{\gamma}^{1})\cdot_{\gamma}\underline{\varphi}(f_{\gamma}^{2}))(\varphi)\doteq\underline{\varphi}(f_{\gamma}^{1}f_{\gamma}^{2})(\varphi),\,\forall\varphi\in
X$

$=\int_{M}\varphi f_{\gamma}^{1}f_{\gamma}^{2}\boldsymbol{\epsilon}(\varphi).$

In the limit $\epsilon\,\longrightarrow 0$, where the assignment
$f_{\gamma}\,\longmapsto\underline{\varphi}(f_{\gamma})$ is injective (the
term $\varphi$ in the integral can be handled by analogous arguments as those
in Proposition 2.1), we get an algebra
$\left\\{\underline{\varphi}(f_{\gamma}k_{0})\right\\}_{f_{\gamma}\in
C_{0}^{\infty}(M);\cdot_{\gamma}}$, for each $k_{0}\in C^{\infty}(\gamma)$
($k_{0}$ is non-zero for each point in $\gamma$; all algebras for different
$k_{0}$ are isomorphic and equivalent to $C^{\infty}(\gamma)$), which, since
the solution $\varphi$ such that $k_{0}=\varphi^{-1}$ is assumed to be non-
vanishing on $\gamma$, is then bijectively mapped to $C^{\infty}(\gamma)$ via
$R_{\varphi}(\underline{\varphi}(f_{\gamma}k_{0}))=f_{\gamma}$; but, since all
the algebras are isomorphic, this means that the relational representation
indeed induces the algebra $C^{\infty}(\gamma)$ of the curve $\gamma$ from a
_single_ algebra of phase space functions for _any_ of those solutions
$\varphi$$.\,\square$

Consider some _subset_ of $\mathrm{Sol}.$ in which all metrics have
_conformally-equivalent_ causal structures (besides this requirement, there’s
freedom for choosing any such subset.) Thus, while we have many different
metrics, at least we have a _single_ causal structure $\mathcal{C}$. Thus, the
only variation we actually have now is that of the _conformal factor_ among
the different metrics. It’s _this_ variation the one we will try to quantize.
Nevertheless, the equations are still non-linear in that factor. For this, we
will consider only the linear part for the quantization; that is, the result
will be only kinematical, since the non-linear coupling is being ignored. In
this way, what we are doing is to consider quantum spacetime geometries which
are “benign” and “semi-classical” enough such that they have a defined causal
structure and which remains more or less the same among the different
geometries. Furthermore, they are also such that the conformal factor doesn’t
vary non-linearly among them. Thus, the geometries are almost classical, the
only variation is a “residual” one in the conformal factor, which is linear,
and hyperbolic with respect to the causal structure $\mathcal{C}$. It’s only
this variation that is quantized below.

$\mathbf{Definition\;2.15}$: A _causal set_(see, e.g., [8] and references
therein) is a partially ordered set $\left(C,\leq_{\mathcal{C}}\right)$ such
that:

  * i)

for all $x\in C,$ $x\leq_{\mathcal{C}}x$ (Reflexive);

  * ii)

for all $x,y\in C,$ if $x\leq_{\mathcal{C}}y$ and $y\leq_{\mathcal{C}}x$, then
$x=y$ (Antisymmetric);

  * iii)

for all $x,y,z\in C$, if $x\leq_{\mathcal{C}}y$ and $y\leq_{\mathcal{C}}z$,
then $x\leq_{\mathcal{C}}z$ (Transitive);

  * iv)

for all $x,z\in C$, $\mid\left\\{y\in
C,\,x\leq_{\mathcal{C}}y\leq_{\mathcal{C}}z\right\\}\mid<\aleph_{0}$ (Locally
finite)$.\,\blacksquare$

$\mathbf{Definition\;2.16}$: Let $\leq_{\mathcal{C}}$ be the _causal partial
order_ on the points in $M$ induced by the causal structure $\mathcal{C}$.
Now, consider a graph defined on _all_ of the _spacetime_ manifold $M$;
furthermore, on each $\varSigma_{t}$, it gives one of the graphs considered
earlier in the discussion of area, while the nodes of each of these graphs for
_different_ $\varSigma_{t}$ and $\varSigma_{t^{\prime}}$ are connected by
timelike/causal (with respect to $\leq_{\mathcal{C}}$) edges (e.g., the
previous curve $\gamma$ could be one of them), while the edges of the graphs
in each $\varSigma_{t}$ are seen as spacelike separated (i.e., the nodes are
not related by $\leq_{\mathcal{C}}$.) We denote each node as $P_{i}$, for a
countable666Note that this is not a capricious, ad-hoc discretization, even
when we are indeed puting it by hand, since it’s justified by the physical and
mathematical arguments given in all previous sections at various points of the
discussion. index $i$. Thus, the collection of _elements_ (it’s more
convenient to call them by this term rather than points)

$C\doteq\left\\{P_{i}\right\\}_{i\in\mathbb{N}}^{\leq_{\mathcal{C}}}$

forms a causal set. Consider the edges $\mathrm{e}_{P_{N_{1}}}$ and
$\mathrm{e}_{P_{N_{2}}}$ such that $\mathrm{e}_{P_{N_{2}}}\subset
J^{+}(\mathrm{e}_{P_{N_{1}}})$. We omit the endpoints and (semi) characterize
$\mathrm{e}_{P_{*}}$ simply by its initial point $P_{*}$ (that is,
$\mathrm{e}_{P_{*}}$ represents _any_ of the edges with initial point
$P_{*}$.) We take the variables
$\underline{\varphi}(\chi_{\mathrm{e}_{P_{*}}})$ (for which
$supp\left(\chi_{\mathrm{e}_{P_{*}}}\right)=\mathrm{e}_{P_{*}}$), where
$\chi_{\mathrm{e}_{P_{*}}}$ is the characteristic function of the set
$\mathrm{e}_{P_{*}}$, and the algebra $\mathcal{A}_{TQG}$ that they linearly
generate (with real coefficients), and promote the labelings to elements
$P_{*}$ from an _abstract_ , countably infinite causal set
$C$$.\,\blacksquare$

$\mathbf{Lemma\;2.6}$: The variables of the proof of Propostion 2.5,
$\underline{\varphi}(f_{\gamma}k_{0})$ (considering the set of _all_ the
elements from _all_ the algebras
$\left\\{\underline{\varphi}(f_{\gamma}k_{0})\right\\}_{f_{\gamma}\in
C_{0}^{\infty}(M);\cdot_{\gamma}}$ that were defined), subjected to the
process of Definition 2.2, result in the variables of Definition 2.16 (the
proof is identical to that of Lemma 2.3 [12])$.\,\square$

Thus, in line with the principle of striping the continuum, this is equivalent
to replacing $f_{\mathrm{e}_{P_{*}}}$ by $k\chi_{\mathrm{e}_{P_{*}}},$ where
$\chi_{\mathrm{e}_{P_{*}}}$ is the characteristic function of the set
$\mathrm{e}_{P_{*}}$ in $M$ and $k\in\mathbb{R}$. Note how this makes the
limit $\epsilon\,\longrightarrow 0$ unnecessary since
$\chi_{\mathrm{e}_{P_{*}}}$ doesn’t change its value along $t$, and then these
type of functions on the finite segment $\mathrm{e}_{P_{*}}$ can be taken as
the “true general functions on the infinitesimal curve segment
$\mathrm{d}\mathrm{e}_{P_{*}}$.” Of course, $\chi_{\mathrm{e}_{P_{*}}}$ is not
a smooth function, but this will not be relevant. That is, associated to
$P_{*}$, we get a $1-$dimensional abstract vector space

$\mathcal{A}_{P_{*}}\doteq\left\\{k\underline{\varphi}(P_{*})\right\\}_{k\in\mathbb{R}}\cong\mathbb{R}$

generated by the sole element $\underline{\varphi}(P_{*})$. Now, of course, if
we quantize this we get just a commutative algebra, since
$\left\\{\underline{\varphi}(f),\underline{\varphi}(f)\right\\}=-\Omega(Ef,Ef)=0$
(because $\Omega$ is a symplectic form.)

Nevertheless, we cannot consider $\mathcal{A}_{P_{*}}$ alone, because

$\mathrm{e}_{P_{N_{2}}}\bigcap
J^{+}(\mathrm{e}_{P_{N_{1}}})\neq\textrm{\O}\;\mathrm{and}\;\left\\{\underline{\varphi}(f_{\mathrm{e}_{P_{N_{1}}}}),\underline{\varphi}(f_{\mathrm{e}_{P_{N_{2}}}})\right\\}=-\Omega(Ef_{\mathrm{e}_{P_{N_{1}}}},Ef_{\mathrm{e}_{P_{N_{2}}}})\neq
0.$

But now, by the linearity of both $\Omega$ and $E$, and remembering that we
replace $f_{\mathrm{e}_{P_{*}}}$ by $k\chi_{\mathrm{e}_{P_{*}}}$, and that
$\mathcal{A}_{P_{*}}\cong\mathbb{\mathbb{R}}$, then it’s clear that, when
passing to the abstract labelings, the symplectic form $\Omega$ becomes simply
a symplectic form $\widetilde{\Omega}$ on the $2-$dimensional space
$\mathbb{\mathbb{R}}^{2}$ and characterized by a simple $2\times 2$ skew
matrix of the form

$\left(\begin{array}[]{cc}0&\theta(P_{N_{1}},P_{N_{2}})\\\
-\theta(P_{N_{1}},P_{N_{2}})&0\end{array}\right),$

where

$\theta(P_{N_{1}},P_{N_{2}})\doteq\Omega(E\chi_{\mathrm{e}_{P_{N_{1}}}},E\chi_{\mathrm{e}_{P_{N_{2}}}}),$

thing which greatly simplifies the problem at the mathematical level.

