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\begin{document}
\begin{titlepage}
\begin{center}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\ten Centre de Physique Th\'eorique\footnote{
Unit\'e Propre de Recherche 7061} - CNRS - Luminy, Case 907}
{\ten F-13288 Marseille Cedex 9 - France }
\vspace{1.5 cm}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
{\twelve LECTURES\footnote{
To appear in the proceedings of the ``First Canibbeau Spring School of
Mathematics and Theoretical Physics'' - Cambridge University Press,
Eds. R.~Coquereaux, M.~Dubois-Violette.
} ON ALAIN CONNES' \\
NON~COMMUTATIVE GEOMETRY \\
AND APPLICATIONS TO FUNDAMENTAL INTERACTIONS}
\vspace{0.3 cm}
{\bf Daniel KASTLER}
\vspace{3,5 cm}
{\bf Abstract}
\end{center}
We introduce the reader to Alain Connes non commutative differential
geometry, and sketch the applications made to date to (the
lagrangian level of) fundamental physical interactions.
\vspace{4 cm}
\noindent Key-Words : non-commutative geometry, fundamental
interactions, standard model.
\bigskip
\noindent Number of figures : 1
\bigskip
\noindent July 1994
\noindent CPT-94/P.3055
\bigskip
\noindent anonymous ftp or gopher: cpt.univ-mrs.fr
\end{titlepage}
\setcounter{chapter}{1}
\centerline{Daniel Kastler}
\begin{center}
{\bf Lectures on Alain Connes' non commutative geometry \\
and applications to fundamental interactions} ~~\\
\end{center}
\bigskip
\centerline{\bf Contents}
\begin{itemize}
\item[A.] The Dirac operator as the carrier of classical differential
geometry
\item[B.] Non-commutative differential geometry via $K$-cycles
\item[C.] The two-point algebra: Higgs bosons in a nutshell
\item[D.] Equivalence of the quantum and the classical Yang-Mills
algorithms
\item[E.] The standard model of elementary particles
\begin{itemize}
\item[\S 1] Strategy
\item[\S 2] The electroweak inner $K$-cycle
\item[\S 3] The electroweak sector
\item[\S 4] Appending the chromodynamics sector
\item[\S 5] Non-commutative Poincar\'e duality. Biconnections
\item[\S 6] The modularity condition
\item[\S 7] The fermionic action
\end{itemize}
\item[F.] The quantum version of classical conformal manifolds
\item[G.] Fractals
\item[H.] Essays on gravitation
\begin{itemize}
\item[\S 1] The Dirac operator and gravitation
\item[\S 2] The quantum Polyakov action in two dimensions
\item[\S 3] The quantum Polyakov action in four dimensions
\end{itemize}
\item[I.] Conclusion and outlook
\end{itemize}
\newpage
These notes are an amplification of the four lectures delivered at the
Guadeloupe school. They attempt to provide an introduction to Alain
Connes' non-commutative geometry-analysis,\footnote{
In a modern sight geometry and
analysis have merged into one discipline: the study of manifolds and
the operators on their vector bundles. This coalescence is in fact
enhanced by Alain Connes' ``quantization'' of geometry-analysis.
} and applications to
fundamental physical interactions.\footnote{
We do not treat in these lectures
applications to solid-state physics (i.e. as of today to the Hall
effect).
} As of today, these applications consist in designing
methods based on non-commutative geometry for constructing lagrangian
actions for the microworld and the cosmos within a conceptual
mathematical perspective.
For the micro-world (standard model of elementary particles) the
mathe\-matical emphasis is on the ``non-commutative De~Rham complex'',
set up of the ``quantum Yang-Mills algorithm'' on which Alain Connes
grounds his reinterpretation of the standard model (technically
resting on the notion of $d^+$-summable $K$-cycle - see section~B). We
discuss this astounding interpretation of the full standard model in
some detail (including a display of formulae and a sketch of the
lengthy calculations in their up-to-date version\footnote{
This last version was preceded by (i) a first essay on the electroweak
theory using the algebra $C^{\infty} (M) \oplus C^{\infty} (M)$
together with a vector bundle taylored so as to obtain the desired
symmetry breaking and (ii) a first version of the present theory using
formal quantum forms and leading to adynamical fields, see~[1], [2],
[5] I and II.
}). As the reader will see, the set-up for the full
standard model including chromodynamics interprets the Higgs bosons as
a fifth gauge boson\footnote{
of a ``discrete'' type not accessible to
usual differential geometry
}; and suggests an
enlargement of the notion of $K$-cycle enacting ``non-commutative
Poincar\'e duality'' based on a ``Poincar\'e dual pair of algebras'' -
a basic concept of non-commutative geometry\footnote{
Likely to become
fundamental in mathematics as a piece of recognition of the basic
nature of ``non-commutative manifolds'' - in particular opening the
way to ``non-commutative characteristic classes''.
} discovered by
Alain Connes {\it whilst looking at the standard model}, where
``Poincar\'e duali\-ty'' correspond to the electroweak-chromodynamics
duality: a beautiful instance of a crucial stride pertaining jointly
to fundamental mathematical and physical features.
In addition to introducing to Alain Connes' version of the standard
model, these notes attempt to sketch the developments of the last year
which inaugurated
\begin{itemize}
\item[---] on the one hand the differential geometry of fractals (our
sketchy section~E hopes to give the reader a ``flavour''),
\item[---] on the other hand the beginning study of gravitation by
techniques of non-commutative geometry: namely (i) the provocative
fact that the Dirac operators codes the data of the Einstein-Hilbert
action (\S 1 of section~H); (ii) a quantum version of the Polyakov
action for strings (\S 2 of section~H), and a transcription of the
latter to four-dimensional conformal manifolds yielding a quadratic
action possibly related to gravitation (\S 3 of section~H).
\end{itemize}
\noindent The two last of the latter subjects use a
different kind of non-commutative differential geometry based on the
notion of $p$-summable Fredholm module (see section~F). The
differential geometry of fractals also uses Fredholm modules as
sketched in section~G. Paragraph \S 1 of section~H merely appeals to
the notion of Wodzicki residue (the unique trace on PDOs, related to
the Dixmier trace). Paragraph \S 5 of section~E, devoted to
non-commutative Poincar\'e duality, is placed there because of its
application to the standard model - conceptually, though, it is a
continuation of section~B.
