# Noncommutativity and Physics: A non-technical review

Ali H. Chamseddine1,4 Alain Connes2,3 and Walter D. van Suijlekom5
1Physics Department  American University of Beirut  Lebanon
2College de France  3 rue Ulm  F75005  Paris  France
3I.H.E.S. F-91440 Bures-sur-Yvette  France
4Ludwig Maximilian University  Theresienstrasse 37  80333  Munich  Germany
5Institute for Mathematics  Astrophysics and Particle Physics  Radboud
University Nijmegen  Heyendaalseweg 135  6525 AJ Nijmegen  The Netherlands.

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###### Abstract

We give an overview of the applications of noncommutative geometry to physics.
Our focus is entirely on the conceptual ideas, rather than on the underlying
technicalities. Starting historically from the Heisenberg relations, we will
explain how in general noncommutativity yields a canonical time evolution,
while at the same time allowing for the coexistence of discrete and continuous
variables. The spectral approach to geometry is then explained to encompass
two natural ingredients: the line element and the algebra. The relation
between these two is dictated by so-called higher Heisenberg relations, from
which both spin geometry and non-abelian gauge theory emerges. Our exposition
indicates some of the applications in physics, including Pati–Salam
unification beyond the Standard Model, the criticality of dimension 4, second
quantization and entropy.

###### Contents

  1. 1 Introduction
  2. 2 Noncommutativity generates time
  3. 3 Discrete and continuous variables
  4. 4 Spectral paradigm of geometry, the line element
  5. 5 Spectral paradigm of geometry, the algebra
  6. 6 Dimension $4$
  7. 7 The second quantized level

##  1 Introduction

Our contribution to this volume on "noncommutativity and physics" will
describe the key role of the transition from commutative to noncommutative
algebra starting from Heisenberg’s discovery of matrix mechanics. The
conceptual reason for the power of this transition is that the encoding by
noncommutative algebra retains more information than its commutative
counterpart. We are in fact very well acquainted with this fact when we write
words. Writing respects the order of the letters and this allows one to encode
information in a very effective manner. Passing to the commutative ignores the
order of letters and equates words whose letter content is the same as happens
in anagrams. The nuance between the noncommutative and the commutative is the
same as in the game of scrabble where the same set of letters might correspond
to quite different words. In a suggestive manner one can view quantization as
the act of lifting from the commutative (semiclassical) shadow to the
noncommutative real world. In Section 2 we explain in which sense
noncommutative spaces are dynamical, and possess a canonical time evolution.
This fact is at the root of noncommutative geometry, leaving the static case
to ordinary geometry. In Section 3 we explain how noncommutativity of real
variables is the key for the coexistence of discrete and continuous variables.
They do coexist in the quantized calculus which gives a perfect stage for
infinitesimal variables and allows for the coexistence of the discrete with
the continuum precisely by the noncommutativity of the actors (the operators)
on this stage. The irruption of noncommutativity in physics described above
had a strong impact on mathematics which led to a reconstruction of geometry
inside the Hilbert space formalism of quantum mechanics. Encoding geometry in
this formalism has two key ingredients, the line element $ds$ whose
incarnation as the Dirac propagator is described in Section 4, and the algebra
of coordinates where once again noncommutativity enters and as explained in
Section 5 allows one to encode even ordinary spaces at a much lesser price
than with commutative algebra. We have understood the minimal amount of
noncommutativity needed to obtain spin 4-manifolds, and this lead to our Pati-
Salam extension of the standard Model. The special role of dimension 4 is
described in Section 6. The non-abelian gauge theory are witnesses of the
slight amount of noncommutativity required in the encoding of 4-manifolds. The
action principle that leads to the Einstein-Hilbert gravitational action
coupled with the above extension of the standard Model is the spectral action
principle. It acquires the meaning of an entropy when working at the second
quantized level, and we close this short paper in Section 7 by explaining the
potential role of second quantization for the spectral paradigm of geometry.

### Acknowledgements

The work of A. H. C is supported in part by the National Science Foundation
Grant No. Phys-1912998 and the Arnold Sommerfeld Center at LMU.

