$\mathcal{H}_{v}$’s is

$\displaystyle\pi(a)_{v_{2}}^{v_{1}}=a_{i(v_{1})}\delta_{v_{2}}^{v_{1}}:\mathcal{H}_{v_{1}}\to\mathcal{H}_{v_{2}}$
(4.2.3)

(where $\delta_{v_{2}}^{v_{1}}$ is the Kronecker symbol). In the real case,
for any $a=\operatorname{\textstyle\oplus}_{i=1}^{r}a_{i}\in\mathcal{A}$, any
$b=\operatorname{\textstyle\oplus}_{j=1}^{r}b_{j}\in\mathcal{A}$, and any
$\psi_{v}\in\mathcal{H}_{v}$, one has
$ab^{\circ}\psi_{v}=a_{i(v)}b_{j(v)}^{\circ}\psi_{v}\in\mathcal{H}_{v}$ ($\pi$
omitted) where $b_{j(v)}^{\circ}$ acts on the right factor of
$\mathbb{C}^{\lambda(v)}\otimes\mathbb{C}^{\rho(v)\circ}$ (see footnote
4.2.2). A similar relation holds on
$\mathbb{C}^{n_{i}}\otimes\mathbb{C}^{\mu_{ij}}\otimes\mathbb{C}^{n_{j}\circ}$.

In the even case, $\gamma$ is determined as the multiplication by the
decoration $s(v)=\pm 1$ on $\mathcal{H}_{v}$. The real operator $J$ is
reconstructed by the family of maps

$\displaystyle
J_{v}\vcentcolon=\epsilon(v,d)\,J_{0}\widehat{\kappa}_{v}=\epsilon(v,d)\,\widehat{\kappa}_{v}J_{0}:\mathcal{H}_{v}\to\mathcal{H}_{\kappa(v)}$

or, equivalently, with $i=i(v)$ and $j=j(v)$, and any
$\xi_{i}\otimes\eta_{j}^{\circ}\in\mathcal{H}_{v}$,

$\displaystyle
J(\xi_{i}\otimes\eta_{j}^{\circ})=\epsilon(v,d)\,\bar{\eta}_{j}\otimes\bar{\xi}_{i}^{\circ}\in\mathcal{H}_{\kappa(v)}.$

In other words,
$J_{v_{2}}^{v_{1}}=\epsilon(v_{1},d)\delta_{\kappa(v_{1})}^{v_{1}}\,J_{0}\widehat{\kappa}_{v_{1}}$.

The Dirac operator is reconstructed by the decorations $D_{e}$ of the edges
$e\in\Gamma^{(1)}$. All the properties of a (even) real spectral triple are
encoded in the relations required above. One has
$J_{\kappa(v)}J_{v}=\epsilon(\kappa(v),d)\epsilon(v,d)\widehat{\kappa}_{\kappa(v)}\widehat{\kappa}_{v}J_{0}^{2}=\epsilon$.
In the same way,
$D_{\kappa(e)}J_{v_{1}}=\epsilon^{\prime}\epsilon(v_{1},d)\epsilon(v_{2},d)\,\widehat{\kappa}_{v_{2}}J_{0}D_{e}J_{0}\widehat{\kappa}_{\kappa(v_{1})}(\epsilon(v_{1},d)\,\widehat{\kappa}_{v_{1}}J_{0})=\epsilon^{\prime}\epsilon(v_{2},d)\,\widehat{\kappa}_{v_{2}}J_{0}D_{e}$
while
$J_{v_{2}}D_{e}=(\epsilon(v_{2},d)\,J_{0}\widehat{\kappa}_{v_{2}})D_{e}$, so
that $JD=\epsilon^{\prime}DJ$. Finally,
$J_{v}\gamma_{v}=s(v)\epsilon(v,d)J_{0}\widehat{\kappa}_{v}$ while
$\gamma_{\kappa(v)}J_{v}=s(\kappa(v))\epsilon(v,d)J_{0}\widehat{\kappa}_{v}$
so that $J\gamma=\epsilon^{\prime\prime}\gamma J$.

Let us describe the Dirac operator for the decomposition
$\mathcal{H}=\operatorname{\textstyle\oplus}_{i,j=1}^{r}\widehat{\mathcal{H}}_{\mathbf{n}_{i}\mathbf{n}_{j}}$.
Introduce an orthonormal basis for each $\mathbb{C}^{\mu_{ij}}$ and label all
these basis vectors in the union of all the $\mathbb{C}^{\mu_{ij}}$’s as
$\\{\sigma_{v}\\}_{v\in\Gamma^{(0)}}$: for any
$v\in\Gamma^{(0)}_{\mathbf{n}_{i}\mathbf{n}_{j}}$, $\sigma_{v}$ is an element
of an orthonormal basis of $\mathbb{C}^{\mu_{ij}}$. We use the identification
$\mathcal{H}_{v}=\operatorname{Span}\\{\xi\otimes\sigma_{v}\otimes\eta^{\circ}\mid\xi\in\mathbb{C}^{\lambda(v)},\eta^{\circ}\in\mathbb{C}^{\mu(v)\circ}\\}$.
For any $e=(v_{1},v_{2})\in\Gamma^{(1)}$ with
$v_{1}\in\Gamma^{(0)}_{\mathbf{n}_{i_{1}}\mathbf{n}_{j_{1}}}$ and
$v_{2}\in\Gamma^{(0)}_{\mathbf{n}_{i_{2}}\mathbf{n}_{j_{2}}}$, define
$\widehat{D}_{e}:\widehat{\mathcal{H}}_{\mathbf{n}_{i_{1}}\mathbf{n}_{j_{1}}}\to\widehat{\mathcal{H}}_{\mathbf{n}_{i_{2}}\mathbf{n}_{j_{2}}}$,
for any $\xi\in\mathbb{C}^{n_{i_{1}}}$ and
$\eta^{\circ}\in\mathbb{C}^{n_{j_{1}}\circ}$, as

$\displaystyle\widehat{D}_{e}(\xi\otimes\sigma_{v}\otimes\eta^{\circ})$
$\displaystyle=\begin{cases}0&\text{if $v\neq v_{1}$}\\\
(D_{L,e}^{(1)}\xi)\otimes\sigma_{v_{2}}\otimes(D_{R,e}^{(2)}\eta^{\circ})&\text{if
$v=v_{1}$}\end{cases}$

Then define

$\displaystyle
D_{i_{2}j_{2}}^{i_{1}j_{1}}\vcentcolon=\sum_{\begin{subarray}{c}e=(v_{1},v_{2})\in\Gamma^{(1)}\\\
v_{1}\in\Gamma^{(0)}_{\mathbf{n}_{i_{1}}\mathbf{n}_{j_{1}}},v_{2}\in\Gamma^{(0)}_{\mathbf{n}_{i_{2}}\mathbf{n}_{j_{2}}}\end{subarray}}\widehat{D}_{e}:\widehat{\mathcal{H}}_{\mathbf{n}_{i_{1}}\mathbf{n}_{j_{1}}}\to\widehat{\mathcal{H}}_{\mathbf{n}_{i_{2}}\mathbf{n}_{j_{2}}}$

The operators $D_{i_{2}j_{2}}^{i_{1}j_{1}}$ can be collected in an operator
$D:\mathcal{H}\to\mathcal{H}$ which is self-adjoint.

One can write a specific version of (11.6.2) for the decomposition
$\mathcal{H}=\operatorname{\textstyle\oplus}_{v\in\Gamma^{(0)}}\mathcal{H}_{v}$
in terms of the operators $D_{e}$ for $e\in\Gamma^{(1)}$. For any
$\bm{\omega}\in\mathbf{\Omega}^{n}_{U}(\mathcal{A})\subset\mathfrak{T}^{n}\mathcal{A}$
which decomposes along a sum of typical terms
$\operatorname{\textstyle\oplus}_{i_{1},\dots,i_{n-1}=1}^{r}\big{(}a^{0}_{i}\otimes
a^{1}_{i_{1}}\otimes\cdots\otimes a^{n-1}_{i_{n-1}}\otimes
a^{n}_{j}\big{)}_{i,j=1}^{r}\in\mathfrak{T}^{n}\mathcal{A}$ and any
$v_{0},v_{n}\in\Gamma^{(0)}$, one has

$\pi_{D}(\bm{\omega})_{v_{0}}^{v_{n}}=\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\text{all
terms at the}\\\ \text{$(i(v_{0}),i(v_{n}))$ entry in
$\bm{\omega}$}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\operatorname{\textstyle\sum}_{v_{1},\dots,v_{n-1}\in\Gamma^{(0)}}a^{0}_{i(v_{0})}D_{(v_{1},v_{0})}a^{1}_{i(v_{1})}D_{(v_{2},v_{1})}\cdots\\\
\cdots
a^{n-1}_{i(v_{n-1})}D_{(v_{n},v_{n-1})}a^{n}_{i(v_{n})}:\mathcal{H}_{v_{n}}\to\mathcal{H}_{v_{0}}$
(4.2.4)

In this formula, one supposes $D_{(v_{i+1},v_{i})}=0$ when
$(v_{i+1},v_{i})\not\in\Gamma^{(1)}$.

A complete basis for any $i,j=1,\dots,r$, let $\\{\sigma_{ij}^{p}\\}_{1\leq
p\leq\mu_{ij}}$ be an orthonormal basis of $\mathbb{C}^{\mu_{ij}}$ (for
instance as in Prop. 4.2.9 or 4.2.10), to which we associate the irreps
$\mathcal{H}_{v}$ defined as in (4.2.1) for any
$v=(i,p,j)\in\Gamma_{\mathbf{n}_{i}\mathbf{n}_{j}}^{(0)}$. One can then fix an
orthonormal basis
$\\{e_{ij,\alpha}=\xi_{i,\alpha}^{(1)}\otimes\eta_{j,\alpha}^{\circ(2)}\\}_{1\leq\alpha\leq
n_{i}n_{j}}$ (sumless Sweedler-like notation) of
$\mathbb{C}^{n_{i}}\otimes\mathbb{C}^{n_{j}\circ}$. Let

$\displaystyle\widetilde{\Gamma}_{\mathbf{n}_{i}\mathbf{n}_{j}}^{(0)}$
$\displaystyle\vcentcolon=\Gamma_{\mathbf{n}_{i}\mathbf{n}_{j}}^{(0)}\times\\{1,\dots,n_{i}n_{j}\\}$
and $\displaystyle\widetilde{\Gamma}^{(0)}$
$\displaystyle\vcentcolon=\cup_{i,j=1}^{r}\widetilde{\Gamma}_{\mathbf{n}_{i}\mathbf{n}_{j}}^{(0)}$

Then for any
$\tilde{v}=(v,\alpha)\in\widetilde{\Gamma}_{\mathbf{n}_{i}\mathbf{n}_{j}}^{(0)}$,
let
$e_{\tilde{v}}\vcentcolon=\xi_{i,\alpha}^{(1)}\otimes\sigma_{ij}^{p}\otimes\eta_{j,\alpha}^{\circ(2)}\in\mathcal{H}_{v}$.
The family $\\{e_{\tilde{v}}\\}_{\tilde{v}\in\widetilde{\Gamma}^{(0)}}$
defines an orthonormal basis of $\mathcal{H}$. We define
$v:\widetilde{\Gamma}^{(0)}\to\Gamma^{(0)}$ as $v(\tilde{v})=v$ for
$\tilde{v}=(v,\alpha)$.

##  Part II Noncommutativity, Spacetime, and Noncommutative Geometry

###  Chapter 5 General Outline

Now that some conceptual and mathematical grounds have been established to
understand the origin of NCG and how it works on the mathematical side, let us
try to see which meaning can be given to this noncommutativity, as well as for
the possible NCG that could accompany it. Since this part is not essential to
understanding the technical developments which follow and present assertions
of a different nature than the other three parts, the reader can directly move
on to part III.

Welcome to those readers who have been willing to keep reading, before going
any further, I would like to warn you that the ideas developed here are
personal ideas, not developed in the literature for most of them, so there may
not be a consensus about them. However, I have tried to make my statements as
clear as possible, demonstrate them as much as possible, and support my
statements with quotations to connect them to the literature and other
author’s thoughts, whenever possible.

I believe that a full-fledged mathematical physicist is both someone who can
understand and develop the mathematical structures that define our
representations of reality, and make the theory clear at that level. But I
think that she/he must also be someone who understands how the mathematical
objects in the theories relate to observables, concrete physical objects, and
how the intuitive/ontological picture we have of the world and the one we get
from its actual mathematical representation go on together. These are not
passive philosophical claims, because these clarifications are essential to
guide the physicist’s vision and work in a meaningful way. Many great
scientists like Einstein, Bohr having done huge and fundamental contributions
to physics are examples of researchers who elaborate theories using something
outside the scope of pure mathematical thinking, with through experiments,
philosophical principles about nature, intuitive feelings… That is why this
part is for me a very important and needed part that can be considered as an
attempt to complete the mathematical presentation made in the first part.

Attempting to give an ontology or an intuitive explanation of what NCG is,
seems not to be an easy task. A partial attempt in this direction was done in
(Huggett, 2018). Intuitive objects such as the notion of a point no longer
make sense. But can we think of anything outside our usual spatial
representations? Can we think of a structure without assuming a set of points
that supports our intuition? The NCG forces us to try to understand the origin
of the notion of point which appears so precious to our minds.

There is a certain line of interpretation of NCG, especially developed by A.
Connes. These are mainly based on wave considerations, with the wave number
$k$ eigenvalue of the Laplacian formed by the Dirac operator, with distance
thought as linked with wave nature for example. This branch of interpretation
happens to be based on a formalism that reveals part of reality, and which is
formally equivalent to matrix mechanics as Shrödinger showed. In what follows,
I will try to highlight the contributions that this dual approach of matrix
mechanics can provide to the interpretation of NCG.