Before starting, I should stress that my subject is -
of course - Alain Connes' intellectual property. My justification for
commenting it is that I have performed in parallel to Connes and Lott
the lenghty computations of the quantum-geometry Yang-Mills action for
the full standard model (coupled electroweak and chromodynamics
sectors) - both in the original manner where one has to deal with
adynamical fields, and by means of the superior new technique using
the genuine quantum forms ({\sl quantum De Rham complex}).
The key device in Alain Connes' non-commutative geometry consists in
replacing classical differentials\footnote{
We denote in this lecture
the exterior differential by a bold-faced straight ${\bf d}$
}
$$
{\bf d}a~,~~~~~~~~~~ a \in C^\infty (M)~, \eqno (\star)
$$
attached to a Riemannian compact manifold $M$ with a spin structure,
by Hilbert space operators
$$
[D\, , \, \ul{a}]~, ~~~~~ a \in A~, \eqno (\star \star)
$$
resp.
$$
[F, \ul{a}]~, ~~~~~~ a \in A~, \eqno (\star \star \star)
$$
where $A$ is a now generally a non-commutative algebra (the points of
the ``space'' $M$ have disappeared), and where~$(\star \star)$,
resp.~$(\star \star \star)$, are commutators between Hilbert-space
representative $\ul{a}$ of the $a \in C^\infty (M)$ with an
(appropriately defined) generalized Dirac operator $D$, resp.\ a
``phase" $F$ (the two possibilities correspond to two different kinds
of non-commutative geometries, in essence metric, resp.\ conformal).
Of course this scheme has to be implemented, given the algebra $A$ (a
dense subalgebra of a C$^{\star}$-algebra) by a specification of the
relevant $A$-modules $(H,D)$ (resp. $(H,F)$), $H$ a ($\bZ/2$-graded)
Hilbert space carrying a bounded representation of $A$, and $D$ (resp.
$F$), a generalized Dirac operator (resp.\ a phase) - the reader will
find the precise specification of $D$ below in section~B under the
name ``$d^+$-summable $K$-cycle" (respectively that of F as
``$p$-summable Fredholm module'' in section~F). Defining, in this
manner, the differential geometry of an algebra $A$ produces an
extension of the usual riemannian (respectively conformal) geometry
with the following powerful features:
\begin{itemize}
\item[-] applied to the classical case of $A = C^\infty (M)$
theories of the
type~$(\star \star)$ reproduce the usual geometry of riemannian
manifolds, comprising (modulo a slight extension of the general
theory) a generalized Poincar\'e duality - whilst theories of the
type~$(\star \star \star)$ reproduce the usual geometry of conformal
manifolds
\item[-] this general scheme covers the case of
non commutative algebras (small enough so their cohomological
dimension is finite);
\item[-] it also covers the case of spaces with non-integer
(Hausdorff) dimension: fractals, whose commutative algebras of
functions have a non-integer cohomological dimension.
\end{itemize}
This guided tour is organized as follows. We first describe
(section~A) the quantum version of classical spin$^c$ riemannian
geometry, showing how one reconstructs from it the classical De Rham
complex and volume form. Extracting the quintessence thereof, we then
formulate (section~B) non-commutative geometry for a cohomologically
finite-dimensional (non-commutative) algebra, a theory based on the
notion of ``$d^+$-summable $k$-cycle''. We then present several
examples: in section~C the example of the two-point algebra already
exhibiting embryonal Higgs bosons; in section~D the quantum version of
Yang-Mills for electrodynamics; in section~E a sketch of the
differential geometry of fractals (= spaces of non-integer Hausdorff
dimension). This line culminates, in section~E, with an exposition
(with partial proofs) of Alain Connes' vision of the standard model of
elementary particles. The second line of the lectures begins in
section~F with a description of ``$p$-summable Fredholm modules'' and
their use for the description of the quantal perspective on classical
conformal manifolds. We then give examples based on this notion: in
section~G a sketch of the differential geometry of fractals and in
section~H \S \S 2 and 3 a discussion of the quantum Polyakov action.
\S 1 of section~H uses merely, as already mentioned the notion of
Wodzicki residue.
\bigskip
\bigskip
\appendix
\setcounter{equation}{0}
\setcounter{chapter}{1}
\noindent {\bf A.~~~ The Dirac operator as the carrier of classical
differential geometry}
\vglue 0,4truecm
In what follows $M$ is a spin Riemannian compact manifold of dimension
$d= 2m$. $H$ is the Hilbert space $H = L^2 (S_M)$ of square integrable
spinor fields ($S_M$ the spinor bundle of $M$) acted upon by the
Atiyah-Singer-Lichnerowicz Dirac operator
\beq
D = \gamma^\nu (\partial_\nu + \sigma_\nu)~,
\eeq
and by the elements of $A = C^\infty (M)$ acting multiplicatively on
the spinor fields:
\beq
(\ul{a} \psi)(x) = \ul{a} (x) \psi (x)~, ~~~~~~~~~~~~~a \in A~,
{}~~~~\psi \in H .
\eeq
We note the following structural features, later to be extracted as
axioms:
\begin{itemize}
\item[(i)] $H$ is $Z/2$-graded: $H = H^0 \oplus H^1$; \footnote{
The grading
operator is chirality $\chi = \gamma^{d+1}$ (half-spinors).
}
\item[(ii)] $A \ni a \rgh \ul{a} \in B (H)$ is a representation of the
algebra $A$ by bounded even operators on $H$ (in fact, a
$*$-representation);
\item[(iii)] $D$ is an (unbounded) odd self-adjoint operator;
\item[(iv)] all
commutators $[D, \underline a]$, $a \in A$, are bounded;
\item[(v)] $D^{-1}$ is a compact
operator, in fact belonging to the ideal $L^{d+} (H)$ (essentially,
one has
$D^{-d} \in L^{1+})$: to simplify, we assume $D$ without zero-modes.