##  2 Noncommutativity generates time

The irruption of noncommutativity in physics can be traced back to this night
of June 1925 when, around three in the morning, W. Heisenberg while working
alone in the Island of Helgoland in the north sea, discovered matrix mechanics
and the noncommutativity of the phase space of microscopic mechanical systems.
Until that day all the physics manipulations of observable quantities were
done within commutative algebra and the discovery that fundamental observables
such as position and momentum could fail to obey the elementary commutative
law is a fundamental turning point which is also at the origin of the
mathematical theory of noncommutative geometry. Indeed at the mathematical
level it showed the relevance in geometry of spaces whose coordinates do not
commute. After the formulation of Heisenberg’s discovery as matrix mechanics
by Born and Jordan in their 1925 paper, von Neumann reformulated quantum
mechanics using Hilbert space operators and went much further with Murray in
identifying “subsystems" of a quantum system as “factorizations" of the
underlying Hilbert space ${\mathcal{H}}$. They discovered unexpected
factorizations which did not correspond to tensor product decompositions
${\mathcal{H}}={\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$ and developed the
theory of factors which they classified into three types. Type I corresponds
to factors $M$ associated to ordinary tensor product decompositions
${\mathcal{H}}={\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$ and such factors are
formed of those operators of the form $T\otimes 1$. Factors of type II are
those which posses a trace and factors of type III are those which are neither
of type I or II. The Tomita-Takesaki theory [39] extended to factors the
correspondence which exists in the type I case or in the type II case using
the trace, between the Boltzmann-Gibbs state $\phi$ and the Heisenberg
evolution $\sigma_{t}$ of observables, both expressed in terms of the
Hamiltonian $H$

$\phi(A)={\rm Tr}(A\exp(-\beta H))/{\rm Tr}(\exp(-\beta
H))\to\sigma_{t}^{\phi}(A)=\exp(itH)a\exp(-itH)$

The starting point of the classification of factors and the reduction of type
III to type II was the discovery in [20] that the evolution
$\sigma_{t}^{\phi}\in{\rm Aut}({\mathcal{M}})$ is in fact independent of the
choice of the (faithful normal) state $\phi$ on the factor ${\mathcal{M}}$
provided one divides the group ${\rm Aut}({\mathcal{M}})$ of automorphisms of
the factor ${\mathcal{M}}$ by those automorphisms $\alpha$ which exist as a
trivial consequence of noncommutativity, i.e. those of the form, for some
unitary $U\in{\mathcal{M}}$,

$\alpha(A)=UAU^{*},\forall A\in{\mathcal{M}}$

Such automorphisms are called "inner" and they form a normal subgroup ${\rm
Int}({\mathcal{M}})\subset{\rm Aut}({\mathcal{M}})$ of the group of
automorphisms of ${\mathcal{M}}$. This showed that factors ${\mathcal{M}}$
admit a canonical time evolution [20, 22] i.e. a canonical homomorphism

${\mathbb{R}}\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}{\rm
Out}({\mathcal{M}})={\rm Aut}({\mathcal{M}})/{\rm Int}({\mathcal{M}})$

by showing that the class of the modular automorphism $\sigma_{t}^{\varphi}$
in ${\rm Out}({\mathcal{M}})$ does not depend on the choice of the faithful
normal state $\varphi$. The above uniqueness of the class of the modular
automorphism [20] drastically changed the status of the two invariants which
had previously been introduced in [18], [19] by making them computable. The
kernel of $\delta$, $T(M)={\rm Ker}\delta$ forms a subgroup of ${\mathbb{R}}$,
the periods of $M$, and many non-trivial non-closed subgroups appear in this
way. The fundamental invariant of factors is the modular spectrum $S(M)$ of
[18]. Its intersection $S(M)\cap{\mathbb{R}}_{+}^{*}$ is a closed subgroup of
${\mathbb{R}}_{+}^{*}$, [27], and this gave the subdivision of type III into
type IIIλ $\iff$ $S(M)\cap{\mathbb{R}}_{+}^{*}=\lambda^{Z}$. Subsequently, the
classification of factors of type IIIλ, $\lambda\in[0,1)$ was shown to be
reduced to that of type II and automorphisms

$M=N\rtimes_{\theta}{\mathbb{Z}},\ \ N\ \textbf{type}\ {\rm II}_{\infty},\
\theta\in{\rm Aut}(N)$