NCG historically comes from the formalism used to describe QM, operator
algebras acting on Hilbert spaces through the Gelfand-Naimark theorem. The NC
of observables in QM is seen as a kind of validation of the potential
relevance of the use of NCGs in physics. Usual NCGs are thought to correspond
to the notion of a “space” underlying an algebra of NC observables considered
as continuous functions on this “space”. Then, if there is an interpretation
for the NC of observables in QM, this can lead to an interpretation for the
corresponding NCG. There are two ways of looking at the QM formalism, more
precisely concerning the $\psi$ state. These two are the epistemic view and
the ontic view. The first one postulates that $\psi$ is not relative to an
objective physical entity, but to a potential knowledge, the second one says
that $\psi$ corresponds to a real physical state of nature. If QM is an
epistemic theory, then there are no objective geometrical consequences to be
determined from its formalism, because it does not refer to anything
substantial. But if the theory is ontic, then (via Gelfand Naimark theorem and
the equivalence of categories between algebra and geometry highlighted in the
first part) there must be geometric consequences. This is why I will often
talk about QM and the ways of interpreting it, to make sense of what I will
say in section 7.6 and chapter 9. We will see that the interpretation we have
of the state (done in section 7.5) permits us to deduce the interpretation of
the NC in QM (done in 7.6), and finally to deduce an interpretation of the
deduced NCG (in chapter 9). This will be supported by the physical arguments
given in chapter 8 inducing that only observables can lead to the space-time’s
geometric representation and that these observables must therefore be taken to
be more fundamental than space-time for the experimental physicist and
thinker.

First of all, in chapter 6 we will have a general discussion about what we
mean by NC, NCG, and why NCG remains largely unknown outside. I will then
define some important notions on which the following statements will be based.
To introduce some useful terminology, we will first define what is a physical
theory in chapter 7, and what are the elements of reality. This will be
necessary for the sections 7.5 and 8.1 where more advanced considerations will
come into play in the discussion. Then, I will give arguments to support these
different points:

  1. (I)

The formalism of $C^{*}$-algebras acting on Hilbert spaces encompasses both
Classical and QM, thus offering a framework to think about their unification
(section 7.3).

  2. (II)

There are strong indications that the QM formalism refers directly to ontic
states, and should therefore be considered seriously as reflecting something
objective i.e. this is not just an algorithm that leads to good predictions
(section 7.5).

  3. (III)

That point (II) induces that NC is linked to non-passivity of the measurement
process, and therefore to contextual state change (section 7.6).

  4. (IV)

Observables are first to the notion of space-time. The space-time must not be
thought of otherwise than as a deduction of the observables that we collect
(section 8.1).

  5. (V)

Giving objective existence to space-time structures seems to be misleading
(section 8.2).

  6. (VI)

Geometry can be entirely algebraized, this procedure culminating with the
Gelfand-Naimark theorem and the birth of NCG. Leading to the fact that the
structures of the algebra of observables and space-time contain just as much
information, and are equivalent (sections 8.3 and 8.4).

  7. (VII)

Because the formalism and observables of QM reflect something objective
according to point (II), observables are prior to space-time structures trough
points (IV), (V) and (VI). We then have to understand what are the geometric
consequences behind this formalism. The first one concerns the origin of the
notion of point, the second one of connectedness, and the last one is about a
length scale at which the notion of point no longer stands, opening the door
to NCG (chapter 9).

  8. (VIII)

Considering such a $\psi$-ontic interpretation, using (III) for the
consequence on the interpretation of the NC of observables, this implies that
a natural interpretation of the NCG can be that of a measured geometry, this
measurement process being non-passive, changing the underlying notion of
“point” coming from the pure state undergoing change (section 9.5).

This can be viewed as three blocks, the first one containing points (I), (II),
and (III) is about the ontic meaning of the formalism of QM and the
interpretation of the NC. The second includes points (IV), (V) and (VI)
defends the fact that observables are prior to space-time representation, and
that they contain all the information we need. The last one with points (VII)
and (VIII) proposes to set up the consequences of the assumption of the first
two blocks, when their respective implications are put together.

###  Chapter 6 Why NCG remains in the Shadow for Physicists?

It is surprising to note how unknown NCG is to researchers in theoretical
physics, and to what extent the minimum basis of understanding of the subject
remains out of reach for many of them. Indeed, it is not easy to enter NCG,
the language used is very limited to the mathematical one, and few efforts are
made to go deeper in the understanding. The specification of the supposed kind
of NC with which NCG deals with, is not clarified anywhere to my knowledge
(which I will try to do in section 6.2 and afterward). Even worse, the terms
chosen to talk about NCG are often extensions of the terms of usual geometry,
which have no more sense in this new context, as we will see in section 6.3.
What is a NCG? How to represent it? And why is it a relevant extension of the
geometric framework? Few simple answers are given to these elementary
questions. It is not enough to say that two observables do not commute and
that mathematically this is written $[a,b]\neq 0$ to have a complete intuition
of what NCG is. One of the main problems in giving an intuition of the NCG
lies in the notion of point, which loses all meaning in NCG. Yet, this notion
is the support of our intuition of “tangible” things, of our representations
of reality. So how do we go without it?

####  6.1 Are Sets and their Elements Primary Objects?

> I know that the great Hilbert said “We will not be driven out of the
> paradise Cantor has created for us” and I reply “I see no reason for walking
> in!”
>
> R. Hamming

Set theory is currently regarded as a foundation of mathematics. Most fields
of mathematics are based on the notion of a set of objects whose relations and
operations are studied. The elements can be called points, vectors, functions…
This way of proceeding, by identifying objects, and looking at the notions
they have between them, is embedded in the innate functioning of our minds and
provides us with a grid for reading the world, through different systems of
representations. When it comes to representations aimed at describing the
“objective” phenomena of the physical world, we formulate physical theories
whose symbolic objects are related to physical objects, phenomena and events.
To describe our space-time, the representation is that of a space-time where
the set is called space, and the elements are called points, then points
events. The same is true for many physical theories, where the notions of set
and element manifest themselves in different representations. But what is the
proof that this reading grid offers a complete picture of reality?

A hint of the insufficiency of this general framework of thinking is given by
the non-commutativity of observables in QM. For instance, if we consider an
operational way to define the elements of a set, according to the
determination of their properties (the experimental way being the only one
accessible to us), and that these properties no longer commute during this
determination, it will no longer be possible to separate this set into sub-
sets corresponding to these properties. Let’s take a simple example to
illustrate this. If we consider that our set is a box and its “elements” are
particles that can have two shapes (square and spherical) corresponding to the
observable $S$ for shape, and two colors (red and blue) for the observable
$C$, then if $[C,S]\neq 0$, it will not be possible to partition this set into
for example the set of red and spherical particles, the other set blue and
square… This is not possible because these particles cannot have the status of
set elements for the properties defined by $S$ and $C$ jointly, and thus, the
procedure for creating sets by specifying the properties of their elements
becomes meaningless, as we no longer obtain sets.

Thus, if the observables do not commute in QM, then we cannot classify
“entities” as elements of a set for these attributes taken together, as we
only have access to observables, and we are not allowed to say more than what
observables tell us. Thus as shown in section 7.2, when partitioning the
physical reality according to the two properties deduced from these
observables, the notion of phase space will no longer be appropriate. This is
intimately linked to the notions of realism/anti-realism, and therefore to the
notion of the element of reality which will be presented in section 7.5. We
will see, with the arguments of section 8.1 and chapter 9 that for the portion
of these observables that leads to geometrical representations of the world,
the usual geometrical framework of space with points is no longer consistent.
In this line of thought, another reason resulting from the formalism of QM, in
disagreement with the notion of element and thus of point, will be presented
in section 9.2.

####  6.2 Which NC are we talking about?

What do we mean by non-commutativity? There is the answer of the
mathematician, and the one of the physicist, which are in some way connected.

For the mathematician, two notions of noncommutativities are encountered:

  * •

The one of the groups, we will say non-abelian for groups. Most often, the
concept of group is accompanied by the concept of transformations of “spaces”.

  * •

The one of algebras, we will say non-commutative for algebras. There is not
necessarily the idea of transformation.

As mentioned before, only NC of algebras is connected to NCG and more
generally to NC mathematics. The one of group is usually thought as linked to
transformation, and if we consider a non-abelian group whose algebra of
function is commutative, then it will belong to commutative mathematics. This
is important to note that no mention of NC coordinates was given as input to
define what is NCG, we only talk about NC of observables.

In the same way, the physicist encounters many non-commutativities in his
everyday life. Some of them are thought to be normal (classical), and others
to be not. But what distinguishes normal non-commutativities from those that
could be attributed to an NCG for example? I haven’t found any clarification
on this important subject. I will try to give one in what follow.

Indeed, a question that naturally arises is: “of which non-commutativity are
we talking about?” There are various kinds of non-commutativities, in the
present state of affairs, when they correspond to physical theories, these can
be classified into two categories:

  1. 1.

Process-like NCs in space and time (Type 1 NCs): NCs of rotations,
irreversible processes, operations on solids, an example given by Connes being
the operations of closing a bottle and filling it. These have extensions in
space-time.

  2. 2.

NCs which are not attributed to processes in space and time (Type 2 NCs): they
are perceived by the physicist as belonging to processes without any spatial
and temporal extension. They are thought (by many of them) as intrinsic. These
are the NC of observables in QM, of all quantized observables, or of non-
abelian gauge theories. These do not have extensions in space-time.

If this first category corresponds to classical intuition, it is to the second
that the special character of the NCG is clearly attributed, because these
observables can well be, within the framework of these models, considered as
functions with values on “points” (in quantum mechanical models), thus without
extension in space-time. We can see that the categories for the physicist are
linked to the ones of the mathematician since group action is thought of as a
kind of transformation in physics, and that algebras are connected to
observables in physics, usually not thought of as transformation. We will see
in sections 7.5 and 7.6 that according to Bohr, the NC in QM does not
represent a process in space and time. And that he is probably at the origin
of this intuition concerning the NC of observables in QM, as not being seen as
processes in space-time.

We will see in section 7.6, that if Type 1 NCs are fully compatible with the
realistic thesis (ontic interpretation) of the state $\psi$, on the other
hand, type 2 NCs induce an anti-realistic thesis regarding the nature of the
symbolic state, and thus (according to the PBR theorem that will be presented)
of physical reality by rejecting the idea that there is an element of physical
reality, showing how NC and realism are intimately connected. We will discuss
to which extent the anti-realist hypothesis is actually tenable in section
7.5. In section 7.6, we will see that type 2 NCs are in fact type 1 NCs if one
adopts an ontic interpretation of the state in QM.

####  6.3 The Problem of the Words used in NCG

> The limits of my language mean the limits of my world.
>
> L. Wittgenstein

The use of words that make good sense is important when trying to understand
new pieces of knowledge. Physics is not only about equations and mathematical
structures, but about common language words and interpretations, which guide
our intuition and the elaboration of mathematical structures that are relevant
to physics. Words carry a heavy load of cultural connotations, and conceptual
associations, both conscious and unconscious. If we consider any conceptual
framework (it could be a physical theory), it is associated with a set of
words, subjective representations, and epistemological concepts that provide
us with a ’satisfactory’ understanding of that framework. When considering an
attempt to extend this framework, care must be taken with this set of words
associated with the old framework, when using them to understand the new one,
as some of these concepts may lose their meaning in the process. As has often
happened in the history of physics, new words were invented to accompany new
perceptions whose relevance became obvious.

However, some physicists who have left an undeniable mark on the history of
physics, such as Bohr, have helped to mark a philosophical break with this
tendency to create new words to understand physics, particularly in quantum
physics. Indeed, as J-M. Lévy-Leblond says in (Lévy-Leblond, 2000) Bohr
considered the forms of expressions of classical physics to be unsurpassable,
and considered the desire to find forms of expression specific to the quantum
domain unattainable and condemnable. For him, it was an impossibility in
principle. I find this position extremely difficult to justify, as it appears
both surprising and dogmatic. It is difficult to understand why, unlike many
great physicists in history who have encountered counter-intuitive phenomena,
Bohr could allow himself to create and spread the idea that these counter-
intuitive characteristics were in fact fundamental principles (such as the
principle of complementarity), expressible in the usual language of classical
physics, as counter-intuitive as they appear to us. Despite his influence,
physicists of the following generations have nevertheless created new words,
specific to quantum physics, such as coherence, entanglement, beable… But the
influence of this epistemological break continues to strongly influence
physics, and in particular it’s current teaching, where there is no longer any
question of having an understanding other than mathematical, the rest of the
modes of understanding knowledge being quite widely considered as subjective
and illusory.

The creation of new words is a constitutive process of scientific knowledge,
which must accompany the emergence of new notions. This is why, starting from
the usual geometrical framework, and all the words and concepts that surround
it, such as the notion of point, distance, coordinate, and differential
structure… and going towards the extension of this framework which is NCG, it
will be necessary to pay particular attention to the words chosen. I have
noticed that some words commonly used in NCG do not really make sense in this
framework, and thus disturb its comprehension. Here are some of them:

  * •

The notion of point, which remains present in the notion of finite $n$-point
space, does not seem adequate. These terms designate the finite Hilbert space
on which the algebra is represented, but as we have seen with the Gelfand-
Naimark theorem, these elements only acquire the status of points for
commutative algebras. The fact that the elements in Hilbert spaces are
intuitively considered as points comes from the status they have in their use
in QM, as classical outcomes, where they only acquire point status at the end
of the measurement process. One suggestion is to call these elements pre-
points, reserving point status for them when the algebra living on them is or
becomes commutative.