\end{itemize}
Interposed comment about $L^{1+} (H)$ and $L^1 (H)$: the familiar
ideal $L^1 (H)$ of trace-class operators consists of the compact
operators $T$ such that $|T|$ has eigenvalues $\lambda_n$ with ${\dsp
\sum_{n =!}^\infty
\lambda_n < \infty ~ (=Tr |T|)}$. ~~$L^{1+} (H)$ is ``just about
above" $L^1 (H)$: it consists of compact operators $T$ with $|T|$
s.t. ${\dsp \sum_{n=1}^N
\lambda_n =O ~(\log N)}$. Applying a dilation-invariant mean $w$ to
the bounded function ${\dsp \sum_{n =1}^\infty \lambda_n / \log (N
+1)}$ on {\bf N} then yields the {\sl Dixmier trace} $Tr_w$, whose
definition domain is $L^{1+}$ (analogously to the familiar fact that
$L^1 (H)$ is the definition domain of the usual trace)\footnote{
Strictly
speaking there are as many Dixmier traces $T_w$ as dilation invariant
means $w$; the latter however coincide in all practical applications
- we thus refer to them as the ``the Dixmier trace".
}.
The fact that the operator $D$ encodes the information of the smooth
structure is a priori shown by the provocative (easy to prove) result
that the geodesic distance $d (p,q)$ between two points $p,q$ of $M$
is given by
$$
d (p,q) = \sup \{ | a (p) - a (q)|\,;\ a \in A,\ \| [ D, \ul{a}] \|
\leq 1 \}~. \eqno(\mbox{A.0})
$$
{}~~
We now proceed to the explicit construction of the De Rham complex
$\Omega (M)$
by means of the Dirac operator. (We recall that ${\dsp \Omega (M) =
\sum_{n =0}^d
\Omega (M)^d}$ equipped with the exterior differential {\bf d} is an
$N$-graded differential algebra).
For this we need a formal construction pertaining to general unital
algebras $A$: {\it the differential enveloppe of} $A$, namely the
$N$-graded differential algebra $(\Omega A, \delta)$ of ``formal
differential forms" obtained as follows: consider linear combinations
of formal symbols:
\beq
a_0 \delta a_1
\cdots \delta a_n~, ~~~~~a_0, \, a_1, \cdots , a_n \in A~,
\eeq
with $\delta {\bf 1} = 0$ and products $a_0 \delta a_1 \ldots \delta
a_n \cdot b_0
\delta b_1 \cdots \delta b_n$ obtained by
shuffling $b_0$ to the left by use of the Leibniz rule for the symbol
$\delta$: requiring this provides us (easy verification) with an
$N$-graded algebra (grading given by the number of differentials).
Moreover the rule
\beq
\delta (a_0 \delta a_1 \cdots \delta a_n) = 1 \delta a_0 \delta a_1
\cdots \delta
a_n
\eeq
establishes $(\Omega A, \delta)$ as an $N$-graded differential
algebra\footnote{
The requirement that $\delta$ be a differential (=
derivation of vanishing square) compels to definition (A.4) and
interprets (A.3) as the product of $a_0$ and the $\delta a_1 , \ldots
, \delta a_n$.
}. Mind that $\Omega A$ is a formal object carrying no
real information: for instance its cohomology vanishes: $\delta \om =
0, \omega \in
\Omega A$, implies that $\om = \delta
\psi$ for some $\psi$.
We now consider $\Omega A$ for $A = C^\infty (M)$ and will use it for
coding information via the Dirac operator $D$. For this we set
\beq
\pi_D (\om) = (-i)^n \underline{a}_0 [D, \underline{a}_1] \cdots [D,
\underline{a}_n]~, ~~~~~~\om = a_0 \delta a_1 \cdots \delta a_n \in
\Omega A
\eeq
and note that we thus obtain a bounded *-representation $\pi_D$ of
$\Omega A$ {\sl as a $^{\star}$algebra}\footnote{
The *-operation of
$\Omega A$ is defined as the extension of the *-operation of $\Omega
A$ such that $(\delta \omega)^* = (-1)^n \delta (\omega^*),\ \omega
\in \Omega A^n$.
} - not as a differential
algebra! - indeed (A.4) consists in replacing the derivation $\delta$
by the derivation $-i [D,
\cdot]$.\footnote{
$\pi_F (\omega)$ with $F^2 = 1$ would yield a
representation of $\Omega A$ as a differential algebra. However $D$ is
not of square one!
} We are now in a situation where we have a
differential algebra $\Omega A$ represented on Hilbert space as an
algebra, a situation leading naturally to a differential algebra by
the (easily established fact) that, with $K$ the kernel of $\pi_D$ -
an ideal of the algebra $\Omega A$ - the set $K + \delta K$ is again
an ideal, but now a {\sl differential ideal}. In fact we need a small
step more, because we want to retain the $\bN$-grading (in addition to
the differential): so replace $K$ by $K^* = \oplus_n K^n,\ K^n = K
\cap \Omega A^n$, which is a graded ideal such that $K^* + \delta K^*$
is stable under $\delta$ and graded. We thus define
\beq
\Omega_D A = \Omega A / \left( K^* + \delta K^*\right)~.
\eeq
This now yields, together with $\delta$ which passes to the quotient,
a differential algebra, amazingly {\it isomorphic as such to the De
Rham complex}:
\beq
(\Omega_D A, \delta) \simeq (\Omega (M), {\bf d})~.
\eeq
We constructed the De Rham complex by means of the Dirac operator!
For the technical computation of $\Omega_D A$ it is useful to notice
that one has in grade $n$:
$$
\Omega_D A^n = \pi_D (\Omega A^n) / \pi_D (\delta K^{n - 1})
\eqno(\mbox{A.7a})
$$
We now describe the volume-form (the device for integration). Since
$D^{-d}
\in L^{1+}$, since $\pi_D (\om)$ is bounded, and since $L^{1+}$ is an
ideal, we may consider
\beq
\tau_D (\om) = Tr_w (D^{-d} \pi_D (\om))~,~~~~~~\om \in \Omega A~,
\eeq
thus obtaining a trace on $\Omega A$ (by inserting in addition under
$Tr_w$ the grading involution $\chi$, we would get the Euler
characteristics).\footnote{
The fact that $\pi_D$ is a trace stems from the fact
that $[D^{-1}, \pi_D (\om)] \in L^1 (H)$ (commutators are smoothing!),
and that
$Tr_\om$ vanishes on $L^1 (H)$.
}
In our practical calculations we shall make use of the fundamental
fact that for
$P$ a pseudodifferential operator of order $-d$, the Dixmier trace
coincides with the {\sl Wodzicki residue}, namely
\beq
{\dsp Tr_w (P) \cong cst \int tr (\sigma_P (x, \xi)) d x
\delta (|\xi|^2 -1) d \xi}
\eeq
where $tr$ is the trace on the fiber and $\sigma_P$ the principal
symbol of $P$.