In the case IIIλ, $\lambda\in(0,1)$, $N$ is a factor and the automorphism
$\theta\in{\rm Aut}(N)$ is of module $\lambda$ i.e. it scales the trace by the
factor $\lambda$. In the III0 case, $N$ has a non-trivial center and, using
the restriction of $\theta$ to the center, this gave a very rich invariant, a
flow, which was used in 1972 [21] to show the existence of hyperfinite non
ITPFI factor. Only the case III1 remained open in [22] and was solved later by
Takesaki [38] using crossed product by ${\mathbb{R}}_{+}^{*}$.

##  3 Discrete and continuous variables

One of the really new totally unexpected features of quantum mechanics is the
fact that outcomes of microscopic experiments can not be repeated. Even if you
consider just a single slit experiment and you shoot electrons or photons
through a slim slit of size comparable to the wavelength of the particles, the
exact location where the particle will land on a target on the other side of
the slit is something that cannot be reproduced.

Figure 1: Quantum random number generator

What one can predict is the probability of the particle arriving somewhere but
arriving at some fixed spot is something which, from the principles of quantum
mechanics, cannot be reproduced. This means that there is some fundamental
randomness which is inherent to quantum physics and has the potential of
providing true random numbers. It thus has become the preferred option for the
scientific applications requiring randomness (Figure 1).

On the mathematical side of things, there is a related question that in fact
goes back to Newton, which is simply _what is a real variable?_. If you ask a
mathematician for his view on this question the most likely answer you will
get is that _a real variable is given by a set $X$ and a map from this set $X$
to the real numbers_. However, if you think more deeply about it, you will
find out that this answer is very unsatisfactory. Indeed, it means that one
can not have coexistence of continuous variables, namely variables which can
take a continuous range of possible values, and discrete variables, namely
variables which can only affect a countable set of possible values, say, with
finite multiplicity for each of them. The reason for this is that if you have
a discrete variable then the original set $X$ that you are dealing with will
have to be countable, which then implies that it does not allow for continuous
variables.

The amazing answer that mathematics provides, but which would not have been
detected if it were not for the formalism of quantum mechanics by von Neumann,
is that a real variable is just a self-adjoint operator in a Hilbert space. As
a matter of fact, there is only one (separable) Hilbert space which is the
Hilbert space with countable basis and one sees that this Hilbert space has
variables with discrete spectrum. Indeed, take a description of Hilbert space
by giving a countable orthogonal basis and take an operator which is diagonal
in that basis. This operator has countable spectrum but it coexists with
operators which have continuous spectrum. In fact one could have described
that same Hilbert space as being the space of square integrable functions on
an interval and of course one will have continuous variables there, given by
multiplication operators. The beauty of this formalism is that continuous and
discrete variables do coexist. All the properties of real variables are there
because if one has a self-adjoint operator it has a spectrum which is composed
of the possible values of the variable and it has spectral multiplicity which
is the number of times a value can be affected.

Real variable $f:X\to{\mathbb{R}}$ | Self-adjoint operator $H$ in Hilbert space
---|---
Range $f(X)\subset{\mathbb{R}}$ of the variable | Spectrum of the operator $H$
Composition $\phi\circ f$, $\phi$ measurable | Measurable functions $\phi(H)$ of self-adjoint operators
Bounded complex variable $Z$ | Bounded operator $A$ in Hilbert space
Infinitesimal variable $dx$ | Compact operator $T$
Infinitesimal of order $\alpha>0$ | Characteristic values $\mu_{n}(T)=O(n^{-\alpha})$ for $n\to\infty$
Algebraic operations on functions | Algebra of operators in Hilbert space
Integral of function $\int f(x)dx$ |  $\displaystyle{\int\\!\\!\\!\\!\\!\\!-}T=$ coefficient of $\log(\Lambda)$ in ${\rm Tr}_{\Lambda}(T)$
Line element $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ |  $ds=\bullet\\!\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\bullet$ : Fermion propagator $D^{-1}$
$d(a,b)=\,{\rm Inf}\,\int_{\gamma}\,\sqrt{g_{\mu\,\nu}\,dx^{\mu}\,dx^{\nu}}$ | $d(\mu,\nu)=\,{\rm Sup}\,|\mu(A)-\nu(A)|,\mid\ \|[D,A]\|\leq 1.$
Riemannian geometry $(X,ds^{2})$ | Spectral geometry $({\mathcal{A}},{\mathcal{H}},D)$
Curvature invariants | Asymptotic expansion of spectral action
Gauge theory | Inner fluctuations of the metric
Table 1: The ‘spectral’ dictionary