$\mathbf{Definition\;2.17}$: We define:
$\mathcal{F}_{Dis.}\equiv\mathcal{A}_{ST.}\equiv\mathcal{A}_{P_{N_{1}}P_{N_{2}}}\subset\mathcal{A}_{TQG}$,
generated by the $\underline{\varphi}(f_{\mathrm{e}_{P_{N_{1}}}}),$
$\underline{\varphi}(f_{\mathrm{e}_{P_{N_{2}}}})$. Now, following Definition
2.2, the quantum algebra
$\widehat{\mathcal{A}}_{ST.}\equiv\mathcal{\widehat{A}}_{P_{N_{1}}P_{N_{2}}}$
will be the _exponentiated_ (recall that the passing from the Heisenberg to
the Weyl algebra is mandatory for quantization in QFT [1]) _quantum_ algebra
for the two quantized edges $\widehat{\mathrm{e}}_{P_{N_{1}}}$ and
$\widehat{\mathrm{e}}_{P_{N_{2}}}$, and generated, then, by two abstract
elements, $u_{1}$ and $u_{2}$, which satisfy the relation

$u_{1}u_{2}=e^{2\pi i\theta(P_{N_{1}},P_{N_{2}})}u_{2}u_{1}.\,\blacksquare$

$\mathbf{Example\;2.4}$: This algebra is well known in NCG [3], and is called
the NC $2-$Torus777Hence the name TQG for this proposal, i.e., _Toroidal
Quantum Gravity_. “ _Toral_ ” can be used, too., $\mathbb{T}_{\theta}^{2}$,
with deformation parameter $\theta$, since it corresponds to the deformation
of the algebra of the classical Torus $\mathbb{T}^{2}$ (defined as the set
$\left[0,1\right]^{2}$ with opposite sides of the square identified, or as
$\mathbb{T}^{1}\times\mathbb{T}^{1}$, where $\mathbb{T}^{1}$ is the unit
circle.) More precisely, we consider the universal $C^{*}-$algebra
$A_{\theta(P_{N_{1}},P_{N_{2}})}$ defined by these generators. For
$(r_{1},r_{2})=r\in\mathbb{Z}^{2}$, define the polynomials $u^{r}\doteq e^{\pi
ir_{1}r_{2}\theta(P_{N_{1}},P_{N_{2}})}u_{1}^{r_{1}}u_{2}^{r_{2}}$, then
$A_{\theta(P_{N_{1}},P_{N_{2}})}\ni a=\sum_{r\in\mathbb{Z}^{2}}a_{r}u^{r}$,
where $a_{r}\in\mathbb{C}$ and the sum converges in the $C^{*}$ norm
$\parallel\cdot\parallel_{C^{*}}$. The NC $2-$Torus is the _dense_ , _smooth_
$*-$subalgebra, pre$C^{*}-$algebra
$C^{\infty}(\mathbb{T}_{\theta}^{2})=\mathcal{\widehat{A}}_{\theta(P_{N_{1}},P_{N_{2}})}$
of $A_{\theta(P_{N_{1}},P_{N_{2}})}$, that is, $r\,\longmapsto a_{r}$ is of
_rapid decay_ (the $C^{*}-$algebra $A_{\theta(P_{N_{1}},P_{N_{2}})}$, of
course, gives the algebra of _continuous_ NC-“functions”.) Now,
$\mathbb{T}^{2}$ forms a Lie group, and we can define an action of this group
on $A_{\theta(P_{N_{1}},P_{N_{2}})}$ by setting, for any $z\in\mathbb{T}^{2}$,

$z\cdot u^{r}\doteq z_{1}^{r_{1}}z_{2}^{r_{2}}u^{r}.$

This action is generated by the following two, commuting derivations:

$\delta_{j}(a)\doteq\left[\frac{\mathrm{d}}{\mathrm{d}t}e^{2\pi
i\phi_{j}}\cdot a\right]\mid_{t=0},\,j=1,2,$

where $e^{2\pi i\phi_{j}}$ belongs to the circle group $\mathbb{T}^{1}$ (i.e.,
complex numbers of norm equal to $1$.) The following trace defines a faithfull
algebraic state on $A_{\theta(P_{N_{1}},P_{N_{2}})}$:

$\omega(a)\doteq a_{0}.$

Note that

$\omega(a^{*}a)=\sum_{r\in\mathbb{Z}^{2}}\mid a_{r}\mid^{2}\leq\parallel
a\parallel_{C^{*}}^{2}.$

With it, we can form the GNS representation $\pi_{\omega}$ [5] of the algebra
(that is,
$\left(a,b\right)_{\mathcal{H}_{\omega}}\doteq\omega(a^{*}b)=\left(a^{*}b\right)_{0}=\sum_{r\in\mathbb{Z}^{2}}a_{r}^{*}b_{r}$.)
We denote as $\underline{a}$ to the algebra elements when seen as elements in
the Hilbert space $\mathcal{H}_{\omega}$ of the representation (whose cyclic
vector is $\underline{I}$ and is irreducible) and the same for the derivations
when seen as acting on the algebra representation. We can now define a Dirac
operator by setting (on the dense domain
$\underline{\mathcal{\widehat{A}}}_{\theta(P_{N_{1}},P_{N_{2}})}$)

$D_{P_{N_{1}}P_{N_{2}}}\doteq-i\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-\frac{1}{2}}\left[\sigma_{1}\underline{\delta}_{1}+\sigma_{2}\underline{\delta}_{2}\right],$

where $\sigma_{1},\sigma_{2}$ are the first two Pauli matrices. In this way,
one can see that

$\left(\pi_{\omega}\left[A_{\theta(P_{N_{1}},P_{N_{2}})}\right],\mathcal{H}_{\omega},D_{P_{N_{1}}P_{N_{2}}}\right)$

forms a $2-$dimensional spectral triple (the eigenbasis of
$D_{P_{N_{1}}P_{N_{2}}}^{2}$ is given by
$\left\\{\underline{u}^{r}\right\\}_{r\in\mathbb{Z}^{2}}$ tensored with the
canonical basis of $\mathbb{C}^{2}$, with eigenvalues
$\lambda_{r}=\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-1}4\pi^{2}r\cdot r$ of
multiplicity $M_{r}=2$.) Furthermore (after some adequate normalization),

$\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})\doteq\mathrm{tr}^{+}(\mid
D_{P_{N_{1}}P_{N_{2}}}\mid^{-2})$

$=\mathrm{a}(P_{N_{1}},P_{N_{2}}).$

See [3] for more details$.\,\blacksquare$

$\mathbf{Remark\;2.10}$: Note that this _doesn’t_ mean that the geometry of
spacetime at the quantum level is simply a NC Torus, since the physical
interpretation of the algebra elements that we made before doesn’t lead to
this. Indeed, for $\theta=0$, $u_{1}$ and $u_{2}$ generate a classical,
continuum Torus, while the
$\underline{\varphi}(\chi_{\mathrm{e}_{P_{N_{1}}}})$ and
$\underline{\varphi}(\chi_{\mathrm{e}_{P_{N_{2}}}})$ generate what’s left of
the algebra of the edges _after we eliminated the continuum_. Thus, the Torus
geometrical interpretation from NCG will not be relevant here, and we are just
using/borrowing its algebra for our own purposes and particular
interpretations$.\,\blacksquare$

$\mathbf{Remark\;2.11}$: The physical interpretation of
$\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})$
_when_ $P_{N_{2}}$ _is the endpoint_ of $\mathrm{e}_{P_{N_{1}}}$ is888We also
take $\mathrm{e}_{P_{N_{1}}}$ as _future-directed (fd) timelike_ and
$\mathrm{e}_{P_{N_{2}}}$ as _spacelike_ (with respect to the causal partial
order.) as the quantized riemannian area of the coordinate surface
$\mathcal{R}_{N_{1}N_{2}}$ of
$\mathrm{e}_{P_{N_{1}}}\times\mathrm{e}_{P_{N_{2}}}$ in coordinate space (that
is, we parallelly propagate $\mathrm{e}_{P_{N_{1}}}$ along
$\mathrm{e}_{P_{N_{2}}}$, then set $P_{N_{2}}$ as having coordinates
$(x_{1},x_{2})=(1,0)$ and the endpoint of $\mathrm{e}_{P_{N_{2}}}$ as $(1,1)$,
and then map the whole surface to a rectangle $\mathcal{R}_{N_{1}N_{2}}$ in
coordinate space). Of course, we lose the lorentzian character, but this is
not a problem since that is already being taken into account by the causal
structure $\mathcal{C}$, but this riemannian metric still gives information
about the duration and lenght of the _finite $2-$dimensional process_
described by the spacetime surface
$\mathrm{e}_{P_{N_{1}}}\times\mathrm{e}_{P_{N_{2}}}$, that is, a process which
happens to the whole edge $\mathrm{e}_{P_{N_{2}}}$ for the duration of
$\mathrm{e}_{P_{N_{1}}}$. In this way, given that, in order to take into
account the causal relation between the two edges, we need to consider the
combined quantized algebras of each one (which in turns gives a noncommutative
algebra due to the causal relation) and that the nontrivial999The
$1-$dimensional Dirac operators of each edge (which give the usual classical
metric to curves) lifted to the full algebra do not count since their action
is trivial on the part corresponding to the algebra of the other edge. Dirac
operator on it forms a triple of dimension $2$, then this leads us to take the
(somewhat expected) view in which