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{2}
\noindent {\bf B.~~~ Non-commutative differential geometry via
$K$-cycles}
\vglue 0,4truecm
Let now $A$ be a (possibly non-commutative) real or complex
$^*$-algebra. What we have seen above now justifies the claim that
one will endow $A$ with a metric differential structure by specifying
a $K$-cycle defined as follows:
{\bf Definition}~~~ A $d^+$-{\sl summable} $K$-{\sl cycle} $(H,D)$ of
$A$ consists
of the following data:
\begin{itemize}
\item[(i)] a $\bZ/2$ graded Hilbert space $H = H^0 \oplus H^1$ (with
grading involution $\chi$, $\bone$ on $H^0$, $-\bone$ on $H^1$);
\item[(ii)] a
$^*$-representation $a \rgh {\underline a}$ of $A$ by even bounded
operators on $H$ ~~$({\underline a} \chi = \chi {\underline a}$ for
each $a \in A$);
\item[(iii)] a self-adjoint
odd (unbounded) operator $D$ ~~$(D \chi = - \chi D)$ such that
\item[(iv)] $[D, {\underline a}]$ is a bounded operator for all
${\underline a} \in A$
\item[(v)] $D^{-1}$ is a compact operator\footnote{
We assume for simplicity that
$D$ has no zero modes. Otherwise one should consider $(D^2 +
m^2)^{\frac{1}{2}}$, instead $m >0$. To be quite exact (vi) should be
replaced by the somewhat more stringent requirement $D^{-1} \in L^{d+}
(H)$. The notation $(H,D)$ implies that
$H$ is a $\bZ/2$-graded left $A$-module (via $a \rgh \ul{a}$).
};
\item[(vi)] $D^{-d} \in L^{1+} (H)$
\end{itemize}
Given such a $d$-summable $K$-cycle $(H,D)$, the representation
$\pi_D$ of
$\Om A$ is defined as in (A.4), leading as above to the ${\bN}$-graded
differentiable algebra $\Om_D A$ (cf. (A.5), (A.5a)) which is
naturally called the {\sl non commutative De Rham complex of} $A$.
``Integration" is then defined by means of the trace $\tau_D$ of $\Om
A$ as given in (A.8).
At this point we dispose of all we need to define a ``non commutative
Yang-Mills action". For simplicity and because this is the case at
hand for electrodynamics and for the standard model, we first describe
this notion in the case where the bundle (or rather its module of
sections) is given by the algebra itself. In that case ({\sl metric})
{\sl connections} are definable as odd derivations $\nabla$ of the
quantum De Rham complex $\Om_D A$, thus of the form $\delta + \rho$,
with $i
\rho$ {\sl the connection one-form (quantum potential)} a
(self-adjoint) element of
$\Om_D A^1$. The corresponding {\sl curvature} is then the
endomorphism\footnote{
of the right $\Om_D A$-module $\Om_D A$.
} $\nabla^2$,
where, as one immediately checks, $\nabla^2 \om = (\delta \rho +
\rho^2) \om$,
$\om \in \Om A$: $\theta = \delta \rho + \rho^2 \in \Om_D A^2$ is the
{\sl curvature two-form}. The {\sl Yang-Mills action} is then
naturally defined as
\beq
\mbox{YM}~(\rho) = \tau_D (\theta^2) = Tr_w \, (D^{-d} \pi_D
(\theta^2))
\eeq
The case of an arbitrary bundle, yielding\ in the non commutative case
an arbitrary finite projective (right) $A$-module\footnote{
We remind that the
modules of sections of vector bundles over a manifold $M$ are
precisely the finite-projective modules over $C^\infty (M)$. For a
non-commutative $A$ the role of (sections of) vector bundles is thus
played by the finite projective modules over $A$.
} $E$, is hardly more
complicated. One considers the set of $E$-valued quantum forms:
\beq
E_\Om = E \otimes_A \Om_D A~,
\eeq
a $\bN$-graded right module over the algebra $\Om_D A$, and defines
the connections $\nabla$ as the grade-one graded $\delta$-derivations
of this module, i.e.\ the $\bC$-linear $\nabla : E_\Om \rgh E_\Om$
fulfilling\footnote{
We recall that a classical connection looked at
in the guise of the corresponding {\sl exterior covariant derivative
} $\nabla$ is a grade-one graded derivation of the
$\Om (M)$-module $E_\Om = E \otimes_A \Om (M)$, restricting on $E$ as
$(\nabla
\eta) (\xi) = \nabla_{\xi} \eta, \, \eta \in E,\, \nabla \xi$.}
\beq
\nabla (X \om) = (\nabla X) \om + (-1)^{\partial X} \delta \om~,
{}~~~~~~~~ X \in E_\Om, ~~ \om
\in \Om_D A~.
\eeq
The curvature is then $\nabla^2 \in \mbox{End}_{\Om A} E_\Om$, thus
implemented by a quantum 2-form $\theta$ with values in
$\mbox{End}_{\Om A} E_\Om$. The Yang-Mills action is then given by
(B.1) when one replaces the trace $\tau_D$ of
$\Om_D A$, by the trace of End$_{\Om (A)}$ $E_\Om$ given by
\beq
\tilde{\tau}_D (X \Phi) = \tau_D ( \Phi (X))~,
{}~~~~~~~~~~~ X \in E_\Om ~, ~~\Phi \in E_\Om^*~,
\eeq
where $\Phi$ is the one-rank endomorphism given by $(X \Phi) U = \Phi
(U) X, ~ U \in E_\Om$ (observe that $E$ being projective-finite, the
same holds for $E_\Om$ whose endomorphisms are thus linearly generated
by one-rank operators).
We end this section by a word of caution. What we have said will hold
only for algebras {\sl possessing $d^+$-summable $K$-cycles}, i.e.\
(by definition) having a {\sl finite cohomological dimension}, that is
``small algebras" (algebras of loop spaces, or of field theoretic
operators, are ``big", i.e.\ cohomologically infinite-dimensional).