All this fits very well with reality in the sense that the quantum variables
are operators in Hilbert space; the new key fact is that the discrete
variables do not commute with the continuous variables. If they would commute,
they would both be functions defined on the same space $X$, which is not
possible.

Summarizing, continuous and discrete variables coexist as operators in Hilbert
space, but as such they necessarily do not commute. It is precisely this lack
of commutativity which is the new ingredient at the core of the quantized
calculus and which renders the framework of non-commutative geometry
effective.

##  4 Spectral paradigm of geometry, the line element

What has happened in the process which led to understand the emergence of
geometry from the quantum is quite instructive. When Heisenberg found his
commutation relations involving $P$ and $Q$ (momentum and position) there was
already quite a hint, a bit of truth, in it, in the sense that when you take
the spectrum of either $P$ or $Q$ you find a real line. The other operator is
a differentiation operator and it gives a geometrical structure for that line.
However, the way things evolved historically from these commutation relations,
is that they were interpreted not as a first hint of "geometry from Hilbert
space operators" but rather in terms of Lie group representations. Of course,
group representations, when applied to e.g. the Poincaré group give a
beautiful conceptual notion of particle and the theory unveiled big pieces of
beautiful landscape. However, finite-dimensional Lie groups can hardly lead us
to the arbitrary geometries that we observe in gravity and to the variables
that we have in gravity. Simple Lie groups are like isolated diamonds but
which simply do not allow for this enormous variability needed in the
geometrical description of gravity.

The development of noncommutative geometry has shown that it is possible to
have geometry emerging from purely Hilbert space considerations but in order
to do that one applies representation theory to far more elaborate forms of
Heisenberg’s commutation relations involving $P$ and $Q$. In order to obtain
these relations one needed to take a step back and understand how to give more
flexibility to both $P$ and $Q$. The additional flexibility for $Q$ was quite
hard to come by and is discussed in Section 5 below. The additional
flexibility for $P$ was not so hard to find. In fact, it was found already by
Paul Dirac in [31] when he realised how to assemble several momenta together
to form a single expression, a single operator, that actually contains in
itself all the components of the momenta. This is the Dirac operator,
expressed in terms of gamma matrices. Thus the understanding of how to give
more flexibility to $P$ stems from Dirac’s work and, even more, this
understanding is thoroughly grounded in physics and in the understanding of
geometry, in particular in the measurement of lengths, as we will now explain.

Many formalisms of geometry start with the Riemannian paradigm as a
prerequisite, that is, the idea that geometry is given by the measurement of
lengths and this measurement of lengths is actually governed locally by simply
prescribing the square of the line element
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$. It turns out that this idea was even
questioned by Riemann himself! He wrote in his inaugural lecture on the
foundation of geometry that it’s questionable whether the texture of space (or
spacetime) will obey this “Riemannian paradigm" at any scale, the reason being
that the notions of light ray and of solid body on which his intuition was
grounded would cease to make sense for very tiny scales.

At the end of the 18th century the desire to unify the measurement of
distances led to a concrete realization of a unit of length which was called
“mètre étalon" and was conserved near Paris in the form of a platinum bar.
Later, the relevance of this choice was put into question in the early 1920’s
because people found out that the mètre étalon was actually changing length
which of course was very problematic. They found this out because they
measured the mètre étalon by comparing it with the wavelength of a fixed
atomic transition of Krypton. The outcome of this observation, many years
later, was that physicists shifted the definition of the unit of length from a
platinum metal bar to the wavelength of a certain transition of Krypton and
later, eventually, to an hyperfine transition of Caesium.