_spacetime events at a quantum level (which we denote as
$\mathsf{e}_{N_{1}N_{2}}$, for the one corresponding to $P_{N_{1}},P_{N_{2}}$)
are given by $2-$dimensional processes, i.e., processes that happen to a thing
of finite length in a finite amount of proper time; furthermore, for a given
quantum spacetime (in the sense defined here), the passing from one event to
another is evidently discretized_101010This applies only to individual quantum
spacetimes. The passing from one to another may be given by a continuous
change in $\mathrm{a}(P_{N_{1}},P_{N_{2}})$ (the points fixed.) Nevertheless,
since in LQG the values of length are discretized, this probably means that
$\mathrm{a}(P_{N_{1}},P_{N_{2}})$ is just the product of two different
discrete variables, and then the mentioned change would also be discrete
because of this._since the graph is countably infinite_ $.\,\blacksquare$

$\mathbf{Lemma\;2.7}$: There exists a state
$\underline{\psi}_{N_{1}N_{2}}\in\mathcal{H}_{\omega}$ such that

$\left(\underline{\psi}_{N_{1}N_{2}},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi}_{N_{1}N_{2}}\right)_{\mathcal{H}_{\omega}}=\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-1}=\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})^{-1}.$

_Proof_ : since the triple is $2-$dimensional, then
$\sum_{k=1}^{N}M_{k}\lambda_{k}(\mid D_{P_{N_{1}}P_{N_{2}}}\mid^{-2})$ must
diverge as $\sum_{k=1}^{N}\frac{1}{k}\sim\mathrm{ln}\,N$ when
$N\,\longrightarrow\infty$. This is similar to the inverse of the harmonic
oscillator Hamiltonian in standard QM, for which

$\psi_{s}(x)\doteq\pi^{-\frac{1}{4}}e^{-(x-s)^{2}/2},\,s\in\mathbb{R},$

is a coherent/Gaussian state. This state can be written in terms of the
eigenbasis
$\left\\{\varphi_{k}\right\\}_{k\in\mathbb{N}\bigcup\left\\{0\right\\}}$ of
the Hamiltonian as

$\psi_{s}(x)=e^{-s^{2}/2}\sum_{k=0}^{\infty}\sqrt{\frac{\left(s^{2}/2\right)^{k}}{k!}}\varphi_{k}(x).$

This suggests to define the following state in $\mathcal{H}_{\omega}$:

$\underline{\psi}_{s}\doteq\frac{\left(e^{s^{2}/4}-1\right)^{-1}}{2}\sum_{\underset{r_{1}\neq
0;r_{2}\neq 0}{r\in\mathbb{Z}^{2}}}\sqrt{\frac{\left(s^{2}/4\right)^{\mid
r_{1}\mid}}{\mid r_{1}\mid!}}\sqrt{\frac{\left(s^{2}/4\right)^{\mid
r_{2}\mid}}{\mid r_{2}\mid!}}\underline{u}^{r}.$

First we need to check if this series defines a state in the first place.
Indeed111111Note that
$\underline{a}_{(0,0)}^{\underline{\psi}_{s}}=(\underline{\psi}_{s},\underline{u}^{(0,0)})_{\mathcal{H}_{\omega}}=0$,
where $\underline{u}^{(0,0)}=\underline{I}$, and
$\underline{a}_{(r_{1},0)}^{\underline{\psi}_{s}}=\underline{a}_{(0,r_{2})}^{\underline{\psi}_{s}}=0,\,\forall
r\in\mathbb{Z}^{2}$.:

$\parallel\underline{\psi}_{s}\parallel_{\mathcal{H}_{\omega}}^{2}=\frac{\left(e^{s^{2}/4}-1\right)^{-2}}{4}\sum_{\underset{r_{1}\neq
0;r_{2}\neq 0}{r\in\mathbb{Z}^{2}}}\frac{\left(s^{2}/4\right)^{\mid
r_{1}\mid}}{\mid r_{1}\mid!}\frac{\left(s^{2}/4\right)^{\mid r_{2}\mid}}{\mid
r_{2}\mid!}$

$=1,\,\forall s\in\mathbb{R}.$

Furthermore,

$\left(\underline{\psi}_{s},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi}_{s}\right)_{\mathcal{H}_{\omega}}=\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-1}8\pi^{2}\frac{\left(e^{s^{2}/4}-1\right)^{-2}}{4}\times$

$\sum_{\underset{r_{1}\neq 0;r_{2}\neq
0}{r\in\mathbb{Z}^{2}}}\frac{\left(s^{2}/4\right)^{\mid r_{1}\mid}}{\mid
r_{1}\mid!}\frac{\left(s^{2}/4\right)^{\mid r_{2}\mid}}{\mid
r_{2}\mid!}(r_{1}^{2}+r_{2}^{2});$

noting that

$r_{*}^{2}s^{2\mid
r_{*}\mid}=\frac{s^{2}}{4}\frac{\mathrm{d}^{2}}{\mathrm{d}s^{2}}s^{2\mid
r_{*}\mid},$

then

$\sum_{\underset{r_{1}\neq 0;r_{2}\neq
0}{r\in\mathbb{Z}^{2}}}\frac{\left(s^{2}/4\right)^{\mid r_{1}\mid}}{\mid
r_{1}\mid!}\frac{\left(s^{2}/4\right)^{\mid r_{2}\mid}}{\mid
r_{2}\mid!}(r_{1}^{2}+r_{2}^{2})$

$=\frac{s^{2}}{2}\sum_{\underset{r_{1}\neq 0;r_{2}\neq
0}{r\in\mathbb{Z}^{2}}}\left(\frac{\mathrm{d}^{2}}{\mathrm{d}s^{2}}\frac{\left(s^{2}/4\right)^{\mid
r_{1}\mid}}{\mid r_{1}\mid!}\right)\left(\frac{\left(s^{2}/4\right)^{\mid
r_{2}\mid}}{\mid r_{2}\mid!}\right)$

$=e^{s^{2}/4}s^{2}\left(\frac{s^{2}}{2}+1\right)\left(e^{s^{2}/4}-1\right);$

in this way:

$\left(\underline{\psi}_{s},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi}_{s}\right)_{\mathcal{H}_{\omega}}=\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-1}\pi^{2}\frac{\left(e^{s^{2}/4}-1\right)^{-1}}{2}\times$

$e^{s^{2}/4}s^{2}\left(\frac{s^{2}}{2}+1\right).$

Now, consider the following equation:

$\pi^{2}\frac{\left(e^{s^{2}/4}-1\right)^{-1}}{2}e^{s^{2}/4}s^{2}\left(\frac{s^{2}}{2}+1\right)=1.$

It has a _single_ solution, given by the _unique_ limit $s\longrightarrow 0$.
We denote $\underline{\psi}_{s\longrightarrow 0}$ as
$\underline{\psi}_{N_{1}N_{2}}$. Thus121212In the _dual_ space, we get
$\left(\underline{\psi^{*}}_{N_{1}N_{2}},D_{P_{N_{1}}P_{N_{2}}}^{-2}\underline{\psi^{*}}_{N_{1}N_{2}}\right)_{\mathcal{H}_{\omega}^{*}}=\mathrm{a}(P_{N_{1}},P_{N_{2}})$,
where the limit $s\longrightarrow 0$ _do_ commutes with
$D_{P_{N_{1}}P_{N_{2}}}^{-2}$ because the latter is compact, and therefore
bounded (i.e., continuous.):

$\left(\underline{\psi}_{N_{1}N_{2}},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi}_{N_{1}N_{2}}\right)_{\mathcal{H}_{\omega}}=\mathrm{a}(P_{N_{1}},P_{N_{2}})^{-1}=\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})^{-1}.\,\square$

$\mathbf{Remark\;2.12}$: In this way, at least formally131313For operators
$\pi_{\omega}(a)$ either in the continuous or smooth algebras, then the
expression
$\mathcal{F}_{a}(\omega_{\underline{\psi}_{N_{1}N_{2}}})\doteq\omega_{\underline{\psi}_{N_{1}N_{2}}}(a)\doteq\left(\underline{\psi}_{N_{1}N_{2}},\pi_{\omega}(a)\underline{\psi}_{N_{1}N_{2}}\right)_{\mathcal{H}_{\omega}}$
_does_ define a genuine pure algebraic state (recall that pure states are in
$1-1$ correspondence with the points $x$ of the spacetime $M$ in the classical
case and $\mathcal{F}_{f}(\omega_{x})\doteq\omega_{x}(f)=f(x)$ is just the
Gelfand transform [5], where $\mathcal{F}_{f}$ corresponds to the continous
function $f$ on $M$.), we can see $\underline{\psi}_{N_{1}N_{2}}$ as a “pure
algebraic state” $\omega_{\underline{\psi}_{N_{1}N_{2}}}$ which acts as

$\omega_{\underline{\psi}_{N_{1}N_{2}}}(D_{P_{N_{1}}P_{N_{2}}}^{2})\doteq\left(\underline{\psi}_{N_{1}N_{2}},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi}_{N_{1}N_{2}}\right)_{\mathcal{H}_{\omega}}.$

Therefore, we can set:

$\mathcal{F}_{D_{P_{N_{1}}P_{N_{2}}}^{2}}(\omega_{\underline{\psi}_{N_{1}N_{2}}})\doteq\omega_{\underline{\psi}_{N_{1}N_{2}}}(D_{P_{N_{1}}P_{N_{2}}}^{2})=\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})^{-1}.$