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{3}
\noindent {\bf C.~~~The two-point algebra: Higgs bosons in a nutshell}
\vglue 0,4truecm
We now look at the simplest possible example $A = \bC \oplus \bC$
(besides the trivial $\bC$ itself!). An element of $A$ is of the form
$a = (f,g)$, $f,g \in
\bC$. On the discrete object $A$ we define the differential geometry
by the following $K$-cycle $(H,D): H = \bC^N \oplus \bC^N$ where the
integer $N$ prefigures the number of fermion families. Linear
endomorphisms of $H$ are thus described by $2 \times 2$ matrices with
entries in $M_N (\bC)$. In this sense we have the grading involution
\beq
\chi = \left(
\begin{array}{cc}
{}~~~\bone_N~~~ & 0 \\
& \\
0 & - \bone_N
\end{array} \right)~,
\eeq
and we define\footnote{
In these expressions we left implicit the identity
$\bone_2$ of $\bC^2$.
}
\beq
{\underline a} = \left(
\begin{array}{cc}
{}~~~f \bone_N~~~ & 0 \\
& \\
0 & g \bone_N
\end{array} \right)~, ~~~~~~~~~~{\underline a} = (f,g) \in A
\eeq
and
\beq
D = \left(
\begin{array}{cc}
{}~~~0~~~ & M^* \\
& \\
M & 0
\end{array} \right)
\eeq
where the $N \times N$ matrix $M$ prefigures the fermionic mass
matrix. For $\om = {\dsp \sum_i a_0^i \delta a_1^i}$, $a_0^i = (f_0^i,
g_0^i)$, $a_1^i = (f_1^i, g_1^i)$ one then easily finds that
\beq
\pi_D (\om) = \left(
\begin{array}{cc}
0 & {\dsp - \sum_i f_0^i (f_1^i - g_1^i) M^*} \\
& \\
{}~~{\dsp \sum_i g_0^i (f_1^i - g_1^i) M} ~~ & 0
\end{array} \right)
\eeq
and
\beq
\pi_D (\delta \om) = \sum_i (f_0^i - g_0^i) (f_1^i - g_1^i) ~\left(
\begin{array}{cc}
{}~M^* M~ & 0 \\
& \\
0 & MM^*
\end{array} \right)
\eeq
This implies that $K^1$ consists of the $\om$ for which ${\dsp \sum_i
f_0^i (f_1^i - g_1^i) = \sum_i g_0^i (f_1^i - g_1^i) = 0}$, these are
such that $\pi_D (\delta \om)$, hence $\pi_D (\delta K^1)$ vanishes.
Since $K^0$ vanishes, one has thus $\Om_D A^1 = \pi_D (\Om A^1)$ and
$\Om_D A^2 = \pi_D (\Om A^2)$ (cf. (A.5a)). The metric connection
one-forms are the antihermitic elements of $\pi_D (\Om A^1)$, thus of
the form
\beq
\rho = \left(
\begin{array}{cc}
0 & - \hbar M^* \\
& \\
{}~~hM ~~ & 0
\end{array} \right)~, ~~~~~h \in \bC~,
\eeq
with, by (C.5):
\beq
\delta \rho = -(h + \hbar) ~ \left(
\begin{array}{cc} ~M^*M~~~ & 0 \\
& \\
0 ~~~ & MM^*
\end{array} \right)~,
\eeq
and
\beq
\rho^2 = -\hbar h ~ \left(
\begin{array}{cc} ~M^*M~~~ & 0 \\
& \\
0 ~~~ & MM^*
\end{array} \right)~,
\eeq
thus with curvature
\beq
\theta = - (\hbar h + \hbar + h) ~\left(
\begin{array}{cc} ~M^*M~~~ & 0 \\
& \\
0 ~~~ & MM^*
\end{array} \right)~.
\eeq
Setting $h + 1 = \phi$ one has $\hbar h + h + \hbar = | \phi|^2 - 1$.
In this finite case the Dixmier trace reduces to the ordinary trace
$Tr$. The Yang-Mills action is thus
\beq
Tr (\theta^2) = 2 [|\phi|^2 - 1]^2 ~Tr\, (M^* M)^2
\eeq
displaying the characteristic form of a Higgs potential.
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{4}
\noindent {\bf D.~~~\begin{tabular}[t]{l} Equivalence of the quantum
and the classical Yang-Mills \\
algorithms \end{tabular} }
\vglue 0,4truecm
We formulated the ``quantum Yang-Mills algorithm" for a general
(cohomologically finite) $A$ in (B.1). We now apply this to
electrodynamics, where $A = C^\infty (M)$, and $D$ is the usual Dirac
operator, and show that this merges into the usual Yang-Mills. The
computation is effected via the analytic form (Wodzicki residue)
(A.9) of $Tr_w$.\\ Let $\om = {\dsp \sum_{i=1}^m a_0 \delta a_1^i \in
\Om A^1}$, $a_0^i$, $a_1^i \in C^\infty (M)$, yielding the quantum
potential
\beq
\rho = \pi_D (\om) = \sum_{i=1}^m {\underline a}_0^i {\underline
\gamma^\mu \partial_\mu a_1^i} =
\gamma (A)
\eeq
where
$A$ is the classical potential\footnote{
Since the representation
$a \rgh {\underline a}$ is faithful, we have $K^0 = 0$, hence $\Om_D
A^1 = \pi_D (\Om A^1)$. $\om$ should be taken self-adjoint, resulting
in $A$ being real (we do not discuss reality conditions).
}
\beq
A = \sum_i a_0^i {\bf d} a_1^i
\eeq
With $\gamma({\bf d} a) = \gamma^{\mu} \partial_{\mu} a$, we then have
that\footnote{
$\gamma$ denotes the canonical map: ${\cal T} (T^M) \rgh
\mbox{Cliff} (T^M)$.
}
\beqn
\pi_D (\delta \om) & = & \sum_{i = 1}^m \gamma ({\bf d} a_0^i) \gamma
({\bf d} a_1^i) =
\gamma (\sum_i {\bf d} a_0^i \otimes {\bf d} a_1^i) \\
& = & \gamma \left( \frac{1}{2} {\bf d} a_0^i
\wedge {\bf d} a_1^i + \frac{1}{2} \sum_{i =1}^m {\bf d} a_0^i \vee
{\bf d} a_1^i
\right) \nonumber
\\
& = & \gamma \left( \frac{1}{2} {\bf d} A \right) + \sum_i ({\bf d}
a_0^i, \, {\bf d} a_1^i )
\nonumber
\eeqn
and
\beq
\pi_D (\om^2) = \gamma (A \otimes A) = (A,A) = A_{\mu} A^{\mu}
\eeq
We have to evaluate the curvature $\rho^2 + \delta \rho$ as an element
of $\Om_D A^2 \simeq \Om (M)^2$. For this we use the fact here stated
without proof that the quotient modulo $\pi_D (\delta K^1)$ of a
Clifford element $\gamma (T)$, $T$ a tensor, is obtained by expressing
$\gamma (T)$ in terms of antisymmetric tensors and retaining only the
part of highest order. This rule leads to the fact that ${\dsp \delta
\rho = \gamma \left( \frac{1}{2} F \right) }$, where $F = {\bf d} A$
and $\rho^2 = 0$.