If you think more deeply about this, you find that the reason why their
classical unit of length needed to be localized was because first it should be
quite small, since it is supposed to represent $ds$, and because it commuted
with the coordinates it had to be localized somewhere (as it happened this was
near Paris). However, the new unit of length which is given by a wavelength of
the hyperfine transition of Caesium111It is called a unit of time using the
speed of light as the conversion factor, is in fact of spectral nature and is
no longer commuting with the coordinates. As a matter of fact, in line with
the above discussion, it involves the Dirac operator, or, rather its inverse,
the Dirac propagator. Obviously, because it does not commute with the
coordinates, it does not need to be localised. Moreover, if one wants to unify
the metric system in our galaxy it is clear that such a spectral definition
should be used as a unit of length. Indeed, it is much more practical to tell
people from nearby stars that our unit of length is a certain transition in
the helium or hydrogen spectrum rather than to tell them that they need to
come to Paris and compare their unit with the metal bar which is located
there.

Besides the physical standpoint that this is a much better definition of the
unit of length, mathematically speaking it implies that one actually replaces
the $ds^{2}$ of Riemannian geometry by a very subtle square root of it, where
the square root is taken through the Clifford algebra and where the
infinitesimal line element which was formulated in terms of infinitesimal
variables by Riemann is replaced by an infinitesimal which is a suitable
operator in Hilbert space (see Table 1). Note that this notion of
infinitesimal was in fact predicted by Newton in the sense that he said
explicitly that an infinitesimal should not be a number but instead it should
be a variable! In fact, Newton gave the definition of an infinitesimal
variable and, when you translate it in terms of the understanding of variables
using Hilbert space operators, it gives precisely what we call a compact
operator. These compact operators have exactly all the properties you would
dream of for infinitesimals, they form an ideal among operators etc. etc…

Now, when you think about the line element for spacetime, you find out that
the mathematical formulation consisting of the inverse of the Dirac operator,
in the language of physics, is encoded by what is usually referred to as the
the fermion propagator. This fermion propagator enters Feynman graphs as the
internal legs involving fermions. When you look at quantum field theory
textbooks à la Feynman, you find that this propagator is a small tiny line
joining two very nearby events:

This qualitative appearance of an infinitesimal line element becomes even more
striking when one realizes that in quantum field theory, this line element
actually acquires quantum corrections by being dressed. This means that if you
take from the beginning the correct version of the line element for geometry,
you soon understand that there are quantum corrections to the measurement of
lengths, to the geometry, which are given by the dressing of the fermion
propagator. Moreover, gauge fields appear as additional terms in the Dirac
operator and this embodies exactly the intuition of Riemann that in case his
paradigm would fail at small distances the geometry should be based on the
forces that hold things together!

##  5 Spectral paradigm of geometry, the algebra

The new spectral paradigm of geometry is encoded by Spectral triples
$({\mathcal{A}},{\mathcal{H}},D)$ where the unbounded operator $D$ encodes the
analogue of the Dirac operator as described in Section 4 and where
${\mathcal{A}}$ is an algebra of operators in the Hilbert space
${\mathcal{H}}$. The algebra ${\mathcal{A}}$ is not assumed to be commutative
and its inner automorphisms (as explained in Section 2) will correspond to the
internal symmetries in physics. Because of its flexibility, this new paradigm
provides the needed tool to refine our understanding of the structure of
physical space in the small and to “seek the foundation of its metric
relations outside it, in binding forces which act upon it". The main idea,
described in details in [28], is that the line element now embodies not only
the force of gravity but all the known forces, electroweak and strong, appear
from the spectral action and the inner fluctuations of the metric. This
provides a completely new perspective on the geometric interpretation of the
detailed structure of the Standard model and of the Brout-Englert-Higgs
mechanism. One gets the following simple mental picture for the appearance of
the scalar field: imagine that the space under consideration is two sided like
a sheet $S$ of paper in two dimensions. Then when differentiating a function
on such a space one may restrict the function to either side $S_{\pm}$ of the
sheet and thus obtain two spin one fields. But one may also take the finite
difference $f(s_{+})-f(s_{-})$ of the function at the related points of the
two sides. The corresponding field is clearly insensitive to local rotations
and is a scalar spin zero field. This, in a nutshell, is how the Brout-
Englert-Higgs field appears geometrically once one accepts that there is a
“fine structure" which is revealed by the detailed structure of the standard
model of matter and forces. Eventually, this allows one to uncover the
geometric meaning of the Lagrangian of gravity coupled to the standard model.
This extremely complicated Lagrangian is obtained from the spectral action
developed in [1] which is the only natural additive spectral invariant of a
noncommutative geometry.