We can interpret this in the sense that
$\mathcal{F}_{D_{P_{N_{1}}P_{N_{2}}}^{2}}$ represents the inverse area
“function” and that there’s just a single point
$\omega_{\underline{\psi}_{N_{1}N_{2}}}$ in which its value is defined. After
we introduce more events (and when they _commute_ with each other), we can
lift the different $\mathcal{F}_{D_{P_{N_{1}}P_{N_{2}}}^{2}}$ into a function
$\mathcal{F}_{\mathrm{area}_{N_{1}N_{2}}}$ “on” the algebra of _all_ the
considered points and which can be interpreted as the analogue of a “
_characteristic function_ ” for the spacetime “point”
$\omega_{\underline{\psi}_{N_{1}N_{2}}}$ (although, the value of the function
is
$\mathrm{area}_{D_{P_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{N_{1}N_{2}})^{-1}$
rather than just $1$, that is, the event $\mathsf{e}_{N_{1}N_{2}}$ is marked
or given a physical interpretation _in a relational manner_ by a property of
the quantum Gravitational Field.) In the classical case, its analogue would
be141414Note that $\chi_{\left\\{P\right\\}}$ is also _not_ in the algebra of
_continuous_ or _smooth_ functions either, but is still a function on the
points $x$ in the most basic sense of the term. Note that
$\chi_{\left\\{P\right\\}}(P^{\prime})=\delta_{P}(\left\\{P^{\prime}\right\\})$,
where $\delta_{P}$ is the Dirac measure on $M$ associated to $P$ and
$\left\\{P^{\prime}\right\\}$ is a singleton.
$\phi_{j}(P)\chi_{\left\\{P\right\\}}:M\,\longrightarrow\mathbb{R}$, where
$\phi_{j}(P)\chi_{\left\\{P\right\\}}(x)=0$ if $M\ni x\neq P$,
$\phi_{j}(P)\chi_{\left\\{P\right\\}}(x)=\phi_{j}(P)$ if $x=P$, and
$\phi_{j}(P)$ are coordinates based on properties of the field (like distances
and proper times)$.\,\blacksquare$

The next step now is to add more events. To this end, consider the previous
event $\mathsf{e}_{N_{1}N_{2}}$, a fd _timelike_ edge
$\overline{\mathrm{e}}_{P_{N_{2}}}$ whose _initial_ point is $P_{N_{2}}$
(recall that $P_{N_{2}}$ is _also_ the initial point of
$\mathrm{e}_{P_{N_{2}}}$, but this edge is spacelike), and the event
$\mathsf{e}_{\overline{N}_{2}N_{3}}$ generated by
$\overline{\mathrm{e}}_{P_{N_{2}}}$ and the spacelike edge
$\mathrm{e}_{P_{N_{3}}}$, where $P_{N_{3}}$ is the endpoint of
$\overline{\mathrm{e}}_{P_{N_{2}}}$ (for notational consistency, from now on
we will denote the previous timelike edge $\mathrm{e}_{P_{N_{1}}}$ as
$\overline{\mathrm{e}}_{P_{N_{1}}}$, and the event $\mathsf{e}_{N_{1}N_{2}}$
as $\mathsf{e}_{\overline{N}_{1}N_{2}}$, even when we are not considering any
spacelike edge from $P_{N_{1}}$.)

$\mathbf{Definition\;2.18}$: For _two events_ , we have $4$ generators,
namely, $u_{1}$, corresponding to $\overline{\mathrm{e}}_{P_{N_{1}}}$,
$u_{2}$, to $\mathrm{e}_{P_{N_{2}}}$, $u_{3}$, to
$\overline{\mathrm{e}}_{P_{N_{2}}}$, and $u_{4}$, to $\mathrm{e}_{P_{N_{3}}}$.
In this way, the commutation relations are given by

$u_{1}u_{2}=e^{2\pi i\theta(\overline{P}_{N_{1}},P_{N_{2}})}u_{2}u_{1},$

$u_{3}u_{4}=e^{2\pi i\theta(\overline{P}_{N_{2}},P_{N_{3}})}u_{4}u_{3},$

$u_{1}u_{3}=e^{2\pi
i\theta(\overline{P}_{N_{1}},\overline{P}_{N_{2}})}u_{3}u_{1},$

$...\,etc.$

We can recognize the first two as the algebras
$\mathcal{\widehat{A}}_{\mathsf{e}_{\overline{N}_{1}N_{2}}}$,
$\mathcal{\widehat{A}}_{\mathsf{e}_{\overline{N}_{2}N_{3}}}$, corresponding,
respectively, to the events $\mathsf{e}_{\overline{N}_{1}N_{2}}$,
$\mathsf{e}_{\overline{N}_{2}N_{3}}$; we denote the one generated by the third
relation as
$\mathcal{\widehat{A}}_{\overline{P}_{N_{1}}\overline{P}_{N_{2}}}$$.\,\blacksquare$

$\mathbf{Example\;2.5}$: These four generators form what is actually known as
the _NC $4-$Torus_ [3], that is, the smooth subalgebra
$C^{\infty}(\mathbb{T}_{\Theta}^{4})$ of the universal $C^{*}-$algebra
$A_{\Theta}$ defined by

$A_{\Theta}\ni a\doteq\sum_{r\in\mathbb{Z}^{4}}a_{r}u^{r},$

$u^{r}\doteq e^{\pi
i\sum_{\underset{j<k}{j,k=1}}^{4}r_{j}\Theta_{jk}r_{k}}u_{1}^{r_{1}}u_{2}^{r_{2}}u_{3}^{r_{3}}u_{4}^{r_{4}},$

$\Theta\doteq\left(\begin{array}[]{cccc}0&\theta(\overline{P}_{N_{1}},P_{N_{2}})&\theta(\overline{P}_{N_{1}},\overline{P}_{N_{2}})&\theta(\overline{P}_{N_{1}},P_{N_{3}})\\\
-\theta(\overline{P}_{N_{1}},P_{N_{2}})&0&\theta(P_{N_{2}},\overline{P}_{N_{2}})&\theta(P_{N_{2}},P_{N_{3}})\\\
-\theta(\overline{P}_{N_{1}},\overline{P}_{N_{2}})&-\theta(P_{N_{2}},\overline{P}_{N_{2}})&0&\theta(\overline{P}_{N_{2}},P_{N_{3}})\\\
-\theta(\overline{P}_{N_{1}},P_{N_{3}})&-\theta(P_{N_{2}},P_{N_{3}})&-\theta(\overline{P}_{N_{2}},P_{N_{3}})&0\end{array}\right).$

Of course, the subalgebra of elements $a$ such that
$a_{(r_{1},r_{2},r_{3},r_{4})}=0$, for _all_ $r_{3}\neq 0$, $r_{4}\neq 0$, is
isomorphic to $\mathcal{\widehat{A}}_{\mathsf{e}_{\overline{N}_{1}N_{2}}}$,
etc. Again, one can define the action of $z\in\mathbb{T}^{4}$ as

$z\cdot u^{r}\doteq
z_{1}^{r_{1}}z_{2}^{r_{2}}z_{3}^{r_{3}}z_{4}^{r_{4}}u^{r},$

and the associated derivations

$\delta_{j}(a)\doteq\left[\frac{\mathrm{d}}{\mathrm{d}t}e^{2\pi
i\phi_{j}}\cdot a\right]\mid_{t=0},\,j=1,2,3,4;$

also the algebraic state

$\omega(a)\doteq a_{0},$

its GNS representation, and, finally, the Dirac operator there151515Where
$\gamma_{j}$ are the generators of the action of the Clifford algebra
$\mathbb{C}\mathrm{l}(\mathbb{R}^{4})$ on $\mathbb{C}^{4}$.:

$D_{\varTheta}\doteq-i\left[\gamma_{1}\underline{\delta}_{1}+\gamma_{2}\underline{\delta}_{2}+\gamma_{3}\underline{\delta}_{3}+\gamma_{4}\underline{\delta}_{4}\right].$

The generalization of all this to a finite number $n\in\mathbb{N}$ of events
should be obvious by now. Of course, the algebra will be given by a NC
$2n-$Torus$.\,\blacksquare$

$\mathbf{Remark\;2.13}$: Note that this NC $2n-$Torus _doesn’t_ represent
quantum spacetime (to start, it would be $2n-$dimensional!), what it actually
represents is _quantum phase space_. The reason for this is that the
relational representation only works for infinitesimal (in time) events at the
classical level, while the time separation between two timelike edges fails to
be of this type. Thus, the situation is the following: the full NC $2n-$Torus
quantum algebra is the algebra of phase space, and, if we restrict to the sub-
algebras corresponding to the NC $2-$Tori that define our individual quantum
events, then we can now apply the relational representration on each of them
and obtain a piece of quantum spacetime there. Furthermore, since now the
Field portion that defines the events is _finite_ , then these events can be
seen as states in the quantum phase space too (more precisely, the _spacetime
support_ , represented by vectors like $\underline{\psi}_{N_{1}N_{2}}$, of
these states, which, in combination with a Dirac operator, do give the state),
unlike the classical case and the infinitesimal portions there. Nevertheless,
the collection of all these geometries _doesn’t_ form something that
represents a metric on “the union of these regions” (in particular, the
previous $D_{\varTheta}$ cannot be interpreted as this desired metric.) Note
that if we change the value of the area of event
$\mathsf{e}_{\overline{N}_{2}N_{3}}$, i.e., changing $D_{P_{N_{2}}P_{N_{3}}},$
then it must be considered to be a _different_ event/state and its spectral
triple _direct summed_ (at the unexponentiated level) with the one of the
previous value in order to have a phase space describing both$.\,\blacksquare$

We go back now to the parameter $\theta(\overline{P}_{N_{1}},P_{N_{2}})$. We
flat in space the spacetime cylinders to spacetime sheets in the $xt$
spacetime plane and then, after integration of delta functions on the
remaining two spatial variables, we have:

$\theta(\overline{P}_{N_{1}},P_{N_{2}})=\int_{\mathrm{e}_{P_{N_{2}}}\bigcap
J(\overline{\mathrm{e}}_{P_{N_{1}}})_{xt}}\chi_{\mathrm{e}_{P_{N_{2}}}}(x,t)(E\chi_{\overline{\mathrm{e}}_{P_{N_{1}}}})(x,t)\,\sqrt{-g}\mathrm{d}x\mathrm{d}t;$

now, since $\mathrm{e}_{P_{N_{2}}}\bigcap
J^{-}(\overline{\mathrm{e}}_{P_{N_{1}}})_{xt}=\emptyset$, then

$\theta(\overline{P}_{N_{1}},P_{N_{2}})=\int_{\mathrm{e}_{P_{N_{2}}}\bigcap
J^{+}(\overline{\mathrm{e}}_{P_{N_{1}}})_{xt}}\chi_{\mathrm{e}_{P_{N_{2}}}}(x,t)(R\chi_{\overline{\mathrm{e}}_{P_{N_{1}}}})(x,t)\,\sqrt{-g}\mathrm{d}x\mathrm{d}t.$

Consider $\mathrm{e}_{P_{N_{2}}}$ to be of coordinate _width_ $\delta$ and
small coordinate _time extension_ $\epsilon^{\prime}$,
$\overline{\mathrm{e}}_{P_{N_{1}}}$ to be of _time extension_
$\epsilon\geq\epsilon^{\prime}$ and small _width_ $\delta^{\prime}\leq\delta$,
and the wave to fall to zero abruptly only when it gets close to its edges
(remaining constant and just equal to $1$ inside them). Since that integral is
_only_ valid at the classical level, the relevant informtion for the quantum
theory must be judiciously extracted from it. If we use riemannian normal
coordinates around $P_{N_{1}}$, then $\sqrt{-g}\mid_{P_{N_{1}}}=1$;
furthermore, in the quantum theory, the _minimal_ possible non-zero length is
given by the length of $\mathrm{e}_{P_{N_{2}}}$, while the _minimal_ possible
non-zero time is given by the time of $\overline{\mathrm{e}}_{P_{N_{1}}}$, and
this implies $\epsilon=\epsilon^{\prime}$ and $\delta^{\prime}=\delta$. Thus:

$\theta(\overline{P}_{N_{1}},P_{N_{2}})=lim_{\epsilon,\delta\,\longrightarrow
0}\int_{[0,\epsilon]}\int_{[0,\delta]}\mathrm{d}x\mathrm{d}t.$

Thus, at the quantum level, all this suggests to take:

$\mathbf{Definition\;2.19}$:
$\theta(\overline{P}_{N_{1}},P_{N_{2}})\doteq\mathrm{area}_{D_{\overline{P}_{N_{1}}P_{N_{2}}}}(\widehat{\mathcal{R}}_{\overline{N}_{1}N_{2}})=\mathrm{a}(\overline{P}_{N_{1}},P_{N_{2}}).\,\blacksquare$

This choice makes the formalism self-contained, since
$\mathrm{a}(\overline{P}_{N_{1}},P_{N_{2}})$ is, initially, just a parameter
that enters in the definition of $D_{\overline{P}_{N_{1}}P_{N_{2}}}$. An
analogous argument also suggests to take:

$\mathbf{Definition\;2.20}$:
$\theta\doteq\theta(\overline{P}_{N_{1}},\overline{P}_{N_{2}})\doteq\mathrm{a}(\overline{P}_{N_{2}},P_{N_{3}}),$
and
$\theta(\overline{P}_{N_{2}},P_{N_{3}})\doteq\mathrm{a}(\overline{P}_{N_{2}},P_{N_{3}}).\,\blacksquare$

$\mathbf{Definition\;2.21}$: Consider now the curve
$\gamma_{\overline{N}_{2}\overline{N}_{n}}\doteq\overline{\mathrm{e}}_{P_{N_{2}}}\bigcup...\bigcup\overline{\mathrm{e}}_{P_{N_{n}}}$
($n\geq 2$.) Thus, by the basic properties of the integral:

$\theta(n)\equiv\theta(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}})\doteq\sum_{j=2}^{n}\mathrm{a}(\overline{P}_{N_{j}},P_{N_{j+1}})$

$=\mathrm{a}(\overline{P}_{N_{2}},P_{N_{3}})+...+\mathrm{a}(\overline{P}_{N_{n}},P_{N_{n+1}}),$

and, assuming that an event
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$ with the area
in the left hand side below exists161616Note that this event _is not_ the
composition of the events that comprise the previous curve, since that’s not
an infinitesimal displacement in the classical case. That is, the event in
consideration is such that just shares, numerically, the same area value as
that composition. If the area of the events is discretized as $a_{m}=ma_{0}$,
where $m\in\mathbb{N}$, then the existence of
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$ is guaranteed.
in the phase space,

$\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}})\doteq\theta(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}}).\,\blacksquare$

$\mathbf{Remark\;2.14}$: What this means is that we can now consider a process
in which one goes from $\mathsf{e}_{\overline{N}_{1}N_{2}}$ to the start of
$\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}$, with fixed $n\in\\{2,3,...\\}$ (that
is, we have $n\in\mathbb{N}$ events), but _without_ “making a pause” in what
would be the “intermediate events”: that is, if any of the intermediate events
happens, then $\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$
_cannot_ happen after it, and viceversa,
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$ is as _an
elementary, non-compound_ event. This new event now takes the role of
$\mathsf{e}_{\overline{N}_{2}N_{3}}$ in the previous scheme with only _two_
“glued” events, but now taking
$\theta\equiv\theta(n)\equiv\theta(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}})$.

Figure 1: Timelike Curve in Quantum Spacetime.

In this way, as $n$ advances, we can interpret this in the sense that the
event $\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}$ is more “far away” from
$\mathsf{e}_{\overline{N}_{1}N_{2}}$ in terms of proper time along the quantum
timelike curve defined by the succession of the events
$\mathsf{e}_{\overline{N}_{1}N_{2}}$ to
$\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}$.

Now, if we _experience_ an _elementary_ process, then, since there aren’t any
“instants” of time “in the middle”, we would simply age a _finite_ amount of
time _abruptly_ ; that is, there’s a _change_ , which, _by itself_ , doesn’t
introduce any _extra_ proper time, the latter, instead, being fully accounted
by the metric of the event and given “all at once”, as a photon gives all of
its finite energy to an electron in an atom all at once. In the case of the
previous elementary, “would be compound” events
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$, if there’s
something like a particle in some of the intermediate events, then, if the
system undergoes the elementary process
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$, it _never
interacts_ with the particle, it just “tunnels it”$.\,\blacksquare$

Before continuing, we will take
$\theta(\overline{P}_{N_{1}},\overline{P}_{N_{2}})\equiv\theta$, in order to
simplify the notation.

In the classical case, the time evolution in phase space is a map
$t\,\longmapsto\phi_{t}^{*}s_{0}$. Here we define it in the same way, but now,
since the states/events are pure states in a Hilbert space, we can also form
linear combinations among them, and this invariably will bring typical quantum
behaviour to the system (in a classical commutative algebra, the only pure
states are the Dirac measures on the corresponding topological space, and
linear combinations give mixed states, which cannot be interpreted as events
in that case.) In the limit $\theta\,\longrightarrow\infty$ (we enter the
macroscopic realm there), we will take a suitable boundary condition so that
we recover the classical-like evolution curve.

$\mathbf{Definition\;2.22}$: Consider now the real valued functions
$G(\theta)$ and $g(G(\theta))$. Then, the
$\underline{\widetilde{\psi}^{\theta}}_{\overline{N}_{2}N_{3}}$ _corresponding
to the process_
$\mathsf{e}_{\overline{N}_{1}N_{2}}\bigcup\mathsf{e}_{\overline{N}_{2}N_{3}}$
(i.e., the event $\mathsf{e}_{\overline{N}_{2}N_{3}}$, but now _seen as the
endpoint of the considered path_ 171717What this means is that, while the
process is certainly
$\mathsf{e}_{\overline{N}_{1}N_{2}}\bigcup\mathsf{e}_{\overline{N}_{2}N_{3}}$
and then it cannot be represented by the state
$\underline{\psi}_{\overline{N}_{2}N_{3}}$, once it happens we are only
interested in its endpoint when trying to consider the probability for the
next transition after this process. The endpoint is, of course, the elementary
process $\underline{\psi}_{\overline{N}_{2}N_{3}}$, and that’s why we will say
that the initial $\underline{\psi}_{\overline{N}_{1}N_{2}}$ jumps to the final
$\underline{\psi}_{\overline{N}_{2}N_{3}}$, _via_
$\mathsf{e}_{\overline{N}_{1}N_{2}}\bigcup\mathsf{e}_{\overline{N}_{2}N_{3}}$,
and only use $\underline{\psi}_{\overline{N}_{2}N_{3}}$ as the initial state
for the next transition. defined as
$\mathsf{e}_{\overline{N}_{1}N_{2}}\bigcup\mathsf{e}_{\overline{N}_{2}N_{3}}$,
or, equivalently, as the start of $\mathsf{e}_{\overline{N}_{3}N_{4}}$), is
given by

$\underline{\widetilde{\psi}^{\theta}}_{\overline{N}_{2}N_{3}}\doteq\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})\doteq
g(G(\theta))\underline{\psi}_{\overline{N}_{1}N_{2}}+G(\theta)(\underline{\psi}_{\overline{N}_{2}N_{3}}+\underline{I}),$

where $\phi_{\theta}$ is _continuous_ , with

$G(0)=0,\,G^{\prime}(0)\neq 0,\,g(0)=1.\,\blacksquare$

Thus,

$\underline{\widetilde{\psi}^{0}}_{\overline{N}_{2}N_{3}}=\phi_{0}(\underline{\psi}_{\overline{N}_{1}N_{2}})=\underline{\psi}_{\overline{N}_{1}N_{2}},$

that is, if $\theta=0$, then it makes no sense to distinguish between both
events, they must be the same; furthermore, in light of this, the process and
path for the classical case, in which $\theta\,\longrightarrow\infty$, should
correspond to
$\underline{\widetilde{\psi}^{\infty}}_{\overline{N}_{2}N_{3}}=\underline{\psi}_{\overline{N}_{2}N_{3}}$.