Inserting this in (A.9), where ${\dsp \theta = \gamma \left(
\frac{1}{2}F
\right)}$ then yields the classical action. (The principal symbol
$|\xi|^{-4}$ of
$D^{-4}$, 1 on the unit sphere, drops out of the calculation).
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{5}
\noindent {\bf E.~~~ The standard model of elementary particles}
\vglue 0,4truecm
\noindent This section describes the method and the results, giving in
detail some of the computations, but omitting the lengthiest, for
which we refer to the paper~[5]III which we are following here.
\bigskip
\noindent {\bf \S 1 ~~ Strategy}
{}~~~~
We have shown in section $E$ that
for electrodynamics the quantum Yang-Mills action is another version
of the classical Yang-Mills action\footnote{
A piece of the general fact that non
commutative geometry applied to the classical case yields the usual
smooth structure.
}. On the other hand the (quantum) Yang-Mills action
of the two-point algebra in section~C had produced the Higgs
phenomenon as stemming from a ``discrete connection". Those two facts
can now be combined thanks to an important property of the quantum
Yang-Mills formalism: given two algebras $A'$ and $A''$ equipped with
respective
$K$-cycles $(H', D')$, $(H'', D'')$, there is indeed a canonical
way for obtaining a tensor product $K$-cycle $(H,D)$ of the tensor
product algebra $A = A' \otimes A''$: namely\footnote{
This is the
``exterior product" of Kasparov's $KK$-theory, $\hat{\otimes}$
denoting a skew tensor-product.
}
\beq
\left\{
\begin{array}{l} H = H'
\otimes H'' \\ ~~ \\
\chi = \chi' \otimes \chi^{''} \\
{}~~ \\
\underline{a' \otimes a^{''}} = {\underline a}' \otimes {\underline
a}^{''} \\
{}~~ \\
\!\!\begin{array}{cl}
D & = D' \hat{\otimes} \bone^{''} + \bone' \hat{\otimes} D^{''} \\
& \\
& = D' \otimes \bone^{''} + \chi' \otimes D^{''}
\end{array}
\end{array} \right.
\eeq
The strategy is now clear, at least for the electroweak sector of the
standard model. The passage from electrodynamics to the electroweak
synthesis occurs through the passage $U (1) \rgh U(1) \times SU (2)$.
This calls for tensoring as in (E.1) a finite $U (1) \times SU (2)$
Yang Mills theory (based on the algebra
$\bC \oplus \bH$ analogous to the former $\bC \oplus \bC$ having
gauge group $U (1) \times U (1)$) with the electrodynamics Yang-Mills
based on $C^\infty (M)$ and the Dirac operator.
Of course one has to specify the choice of differential geometry for
$\bC \oplus \bH$ - i.e.\ the relevant ``inner $K$-cycle" (appropriate
analogue of the
$K$-cycle of $\bC \oplus \bC$ described in section~C). The choice of
$\bC + \bH$ was dictated by its gauge group $U (1) \times SU (2)$
emerging from the phenomenology of weak interactions. As we now
describe, the inner $K$-cycle is likewise dictated by phenomenology.
\bigskip
\noindent {\bf \S 2 ~~ The electroweak inner $K$-cycle}
{}~~~~
The algebra $\bC \oplus \bH$ consists of pairs $(p,q)$, $(p \in \bC)$,
$q \in \bH$. For reasons which appear later it is convenient to
identify the complex number $p$ with a diagonal quaternion\footnote{
We recall that quaternions are $2
\times 2$ complex matrices of the form ${\dsp \left(
\begin{array}{cc}
\ap &
\beta \\
- \overline{\beta} & \overline{\ap}
\end{array} \right)}$, $\ap, \,
\beta \in \bC$.}
\beq
\bC \ni p \leftrightarrow \left(
\begin{array}{cc} \overline{p} & 0 \\
0 & p
\end{array} \right) \in \bH_{diag}
\eeq
The electroweak inner $K$-cycle $(H_f, D_f)$ ($f$ for fermion) is now
the direct sum $(H_l \oplus H_q, D_l \oplus D_q)$ of a leptonic
$K$-cycle $(H_l, D_l)$ and a quarkonic $K$-cycle $(H_q, D_q)$. We
begin with the description of the latter, from which the former is
obtained by ``splitting a corner" ({\sl leptonic reduction}). The
(finite) dimensional Hilbert space $H_q$ is the tensor product
\beq
H_q = ( \build{\bC_R^2}_{u_R d_R}^{} ~~
\oplus ~~ \build{\bC_L^2}_{u_L d_L}^{} ) ~~
\otimes \bC^N~,
\eeq
where the factor $\bC^N$ stands for $N$ fermion families, $R$ and $L$
stand for right, resp.\ left, and we indicated the particle - content,
$u$ and
$d$ standing for upper, resp.\ lower quarks. We now specify the
grading, the representation of $\bC \oplus \bH$, and the generalized
Dirac operator: they are:\footnote{
We describe elements of
End$_{\bC} H_q$ as $2 \times 2$ matrices with entries in $M_2 (\bC)
\oplus M_N (\bC)$; and use (E.2).
}
\beq
\chi_q = \left(
\begin{array}{cc}
\bone_2 \otimes \bN & 0 \\
& \\
0 & - \bone_2 \otimes \bone_N
\end{array} \right)
\eeq
\beq
\pi_q (p,q) = \left(
\begin{array}{cc}
p \otimes \bone_N & 0 \\
& \\
0 & q \otimes \bone_N
\end{array} \right)
\eeq
\beq
D = \left(
\begin{array}{cc}
0 & M_q^* \\
& \\
M_q & 0
\end{array} \right) \quad \mbox{with} \quad M_q = \left(
\begin{array}{cc} M_u & 0 \\
& \\
0 & M_d
\end{array} \right)
\eeq
{}~~\\
where $M_u$, resp.\ $M_d$, are the mass matrices for upper,
resp.\ lower quarks. Notice that these definitions are natural:
$(p,q)$ is even and $D$ odd as they should. For definition (E.5) we
have the following guidance: since the bundle is the algebra itself,
the gauge group is the group of unitaries of the latter. Now $(p,q)$
is unitary for $p\in S^1$ whilst $q$ is a unitary quaternion (i.e.\ an
element of $SU (2)$).