  * •

In mathematics, a space is a set with remarkable additional structures,
allowing us to define objects analogous to those of usual geometry. The
elements can be called points, vectors, functions… The notion of space does
not make sense in NCG, since we can no longer think of this reality as based
on the precious notions of elements of a set, which are points in the
geometry. The Hilbert space does not have the status of a set (of points) from
the point of view of the algebra, it only has one for the classical observer
who perceives well-defined outcomes, there is thus an error in terminology.

  * •

In NCG, coordinates are replaced by generators of the algebra. As they do not
commute they cannot be simultaneously diagonalized and the space disappears.
But people continue to name these generators coordinates. Take a look at the
meaning of the word coordinate, this comes from the Latin “co” for “together”
with “ordinare” for arranging or put in order. Therefore, the meaning of
coordinate can be seen as “put in relative order objects together” potentially
with numbers. In NCG, neither objects/points nor order relations are given in
a non-contextual way because of the NC of the observables providing such
numbers used for this ordering. Therefore, this word has no meaning in NCG.

The first two points are deductions from the arguments presented in section
6.1 concerning the limitations of the concepts of set and set’s elements
(space and points respectively).

There are other obstacles to the conceptual understanding of NCG, and this at
the basis of its study, as mentioned in section 6.2, about which NC is
concerned. In addition, regarding the status of NCG in physics, most physics
researchers seem to consider, unconsciously at least, that NCG is a theory,
just like loop quantum gravity or string theory. However, NGC is not a theory,
it is a framework of mathematical thoughts, in the same way as Riemannian
manifolds were. General relativity has not been called a theory of Riemannian
manifolds, it is a theory whose mathematical basis is based on this
mathematical formalism. So it seems natural to me that researchers who want to
set up a theory of quantum gravitation should try to acquire some knowledge of
NCG as a framework, in order to try to understand how to quantify space-time
in a relevant way. NCG should not be considered as a theory, and therefore a
rival theory, but rather as representing a mathematical framework for
formulating such theories. Because theories that want to go beyond general
relativity must belong to a formalism that is broader than that of Riemannian
manifolds while having links with it since the theory of general relativity
must be found in the limiting cases of large scales and small energy
densities.

The problem is probably even deeper than that, connected to the very structure
of our language. In everyday life, we identify objects, which can be thought
of as elements of a set, and we attribute specific words to them. But if the
notion of element/object is no longer relevant, is it still possible to put
words on things? If, as with many words, they refer in their definitions to
properties of the thing referred to, then to the extent that these properties
become contextual, the usual words lose their meaning. I don’t think there is
anything in the structure of our language to capture in essence the nature of
an object whose properties are revealed contextually. But as always, in
science, we can create new words, and new meanings, to improve and extend our
representations of reality.

In section 9.5 I will bring a personal answer resulting from the arguments
collected in the different sections which will follow, as to how one can
represent the NCG, and how one can come to get rid of the notions of points,
space, and coordinates, by changing one’s perspective, this change of
perspective being among other things allowed by the theorem of Gelfand-Naimark
via the relation 1.2.1. I would finish this section by saying that in my
opinion, the words’ problem extends well beyond NCG, and contaminates many
fields of the current research in theoretical physics.

###  Chapter 7 NC and Physical Theories

In QM, three phenomena can be considered as really new. The principle of
superposition, entanglement, and the contextuality due to the NC of
observables. All of these are due to the replacement (in QM) of functions by
operators. In this chapter, we will try to understand how NC can be
understandable in the context of physical theories (about nature), which
change occurs at the level of the formalism, and what can this tells us about
nature, in the context of QM.

####  7.1 General Structure of Physical Theories

> Experience without theory is blind, but theory without experience is mere
> intellectual play.
>
> I. Kant

A physical theory consists of two things: a mathematical structure
$\mathfrak{M}$ and a mapping of the elements of this structure to the
operational measurements process of the corresponding elements in nature.
Given a physical state $ER$ (an element of reality), our representation in the
mathematical structure $\mathfrak{M}$ will be called the symbolic state
$\pi_{\mathfrak{M}}(ER)$. The measurement of an $ER$ is the determination of
the magnitude of something through the comparison of this unknown quantity
associated with the $ER$ with a reference quantity of equivalent nature
(carried by the element of reality of the object being used as a measuring
device), known as the measurement unit. The informational content we get from
the measurement of an $ER$ is called an observable $O(ER)$. The physical
theory must give the link between $\pi_{\mathfrak{M}}(ER)$ and the measurement
process of the physical state $ER$ such represented:

$\displaystyle\pi_{\mathfrak{M}}(ER)\overset{}{\longrightarrow}\text{Measurement
of }ER\longrightarrow O(ER)$

If there is no such measurement process, the fact that
$\pi_{\mathfrak{M}}(ER)$ is representing something real may be questionable.
We will call $Mes(ER)$ the measurement process of the $ER$, and
$\pi_{\mathfrak{M}}(Mes(ER))$ it’s representation in the mathematical
structure $\mathfrak{M}$. $O(ER)$ is the only thing that we see (as
experimenters), $ER$ and $Mes(ER)$ being out of reach. As the measurement
process corresponds to an interaction, it can change the element of reality:

$\displaystyle ER\overset{Measurement}{\longrightarrow}ER^{\prime}$

If the mode of observation of this $ER$ is non-passive, that is to say, that
it disturbs the $ER$, then, we will have to distinguish two notions of
physical state, the one which would correspond to the physical state before
measurement and the one after the process, which we will call
$F(ER)=ER^{\prime}$. $F(ER)$ can be seen as what we see, and $ER$ as “what is”
(before measurement). A physical theory can then be considered as describing
what we observe, not what is, but we often do this through shortcuts, because
in our classical world, these two categories are confused. We can complete
this picture by specifying how and by which apparatus/observer the measurement
is done since observational content is both a function of the $ER$ and of the
measurement apparatus. We will encode all these data into one symbol, “$a$”
for apparatus (and then for observable in the framework of QM as we already
see in chapter 3). The previous notations become $Mes_{a}(ER)$, $O_{a}(ER)$
and $F_{a}(ER)$.

The act of (classical) measurement should be seen as an interaction between
physical objects and is therefore the restriction to classical observers of a
more complete picture of the interaction pattern underlying the measurements
process. This interaction extracts some information about the physical state,
this information taking the form of $\pi_{\mathfrak{M}}(ER)$. For some
theories, $\pi_{\mathfrak{M}}(ER)$ does not describe directly an $ER$ but the
data we have on it, for example statistical data. It’s also possible that
$\pi_{\mathfrak{M}}(ER)$ can be an incomplete representation of an $ER$, that
some hidden variable exists. It is also possible for some theories to reject
the idea of an existing $ER$.

In the special case of QM, $\pi_{\mathfrak{M}}(ER)$ will be replaced by $\psi$
or $\rho=|\psi\rangle\langle\psi|$, and $O_{a}(ER)$ by $\lambda_{i}$ with a
probability weight $\operatorname{tr}(P_{\lambda_{i}}\rho)$ (or by
$O_{a}^{av}(\psi)=\langle\psi|a|\psi\rangle=\operatorname{tr}(a\rho)$ in
average), for discrete spectrum as defined in (1.1.2). The actual QM theory
does not predict deterministically the outcome. There will be two manners to
interpret its mathematical structure $\mathfrak{M}$. Indeed, ontological
models of QM can be categorized in two ways: the $\psi$-ontic way and
$\psi$-epistemic way. $\psi$-ontic models consider the symbolic state $\psi$
($\pi_{\mathfrak{M}}(ER)$) as referring directly to objective physical
reality, whereas the $\psi$-epistemic ones consider it as simple knowledge of
the observer on it. As we will see in section 7.5, this second kind of
interpretation can be split into two categories, the one that admits the
existence of $ER$, and the one that rejects it. For models admitting the
existence of a physical state ($ER$), we will call $\psi$ its symbolic state’s
representation.

Let’s $a$ and $b$ design two different measurement processes, potentially
linked to different observables (or to the same). Therefore, it is possible to
encounter non-commutative relations like:

$\displaystyle[\,O_{a}\,,\,O_{b}\,](ER)\,\neq\,0.$

This means that
$[\,O_{a}\,,\,O_{b}\,](ER)=O_{a}(F_{b}(ER))-O_{b}(F_{a}(ER))\neq 0$. As said,
the physicist doesn’t know $ER$ (then $F(ER)$) nor the way in which a
classical observation $O(ER)$ is given from this $ER$. Therefore an
interpretation of non-commutativity may concern either the link between $ER$
and $F(ER)$, or the way in which the measurement process can transmit the
information $O(ER)$, in classical information format. We will see in section
7.6, that in the framework of QM, an ontic interpretation of the symbolic
state implies that $ER\neq F_{a\,\text{or}\,b}(ER)$, putting the
interpretation at the level of the link between $ER$ and $F(ER)$, whereas for
epistemic interpretation (defended by Bohr), it is at the level of the
transmission in a classical form that the interpretation must be centered.

The basic mathematical structures $\mathfrak{M}$ corresponding to a huge part
of the physical theories are in the field of algebra. Algebra is the study of
symbols and the rules for manipulating and combining these symbols in
formulas, the basic example being the product. These symbols will correspond
directly or indirectly to elements of reality, and the rules will correspond
to how these elements of reality interact or combine into bigger systems.
Therefore it is interesting to play with different symbols (representations),
or different products, like the universal/tensorial, star, matrix, or inner
products for example.

####  7.2 Classical Phase Space vs. Quantum Configuration Space

In general, a physical theory consists of four components: a convex state
space $S_{ps}$, a real vector space of observables $\mathcal{U}_{\mathbb{R}}$,
a function connecting these two spaces $\langle\,,\rangle\
:S_{ps}\times\mathcal{U}_{\mathbb{R}}\to\mathbb{R}$ allowing to obtain the
expected value of the given observable for the state in question, and finally
laws specifying the dynamics of the theory, such as a Lagrangian, the
Schrödinger or Dirac equation (Landsman, 1998).

In classical mechanics, the phase space $S_{ps}$ is a space of points
interpreted as the pure states of the system. The mixed states are identified
by measurement probabilities on $S_{ps}$. The observables of the theory are
real functions on the phase space $S_{ps}$, most often continuous and bounded.
Thus the space $\mathcal{U}_{\mathbb{R}}$ of observables can be taken to be
equal to $C^{\infty}(S_{ps},\mathbb{R})$, $C_{0}(S_{ps},\mathbb{R})$ or
$C_{b}(S_{ps},\mathbb{R})$. There is a function
$\langle\,,\rangle:S_{ps}\times\mathcal{U}_{\mathbb{R}}\to\mathbb{R}$ between
the state space $S_{ps}$ of measurement probabilities $\mu$ sur $S_{ps}$,and
the space $\mathcal{U}_{\mathbb{R}}$ of observables $f$:

$\displaystyle\langle\mu,f\rangle=\mu(f)=\int_{S}d_{\mu}(\sigma)f(\sigma)\equiv\frac{\sum
A_{n}e^{-\beta E_{n}}}{Z}$

with $f(\sigma)$ the value of the observable $f$ in the state $\sigma$, and
$Z$ the partition function. This function gives the expected value for the
observable $f$. In general, we have
$\langle\mu,f\rangle^{2}\neq\langle\mu,f^{2}\rangle$. For a pure state
$\sigma$, the measure is the Dirac one and we have
$\delta_{\sigma}(f)=f(\sigma)$ and
$\langle\mu,f\rangle^{2}=\langle\mu,f^{2}\rangle$.

In QM, because of the commuting property of the canonically associated
variable of the state space, this one cannot be taken to be a state’s phase
space anymore, it became a configuration space $S_{cs}$. The state space will
then consists of the set of all density matrices $\hat{\rho}$, constructed on
a given Hilbert space $\mathcal{H}$:

$\displaystyle
S_{cs}=\\{\hat{\rho}=|\psi\rangle\langle\psi|\quad,\quad\psi\in\mathcal{H},\quad\|\psi\|=1\\}$

these are the pure states which are in correspondence with the unitary $\psi$
states. Similarly, the observables are the bounded self-adjoint operators on
$\mathcal{H}$. The function linking the states to the observables is given by:

$\displaystyle\langle\hat{\rho},A\rangle=Tr(\hat{\rho}A)$

equal to $(\psi,A\psi)$ for pure states. Thus we have respectively for the
classical case, the statistical physics case and the quantum case:

$\displaystyle\langle\mu,f\rangle=\int_{S}d_{\mu}(\sigma)f(\sigma)\quad\leftrightarrow\quad\langle
Z,A\rangle=\frac{\sum A_{n}e^{-\beta
E_{n}}}{Z}\quad\leftrightarrow\quad\langle\hat{\rho},A\rangle=Tr(\hat{\rho}A)$

The main difference between quantum and classical systems is that for the
quantum, the property
$\langle\hat{\rho},A\rangle^{2}\neq\langle\hat{\rho},A^{2}\rangle$, will
generally be true, just as in the classical case, but it will be true even for
pure states, with equality if $\psi$ is an eigenvector of $A$.

In general, a quantum system is obtained from a classical system by the
quantization process described by the linear application
$Q:\mathcal{U}_{\mathbb{R}}^{0}\to\mathcal{L}(\mathcal{H})$ with
$\mathcal{U}_{\mathbb{R}}^{0}$ belonging to $C^{\infty}(S_{ps},\mathcal{R})$,
and $\mathcal{L}(\mathcal{H})$ the space of self-adjoint operators on
$\mathcal{H}$. The quantization procedure therefore consists in establishing
the relationship:

$\text{Classical observable $f$}\leftrightarrow\text{Quantum observable
$Q(f)$}.$

It is interesting to observe that the set of classical observables is supposed
to constitute $\mathcal{C}^{0}(M)$, where the procedure for expressing this
algebra as being based on $M$ can be provided by the Gelfand Naimark theorem.
Thus, the quantization of an observable belonging to $\mathcal{C}^{0}(M)$ will
no longer be thought of as being based on a manifold $M$ but as based on a
Hilbert space, which makes the processes of QM impossible to visualize as
occurring in the usual space-time. The observables thus quantized cannot be
thought of as living on a manifold. Thus, given a locally compact manifold $M$
of Hausdorff kind, a “quantization” of $M$ consists in giving a Hilbert space
$\mathcal{H}$ and a positive map
$Q:\mathcal{C}^{0}(M)\to\mathcal{B}(\mathcal{H})$. When $M$ is compact, we
want $Q(1_{M})={\text{\char 49\relax}}$. More detail on the quantization
procedure can be found in (Landsman, 2006).