$\mathbf{Definition\;2.23}$: Now that we have two events,
$\mathsf{e}_{\overline{N}_{1}N_{2}}$ and $\mathsf{e}_{\overline{N}_{2}N_{3}}$,
we can calculate the _transition probability_ between the events _when
following a particular path in spacetime given by the process_
$\mathsf{e}_{\overline{N}_{1}N_{2}}\bigcup\mathsf{e}_{\overline{N}_{2}N_{3}}$,
which we define, as

$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)^{\frac{1}{2}}\doteq(\underline{\widetilde{\psi}}_{\overline{N}_{2}N_{3}}^{\theta},\underline{\psi}_{\overline{N}_{1}N_{2}})_{\mathcal{H}_{\omega}}=g(G(\theta))\parallel\underline{\psi}_{\overline{N}_{1}N_{2}}\parallel_{\mathcal{H}_{\omega}}^{2}+$

$G(\theta)(\underline{\psi}_{\overline{N}_{2}N_{3}},\underline{\psi}_{\overline{N}_{1}N_{2}})_{\mathcal{H}_{\omega}}+G(\theta)(\underline{I},\underline{\psi}_{\overline{N}_{1}N_{2}})_{\mathcal{H}_{\omega}}$

$=g(G(\theta)).\,\blacksquare$

In the classical situation, we have

$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}^{Class.}\doteq\mid(\underline{\psi}_{\overline{N}_{1}N_{2}},\underline{\psi}_{\overline{N}_{2}N_{3}})_{\mathcal{H}_{\omega}}\mid^{2}\,\mathrm{or}\,\mid(\underline{\psi}_{\overline{N}_{1}N_{2}},\underline{\psi}_{\overline{N}_{1}N_{2}})_{\mathcal{H}_{\omega}}\mid^{2},$

and then we get only two possible values for the transition probability,
namely, $0$ or $1$, respectively.

$\mathbf{Remark\;2.14}$: The map $\phi_{\theta}$ can be seen as the
equivalent, in this formalism, of (the push-forward of) a $1-$parameter family
of diffeomorphisms, and
$\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})$ can be seen as the
_“potential time evolution” in the phase space picture_ for the current
quantum theory. Nevertheless, the physical interpretation is quite different.
Indeed, in standard quantum physics, a well defined (by the true Hamiltonian
with respect to a background spacetime geometry) and unique time evolution
gives the state of the system at time $t$. Here, we don’t even have a “time
$t$” to begin with: what
$\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})$ actually represents
is, for a given possible time $\theta$, a possible time evolution itself to
it, that is, in this situation, _the different time evolutions (where what
varies among them is the value of $\theta$) themselves are the ones which have
different probabilities of occurring_ for the _always fixed_ pair of “manifold
points”/state supports $\underline{\psi}_{\overline{N}_{1}N_{2}}$ and
$\underline{\psi}_{\overline{N}_{2}N_{3}}$. In this way,
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)^{\frac{1}{2}}$
is the probability that, when the initial event
$\mathsf{e}_{\overline{N}_{1}N_{2}}$ goes to the final
$\mathsf{e}_{\overline{N}_{2}N_{3}}$ (whose end sits at time
$\theta+\theta_{12}$ from $\mathsf{e}_{\overline{N}_{1}N_{2}}$: since we have
only two events here, $\theta$ can only take that value, besides $0$, of
course), it will do it via the time evolution
$\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})$. Note that
$\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})$ _doesn’t_ represent
the state after the evolution, i.e.,
$\underline{\psi}_{\overline{N}_{2}N_{3}}$, it’s more than that,
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)^{\frac{1}{2}}$
is the probability that $\underline{\psi}_{\overline{N}_{1}N_{2}}$ quantum
jumps/transitions to
$\phi_{\theta}(\underline{\psi}_{\overline{N}_{1}N_{2}})$, which is the state
in which a given time evolution (seen as some particular change or process)
defines and in which event $\mathsf{e}_{\overline{N}_{2}N_{3}}$ has
probability $1$ for ocurring, so that time actually acquires the value
$\theta$, and then, since the notion of time evolution as something giving the
state of the system at a defined time can make sense, can be reinterpreted as
an evolution more similar to the one from standard quantum physics, in
particular one which is such that the state at time $\theta$ is
$\underline{\psi}_{\overline{N}_{2}N_{3}}$$.\,\blacksquare$

It remains to determine the functions $g,G$. For this, we will consider the
dynamics as encoded in the spectral action. The spectral action on the _NC
$4-$Torus_ is given by ([9])

$S^{0}=\omega(F_{\mu\nu}F^{\mu\nu}),$

where

$C^{\infty}(\mathbb{T}_{\Theta}^{4})\ni
F_{\mu\nu}\doteq\delta_{\mu}(A_{\nu})-\delta_{\nu}(A_{\mu})+\left[A_{\mu},A_{\nu}\right],$

and $C^{\infty}(\mathbb{T}_{\Theta}^{4})\ni A_{\mu}=-A_{\mu}^{*}$.

$\mathbf{Definition\;2.24}$: We define the following “displaced” version of
the previous spectral action (we restrict $A_{\mu}$ to be only from any of the
two algebras of the two events in consideration, since that’s where the action
of $\phi_{\theta}$ was defined)

$S^{\theta}\doteq\omega(\phi_{\theta}(F_{\mu\nu}F^{\mu\nu}))=g(G(\theta))S^{0}+G(\theta).\,\blacksquare$

$\mathbf{Hypothesis\;2.3}$: We impose its _invariance_ 181818Since the GNS
Hilbert space of the NC $4-$Torus represents phase space, and the elements of
the algebras are vectors/states there, then we can see $S^{0}$ acting on the
events/states
$\underline{\psi}_{\overline{N}_{1}N_{2}},\underline{\psi}_{\overline{N}_{2}N_{3}}$
as a functional acting on the covariant phase and expressed using the
spacetime picture (for the particular flat-metrics given by
$D_{P_{N_{1}}P_{N_{2}}}$ and $D_{P_{N_{2}}P_{N_{3}}}$), in a way very
analogous to the classical Gravitational action
$S_{GR}\left[g\right]=\int_{M}R(g)\boldsymbol{\mathbf{\epsilon}}$ (one would
need to make a conformal perturbation of the flat metric of the NC torus in
order to have a non-zero curvature [15], that’s why the present action
$S^{0}=\omega(F_{\mu\nu}F^{\mu\nu})$ contains only a Yang-Mills term); the
action of $\omega$ can be seen as an analogous to integration. Note that we
can make all these interpretations _only_ thanks to the fact that we have a
spacetime picture. (because the $\phi_{\theta}$ are the analogous of physical
diffeomorphisms) on the variable $\theta$:

$\frac{\mathrm{d}S^{\theta}}{\mathrm{d}\theta}=0.\,\blacksquare$

$\mathbf{Lemma\;2.8}$: Hypothesis 2.3 implies that
$g(G(\theta))_{S^{0}}=1-\frac{1}{S^{0}}G(\theta)$.

_Proof_ :

$\frac{\mathrm{d}S^{\theta}}{\mathrm{d}\theta}=\frac{\mathrm{d}g}{\mathrm{d}\theta}S^{0}+\frac{\mathrm{dG}}{\mathrm{d}\theta}$

$0=\frac{\mathrm{d}g}{\mathrm{d}G}\frac{\mathrm{d}G}{\mathrm{d}\theta}S^{0}+\frac{\mathrm{d}G}{\mathrm{d}\theta}\,\Longrightarrow
0=\frac{\mathrm{d}g}{\mathrm{d}G}S^{0}+1\,\Longrightarrow$

$g(G(\theta))_{S^{0}}=k-\frac{1}{S^{0}}G(\theta),$

with $k=1$, since $G(0)=0$ and $g(0)=1$. In this way:

$g(G(\theta))_{S^{0}}=1-\frac{1}{S^{0}}G(\theta).\,\square$

Now, consider the case for the solution $A_{\mu}^{0}=0$, then
$S^{\theta}(A_{\mu}^{0})$ _cannot_ be invariant on $\theta$, and, in fact
(recall that $G^{\prime}(0)\neq 0$):

$S^{\theta}(A_{\mu}^{0})=G(\theta);$

each value of $\theta$ corresponds to a different physical situation (roughly
speaking, in the semiclassical case, the second event varies its temporal
“distance” with respect to the initial one as $\theta$ varies); thus, the
three action values $S^{\theta_{1}}(A_{\mu}^{0}),$
$S^{\theta_{2}}(A_{\mu}^{0})$, $S^{\theta_{1}+\theta_{2}}(A_{\mu}^{0})$,
correspond to three different physical systems.