The leptonic Hilbert space $H_l$ is
\beq
H_l = ( \build{\bC_R^1}_{e_R}^{} ~~
\oplus ~~ \build{\bC_L^2}_{\nu_L e_L}^{} ) ~~
\otimes \bC^N~,
\eeq
coding the fact that there is no right-handed neutrino: $e_R, \,
\nu_L$ and
$e_L$ respectively stand for right-hand electron (left handed)
neutrino, left-handed electron - where electron stands, generically,
for electron - muon - tau. Notice that (E.7) is analogous to (E.3)
with the first entry chopped-off. Correspondingly, $\chi_l$, $\pi_l$
and $D_l$ are obtained from $\chi_q, \, \pi_q$ and $D_q$ written as $4
\times 4$ matrices with entries in $M_N (C^2)$ by effecting the
changes $u_L \rgh 0$,
$d_R \rgh e_R$, $u_L \rgh \nu_L$, $d_L \rgh 0$, $M_u \rgh 0$, $M_d
\rgh M_e$; and then chopping off the first row and column of the
matrix. This procedure ({\sl leptonic reduction}) allows to evolve the
computation for the leptonic from those for the quarkonic $K$-cycle.
We will thus only treat the latter.
We now describe the quantum one- and two-forms. From (E.5) and (E.6)
we have, successively, for $p , \, \pi \in \bH_{diag}$, $q, \chi \in
\bH$, writing $M_q = M$:
\beq
i \pi_q (\delta (p,q)) = [D, \, \pi_q (p,q) ] = \left(
\begin{array}{cc} 0 &
\bM^* [(q-p) \otimes \bone_N\,] \\
& \\
\left[(p-q) \otimes \bone_N \right]\, \bM & 0
\end{array} \right)
\eeq
(we used the fact that $p \otimes \bone_N$ commutes with $\bM$ and
$\bM^*$)
\begin{eqnarray}
& & i \pi_q (p,q) \delta (\pi, \chi) = \left(
\begin{array}{cc}
0 & \bM^* [p (\chi - \pi)
\otimes \bone_N] \\
& \\
\left[ q(\pi - \chi) \otimes \bone_N \right] \bM & 0
\end{array} \right) \\
& & \nonumber \\
& & \pi_q (\delta (p, q) \delta (\pi, \chi)) = \left(
\begin{array}{c} \bM^*
[(p-q)(\pi - \chi) \otimes \bone_N] \bM ~~~~~ 0 \\
{}~~ \\
0 ~~~~~ [(p-q) \otimes \bone_N] \bM \bM^* [(\pi - \chi)
\otimes \bone_N ]
\end{array} \right) \\
& & \nonumber \\
& & \pi_q ((s,r) \delta (p,q) \delta (\pi, \chi) = \left(
\begin{array}{c}
\bM^* [s (p-q) (\pi - \chi) \otimes \bone_N ]\bM ~~~~~ 0 \\
\\
0~~~~~ [ r (p-q) \otimes \bone_N ] \bM \bM^* [(\pi - \chi)
\otimes \bone_N]
\end{array} \right)
\end{eqnarray}
{}~~
Note here that one has from (E.6a), with $K$ the third basic
quaternion
\beq
\bM \bM^* = \bone \otimes \Sigma - i K \otimes \Delta~, ~~~~\left\{
\begin{array}{l}
{\dsp \Sigma = \frac{1}{2} (M_u M_u^* + M_dM_d^*) } \\
{}~~ \\
{\dsp \Delta = \frac{1}{2} (M_u M_u^* - M_d M_d^*)}
\end{array} \right.
\eeq
{}~~\\
We then conclude that one has for $\omega = {\dsp \sum_i (p_i, q_i)
\delta (\pi_i, \chi_i) \in \Omega A^1_{ew}}$:
\beq
i \pi_q (\om) = \left(
\begin{array}{cc} 0 & \bM^* (Q' \otimes
\bone_N) \\
& \\
(Q \otimes \bone_N) \bM & 0
\end{array} \right)
\eeq
where the couple
$$
\dsp \left\{
\begin{array}{l} {\dsp Q = \sum_i q_i
(\pi_i - \chi_i)} \\
{}~~ \\
{\dsp Q'=\sum_i p_i (\chi_i - \pi_i)}
\end{array} \right.
\eqno (\mbox{E.13a})
$$
{}~~\\
ranges through $\bH \times \bH$; and, for ${\dsp \theta = \sum_i
(s_i, r_i)
\delta (p_i, q_i) \delta (\pi_i, \chi_i)}$:
\beq
\pi_q (\theta) = \left(
\begin{array}{cc}
\bM^* (Q \otimes \bone_N) \bM & 0 \\
& \\
0 & Q' \otimes \Sigma - i Q^{''} \otimes \Delta
\end{array} \right)
\eeq
where the triple
{}~~\\
$$
{\dsp \left\{
\begin{array}{l} {\dsp Q= \sum_i s_i (p_i - q_i)
(\pi_i - \chi_i) } \\
{}~~ \\
{\dsp Q'= \sum_i r_i (p_i - q_i) (\pi_i - \chi_i) } \\
{}~~ \\
{\dsp Q^{''}= \sum_i r_i (p_i - q_i)K (\pi_i - \chi_i) }
\end{array} \right.
}\eqno (\mbox{E.14a})
$$
{}~~\\
ranges through $\bH \times \bH \times \bH$: this is obvious for $Q$
and $Q'$ and will follow for $Q^{''}$ from the forthcoming argument.