In order to comply with Riemann’s requirement that the inverse line element
$D$ embodies the forces of nature, it is evidently important that we do not
separate artificially the gravitational part from the gauge part, and that $D$
encapsulates both forces in a unified manner. In the traditional
geometrization of physics, the gravitational part specifies the metric while
the gauge part corresponds to a connection on a principal bundle. In the
noncommutative geometric framework, $D$ describes both forces in a unified
manner and the gauge bosons appear as inner fluctuations of the metric but
form an inseparable part of the latter. The key point is that non-abelian
gauge fields arise inevitably as a result of the noncommutativity of our
geometric framework.

The noncommutative geometry dictated by physics is then given by the product
of the ordinary $4$-dimensional continuum by a finite noncommutative geometry
$({\mathcal{A}}_{F},{\mathcal{H}}_{F},D_{F})$ which appears naturally from the
classification of finite geometries of $KO$-dimension equal to $6$ modulo $8$
(cf.[5, 3]). The finite dimensional algebra ${\mathcal{A}}_{F}$ which appeared
is of the form

${\mathcal{A}}_{F}=C_{+}\oplus C_{-};\qquad C_{+}=M_{2}(\mathbb{H}),\ \ \
C_{-}=M_{4}(\mathbb{C}).$

The agreement of the mathematical formalism of spectral geometry and all its
subtleties such as the periodicity of period $8$ of the $KO$-theory, is
promising but might still be accidental. Instead, what is most convincing is
the pertinence of this approach when recovering [3] the see-saw mechanism
(which was dictated by the pure math calculation of the model), while being
unaware at the time of its key physics role to provide very small non-zero
masses to the neutrinos, and of how it is “put by hand" in the standard model.
The low Higgs mass then came in 2012 as a possible flaw of the model, but in
[10] we showed the compatibility of the model with the measured value of the
Higgs mass, due to the role in the renormalization of the scalar field which
was already present in [7] but had been ignored thinking that it would not
affect the running of the self-coupling of the Higgs.

In all the previous developments we had followed the “bottom-up” approach,
i.e. we uncovered the details of the finite noncommutative geometry
$({\mathcal{A}}_{F},{\mathcal{H}}_{F},D_{F})$ from the experimental
information contained in the standard model coupled to gravitation. In 2014,
in collaboration with S. Mukhanov [15, 14] we were investigating the purely
geometric problem of encoding $4$-manifolds in the most economical manner in
the spectral formalism. The problem had no a priori link with the standard
model of particle and forces and the idea was to treat the coordinates in the
same way as the momenta are assembled together in a single operator using the
gamma matrices. The great surprise was that this investigation gave the
conceptual explanation of the finite noncommutative geometry from Clifford
algebras! This is described in details in [28] to which we refer. What we
obtained is a higher form of the Heisenberg commutation relations between $P$
and $Q$, whose irreducible Hilbert space representations correspond to
$4$-dimensional spin geometries. The role of $P$ is played by the Dirac
operator as explained before, and the role of $Q$ by the Feynman slash of
coordinates using Clifford algebras. The proof that all spin geometries are
obtained relies on deep results of immersion theory and ramified coverings of
the sphere. The volume of the $4$-dimensional geometry is automatically
quantized by the index theorem and the spectral model, taking into account the
inner automorphisms due to the noncommutative nature of the Clifford algebras,
gives Einstein gravity coupled with the slight extension of the Standard
Model, as a Pati–Salam model. This model was shown in [11, 13] to yield
unification of coupling constants. We refer to the survey [17] for a concise
account of the whole story of the evolution of this theory from the early days
to now.