$\mathbf{Hypothesis\;2.4}$: We propose that the _dynamics_ of these systems
under the $\theta$ variable varies in a _homogeneous_ and _additive_
191919Like proper time defined as an integral over the curve is. fashion, that
is (recall that the sum of actions corresponds to the _composition_ or
_physical sum_ of physical systems),

$S^{\theta_{1}+\theta_{2}}(A_{\mu}^{0})=S^{\theta_{1}}(A_{\mu}^{0})+S^{\theta_{2}}(A_{\mu}^{0})\,\Longrightarrow$

$G(\theta_{1}+\theta_{2})=G(\theta_{1})+G(\theta_{2}).$

If $G$ is continuous, then this implies that

$G(\theta)=\alpha\theta,$

for some constant $\alpha=G^{\prime}(0)\neq 0$ (not that this is valid for
_any_ solution $A_{\mu}$, since $G$ only depends on $\theta$)$.\,\blacksquare$

$\mathbf{Corollary\;2.1}$: The final form, _completely determined by the toral
spectral action and our assumptions_ , of the transition probability becomes

$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)_{\alpha,A_{\mu}}^{\frac{1}{2}}=1-\frac{\alpha}{S^{0}(A_{\mu})}\theta.\,\square$

$\mathbf{Remark\;2.15}$: From the form of the toral algebra
$\mathcal{\widehat{A}}_{\overline{P}_{N_{1}}\overline{P}_{N_{2}}}$, where
$u_{1}u_{3}=e^{2\pi i\theta}u_{3}u_{1}$, it’s clear that a redefinition of
$\theta$ by a displacement $\theta\,\longmapsto\theta+n$, with
$n\in\mathbb{N}$, gives an isomorphic algebra; thus, we can restrict to
$\theta\in\left[0,1\right]$. Furthermore, the map defined by
$u_{1}\,\longmapsto u_{3}$ and $u_{3}\,\longmapsto u_{1}$ establishes an
isomorphism between the toral algebra with $\theta$ and the one with
$1-\theta$; if we combine both of these results, then we get that the family
of toral algebras whose parameter $\theta$ ranges on the subset
$\left[0,\frac{1}{2}\right]$ exhausts _all_ the possible _non-isomorphic_
toral algebras (for the $2-$dimensional case, of course.) In this way, if we
impose the _only_ physically meaningful boundary condition, namely,

$lim_{\theta\longrightarrow\frac{1}{2}}\underline{\widetilde{\psi}^{\theta}}_{\overline{N}_{2}N_{3}}\propto(\underline{\psi}_{\overline{N}_{2}N_{3}}+\underline{I}),$

which is equivalent to asking202020Recall that, in the classical case,
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}^{Class.}\doteq\mid(\underline{\psi}_{\overline{N}_{1}N_{2}},\underline{\psi}_{\overline{N}_{2}N_{3}})_{\mathcal{H}_{\omega}}\mid^{2}=0$.

$lim_{\theta\longrightarrow\frac{1}{2}}g(G(\theta))_{S^{0}(A_{\mu})}^{\alpha}=lim_{\theta\longrightarrow\frac{1}{2}}\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)_{\alpha,A_{\mu}}=0,$

then, for _each_ solution $A_{\mu}$, there’s a _unique_ solution

$g(G(\theta))_{S^{0}(A_{\mu})}^{\alpha_{\frac{1}{2}}}=1-2\theta$

to the _displaced_ dynamics, obtained by taking

$\alpha_{\frac{1}{2}}=2S^{0}(A_{\mu}).$

For this case, the graph of
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\theta)_{\alpha_{\frac{1}{2}},A_{\mu}}$
is, of course, just the following:

Figure 2: Transition Probability for $\alpha_{\frac{1}{2}}$$.\,\blacksquare$

$\mathbf{Remark\;2.16}$: In the _unexponentiated_ algebra, one would like to
have non-isomorphic algebras for _any_ value of the parameter $\theta$ in the
range $[0,\infty)$. Thus, in the _exponentiation_ of the algebra, we made an
_unintended compactification_ 212121Which, of course, is directly related to
the compactness of the classical $2-$torus $\mathbb{T}^{2}$. of $[0,\infty)$
into $\left[0,\frac{1}{2}\right]$. To obtain the actual domain and shape of
the transition probability, we must change the variable $\theta$ to
$\vartheta\in[0,\infty)$, where the passing from $\theta$ to $\vartheta$ is
given by a _bijective conformal stretching_ of $[0,\frac{1}{2})$ into
$[0,\infty)$. The most sensible option seems to be:

$\theta=\frac{1}{2}\mathrm{tanh}\,\vartheta,$

in which case

$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\vartheta)_{\alpha_{\frac{1}{2}},A_{\mu}}=(1-\mathrm{tanh}\,\vartheta)^{2}$

$=e^{-4\vartheta}(1+\mathrm{tanh}\,\vartheta)^{2};$

now, for $\vartheta>>4$, this reduces to222222Since
$(1+\mathrm{tanh}\,\vartheta)^{2}\sim 4,\,\forall\vartheta>>4.$

$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{3}N_{4}}^{start}}(\vartheta)_{\alpha_{\frac{1}{2}},A_{\mu}}\sim
4e^{-4\vartheta}.$

Then the graph becomes:

Figure 3: Transition Probability for $\alpha_{\frac{1}{2}}$ (_conformally
stretched_)$.\,\blacksquare$

The state of the system is given by
$\Psi=\underline{\psi}_{\overline{N}_{1}N_{2}}$, then it has _non-zero_
transition probability for _several_ other processes; that is, upon
_interaction_ with the surrounding matter fields, the initial state
$\underline{\psi}_{\overline{N}_{1}N_{2}}$ can make a quantum jump to the
start of another event $\underline{\psi}_{\overline{N}_{n+1}N_{n+2}}$ via an
elementary, “would be compound” process
$\mathsf{e}_{\gamma_{\overline{N}_{1}\overline{N}_{n}}N_{n+1}}$, which may
even lie in a _different_ curve $\gamma^{\prime}$ (see next figure), and this
is the physical interpretation we give to
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}^{start}}(\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}}))_{\alpha_{\frac{1}{2}},A_{\mu}}$.
We also note that, unlike the classical case, we don’t need to introduce
change in an ad-hoc manner here, since change can be seen as arising from
quantum collapse232323Which, for more precision, we define as the abrupt jump
from one state to another after an interaction, if one accepts that Quantum
Theory is a complete theory, or as the abrupt change in the values, if one
doesn’t accept that and, instead, introduces contextual hidden variables
(which give us the values.) (_now taken as ontologically fundamental_
242424This is the case, for example, in Rovelli’s Relational Interpretation
[10], which, then, seems very well suited for this approach. _and
irreducible_) after an interaction. We take the collapse as the _only_ source
of change and actually _identify_ it with it (in other views, collapse, of
course, implies change, but the converse is not necessary; here we say they
are indeed the same thing.) Thus, the change in the classical theory actually
comes from the fundamental quantum theory, of which the former is a limit.
Furthermore, in light of this, then the argument used to show the necessity of
the collapse in QM can be now used to show the necessity of change, which then
becomes a quantum phenomena and very tied to the characteristic non-
commutativity of quantum properties.

Figure 4: Two Different Curves.

In the actual physical reality, the system is constantly interacting and its
state collapsing. Therefore, its real spacetime trajectory is something like
what’s illustrated in the figure below (note that not all of the intermediate
events in the curves $\gamma_{\overline{N}_{1}\overline{N}_{n}}$ are visited,
i.e., not all the intermediate values of process-area will be visited.) It’s
actually for this trajectory that we can define something like

$m\,\longmapsto\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n(m)}},P_{N_{n(m)+1}})\,\longmapsto\mathsf{e}(\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n(m)}},P_{N_{n(m)+1}}))\equiv\mathsf{e}_{\overline{N}_{n(m)+1}N_{n(m)+2}}^{start},$

as in the classical $t\,\longmapsto$$\tau(t)\,\longmapsto\gamma(\tau(t))$.

Figure 5: Real Spacetime Trajectory of the System.

$\mathbf{Remark\;2.17}$: The so-called _Problem of Time_ can be resolved here
simply by calculating the values of the properties of interest (say, the
spatial curvature) on each of the nodes of
$\mathsf{e}(\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n(m)}},P_{N_{n(m)+1}}))$.
Of course, since quantum collapse is random, we cannot predict with certainty
what’s the trajectory that the system will take, and thus the precise time
evolution of the value for the property being considered. The only thing we
can do is to calculate the probability for each possible time evolution of the
value, by setting:

$\mathcal{P}(m)\doteq\prod_{k=1}^{m}\mathcal{P}_{n(k),n^{\prime}(k)}^{spatial}\mathcal{P}_{\mathsf{e}_{\overline{N}_{n(k)}N_{n(k)+1}}\rightarrow\mathsf{e}_{\overline{N}_{n^{\prime}(k)+1}N_{n^{\prime}(k)+2}}^{start}}(\mathrm{a}(\gamma_{\overline{N}_{n(k)+1}\overline{N}_{n^{\prime}(k)}},P_{N_{n^{\prime}(k)+1}}))_{\alpha_{\frac{1}{2}},A_{\mu}},$

where only the events in
$\mathsf{e}(\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n(m)}},P_{N_{n(m)+1}}))$
are considered for the product$.\,\blacksquare$

$\mathbf{Remark\;2.18}$: Now, since
$\mathcal{P}_{\mathsf{e}_{\overline{N}_{1}N_{2}}\rightarrow\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}^{start}}(\mathrm{a}(\gamma_{\overline{N}_{2}\overline{N}_{n}},P_{N_{n+1}}))_{\alpha_{\frac{1}{2}},A_{\mu}}$
falls exponentially, then the events
$\mathsf{e}_{\gamma_{\overline{N}_{2}\overline{N}_{n}}N_{n+1}}$ for _large_
$n$ are very unlikely to happen. This means that
$\mathsf{e}_{\overline{N}_{1}N_{2}}$ tends to go to a
$\mathsf{e}_{\overline{N}_{n+1}N_{n+2}}^{start}$ which is closer to it in
proper time separation. Physically, this means that time advances as a
succession of instants which are very close to each other in proper time
distance and in which the duration of the instants themselves is very small.
Thus, at the macroscopic scale, this is perceived as a succession of instants,
each of duration zero, which forms a continuum whose subsets have finite
duration, and which monotonically increases (since, upon change, almost all
intermediate steps are visited, and, thus, the system can interact with
whatever thing that resides at those steps), that is, the classical picture of
time. The quantum system at an event is surrounded by a dispersion cloud of
events which can visit next, and the classical proper time is just some
average $<\tau>$ of that dispersion, and the actual quantum transitions
measure how much the actual quantum time _deviates_ from this average
$<\tau>$$.\,\blacksquare$