####  7.3 KvN formalism and the Classical to Quantum Unification

Many physicists believe that the classical world must be thought of as an
emergence of the quantum world, in certain limit cases. For instance, with
phenomena involving large numbers of particles, whose action goes far beyond
$\hbar$ are in which decoherence occurs. Therefore, it seems interesting to
find a formalism that deals with both classical mechanics (i.e. classical
observables) and QM (i.e. quantum observables). It is important to notice that
in the language of $C^{*}$-algebra, these two algebras of observables
correspond to:

  * •

Algebra of continuous functions $\mathcal{C}^{\infty}(M)$ over a particular
topological space (Classical observables)

  * •

Algebras of bounded operators $\mathcal{B}(\mathcal{H})$ on a Hilbert space.

It turns out that such formalism was known very quickly after the development
of QM. Indeed, the formalism of $C^{*}$-algebras and Hilbert spaces seems to
be the most general framework to describe physics. A little-known fact is that
a reformulation of classical physics has been made in this framework so that
it encompasses classical and quantum in the same formalism. Koopman-von
Neumann (KvN) mechanics, first presented by Koopman and von Neumann in 1931,
provides a description of classical mechanics in terms of Hilbert spaces. It
opens up an interesting path for studying quantum-to-classical correspondence.
An elementary point that emerges is that only two distinctions between the
classical and the quantum descriptions are needed to go from classical to the
actual formulation of the quantum formalism. This transition needs to add non-
commutativity between conjugate variables and a collapse to render
observational content from the formalism. A good introduction to the KvN
formalism can be found in (Bondar et al., 2012)

On the algebraic level, such a reconciliation of quantum and classical
concepts would result in particular in a formalism that contains the
explanation of the $\mathcal{B}(\mathcal{H})\,\to\,C^{\infty}(M)$ transition.
The problem is that we have a Hilbert space on the one hand and a manifold on
the other hand. Some physicists justify the fact that we cannot represent
ourselves quantum mechanical phenomena because they don’t live in a manifold
$M$ (space-time) but on a Hilbert space $\mathcal{H}$. However, as we have
seen with the Gelfand-Naimark theorem, there is a way to find the points of
$M$ from the pure states of its algebra, when it becomes commutative. This
last fact seems to offer a tool for the path towards the unification of
classical and quantum, and to recover $M$ from $\mathcal{H}$ and the algebra.
More detail on the quantum to classical correspondence can be found in
(Landsman, 2006).

####  7.4 What is Measurement?

What is our contact with elements of reality? The basic answer is through
measurement. We use to think of measurement and observation as direct
perception of what is, but measurement is an interaction, and this must have
consequences. Indeed empirical knowledge and knowledge in general are not
knowledge of reality in itself, but of our interaction with this reality. A
measurement process can be seen as an interaction between the object we want
to measure and an object of reference, on which, the corresponding
perturbations are interpreted as caused by this interaction. It is therefore
logical to perceive the measurement scheme as an interaction and then a
reading of the information, and not a direct reading of reality as it actually
is. However, since usual classical measurements correspond to interactions
that disturb the object being measured in a negligible way, the intuition that
emerges is that we are observing reality as it is, which is a shortcut to
phenomenal thinking, since the reality is that we are observing a disturbance
on our sensors and that this disturbance is the signature of what the object
is, after this interaction.

This is intuitive insofar as any measurement corresponds to a physical
interaction where the disturbance of the physical quantities of a reference
object (photon, measuring device, etc.) testifies to the presence or of
information on the measured object. This perturbation is linked to conserved
physical quantities, such as energy, momentum, spin, or other quantum numbers.
Thus, any change in the properties of the standard object will necessarily be
accompanied by an equivalent change in the physical state being measured. Or
the minimal element of reality for a measurement apparatus is the photon of
light, which has at least a certain impulse… Thus, the interaction that is
intended to imprint on the mediator a trace testifying to the presence of the
measured object is always accompanied by the negative reflection of this
interaction on the measured particle, thus an alteration of the corresponding
quantities, this is a fact that cannot be contested and which is fully
inscribed in the usual classical thinkings of the world. This argument has a
simple consequence, it is that any measurement acquiring non-trivial
information disturbs the physical state, a proof of this last assertion can be
found in (Busch, 2009), where the author shows that any measurement process
that acquires nontrivial information on a physical object disturbs this
object. This is quite classical reasoning. Then there is no instrument that
leaves unchanged all states of the system unless the associated observable is
trivial.

The fact that measurement in QM cannot be so straightforwardly accompanied by
the idea that it is an observation of the world as it is is at the origin of
the creation of the concept of beable introduced by J. Bell to replace the
notion of observable (Bell et al., 2001).

####  7.5 Realism, Anti-Realism and the PBR Theorem

> Naive realism leads to physics, and physics, if true, shows naive realism to
> be false. Therefore naive realism, if true, is false; therefore it is false.
>
> B. Russell

Many scientists have the conception that QM’s formalism does not describe
reality (only give us experimental results), they have accepted to be
mystified, and resign themselves to say that $\psi$ and $a$ are formal tools
that give the measurement’s result, but without physical meaning, as they do
not directly represent real objects. For them, the only view is that of a
Hilbert space, and physical objects do not live elsewhere than in this
mathematical representation, or perhaps that there are no real physical
objects behind this description…

In this section, we will discuss the interpretations of QM’s formalism. It
will be a question of conceptually partitioning the set of possible
ontological options, according to whether realism is admitted or not, whether
it is at the level of the physical state or of what is observed, and finally
whether the theory is of the $\psi$-ontic or $\psi$-epistemic type. The
motivations and concepts behind these postures will be presented throughout.
Then arguments will be presented to isolate one of the possibilities of this
classification as agreeing with the results of QM’s experiments. This section
is of special significance, as the conclusion reached by these arguments will
be a very important element of the argument presented in section 7.6, and for
the conclusions drawn in chapter 9 on the interpretation of the NCG.

As mentioned before, two views can be adopted regarding the ontological nature
of the objects described by QM, the Realistic (R) and the Non-Realistic (NR)
postures. To define the relevance of the notion of realism, let us start by
mentioning the original criteria that led to the conjecture of the existence
of an element of reality. Two criteria of reality can be defined:

  1. 1.

The Repeatability Criterion (RC): It can be defined as follows (EPR paper
(Einstein et al., 1935)): if without disturbing the system in any way, we can
predict with certainty (with a probability of 1) the value of a physical
quantity, then there is an element of physical reality $ER$ corresponding to
this physical quantity.

  2. 2.

The Kickability Criterion (KC): If a reference system is disturbed in a way
that is not understandable as its free behavior, then we will say that
something physically real has interacted with it, in other worlds, it
possesses a causal power.

Put like that, it is obvious that these realism criteria refer to $O(ER)$,
because it is a question about what we observe. The NR view denies the
existence of objective observed observable $O(ER)$ before the measurement, if
RC is not checked, the R position is thus hardly tenable. The KC is a much
less powerful criterion than the RC, because its negation is not clearly
definable, as the notion of free behavior is difficult to specify, and
sometimes a matter of convention. The non-verification of the KC is therefore
not a direct refutation of the R position.

######  Remark 7.5.1.

By element of reality (or equivalently physical state), we only refer to a
piece of hypothetical information “carried” by the physical entity in
question.

######  Remark 7.5.2.

It is interesting to stress that (RC) assumes that the measurement process
must be non-disturbing. Thus, disproving the RC will require proving that no
disturbance has been caused to the system during the measurement.

######  Remark 7.5.3.

The (KC) is the physical principle at the heart of the very notion of
measurement, as discussed in section 7.4.

Although the notion of realism was initially defined with observational
results, it is important to distinguish between the R/NR thesis which concerns
the physical state ($\psi$ in ontic models) itself (ER), and those that
concern the measurement result ($a\psi$ in ontic models for a given observable
$a$), thus $O(ER)$. We are therefore left with four categories of realism
posture, the one about the physical state, which will be called SR and SNR
thesis (for State Realist or State Non-Realist), and the ones concerning
observations will be called OR and ONR thesis. SNR trivially induces ONR, but
the converse is not true as we will see with the NC observables. The state and
observation views on realism are sometimes confused by different authors, this
is the origin of many misconceptions. One could summarize these two positions
in this way:

  1. 1.

Reality exists independently of our observation, we can consider it
conceptually, and the act of observation is a more or less precise
transmission of this reality, through an interaction which is the measurement
process.

  2. 2.

Reality is what is observed.

######  Remark 7.5.4.

As mentioned in section 7.4, if a theory only knows disruptive measurement
process, therefore, we have to make the distinction between what is, and what
we observe. This is not the case in classical mechanics.

The $OR$ and $ONR$ categories are not really classifying, as they are
secondary in the causal chain of the measurement process, and can be
considered as (classical) deductions of the physical state. Therefore we will
only consider the $SR$ and $SNR$ categories in the classification. It is also
possible to divide the set of theories according to the link between the
possible physical state, and the symbolic state representing it. The theories
concerning QM are thus distributed according to 3 categories:

  * •

$\psi$-ontic theory admitting SR ($\psi$-OSR): $\psi$ is a state that
corresponds in a bi-uniquely way to this ontic physical state.

  * •

$\psi$-epistemic theory admitting SR ($\psi$-ESR): Realism is admitted, a real
physical state exists, but the $\psi$ state only represents information about
this state.

  * •

$\psi$-epistemic theory admitting only SNR ($\psi$-ESNR): Realism is refuted,
there is no ontic physical state, $\psi$ is a kind of information, about
potentialities but there is no defined physical state.

This classification is summarised in the following diagram:

ObservationsRC $\&$ KCORONRSR$\psi$-OSR$\psi$-ESRSNR$\psi$-ESNR C NC
Observable $O$ $\longrightarrow$Physical state $ER$ $\longrightarrow$Link with
$\pi_{\mathfrak{M}}(ER)$ $\longrightarrow$

This diagram is presented as starting on the top from the observations, i.e.
from what an experimenter can be sure of, and going deeper into the
speculations we can make about nature within the theory, first about the
realism of what is observed, then about the possible underlying physical
state, and finally about the different types of symbolic $\psi$ states
describing it, thus offering a classification of possible theories in QM.

Bohr was perhaps the most influential physicist involved in the development,
and especially the interpretation of QM. He was probably the most aware of the
subtleties of QM, and the most invested in trying to develop a consistent
picture of it. So we will try to understand how his views allow us to locate
its position in this diagram, which is of great importance, since he is,
broadly speaking, followed by most of the physicists past and present, with
the Copenhagen interpretation. We will proceed in the logical order of the
diagram, from top to bottom.

The first question concerns his position in terms of realism regarding the
observational results $OR$ and $ONR$. One can unambiguously say that he was in
the $ONR$ category, his principle of complementarity being the incarnation of
it, as this quotation from Bohr in (Bohr, 1950) shows:

A sentence like “we cannot know both the momentum and the position of an
atomic object” raises at once questions as to the physical reality of two such
attributes of the object…

It is interesting to note that his statements are strictly at the level of
$ONR$, without leaving any clue as to its positioning $SR$/$SNR$, as he uses
the term “attributes of the object”. What Bohr disputes is that the reality
criterion can be applied jointly to position and momentum. This dependence on
the context of measurement is called contextuality. One cannot speak of
certain predictions concerning both position and momentum, these are revealed
not as intrinsic attributes of the quantum object, but as relative to the
apparatus context. Therefore, he positions himself at the level of $ONR$. The
main argument which made him adopt such a position is the NC of observables
and the contextuality which is deduced from it, which he interpreted with the
principle of complementarity which he supported (As we shall see in section
7.6) because of his attachment to wave-particle duality, which he considered
as a fundamental duality (Mehra, 1987).

Concerning his positioning regarding the realism of the physical state
($SR$/$SNR$), a lot of confusion has been made about his claims, sometimes by
attributing him ideas that he never had, but also by authors who do not
distinguish between the $ONR$ thesis and the $SNR$ thesis, or between the
claims concerning the link between physical and symbolic states, i.e. in a
more general way between reality and our representation of reality. Indeed, it
is commonly attributed to the defenders of the Copenhagen interpretation, and
thus to Bohr, to deny the existence of an objective reality concerning the
quantum world, thus displaying a $\psi$-ESNR posture, which was a novelty in
history. This can be seen, for example, in this quote from Heisenberg:

“atoms or elementary particles themselves are not real; they form a world of
potentialities or possibilities rather than one of things or facts”.

Similarly, in (Petersen, 1963) A. Petersen refers to the following “shocking”
quotation of Bohr, which was very often attributed to him afterward:

’There is no quantum world. There is only an abstract quantum physical
description.’