Standard Model | Spectral Model
---|---
Higgs Boson | Inner metric(0,1)
Gauge bosons | Inner metric(1,0)
Fermion masses $u,\nu$ | Dirac(0,1) in $\uparrow$
CKM matrix, Masses down | Dirac(0,1) in $(\downarrow 3)$
Lepton mixing, Masses leptons $e$ | Dirac(0,1) in $(\downarrow 1)$
Majorana mass matrix | Dirac(0,1) on $E_{R}\oplus J_{F}E_{R}$
Gauge couplings | Fixed at unification
Higgs scattering parameter | Fixed at unification
Tadpole constant | $-\mu_{0}^{2}\,|{\bf H}|^{2}$
Table 2: The dictionary between the physics terminology and the fine structure
of the geometry

##  6 Dimension $4$

The above approach to the geometry of space-time was not directly motivated by
quantum gravity but it addresses a more basic preliminary question which is to
understand the reason why gravity together with the standard model appear as
the fundamental forces of nature. We have seen in Section 5 that as long as
one allows the encoding of the algebra of coordinates using a minimal amount
of noncommutativity, gravity coupled to the standard model appears naturally
with the non-abelian gauge theories as witnesses of the noncommutative
encoding. We now discuss briefly how dimension $4$ is singled out in this
approach. As explained above the extension of Riemannian geometry beyond its
classical domain, in the framework of Hilbert space operators, provides the
needed flexibility in order to answer the query of Riemann in his inaugural
lecture. From the mathematical standpoint it is the very notion of manifold
which is at centerstage, and when one wants to capture the key properties
fulfilled by manifolds at a global (non-local) level, one finds that the main
one is Poincaré duality. But it is not Poincaré duality in ordinary homology
but rather in a more refined theory called $KO$-homology. A remarkable
interaction with the quantum formalism then appears from the confluence of the
abstract understanding of the notion of manifold from its fundamental class in
$KO$-homology with the realization of cycles in $KO$-homology from
representations in Hilbert space. The final touch on the understanding of the
geometric reason behind gravity coupled to the standard model, comes from the
simultaneous quantization of the fundamental class in $KO$-homology and its
dual in $KO$-theory which gives rise to the higher Heisenberg relation. We now
explain in more details why dimension $4$ plays a special role in this context
[15, 14]. In order to encode a manifold $M$ of dimension $d$ using the above
duality between $KO$-homology and $KO$-theory one needs to construct a pair of
maps $\phi,\psi$ from $M$ to the sphere (of the same dimension $d$) in such a
way that the sum of the pullbacks of the volume form of the (round) sphere
vanishes nowhere on $M$. This problem is easy to solve in dimension $2$ and
$3$ because one first writes $M$ as a ramified cover $\phi:M\to S^{d}$ of the
sphere and one pre-composes $\phi$ with a diffeomorphism $f$ of $M$ such that
$\Sigma\cap f(\Sigma)=\emptyset$ where $\Sigma$ is the subset where $\phi$ is
ramified. This subset is of codimension $2$ in $M$ and there is no difficulty
to find $f$ because $(d-2)+(d-2)<d$ for $d<4$. It is worth mentioning that the
$2$ for the codimension of $\Sigma$ is easy to understand from complex
analysis: For an arbitrary smooth map $\phi:M\to S^{d}$ the Jacobian will
vanish on a codimension $1$ subset, but in one dimensional complex analysis
the Jacobian is a sum of squares and its vanishing means the vanishing of the
derivative which gives two conditions rather than one. Thus dimension $d=4$ is
the critical dimension for the above existence problem of the pair
$\phi,\psi$. Such a pair does not always exist222It does not exist for
$M=P_{2}({\mathbb{C}})$. but as shown in [15, 14], it always exist for spin
manifolds which is the relevant case for the functional integral performed in
Euclidean signature.