$\mathbf{Remark\;2.19}$: Also in relation to this, this point of view may also
shed some light on the so-called _Measurement Problem_ of standard QM,
understood as the _impossibility_ to _explain_ the collapse of the
wavefunction in terms of the standard Schrödinger time evolution (assuming
that the collapse gives definite values and that standard QM is complete.)
Indeed, in our view, the standard classical time, which, among other things,
is used as the external time parameter to define the Schrödinger time
evolution, is only an emergent feature at the macro level, and fuelled at the
micro, fundamental level by an irreducible collapse. Thus, the Schrödinger
time evolution will _break_ when something from the fundamental level _leaks_
to the macroscopic level. And, of course, this is precisely the situation in a
quantum measurement, when the state collapses when certain two systems
interact: the classical time will never be able to explain this process since
the latter intervenes precisely in making possible the quantum, and therefore
also the classical, time. Finally, one can also may be able to avoid spacetime
singularities (the “ _Singularity Problem_ ” of classical GR) in this approach
thanks to the discretization of time and the mentioned “tunneling” effect.
Indeed, the discretization eliminates the possibility for a property to
acquire values which are finite yet arbitrary large as one approaches an event
separated by a finite amount of proper time (either to the past or future)
with respect to the initial one, since there are only a finite number of other
events between them, thus the property reaches a finite maximum value,
possibly at the final event; furthermore, if the property is not defined for
an event, then, of course, it cannot form part of the spacetime, yet, due to
the possibility of tunneling, even if the classical spacetime is inextendible,
there still exists a non-zero transition probability for the system to tunnel
to an extended quantum spacetime, that is, beyond the “boundary” of the
classical spacetime defined by the singularity$.\,\blacksquare$

## 3\. Discussion

In CLQG (Covariant LQG [13]), given the Hilbert spaces and quantum algebras at
the boundary, it seems one can only calculate things like (quantum) properties
of the induced spatial $3-$d metric (areas, volumes, curvature) and the
extrinsic curvature of the hypersurface, and not quantum durations (for
example, in the analysis done for the extrinsic coherent states, the area and
the extrinsic curvature peak at those states, but time is introduced as an
_already_ peaked external parameter.) There are some extensions that introduce
timelike boundaries, but, for a time evolution, we need to be able to
introduce durations also in the _interior_ of the process; furthermore, the
commutation relations for variables whose weight functions have a space _time_
support, and in regions in spacetime that are such that one is in the causal
future of the other, are non-trivial (this is the important result of the so-
called covariant Poisson brackets), and are not accounted for by the _part_ of
the commutation relations that arise _purely_ due to the structure
(kinematical in nature) of the group in the Yang-Mills-like/connection
variables used in these formulations [14], since the dependence over the
causality comes from the hyperbolic character of the dynamics and the
mentioned issue with the supports, and then it’s indifferent if we take the
Yang-Mills/connection group to be $SU(2)$ (spacelike boundary) or $SU(1,1)$
(timelike boundary), that is, this causal part will be present even if the
field variable were a scalar field; as argued in the paper, and from
relational considerations, we consider this a key issue in the quantization of
duration, which makes it quite different from the quantization of the spatial
metric and that it cannot thus be completely obtained by tweaks on the already
existing methods for the latter. In particular, the extensions of CLQG give
the area of a timelike triangle for vectors (one timelike and the other
spacelike) whose origin is at the same vertex (evidently, the causal part is
irrelevant, since the origin of both vectors is the same point, and then only
the group part enters to play there), while TQG describes the same area but
from the point of view of the causal connection between the initial and
enpoint of the timelike vector (thus, the causal part is relevant for that);
both refer to a same event and as a finite process, but TQG gives a more
dynamical characterization of it while CLQG gives252525Note that the dynamical
transition in CLQG between the initial and final points of the previous
timelike vector is not the same thing as the process described by TQG between
them: the transition in CLQG is actually a transition between two processes,
the latter ones each of kinematical area given by the triangles at the points
in question, while the events in TQG describe dynamically (via the causal
relation between the initial and final points) the processes corresponding to
_each_ triangle. The dynamical transitions in CLQG cannot describe this latter
dynamical characterization for each individual triangle because they are
transitions from state to state, triangle to triangle. The transition
described in CLQG corresponds to the transition between events in TQG, but
where each theory describes different aspects of it (in CLQG, is the dynamics
between the kinematics of each event, while in TQG is the dynamics between the
dynamics of each.) We would say that it’s actually TQG the theory that gives
actual physical entity to those events since it describes them as dynamics.
its internal kinematical degrees of freedom. Thus, the corresponding quantum
algebra for duration is a composition of the group and causal aspect of the
classical Poisson bracket. From this causal part, unique to TQG, one gets an
_universal_ decay for the transition that explains the features of duration at
the macro scale (this decay is always there and is independent of the details
of the other parts of the transitions); this part of the transition for
duration is ignored (or considered to have already peaked) in treatments that
rely heavily in coherent states adapted only to the group part.

In classical GR, the _full_ form of a solution is $g_{ab}=n_{a}n_{b}+h_{ab}$,
with $n^{a}=\frac{1}{N}(t^{a}+N^{a})$, where $N$ and $N^{a}$ are,
respectively, the lapse and shift (which also satisfy $N=-g_{ab}n^{a}t^{a}$
and $N^{a}=h_{b}^{a}t^{b};$ in this way, $N$ is interpreted as the “rate of
proper time with respect to coordinate time $t$ as one moves normaly to the
hypersurfaces of the foliation”), so that the proper time becomes
$\tau=\int\sqrt{N^{2}-h_{ab}N^{a}N^{b}}\,\mathrm{d}t$. Now, it’s also known
that the induced metric and the extrinsic curvature (the data given by the
boundary Hilbert spaces and algebras of CLQG) determine a point in phase space
if we were in the classical theory. But this is not enough to generate a
solution, one also needs to specify the lapse and shift $N$ and $N^{a}$ (which
measure how the time function/coordinate $t$ of the foliation $\varSigma_{t}$
of the manifold interacts with the metric information.) Furthermore, from the
previous formulas we can see that the lapse and shift $N$ and $N^{a}$ are
precisely the things that encode the time part/time evolution of the spacetime
__ behaviour of the solution, as well as the proper time (we just can’t have
one without the other.) Therefore, in order to describe time evolution
properly, we must introduce some extra information to CLQG, which is given by
an algebra that can econde the non-trivial commutation relations between
causally connected regions in spacetime in the interior of the process (and
find a way to describe metric properties related to it.) But, even if we have
this, the transition amplitude $W(x_{i},x_{f})$ is not useful for obtaining
the amplitude for the transitions between quantum states of duration, since
it’s based on the path integral, which, as one can see in the canonical
framework, involves integration over all lapse functions $N$, and then
information about the duration part of a _particular_ solution cannot be taken
as being prescribed from there. Thus, the dynamical transitions for duration
must also be provided by different means. TQG provides both of these necessary
ingredients for the adequate description of time evolution in QG (the areas of
the elementary processes in the paper can be interpreted as the information
provided by $N$ and $N^{a}$ in the classical case.)

The probabilty amplitude $\mid W(x_{i},x_{f})\mid^{2}$ of CLQG should,
perhaps, be interpreted as what is denoted as
$\mathcal{P}_{n(k),n^{\prime}(k)}^{spatial}$ here. On the other hand, TQG, _by
providing an actual description of time duration in QG_ , is what actually
justifies the physical interpretation of $W(x_{i},x_{f})$ as a transition
amplitude in CLQG, the latter is a thing which has often been criticized ([2])
due to the integration over all lapse functions on the path integral: if we
delegate the time part to TQG, then this integration can be seen only as a
device of the calculation, since no contradiction arises because the path
integrals are being applied to states that only describe space and to obtain
the pure space part of the transition amplitude.

Thus, we actually see CLQG and TQG as _complementary_ theories.

## 4\. Conclusions

We made a schematic partial quantization of the temporal part in line with the
mentioned ideas. For doing that, we had to introduce a new formalism for QG,
which we call TQG, since it’s heavily based on the NC Tori. This allowed us to
obtain numerous insights about the nature of time, like its discretization,
its regular pace at the macroscopic scale, a solution to the Problem of Time,
and a connection with the Measurement Problem of QM.

TQG is not that much a theory of quantum gravity, but a necessary part of it.
In particular, it provides a basic framework on which to discuss within such a
theory any aspect related to time, time evolution, and duration. In this
proposal, the part of quantum phase space that deals with these aspects is
modeled by an $n-$dimensional noncommutative Torus, where each state on a
NC-2-subTorus there corresponds to an elementary and irreducible quantum
process in a spacetime picture; the NC-Torus arises here because for, say,
three (of four) elements that are causally related, one needs three generators
that do not commute among each other (this comes from the canonical
quantization of the Poisson brackets of phase space properties with weight
functions having causally connected supports), and this cannot be accomodated
just in the product of two NC-2-Torus (that model an individual quantum
elementary process), but one needs a missing third non-commutation between
these two, so to speak, and this can be done in the general $3$ and $4-$NC-
Torus. This type of phase space hasn’t been discussed in any of the current
proposal for quantum gravity theories, despite the fact that often some talk
is informally given ([15, 4]) about the behaviour of quantum time (like its
hypothetical quantum nature, with superpositions, etc.), that, really, can
only be properly justified if one has a model for this phase space.

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