This would put the defenders of Copenhagen, as well as Bohr, in the category
of $SNR$, denying the existence of an objective physical state. However, as
N.D. Mermin says in (Mermin, 2004), Bohr has never published such an
assertion, in any of his articles where he affirms his position on quantum
reality, nor in any of his other writings. Several authors (Folse in (Folse,
1986), and Faye in (Faye, 2012)) argue that Bohr had a realistic view of the
physical state (SR). They say that Bohr always implicitly supported the $SR$
thesis, as this quote from Bohr in (Bohr, 1929) proves:

[…] the extraordinary development in the methods of experimental physics has
made known to us a large number of phenomena which in a direct way inform us
of the motions of atoms and of their number. We are aware even of phenomena
which with certainty may be assumed to arise from the action of a single atom.
However, at the same time as every doubt regarding the reality of atoms has
been removed and as we have gained a detailed knowledge of the inner structure
of atoms, we have been reminded in an instructive manner of the natural
limitation of our forms of perception.

He never went back on these statements. Similarly, in (Folse, 1986) Folse
argues that the reality of atoms has been established by experiments that
directly cause phenomena that inform us of the existence and behavior of
atoms. This is the $KC$ criterion: the proof that there exist real quantum
objects is an experimental fact, the simple fact being that we measure them,
or that we deduce causal consequences from their properties, as Bohr
underlines, this is a proof of the objective existence of these quantum
objects. For Bohr the existence of atomic systems is “the given”, which theory
must describe in a way allowing prediction of those phenomena caused by these
entities. Bohr is then, in fact, clearly convinced about the objective reality
of the physical state. Many authors say that Bohr’s words are confused or even
contradictory concerning his ontic position, but his words on this subject are
always accurate. They are in terms of “objective communication” and not in
terms of “truth”. The confusion comes, in my opinion, from the readers, whose
innate unconscious reflex is to associate with what one observes, the
impression that it is pure reality, without assuming that it is only a piece
of information and that this observation is moreover an interaction. Other
authors like Hacking and Cartwright have supported this view about the reality
of quantum physical states, which they call Entity realism.

Let us come back to the $SNR$ posture, which is justified by its defenders by
saying that $SNR$ represents only information about potentialities and that
this is a complete description of reality i.e. that there is no hidden $ER$
below this. A very simple argument can provide a serious counter-indication to
the $SNR$ posture thus described.

######  Proposition 7.5.5.

The SNR view is not in accordance with the results of quantum mechanical
experiments.

###### Proof

The $SNR$ posture rejects the idea that there is an element of reality
corresponding to the quantum world and therefore carrying the information that
determines the measurement result. However, the repeatability of an identical
measurement made at short time intervals forces one to consider that
information is conserved. So the only way left by this position is, as they
do, to say that this information is on the side of the observer, only our
abstract description, and the information thus symbolized exists, not the
physical state. This is the position defended by Heisenberg in particular. Now
let’s take any quantum experiment, with a $\psi$ state, and two observers
$O_{1}$ and $O_{2}$, these two observers are sensibly the same, they are two
identical machines. We then choose to measure the state with $O_{1}$, we
collect the information by looking at the result of the measurement, then we
remove $O_{1}$, and we do the same experiment with $O_{2}$, immediately after,
and we collect the information again. If there is no $ER$, and $\psi$ only
represents the information we have, on a world of potentialities, that is to
say that there is no information stored in a $ER$, then knowing that $O_{2}$
does not have the information that $O_{1}$ has collected, it should have every
chance to find another result, since this information cannot be stored in a
$ER$. However, the experimental results show repeatability of the collected
information, which is in disagreement with the $SNR$ posture.

We will give more arguments against the $SNR$ view later.

Now, concerning his position on the link between the physical state and the
symbolic state, Bohr was clearly epistemic, as were most of the advocates of
the Copenhagen interpretation. The remarks of the preceding paragraph thus
place him in the $\psi$-ESR position. The epistemic interpretation of the
symbolic state is supported by various physicists, for several reasons:

  * •

Because of the multidimensionality (The fact that we are left with nothing but
a description of a state living in a multidimensional Hilbert space.) of the
$\psi$-state in Hilbert space, which seems not to offer a realistic vision of
the phenomenon as existing in space-time.

  * •

To solve the collapse problem, becoming an actualization of the observer’s
knowledge in the Copenhagen interpretation. For Bohr, the $\psi$ function is a
predictive tool and does not reflect any physical reality. It is therefore not
necessary to say that the subject would provoke any physical process during
the measurement. It merely defines a new form of objectivity, by including the
mention of the experimental context, while refraining from including the human
subject. This new objectivity is thus fully linked to the thesis of
contextuality. In this view (Copenhagen) the ’collapse’ of the wave function
is not a physical process, and it just reflects an update of our information
about the system.

  * •

Because of the EPR paper which emphasizes that the formalism of QM is in
contradiction with the thesis of local realism.

His belonging to the epistemic category is underlined by an important aspect
of his thought, concerning the formalism of which he had a purely symbolic
vision (Bohr, 1950):

The entire formalism is to be considered as a tool for deriving predictions,
of definite or statistical character, as regards information obtainable under
experimental conditions described in classical terms and specified by means of
parameters entering into the algebraic or differential equations of which the
matrices or the wave-functions, respectively, are solutions. These symbols
themselves, as is indicated already by the use of imaginary numbers, are not
susceptible to pictorial interpretation; and even derived real functions like
densities and currents are only to be regarded as expressing the probabilities
for the occurrence of individual events observable under well-defined
experimental conditions.

Therefore as Faye says in (Faye, 2012):

Bohr also considered the state vector to have no ontological status but merely
to be a heuristic device for the calculation of the probability of a specific
outcome of the measurement.

Bohr is definitely on the $\psi$-ESR side. For him and many others, the QM
formalism has no direct correspondence with the reality of real objects, it is
a symbolic representation that is not faithful to reality, allowing to set up
a kind of algorithm enabling to find the results of experiments. Most of the
physicists seem to have this posture, and do not consider $\psi$ as real, but
rather as a kind of information, which is supported by the fact that the
result provided is probabilistic. This point is of importance for the chapter
9, because depending on whether one attributes a concrete character to the
symbolic state or not, speaking of geometrical consequences deduced from the
QM formalism will have a meaning, or will not have any.

It seems, therefore, that the only positions which are relatively tenable are
the $ONR$, the $\psi$-$OSR$, and the $\psi$-$ESR$, the first being non-
disjoint from the other two. Although it is not intuitively obvious, the
$\psi$-$OSR$ and $\psi$-$ESR$’s postures are not only different at the
philosophical level. Indeed the quantum wavefunction $\psi$ is said to be
$\psi$-ontic if any physical/ontic state in the theory is consistent with only
one pure quantum state and $\psi$-epistemic if there exist physical/ontic
states that are consistent with more than one pure quantum state. This point
will be important to formally distinguish these two ontological issues and
will be used in the proof of the PBR theorem which I will present now. This
will be of major significance in order to decide between the postures, i.e.
between the $\psi$-OSR and $\psi$-ESR’s interpretations, and then in section
7.6 to interpret the NC of observable in QM. To finish, in chapter 9 it will
permit to interpret the associated geometry we can infer of such an
observable, according to the arguments that I will present in chapter 8 about
the objective path (trough measurement) we have to test physical theories
about space-time, in the mathematical framework $\mathfrak{M}$ of geometry.

A theorem of capital importance allows us to determine between the $\psi$-OSR
and $\psi$-ESR positions. Indeed, in 2012, a powerful no-go theorem against
$\psi$-epistemic interpretations $\psi$-ESR was found (Pusey et al., 2012).
This is called the PBR theorem (for Pusey, Barrett and Rudolph). This theorem
is considered as being of similar importance to the Bell, and
Bell–Kochen–Specker ones. As these last theorems, PBR theorem is a statement
about hidden variable theories. It constrains hidden variable theories in a
very strong way, almost killing the relevance of their potential existence.

What the PBR theorem shows is that assuming the existence of a physical state
($ER$), the idea that the associated symbolic state
$\pi_{\mathfrak{M}}(ER)=\psi$ does not represent reality in a one-to-one way
(then this physical state), but our (probabilistic) knowledge on it is no
longer tenable. Indeed, if there is such a physical state, the pure symbolic
state that we associate with it corresponds to a definite statement about this
reality, and not to a probabilistic one. $\psi$ is therefore no less “real”
than the analogous objects used in classical theories to designate states of
the system.

More precisely the PBR theorem states that if we assume that:

  1. 1.

A quantum system always does have some objective physical state.

  2. 2.

For any repeatable experimental procedure, there exists a well-defined
probability distribution over the set of possible final outcome states, and a
quantum system can be measured projectively on any basis.

  3. 3.

Statistical independence of experiments, and tendency to tend to the
probability distribution just mentioned when the experiment is repeated many
times.

  4. 4.

Even if two systems are brought together for a joint measurement on an
entangled basis, they may until then be treated as separate systems, with
their own space of states, and prepared so that the respective probability
distributions are independent.

  5. 5.

Quantum predictions are correct.

Then, for a QM description done in a given Hilbert space, taking two
preparation procedures that place the system into two different (symbolic)
pure states, they can’t represent the same physical state. If we have two
quantum symbolic states $\pi_{\mathfrak{M}}(ER_{1})$ and
$\pi_{\mathfrak{M}}(ER_{2})$ with a physical state as an unknown, then:

$\displaystyle\pi_{\mathfrak{M}}(ER_{1})\neq\pi_{\mathfrak{M}}(ER_{2})\qquad\to\qquad
ER_{1}\neq ER_{2}$ (7.5.1)

Two different symbolic pure quantum states are directly linked to different
physical states, whatever the reality in which these physical states take
sense may be. This is a really strong statement, it basically says that the
symbolic quantum states $\psi$ we use in QM are as real as our usual classical
symbolic states, since they refer both to physical states in a faithful and
complete manner. The symbolic quantum state is then ontic, and the only
survivor of this battle and the happy winner is $\psi$-$OSR$.

######  Remark 7.5.6.

In the $\psi$-OSR theories, the wave-particle duality cannot be a fundamental
duality (which is at the basis of Bohr’s philosophy). Indeed, if the aspects
of wave and corpuscular behavior can be associated with two different $\psi$
states (one spread out in the space of positions and the other not), knowing
that these states are in one-to-one correspondence with two physical states,
then these two physical states will be different too. There is not, as Bohr
represents it to himself, a quantum reality, whose properties can be
manifested as undulatory or corpuscular according to our mode of observation,
but two distinct quantum elements of reality. Thus, it is an effective
duality, not a fundamental one, and the principle of complementary that
accompanies it is no longer tenable.

This theorem is of crucial importance for the interpretation of the quantum
mechanical formalism, but it says nothing about $\psi$-ESNR type
interpretations since it is based on the assumption of the existence of a
physical state. I have presented above a strong argument against this posture
in the proposition 7.5.5. This point being important, I wish to give here
other more qualitative arguments against the SNR thesis, some of these
arguments are also arguments against the $\psi$-ESR posture:

  1. 1.

Against $\psi$-ESNR: If QM is the basis of the classical as usually
thought111In the sense that classical theories must be recovered as
approximations of quantum theory, in certain regimes, i.e. large number of
particles, large action, particle dense environment…, then the quantum state
must have the same status of realism as our classical states. I don’t see how
it can be otherwise, we are constituted of particles with quantum behavior!
And it happens that a collapsed particle can find coherences again (Bouchard
et al., 2015). So the fact that its status of realism is going to pass from
realism to nonrealism and then realism by regaining coherences seems to me
doubtful. Another argument comes from the KvN formalism, indeed the implicit
message is that the unification of the classical and the quantum can
eventually be done in a single formalism. So if quantum and classical states
were of different natures, as the $\psi$-ESNR interpretation suggests, then
this unification would not be thinkable. I find this relatively inconceivable.
In connection with the previous argument, the wave function after the collapse
is of the same nature as the one before the collapse in the formalism, the
only difference (in the case of a position measurement for example) is that it
is more localized. So there is no information about potentialities on one side
and a concrete actualized result on the other, it appears strange to say that
the ontic status of $\psi$ can have changed during the measurement process,
going from $\psi$-epistemic to $\psi$-ontic. I add to this the fact that what
we call real objects, are made of these same particles, localized, and
interacting, so how could it be that the ontic status of $\psi$ is different
from that of our machines, of all classical objects? I don’t see how. This
continuity of vision argument is for me a strong reason to believe that $\psi$
designates an entity no less real than any object whose reality is considered
without doubt.

  2. 2.

Against $\psi$-ESNR: The multidimensionality of the Hilbert space in which the
wave function lives is not an argument against realism, it is only that it is
no longer thinkable as a phenomenon in a usual space-time, as mentioned in
7.3.

  3. 3.

Against $\psi$-ESNR: In connection with the previous point, the EPR paper as
well as Bell’s inequalities formulated later cannot be taken as proof that
realism must be abandoned, it is only local realism that is faulted, there are
many other notions of realism, such as structural realism, or ontic structural
realism which remain possible.

  4. 4.

Against $\psi$-ESNR and $\psi$-ESR: As information does not correspond to
something real, then if $\psi$ only describes information, then it cannot act
on the environment. Thus, the second criterion of realism (KC) is an argument
for me that decoherence is proof that the wave function is objective because
it interacts with the particles in the environment wherever it has coherences.
If $\psi$ was only information, it would not interact with the environment.

I have a simple question to ask if we consider that a criterion of existence
for an entity is its ability to interact with its environment, that $\psi$
does not describe anything real, only information, and that information that
is not material cannot interact with the environment. How is it then that a
theory such as decoherence, verified experimentally, is based on the fact that
before the collapse, $\psi$ interacts with all the particles of the
environment contained in $D(\psi)$? Isn’t this a clue that $\psi$ is in itself
a real entity?

  5. 5.

Against $\psi$-ESR: The epistemic interpretation of $\psi$ was promoted to
solve the problem of the collapse postulate, taking it as an actualization of
the observer’s knowledge. This was, together with the wave-particle duality, a
strong reason that led them to think that $\psi$ refers only to information
are potentialities, and that we should abandon the assumption of realism. But
today, the collapse postulate is more commonly considered to be a problem of
the actual QM’s measurement theory.