The higher form of the Heisenberg commutation relation mentioned above
involves in dimension $d$ the power $d$ of the commutator $[D,Z]$ of the Dirac
operator with the operator $Z$ which is constructed (using the real structure
$J$, see [28]) from the coordinates. We shall now explain briefly how this
fits perfectly with the framework of D. Sullivan on Sobolev manifolds, i.e. of
manifolds of dimension $d$ where the pseudo-group underlying the atlas
preserves continuous functions with one derivative in $L^{d}$. He discovered
the intriguing special role of dimension $4$ in this respect. He showed in
[37] that topological manifolds in dimensions $>5$ admit bi-Lipschitz
coordinates and these are unique up to small perturbations, moreover existence
and uniqueness also holds for Sobolev structures: one derivative in $L^{d}$. A
stronger result was known classically for dimensions 1,2 and 3. There the
topology controls the smooth structure up to small deformation. In dimension
4, he proved with S. Donaldson in [32] that for manifolds with coordinate
atlases related by the pseudo-group preserving continuous functions with one
derivative in $L^{4}$ it is possible to develop the $SU(2)$ gauge theory and
the famous Donaldson invariants. Thus in dimension $4$ the Sobolev manifolds
behave like the smooth ones as opposed to Freedman’s abundant topological
manifolds.333for which any modulus of continuity whatsoever is not known. The
obvious question then is to which extent the higher Heisenberg equation of
[15, 14] singles out the Sobolev manifolds as the relevant ones for the
functional integral involving the spectral action.

##  7 The second quantized level

This final section is more speculative and unlike the previous ones it
addresses a fundamental question which is essentially open. This question can
be formulated as follows

> What is the relevance of the many particle formalism of Quantum Field Theory
> to the geometry of space-time?

In fact we have already seen a hint of an answer in Section 4 when we pointed
out that since the line element encoded by the Dirac propagator gets dressed
(as a formal power series in powers of $\hbar$) from the quantum field
corrections, this suggested that the geometry itself, being encapsulated by
the Dirac propagator gets dressed. But the above formalism of geometry
remained at the level of first quantization and the issue of second
quantization cannot be ignored since the quantum corrections to the line
element as explained in Section 4 above are only the tip of the iceberg.
Indeed, the dressing occurs for all the $n$-point functions for fermions, and
not only the two point function.
Another significant hint for the role of the second quantization is the result
of [16] on the second quantization of fermions and the spectral action as an
entropy, The point there is that the action principle which recovers gravity
coupled with the Standard Model is given by the most general additive
functional of spectral geometries and these depend on an a priori arbitrary
even function $f$ of a real variable which defines the action as the trace of
$f(D)$. What we showed in [16], as a first step towards building from quantum
field theory a second quantized version of spectral geometry, is that if one
applies the free functorial quantization to Fermions as Clifford algebras and
uses the Dirac operator to define a time evolution on this Clifford algebra
one discovers that the von Neumann entropy of the Boltzmann-Gibbs equilibrium
states (see Section 2) is the spectral action for a function $f$ deeply
related to the Riemann zeta function. This led us to the first lines of a
dictionary from first to second-quantized as given in Table 3. Note also that
an analogous result to [16] has been obtained in [33] for the bosonic case.

First-quantized | Second-quantized
---|---
Algebra | Action of inner fluctuations on time evolution
Hilbert space | Clifford algebra
Dirac operator | Time evolution for KMSβ-state on Clifford algebra
Spectral action | Entropy of KMSβ-state
Table 3: The dictionary for fermionic second-quantization of spectral triples

From the purely mathematical standpoint this need to pass to a second
quantized higher level of geometry is in a way similar to what has happened in
the development of $K$-theory. The topological $K$-theory as developed by
Atiyah-Hirzebruch based on Bott periodicity leads to the key duality between
$KO$-homology and $KO$-theory is the origin of the higher Heisenberg relation.
As already mentioned in [26], algebraic $K$-theory, which is a vast refinement
of topological $K$-theory, is begging for the development of a dual theory and
one should expect profound relations between this dual theory and the theory
of interacting quanta of geometry. As a concrete point of departure, note that
the deepest results on the topology of diffeomorphism groups of manifolds are
given by the Waldhausen algebraic $K$-theory of spaces and we refer to [34]
for a unifying picture of algebraic $K$-theory.
Finally, the conceptual understanding of renormalization has shown (see [30])
that there is a highly noncommutative symmetry group, the Cosmic Galois group,
which embodies the ambiguities in the renormalization process inherent to the
physics computations of interacting Quantum Field Theory. How this symmetry
group combines with the above impact of physics on geometry, and how
renormalization fits with the spectral action principle are deep open
questions of which only the first few steps have recently been taken [35, 36].
These questions are witnesses of the amazing role of noncommutativity in
physics.

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