  6. 6.

Against $\psi$-ESR: Wave-particle duality was taken to be a proof of the fact
that $\psi$ does not refer directly to a physical state, but to information we
have on it. But as mentioned in remark 7.5.6, which was a deduction of the PBR
theorem, wave-particle duality can no longer be taken as a fundamental
duality. Therefore, wave-particle cannot be taken to validate $\psi$-ESR
interpretations.

The PBR theorem, as well as these arguments, provide a counter indication to
the $\psi$-ESR and $\psi$-ESNR theories, thus putting ahead, as far as this
classification is complete, the $\psi$-OSR theories. The $\psi$-state thus
designates a physical state, the collapse must correspond to a non passive
physical process that remains to be discovered. Therefore, surprising
properties such as noncommutativity and entanglement must be taken seriously
as to what they say about our world, at its deepest level because they concern
the nature of the fundamental constituents of all things in our universe.

####  7.6 Knowing is Changing: NC as an Observer Effect?

> The Very Perception Is Action.
>
> J. Krishnamurti

As mentioned in section 7.3, one of the main discrepancies between QM and
classical mechanics comes from the emergence of observables that do not
commute. This fact is the mathematical origin of the Uncertainty Principle
(UP) which was introduced by Heisenberg in 1927. How we make intuitive sense
of this noncommutativity depends on the interpretation of the wave function we
use. There are two almost equivalent ways to consider the UP, via the wave
approach, or via the matrix approach. These are comparable representations,
but they highlight different aspects of reality.

In wave formalism, it is easy to see to which extent a spread according to the
momentum modes leads to a possible interferential pattern, which allows the
particle to be localized, but, if the wave packet consists of only one $p$
mode, then, just like a planar wave, it is spread all over space, and the
particle is delocalized. There is therefore a relationship between the spread
in position and the one in momentum. It is a wave property, $\psi$ cannot be
localized in both momentum and the position basis at the same time. The bases
of the Fourier modes in position and momentum representations are therefore
connected.

Let’s see to which extent the matrix formalism can complete this picture. In
QM, measurement is taken to collapse the state of the system to one of the
eigenstates. When the state is not an eigenstate of the observable, this
collapse is not a projection, but something like a transformation then a
projection. For an operator $a$, we will call $a_{c}$ the composition of $a$
with the collapse operator (plus normalization, which will not be taken into
consideration here, because it does not change the argumentation). As in
(1.1.2), if we take two (normal) operators $a$ and $b$ with discrete spectrum,
then we obtain their following spectral decomposition:

$\displaystyle a=\sum_{i\in
S_{a}}\lambda_{i}P^{a}_{\lambda_{i}}\qquad\qquad\text{and}\qquad\qquad
b=\sum_{j\in S_{b}}\lambda_{j}P^{b}_{\lambda_{j}}.$

This represents the operation that corresponds to the associated measurement
process in QM. Let us see what NC of $a$ and $b$ means in first time, and what
it implies when we add the collapse postulate i.e. what is the consequence for
$[a_{c},b_{c}]$.

If $\psi$ is in an eigenstate of $a$ then it is not affected by the
measurement of $a$ : $a\psi=\lambda\psi$ with $\lambda$ the eigenvalue, result
of the observation. Otherwise, $\psi$ undergoes a transformation:
$a\psi=\psi^{\prime}$.

The NC of the observables $a$ and $b$ on $\psi$ induces that $\psi$ cannot be
the eigenstate of both simultaneously. Thus, at least one of these two
observables is a non-passive measure of the symbolic state $\psi$.

To illustrate this, let us take $a=\hat{x}$ and $b=\hat{p}$, and assume that
$\psi$ is eigenvector of the operator position:
$\hat{x}|\psi\rangle=x_{0}|\psi\rangle$, and that it is not the case for the
momentum $\hat{p}|\psi\rangle=|\psi^{\prime}\rangle$, thus:

$\displaystyle[\hat{x},\hat{p}]|\psi\rangle=(\hat{x}\hat{p}-\hat{p}\hat{x})|\psi\rangle=(\hat{x}-x_{0}\hat{I})\hat{p}|\psi\rangle=i\hbar|\psi\rangle$

so that
$(\hat{x}-x_{0}\hat{I})\hat{p}|\psi\rangle=(\hat{x}-x_{0}\hat{I})|\psi^{\prime}\rangle\neq
0$ which explicitly means that $|\psi^{\prime}\rangle$ is not an eigenvector
of $\hat{x}$ with eigenvalue $x_{0}$ ($\hat{x}|\psi^{\prime}\rangle\neq
x_{0}|\psi^{\prime}\rangle$). Then it allows us to get a result different from
$x_{0}$ as if the position had changed. The conclusion is the same if we take
$\psi$ to be a linear combination of eigenvectors of $\hat{x}$, and that at
least one of these elements is not an eigenvector of $\hat{p}$.

If we add the collapse in this picture: $a\to a_{c}$ and $b\to b_{c}$ (to
obtain the true measurement process’s induced transformation), then the effect
on $\psi$ is to collapse it onto one of the eigenspaces associated to
$P^{a}_{\lambda_{i}}$ for $a$ or $P^{b}_{\lambda_{i}}$ for $b$. A basic result
of spectral theory is that if you have $[a,b]\neq 0$, then their exists at
least one couple $(i,j)\in(S_{a},S_{b})$ such that
$[P^{a}_{\lambda_{i}},P^{b}_{\lambda_{j}}]\neq 0$. This induces
$[a_{c},b_{c}]\neq 0$ and the previous conclusion works also here.

Then NC of observables (operators in general) is always connected to the non-
passive action of at least one of the two measurement processes on the
symbolic state $\psi$222Note that these conclusions only apply to the symbolic
state, then not directly to the physical one.. An example for the first case
(without collapse) can be the NC of gauge potential in non-abelian gauge field
theories (if we replace $(a,b)$ by $(A_{\mu},A_{\nu})$), and the second one
(with collapse) corresponds to QM’s measurement process. For this last case,
the non-passiveness is not only induced by the NC since the collapse
represents a change of the symbolic state too. We can say that NC adds a
contextual property to this collapse-induced change. We cannot directly
conclude that NC is linked to the non-passiveness of the measurement process
because the previous conclusions are at the level of the symbolic state
$\psi$. Therefore, according to our ontologic posture, we are left with two
ways of interpreting this NC:

  * •

The one coming from $\psi$-ESR interpretation schemes: There is change at the
level of the symbolic state $\psi$, but as it is not directly linked to the
physical state, we are not forced to said that the latter undergoes an
observer effect. It is even possible to suppose that there is no change at all
in this physical state, by postulating that this NC is induced by the fact
that the classical properties in question have no real meaning for the quantum
object, which can thus be, without ambiguity, both a wave and a particle,
whereas this has no meaning in the classical world. This avoids having to
consider discontinuous changes in the physical state in question. It’s a
statement about reality itself. Particles don’t have precisely defined
momentum and position to measure. This is a NC interpretation of type 2. There
is no Observer effect.

  * •

The less represented one coming $\psi$-OSR interpretation schemes: the
revelation of the classical properties has nothing mysterious, it is in
bijection with the real physical state. But here it is not a question of a
single physical state, since this one is in bijection with the symbolic state…
there are thus two physical states, one transiting to the other one during the
measurement, and thus non passivity of the act of measurement! This is caused
by the fact that collapse must be considered to be a physical process which
transforms the physical state in this kind of interpretation, non-
commutativity rendering this process contextual. It’s a statement about what
we can know about reality. The act of measuring position changes the momentum,
and vice-versa. This is an interpretation for type 1 NC. There is an Observer
effect.

Before going further, let’s try to understand how this NC has been
historically interpreted, mainly by Heisenberg, then by Bohr.

Heisenberg’s first explanation of his UP was based on the idea that any
measurement process must change the physical state, and that this leads to
significant perturbation when considering the measurement of quantum states.
Indeed, Heisenberg originally illustrated the intrinsic impossibility of
violating the UP by using the observer effect of an imaginary microscope as an
elementary measuring device. The argument goes as follows. Let us take the
example of the electron that we want to localize in space, the minimal
measurement scheme to localize it implies shining at least one photon on it.
If the electron was a classical object, like a bird in the sky, the photon
will affect its state in a so tiny way that we neglect this fact in all days
life, which leads to the feeling of seeing “what is”. However, this is not
true for quantum objects. Most of the time, the energy of a single photon is
ridiculously small (in comparison to the one of the electron), and affects it
in a negligible way, but it’s not true for very energetic photons.

Let’s explain this measurement scheme more concretely, the microscope
experiment consists in observing how a light ray is deviated by the Compton
effect when it scatters with an electron, the measurement of the deviation
angle permit to determine the localization of the electron with a certain
accuracy. But this ray of light has a certain wavelength $\lambda$, below
which it is not possible to measure the position of the particle, and this
wavelength is linked with its energy by the relation $E=hc/\lambda$. Thus,
improving the accuracy of the measurement of such a position requires to
increase the energy of the photon used. But such a photon is also associated
with momentum $p=hc/\lambda$, and a portion of that momentum will be
transmitted to the particle during the interaction process at play, by
conservation of this quantum number. Therefore, the more you measure the
position with accuracy, the more you change the momentum. The fact that the
momentum of the particle is changed makes the notion of phase space
meaningless, this being at the origin of the introduction of the configuration
space mentioned in section 7.2.

This is a very important argument, it is obviously a purely experimental fact
of QM and it as not to be deduced by any outer logical argument, I will come
back on this later in section 9.3. More details on the microscope experiment
can be founded in Heisenberg book (Heisenberg, 1949), and (Tipler and
Llewellyn, 2012), and (Greenstein and Zajonc, 2006). This phenomenon was
understood by him as a manifestation of the UP. This argument of the non-
passivity of measurement seems quite reasonable, especially since, as said
before, it has been demonstrated in (Busch, 2009), that a measurement that
acquires information on a given system, necessarily disturbs the measured
state. But these argument work at the level of the symbolic state, therefore
we cannot conclude directly like Heisenberg does that it is connected with an
objective observer effect.

Indeed, it was argued by Bohr, that the Heisenberg UP has nothing to do with
an observer effect. According to Bohr, he did not make clear the distinction
between a position measurement merely disturbing the momentum value of the
particle and the more radical idea that momentum is meaningless or undefinable
in a context where the position was measured instead. As mentioned in (Mehra,
1987):

Bohr felt that Heisenberg had not treated the thought-experiments with the
X-ray microscope and the investigations on the Compton effect and resonance
fluorescence light quite properly; his fundamental objection was something
quite different, As Heisenberg recalled: “The main point was that Bohr wanted
to take this dualism between waves and corpuscles as the central point of the
problem.”

It is important to stress the fact that Bohr has no fundamental criticism to
do to Heisenberg’s paper, but its attachment to wave-particle duality and its
epistemic interpretation of $\psi$, which are linked in the large picture in
which he tries to make sense of QM lead him to see things differently than
Heisenberg for this experiment and it’s linked to non-commuting observables.
For Bohr, the wave-corpuscle duality was a fundamental duality, which he made
a consequence of the Complementary Principle (CP), which he proposed as a
replacement for Heisenberg’s UP.

In his view of CP, Bohr does not refer to discontinuous changes of the
corresponding quantities during the measurement process, on the contrary, he
emphasizes the possibility of defining these quantities. He also rejects the
common view (that Heisenberg adopts for example) about the fact that
measurement creates definite results:

The unaccustomed features of the situation with which we are confronted in
quantum theory necessitate the greatest caution as regard all questions of
terminology. Speaking, as it is often done of disturbing a phenomenon by
observation, or even of creating physical attributes to objects by measuring
processes is liable to be confusing, since all such sentences imply a
departure from conventions of basic language which even though it can be
practical for the sake of brevity, can never be unambiguous. (Bohr 1939: 24)

Bohr always stressed that uncertainty relations are first and foremost an
expression of complementarity (Mehra, 1987). This may seem odd since
complementarity is a dichotomic relation between two types of description
whereas the uncertainty relations allow for intermediate situations between
two extreme potentialities (wave and corpuscular behaviors), and therefore
causal chain, contrary to the CP which defends the idea that causal view is a
classical restriction that doesn’t make sense in QM:

A causal description of the process cannot be attained; we have to content
ourselves with complementary descriptions. “The viewpoint of complementarity
may be regarded”, according to Bohr, “as a rational generalization of the very
ideal of causality”

For Bohr, there is no time lapse during which a physical object behaving as a
wave would become a particle or the opposite. He means that these two visions
of things are only valid at the classical level and that the particle does not
carry these properties in itself. Rather, the symbolic $\psi$ state being only
there to give us a sort of classical information useful to understand this
inexpressible reality in a raw way as possessing in an unambiguous way these
(classical) properties called complementary. This is why, in this same line of
thought, he defended that causal thinking was also limited to our classical
understanding of phenomena and that it reached its limit in a very obvious way
in the experiments involving these complementary observables.

In a more general way, Bohr defended the idea that the usual language,
impregnated with the classical vision, was not able to make quantum phenomena
understandable, and that this was probably a fundamental limitation that could
not be overcome for our intuitive understanding of QM within our classical
language. Indeed, in (Bohr, 1950), in order to defend the idea that the NC, in
connection with the principle of complementarity, is not a matter of processes
in space and time, and that therefore the classical causal analysis is no
longer valid, Bohr does not hesitate to say that the ambiguity of such an
interpretation lies in the fact that the classical language, which is
intrinsically causal, cannot describe the reality that lies behind such a NC:

These so-called indeterminacy relations explicitly bear out the limitation of
causal analysis, but it is important to recognize that no unambiguous
interpretation of such relations can be given in words suited to describe a
situation in which physical attributes are objectified in a classical way.

To summarise Bohr’s view on the interpretation of the NC of observables, the
complementary between such phenomena implies that there is only one object
which is making different phenomenal appearances in different experimental
arrangements, which lead to different interactions scheme between the observer
apparatus, and the physical state. There is no causal chain, this physical
state is not changed by the measurement process, it is only the measurement of
certain properties simultaneously that cannot make sense, and different
experimental arrangements will provide complementary evidence about this
unique object. This phenomenon cannot be intuitively represented because the
classical words, which he considers unsurpassable, do not allow us to
intuitively capture its nature.

For me, this is not a sufficient explanation, he did not explicitly show that
it was not a matter of processes in space and time, whereas it was the result
of Heisenberg’s initial demonstration, of which he did not refute the
mathematical content. If he had sought to make this demonstration, he would
have realized that the conservation of quantum numbers during the process of
measurement induces that the perception of an information that testifies to
the presence or properties of an object, during the process of measurement,
necessarily induces the perturbation of the corresponding properties that the
measured object would carry, this argument having been more fully discussed in
section 7.4. On the other hand, he relies on the idea that classical language
is unsurpassable and provides a limitation to this understanding, which seems
questionable from the arguments presented in section 6.3. Furthermore, his
interpretation of NC, which he accompanies with his principle of
complementarity, is not satisfactory, as the use of the concept that
conventional language and associated classical notions cannot deal with the
quantum is not a convincing argument. It is like a black box that allows us to
say that we cannot hope to find a satisfactory explanation for quantum
phenomena, even by creating new words. This makes it acceptable to abandon the
idea of obtaining an intuitive understanding of reality. The concern to obtain
an intuitive and conceptual coherence is no longer put forward. In my opinion,
this idea has strongly polluted the scientific thoughts developed later.
Moreover, as JM. Lévy-Leblond points out (Lévy-Leblond, 2000), the principle
of complementarity was not considered a relevant and convincing idea by
researchers such as Einstein and Schrödinger, and even among his close
collaborators such as Heisenberg and Pauli. Yet it was the basis of Bohr’s
philosophy, and of the commonly supported Copenhagen interpretation of which
he was the principal architect.

As we have seen in section 7.5, there is a much stronger argument to defeat
his view. Indeed, Bohr’s statements and interpretations are based on his
epistemic interpretation of the symbolic state, without which, if he were to
admit an ontic view of formalism, he could not say that there is no
transformation of the state as mentioned above, and worse than that as
mentioned in remark 7.5.6, the wave-corpuscle duality can then no longer be a
fundamental duality, which is at the heart of the interpretation he defends.
These two aspects are then connected to two distinct physical states, and
there are therefore intermediate stages of the physical state to consider.

But, as we see in section 7.5, only $\psi$-OSR interpretations can be retained
to be in accordance with experimental results. So if the argumentation
presented in section 7.5 is correct, the arguments of Bohr cannot be well
founded. Then, NC of QM is of the first kind, linked to process in space and
time, caused by the non-passivity of the act of measurement, then of the same
kind as “classical” ones. However, they differ in that we are unable, as
classical observers, to see beyond these NCs, to see the intermediate stages
of the process. In QM, processes such as collapse, decoherence, the
measurement process, quantum jumps, etc. were initially considered to be
instantaneous and discontinuous. These phenomena of discontinuous jumps from
one symbolic state to another were called quantum jumps, and it was the very
nature of this discontinuity in evolution that made Bohr (among others) prefer
to opt for an epistemic interpretation, as he could not consider that the
physical state could undergo such a discontinuous evolution, which seems
reasonable. Schrödinger, who thought that the wave function is a faithful
representation of reality, directly opposed this idea of quantum jump
(Schrödinger, 1952) by also rejecting the principle of complementarity.
Nevertheless, at the time, nothing could have foreshadowed a possible non-
continuity of the evolution of the symbolic state, and therefore Bohr’s
interpretation could make sense in the light of the knowledge of the time,
even if he could have concluded that the QM formalism could possibly be either
incomplete or that certain developments were missing to draw all the
consequences. Today, it is experimentally demonstrated that decoherence,
quantum jumps (Minev et al., 2019) and therefore the measurement process is
not instantaneous or discontinuous, but continuous and over very short periods
of time. There is as yet no theory explaining collapse in a continuous manner,
but history seems to show that the apparent discontinuities in the evolution
of $\psi$ are gradually disappearing from the scientific landscape, suggesting
the possibility of a strictly continuous evolution for the state (physical and
symbolic).

To summarise the information provided by wave and matrix’s representations of
reality, the wave aspect tells us that the distribution of the state according
to the impulse and position bases are related. In particular for their spread.
And the matrix interpretation clearly shows us how the state will undergo an
evolution for its distribution according to one of these two bases, during the
act of measuring one of the two observables. Thus, we can now see how any non-
commutativity is linked to the non-passivity of the act of measuring one of
the two observables over the symbolic state $\psi$, and then on the physical
one. This can also be seen as a simple consequence of what was said in section
7.4 about the fact that any measurement that obtains nontrivial data is
disruptive for the state.

######  Remark 7.6.1.

According to remark 7.5.2, the negation of observable realism induces proof of
the fact that there is no state change during the process. Bohr skipped this
step.

######  Remark 7.6.2.

According to PBR theorem if there is an ER, it will be directly associated
with a symbolic state in a one-to-one correspondence, then supposing that the
NC of observable is of type 2 lead to the fact that both the symbolic and
physical state has not been changed since it’s not linked to process in space-
time, but as we just see, for $\psi$-OSR interpretations, NC is linked to
state change. Therefore, as mentioned in section 6.2, NC of type 2 and state
realism SR are not compatible. But NC will always be linked to observable
nonrealism (ONR).

We thus see how the NCs of QM and gauge theory formalism arise from the same
fact, the non-passivity of the interaction that is the measurement in QM, or
the non-passivity of the interaction with a boson vector in gauge theory.
Therefore, by the several arguments I have presented so far, I would adopt the
ontic point of view and its implications on the equivalence between
noncommutativity and nonpassivity of the act of measurement in what follows.

###  Chapter 8 Geometry and the Physical Theories of Space-Time

> Geometry is not true, it is advantageous.
>
> H. Poincaré

History shows us that the extension of the geometrical framework used in
physical theory has several times allowed us to better understand natural
processes. The greatest advances in this direction were, in my opinion, the
passage from Euclidean geometry to Riemannian geometry. Each of these steps
was motivated by physical arguments. The first geometry thought to describe
space-time was a geometry of absolute space-time, as a background theater for
events. Then with Galileo, space lost its absolute character by becoming that
of Galilean reference frames. Then came Einstein who made geometry dependent
on the observer, the notions of distances and durations relative to the
latter, and the notion of simultaneity also, thus breaking the idea of an
absolute time. Later, geometry became actively determined by the fields of
matter.

I believe that the time of questioning the geometrical notions and their link
with the concept of space-time is not over. Therefore, we can ask the
following question: is there any reason to think that we need a new extended
geometric picture to understand physical processes and space-time? Several
arguments can be given for an affirmative answer. The purpose of the following
considerations is to explore how the algebraic framework in which NCG was
developed, and the concepts surrounding it may be relevant when we try to go
even further in understanding what space-time is. In what follows, we will
question the epistemological status of geometric descriptions of space-time,
starting by asking how it is measured, and then considering how geometric
concepts describing space-time, for example, have been progressively
algebraized throughout history, this process culminating with the Gelfand-
Naimark theorem and the creation of the NCG’s framework.

####  8.1 Geometry and Space-Time

> Time and space are modes by which we think and not conditions in which we
> live.
>
> A. Einstein, 1944

Geometry is a representation of the mind where the processes of reality are
made intelligible, it aims to represent space-time. But what exactly is
geometry, and what is the relation between its objects $\mathfrak{M}_{st}$ and
those of physics?

$\displaystyle\text{geometric object
}\mathfrak{M}_{st}\overset{?}{\longleftrightarrow}\text{Spacetime structure
}ER_{st}:\mathfrak{M}_{st}\sim\pi_{\mathfrak{M}_{st}}(ER_{st}).$

Is $\mathfrak{M}_{st}$ connected to a space-time element of reality $ER_{st}$?
There is no proof of this, on the contrary, the supposition of the existence
of an $ER_{st}$ leads to problems as we will see in section 8.2.

This question is more complicated than it seems at first sight. For example, a
geometric object can be a distance, a duration, a point, or a coordinate. It
must correspond to some structure of space-time. This means that we have to
find an operational experimental way to deduce what corresponds to the given
geometric object. This is the only objective access we can have to this
structure and giving meaning to these concepts outside of this scope is
probably misleading. For example, if we want to make sense of the concept of
location for a material object, then we have to find an experimental
operational way by which what we refer by this concept can be measured. If
there is no such path, this concept becomes meaningless.

######  Remark 8.1.1.

Note that these considerations on operational ways of measurement of space-
time concepts often concern the use of photons and material objects. We will
try to do the same in section 9.3.

An important observation can then be made, the spatiotemporal (physical)
structures $ER_{st}$ are not measurable. As Poincaré said:

“Experiments only show us the relationships of bodies to each other; none of
them deals, or can deal, with the relationships of bodies to space, or with
the mutual relationships of the various parts of space.”

We cannot directly observe space and its properties. We can only observe
material bodies or light (and other bosons) phenomena. It is only through the
observation of these material bodies and light phenomena that we can obtain
information about the properties of space. Then these corresponding $ER$ are
those of material objects. Therefore, the conclusions we draw about the
properties of space-time concern the use of concepts and experiments about its
material bodies and light experiment.

As I said, these experiments do not give us direct access to these notions,
but to a set of observables $\\{O_{i}\\}$ with relations $\\{R_{ij}\\}$
between them, that allow us to deduce these notions. We must therefore
conclude that observables (of material objects) are (on the operational level)
primary to the notion of space-time.

Spacetime is not an observable.

It is important to stress that these deductions do not represent space-time in
itself, but space-time as experienced by the processes of nature. This
distinction becomes important when these processes start to affect the
structure of space-time, or, more exactly, of the entities whose measurement
makes it possible to deduce it. The experienced space-time becomes different
from the “pre-existing” one. Thus, what we observe is space-time as it is
experienced by material objects. The “measurement of space-time structure” can
then be seen like this:

$\displaystyle\pi_{\mathfrak{M}}(ER)\overset{Measurement}{\longrightarrow}O(ER)\overset{Deduction}{\longrightarrow}\text{idealized
\textquotedblleft space-time structure\textquotedblright
}\pi_{\mathfrak{M}_{st}}(ER).$

Notice that we use $ER$ a material element of reality instead of $ER_{st}$ a
space-time element of reality since the latter are not measurable, only
deducible from measurement on material objects, and then potentially are man-
made creation. A space-time concept can need more than one element of reality
to be deduced.

In physics, we have to let the phenomena speak by themselves and take care not
to allow our conscious representations to replace them, and eventually mislead
us. What about the representation of space-time? We implicitly believe that
space-time exists, but we never measure it directly. It is therefore a man-
made concept whose fundamental nature can be questioned. We can thus ask
ourselves what led us to the sensation/feeling of space-time. Ernst Mach, in
(Mach, 1890) thinks that space is constructed from the association, the
ordering of our sensations (i.e collection of measurements made by the body).
In the same philosophy an intuition about this was given by Poincaré:

The notion of space cannot be an integral part of any of our sensations taken
in isolation. It is only when we observe the order in which these sensations
follow one another that this notion can be born.

In other words, the collection data of observation change (of material
objects) leads to the intuitive feeling of space-time. This set can be written
$\mathbb{O}=\\{\delta O_{a_{i}}(ER^{j})\\}^{i\in I_{O}}_{j\in I_{ER}}$, with
$I_{O}$ the index set of all observer/measurement contexts and $I_{ER}$ the
one of physical states/elements of reality. It’s now important to distinguish
between two kinds of observation change:

  * •

The ones which lead to the elaboration of our geometric intuition of space-
time, we will call the set of all these observable changes $\mathbb{O}_{st}$.

  * •

The ones which do not correspond to this usual geometric description, we will
call the set of all these observable changes $\mathbb{O}_{nst}$ for non-space-
time likable.

With the total set $\mathbb{O}=\mathbb{O}_{st}+\mathbb{O}_{nst}$. I will come
back on this in section 8.3.

The question is then to know how this collection of observables and observable
change leads to the intuitive, then geometric representation we have of our
world, based on concepts like distance, time, point, and coordinates. A first
observation, which will be useful later is that (in many cases, maybe all) the
collection of these observables together with the composition law form an
algebra, and that, as mentioned in chapter 3, the observable variations are
encoded in something called differential structure, which as we have seen in
chapter 3 enabled to complete the geometry to algebra unification. Therefore
the intuition of Poincaré seems to be proved on the mathematical side since it
is implemented in the differential structure, which was the key ingredient to
recover something like geometry in the NCG framework.

An element of response to explain how an impression of spatial representation,
and then of geometry, can emerge from such a collection of observations is
again given by Poincaré, (where he meets Lie and Helmholtz for the basic
intuition). Indeed, for Poincaré, the intuition of transformation groups is
primary to that of geometry. For him, the study of geometry is only the study
of a group (of transformations). To say that Euclidean geometry is the most
suitable means that among the possible groups, the one constituted by the
transformations of Euclidean geometry corresponds the most to our experience
and in particular to the motion of natural solid bodies. As he said:

Like Lie, I believe that the more or less unconscious notion of the continuous
group is the only logical basis of our geometry. Like Helmholtz, I believe
that the observation of the movements of solid bodies is its psychological
origin.

Taking the group of transformations as the fundamental origin of geometric
intuition, he adventured to imagine other worlds, where a different
experience, with the observation of different groups could lead to the
elaboration of other geometries, such as non-Euclidean geometry.

Other researchers agree with him on this intuition. According to Klein,
geometry is a space, or a set of points, with a structure, and the bijective
applications of the space onto itself that preserve the structure form a group
which can be called the group of automorphisms. For Riemann, an automorphism
of a Riemannian space is a distance-preserving application of the space to
itself. We will show in chapter 11 how the automorphisms of the algebra play a
fundamental role in NCG, in order to find on the one hand the usual coordinate
changes of general relativity, and on the other hand the local gauge
transformations.

We thus see how the notion of observable and symmetry group on it is seen as
the fundamental “reason” behind the intuition of space and its geometrization.
As mentioned in section 3.1, these groups of symmetries on the observables
form the automorphism group of the algebra of observables, which is connected
(for infinitesimal automorphisms) to the notion of derivation, and hence of
differential calculus. The argument that groups of symmetries on observables
are more fundamental than the notion of geometry, and that we should deduce
our notion of space accordingly, will be a crucial point in the argument of
the sections 8.3 and 8.4, where it will be a question of defending the
relevance of geometrizing Gauge’s theories, and of discarding the notions of
geometry and space-time, in order to work on these concepts on the side of the
algebra of observables.

However, these statements assume that we are talking about measured space-
time, not space-time in itself. It is not clear then whether our theories
speak of space-time in itself, or of the one we measure. If the way we measure
it does not affect its structure, it is not essential, for all practical
purposes, to differentiate these cases. But if this is not the case, we must
know it, because the two cases will give a different description of what
space-time is. I have found no indication in the literature on special and
general relativities regarding this point of interpretation. But if we go back
to the method used by Einstein and Poincaré to discover special relativity
(Hladik, 2004), we notice that the concepts of duration and distance are
studied according to the operational mode of measurement by the photons of
light, whose speed is independent of any reference frame. This led them to a
deeper understanding of the concepts of duration, distance, and simultaneity.
This is for me a strong indication that relativistic theories are descriptions
of measured space, another indication being that, if this theory is a physical
theory, then it must predict what we observe, then the measured space-time. In
this line of thought, we will test the notion of point in section 9.3, using a
similar measurement scheme, with photon and material bodies.

More details and the origin of the citations used in this section can be found
in (Michel, 2004), (Nabonnand, 2010), and (Bächtold, 2014).

####  8.2 The Point on the Point: is Space-Time Substantial?

> Formerly, people thought that if matter disappeared from the universe, space
> and time would remain. Relativity declares that space and time would
> disappear with matter.
>
> A. Einstein

Whether space-time corresponds to something having tangible
existence/substance (manifold substantialism view of general relativity) or
not is a very difficult question. The theory of ether was supposed to describe
“what fills” space-time as a kind of fluid in which material objects undergo
displacement. But according to Einstein in (Einstein, 2007), the relativities
show us that if there is such an ether, it must not possess any mechanical
properties like velocity for example. This seems to be far away from what we
can think of as something being substantial (i.e. corresponding to an element
of reality).

Let’s suppose that there are such elements of physical reality corresponding
to elementary space-time structures $ER_{st}$. Let’s suppose that this
elementary structure corresponds to what we call space-time points, we will
then call it $ER_{x}$. Let’s try to test whether such an element of reality
can truly be consistent with observations.

As mentioned in section 7.5, the origin of the concept of element of reality
comes from the fact that the RC (repeatability criterion) is respected, i.e.
one or more observers will make predictions in agreement with each others when
they do repeated measurements on the same object at not-so-distant time
intervals. This therefore equally applies to space-time structures. Take the
notion of point and the corresponding element of reality $ER_{x}$. If there is
such a physical notion, independent of any material object, then, a series of
idealized measurements 111Because as said before, space-time is not measurable
since only observation of material objects is possible, and that there are no
material objects to measure in this idealized measurement scheme of this
notion of point by different observers should all coincide with the following
result i.e. that $ER_{x}$ describes a substantive point in space-time with a
probability of 1. But we know that the notion of a point depends on the chosen
reference frame, so different observers will see either a point or
trajectories in space-time. Indeed, any test of the notion of a spatial point
without the notion of a material object of reference is doomed to failure,
because of the physical equivalence of all Galilean reference frames. The
question of the difficulty of conceiving physically an empty space, without
material objects of reference had already been raised by Descartes, a similar
argument using a box containing space as the elementary tool can be found in
(Einstein, 1990).

Similarly as mentioned in section 8.1, spatial structures are not measurable,
only the material objects from which we deduce spatial concepts are. Thus,
from the point of view of the physicist’s thinking, concepts of space-time are
derived from those of observation of material objects and thus are not
necessarily supposed to represent a real physical entity in their own right,
as distinct from what matter is. There are many other arguments against the
idea that space-time is an autonomous physical entity.

A strong argument against giving physical meaning to points ($x$) in general
relativity was given by Einstein with the hole argument. The general
covariance requirement on the Einstein field equations implies that if we
consider a hole in space-time i.e. a region where the stress-energy tensor $T$
vanishes, a change in the coordinate system $x\,\to\,x^{\prime}$ taken to be
only inside the hole will induce different metrics $\tilde{g}$, for only one
$T$. But the new metric is also a solution in the first coordinate system $x$.
Therefore, according to the “points” $x$ of $M$, there can be as many distinct
trajectories as they are different metrics, but for only one matter field
configuration given by $T$. This is problematic if we consider these points as
being true objective (and then distinct) locations. Thus, either we restrict
ourselves in the set of possible coordinate changes and lose the general
covariance, or we lose the deterministic property of the field equations
concerning the trajectories which are its solution. These two solutions are
not acceptable. This can also be seen as a consequence of diffeomorphism
invariance, see (Stachel, 2014; Giovanelli, 2021) for technical details.

Some time later, Einstein finally realized that this argument is based on an
implicit assumption we have on space-time, and therefore on its symbolic
representation as being constituted of points $x$ on which physical properties
like the value of a gravitational field (or equivalently of the $T$, then of
any field associated to $x$) are defined in a meaningful way. By relaxing this
requirement, he makes general covariance and determinism compatible. But how
do we provide meaning to “points” $x$ if they are not therefore intuitively
related to “locations” of physical quantities? How do these coordinates relate
to objective locations?

The answer was given by Einstein as “the point coincidence argument”. This
means that true objective locations are not given by coordinates $x$, but by
crossings of world lines which are solutions of the Einstein fields equations.
Indeed, the only observable structures of space-time are point coincidence
events, interaction of a part of the observer apparatus, and the corresponding
structure. As he said in (Einstein, 1923):

Our space-time verifications invariably amount to a determination of space-
time coincidences. If, for example, events consisted merely in the motion of
material points, then ultimately nothing would be observable but the meetings
of two or more of these points. Moreover, the results of our measurements are
nothing but verifications of such meetings of the material points of our
measuring instruments with other material points, coincidences between the
hands of a clock and points on the clock-dial, and observed point-events
happening at the same place at the same time. The introduction of a system of
reference serves no other purpose than to facilitate the description of the
totality of such coincidences.

He considers this argument as one of its deepest insights. This was a strong
counter indication to the manifold substantialism view since its points are
not referring by their intrinsic structure to locations. Then, the coordinate
systems have no intrinsic meaning. The physical significance of the notion of
space-time point is then dependent on the concept of matter as moving inside
them. The notion of point must therefore be operationally based on such
coincidence, this will be used in section 9.3.

Therefore, is it still possible (according to these arguments) to think of
empty space, with locations, as mentioned above? Going back to the hole
argument, if we say that space-time does exist where there is no matter, we
realize that the only way to specify an adequate notion of point is to invoke
the geodesic crossings, and thus from a physical perspective, to reintroduce
at least one particle into the hole which must interact with our test
particle. Then, if we implement operationally the point coincidence argument,
this means that to implement an objective notion of point, the hole is no
longer a hole since it must contain a particle with which coincidence permits
to set up this notion of point.

So we can see that the idea of thinking of space-time physics as based on a
notion of a manifold $M$ with points leads to paradoxes and strange postulates
about what these points are. One way to get rid of this problem could be to
find a formalism where $M$ does not play a primary role. The formalism of
general relativity consists roughly in supposing the existence of a smooth
manifold $M$ endowed with a metric, on which we consider the set of tensorial
operations that we can do on it, and thus construct equations whose structural
properties encode the relevant structures of $M$. An interesting observation
is that $M$ only comes into play in formalism as a tool to introduce the ring
of smooth functions on $M$ (Geroch, 1972). Everything that follows in the
developments is purely algebraic. It is this remark that led to the
development of Einstein’s algebras, which are exclusively based on this ring.
By taking this object as first, without any reference to a manifold, one can
reformulate general relativity, without being troubled by $M$ and the
paradoxes that come with it when it is taken as input.

As mentioned by N. Huggett in (Huggett, 2018), Earman suggested that taking
the algebra as more fundamental than $M$ and its points would avoid the hole
problem about the meaningfulness of coordinates by virtue of locations. This
was seen as a replacement of manifold substantivalism by algebraic
substantivalism. In this view, points are seen as deductions of matter fields,
and of the elements of the corresponding algebra. The idea that algebra should
be taken as more fundamental than space-time and these points was defended by
J. Earman and in Geroch’s work in an attempt to bring arguments around the
substantivalist-antisubstantivalist debate revived with general relativity and
its conceptual problems.

I cannot conclude from these arguments that only the antisubstantivalist
position makes sense, but insofar as this would be true, and that physical
elements of reality are needed to explain the origin of space-time, then only
the elements of reality offered by material objects can be used to derive such
a space-time structure and it’s representation. As we see, this assertion is
supported by the point coincidence argument. In chapter 9, I will present how
some essential structures at the heart of space-time’s concept such as the
notion of point and continuity between two points can be offered by pure
matter concepts. In section 9.3, I will use these arguments to show the limits
at which such an operationally implemented notion of point failed, opening the
door to NCG… A nice review about the Relationalism versus Substantivalism
debate can be found in (Field, 1984).

####  8.3 Back to what is Observable, Consequence for Gauge Theories

In section 8.1, arguments have been put for emphasizing the primary status of
observables over our usual representation of space-time as an entity in
itself. Let us then try to reproduce the path of creation of the intuition of
space proposed by Poincaré, without presuming anything. A first observation is
that we will acquire, by a set of measurements, a whole state of observables,
and data on their evolution. Our mind then begins to make representations in
which this shapeless mass of observables, as well as their evolutions, can
make sense of these data, according to the symmetries it perceives in the
structure and evolution of these observables. Let’s start with the notion of
point. Usually, points are taken as input in all physical theories, i.e. we
suppose point $x\in M$ as pre-existing entities on which events can occur. But
we have to admit that these notions of point are necessarily reconstructions
from the observables we collect, and concecuntly, these observables have to be
considered as more fundamental than these points. As mentioned in section 1.2,
the Gelfand Naimark theorem provides such a link between observable and points
of an underlying “hypothetical manifold”. This theorem has a deep meaning, it
implements a part of Poincaré’s idea presented in section 8.1, i.e. space-time
geometrical notions have to be deduced from the set of observations we can
collect. Indeed, if we take again the equality 1.2.1, $\hat{f}\in\mathcal{A}$
being such an observable, $x$ a character on $\mathcal{A}$, and $f$ the
inverse Gelfand Transform of $\hat{f}$, then we have:

$\displaystyle\hat{f}(x)=x(f)$

This shows that we can see these two views as equivalent:

  * •

space-time as fundamental $\to\,\hat{f}(x)$: Take the points $x\in M$ as
fundamental, and then interpret measurement results as evaluations of
observable on these points.

  * •

Algebra of observables as fundamental $\to\,x(f)$: Take observables
$\hat{f}\in\mathcal{A}$ as fundamental, and points as special functions
evaluating these observables.

We saw in section 8.2 that the notion of point being a source of problems, the
approach starting from observables (Einstein Algebra) could be considered as
more fundamental. Moreover, as defended in section 8.1, the structures of
space-time are not measurable, only the observables on the matter are. And
finally, as mentioned in sections 7.2 and 7.3, we saw that the first approach
which consists in the evaluation of observables pointwisely is not defined
(because of the NC), whereas the algebra of observables is. This is why I
think the second approach is more fundamental. Then, as mentioned in
(Coquereaux, 1998), the measurement process can be taken as a replacement for
the first given consideration of the geometric background space.

Let us thus resume our initial apprehension of what space-time is, starting
from the observables and their variations as mentioned in section 8.1. As said
above, a given proportion ($\mathbb{O}_{st}$) of these observables (and their
evolutions) leads to our usual representation of space-time, while the other
($\mathbb{O}_{nst}$) can rather be perceived as moving in non-spatialized
dimensions, in purely formal structures, remaining attached to points of
space-time. These non-spatialized evolutions can be for example the evolution
of color at a given point, of a fermionic field according to its different
charge components, of a spin, or simply of any properties of matter without
spatial extension, and manifesting a temporal evolution. In the first case, we
speak of reconstructing notions of trajectories in space-time and in the
second case of evolution in a non-spatial sense. But what determines these
trajectories and evolutions?

The most advanced theoretical view is to consider that trajectories are the
results of the path for which the phase interferences of the wave function of
a particle are the more constructive. It is then the evolution of the phase of
the particle that becomes the primary theoretical object of these notions of
trajectory and evolution. The phase is the evolution parameter taken as an
argument of the function whose variation is considered.

In geometry, the evolution of this phase along paths is encoded by the notion
of parallel transport, implemented by connections. We therefore observe
trajectories, which are derived from the evolution of the phase, which is
parameterized by the connections that encode the parallel transport. More
generally, given the collection of phase observations, we are free to
interpret the infinitesimal irregular changes of the phase in two ways:

  * •