Note that whilst the general quantum one-forms are given in (E.13)
(since $\Om_D A^1_{ew} =
\pi_D (\Om A^1_{ew}$) owing to $K^0 = 0$), the quantum two-forms will
be the classes of elements (E.14) modulo the set $\pi_D (\delta K^1)$,
which we now investigate. The kernel $K^1$ in grade one consists of
the $\om$ in (E.13) fulfilling
\beq
\sum_i q_i (\pi_i - \chi_i) = \sum_i p_i (\pi_i - \chi_i) = 0~,
\eeq
$\pi_D (\delta \om)$ being given by (E.14) where on makes $s_i = r_i$
for all $i$. Looking at (E.14a) we see that the conditions (E.15) then
imply the following shape of the matrix (E.14)
\beq
\left(
\begin{array}{cc}
0 & 0 \\
& \\
0 & - i Q^{''} \otimes \Delta
\end{array} \right) ~~\mbox{with}~ Q^{''} =
\sum_i (p_i - q_i) K (\pi_i - \chi_i)~,
\eeq
with the range of such $Q^{''}$ including the set of ${\dsp \sum_i q_i
K \chi_i}$, $q_i$, $\chi_i$ arbitrary quaternions: since this set is
obviously an ideal of the quaternions, thus the whole quaternions, we
see that one has
\beq
\pi_D (\delta K') = \left\{ \left(
\begin{array}{cc} 0 & 0 \\
& \\
0 & i Q^{''} \otimes \Delta
\end{array} \right)~, ~~~~Q^{''} \in \bH \right\}~,
\eeq
and thus the quantum two-forms are of the type
\beq
\left(
\begin{array}{cc}
\bM^* (Q \otimes \bone_N) \bM & 0 \\
& \\
0 & Q' \otimes \Sigma
\end{array} \right) ~, ~~~~~Q, \, Q' \in \bH~.
\eeq
It is interesting that quotienting through $\pi_D (\delta K^1)$
precisely leaves us with the quaternionic part in (E.14).
\bigskip
\noindent {\bf \S 3.~~ The electroweak sector}
{}~~~~
We now effect the tensor product as described in (E.1), taking
for $(H', D')$ the classical $K$-cycle $(H = L^2 (\delta_M)$,
Dirac operator) of $A = C^\infty (M)$, and taking
for $(H^{''}, D^{''})$ the quarkonic $K$-cycle $(H_q, D_q)$ of
$A_{ew}$ as described in the last section. We thus obtain a
$4^+$-summable $K$-cycle\footnote{
We leave aside, momentarily, the
leptonic $K$-cycle $(H_l, D_l)$, relying on leptonic reduction.
} $(\cD_q, \cH_q = H \otimes H_q)$ of the compound algebra
\beq
\begin{array}{ll}
\cA = & A \otimes A_{ew} = C^\infty (M, \bH_{diag} ) \oplus C^\infty
(M, \bH)
\\
& \\
& = \{ (f,q); ~f \in C^\infty (M , \bH_{diag}), ~q \in C^\infty (M,
\bH) \}
\end{array}
\eeq
Note that, considering now endomorphisms of $\cH_q$ as $2 \times 2$
matrices with entries endomorphisms of $H \otimes C^2 \otimes C^N$,
we have that
\begin{eqnarray}
& & \pi_{\cD_q} (f \otimes (p,q)) = \left(
\begin{array}{cc} f \otimes p \otimes
\bone_N & 0 \\
& \\
0 & f \otimes q \otimes \bone_N \end{array} \right) \\
& & \nonumber \\
& & \cD_q = \left(
\begin{array}{cc}
D \otimes \bone_2 \otimes \bone_N & \gamma^5 \otimes \bM^* \\
& \\
\gamma^5 \otimes \bM & D \otimes \bone_2 \otimes \bone_N
\end{array} \right)
\\
& & \nonumber \\
& & \chi_{\cD_q} = \left(
\begin{array}{cc} \gamma^5 \otimes \bone_2
\otimes
\bone_N & 0 \\
& \\
0 & - \gamma^5 \otimes \bone_2 \otimes \bone_N
\end{array} \right)
\end{eqnarray}
{}~~
For describing the quantum one- and two-forms we are now guided by a
tensor-product structure discussed in detail in [6] for the systems
compound of space-time and a finite dimensional inner space. Stated in
our present case the structure is as follows. The differential
envelope $\Om \cA$ projects homomorphically onto the skew tensor
product $\Om \cA \hat{\otimes}
\Om A_{e w}$ of differential envelopes. The skew product tensor is
perfect at the level of the representation $\pi_{\cD_q}$: one has for
one-forms (formal or quantum)\footnote{
Note that $\pi_{\cD_q}$ is
faithful. One will thus have
$\Om_{\cD_q} \cA^1 = \pi_{\cD_q} (\Om \cA^1)$.
}
\beq
\pi_{\cD_q} (\Om \cA^1) = \pi_D (\Om A^1) \hat{\otimes} \pi_q (A_{e
w}) + \pi_D (A) \hat{\otimes} \pi_{\cD_q} (\Om A^1_{ew})~,
\eeq
i.e.
$$
\begin{array}[t]{rl}
\Om_{D_q} \cA^1 & = \Om_D A^1 \otimes A_{ew} \oplus A \otimes
\Om_{D_q} (A_{ew})~, \\
& \\
(&= \mbox{``}\Om_{D_q} \cA_{[1,0]} \oplus \Om_{D_q} \cA_{[0,1]}
\mbox{"})
\end{array}
\eqno (\mbox{E.23a})
$$
{}~~\\
and for representatives of formal two-forms:
\beq
\begin{array}{rl}
\pi_{\cD_q} (\Om \cA^2) & =\pi_D (\Om A^2) \hat{\otimes} \pi_q
(A_{ew}) + \pi_D (\Om A^1) \hat{\otimes} \pi_q (\Om A_{ew}^1) + \pi_D
(A) \hat{\otimes} \pi_q
(\Omega A_{ew}^2) \\
& \\
( &=\mbox{``}\, \pi_{\cD_q} (\Om \cA_{[2,0]} ) + \pi_{\cD_q} (\Om
\cA^2_{[1,1]} ) + \pi_{\cD_q} (\Om \cA^2_{[0,2]})\, \mbox{"}\,)
\end{array}
\eeq
Furthermore the set by which one has to devide in order to get the
quantum one-forms also tensorially decompose
\beq
\pi_{\cD_q} (\delta K_\cA^1 ) = \pi_D (\delta K_A^1) \otimes \pi_q
(A_{ew}) + \pi_D (A) \otimes \pi_q (\delta K^1_{A_{ew}})
\eeq
This structure then implies the following: the quantum one-forms
$\rho \in
\pi_{\cD_q} (\cA^1) = \Om_{D_q} \cA^1$ are given as follows in terms
of quadruples $i \rho =({\bf a},\,{\bf b},\, H, \, H')$, ${\bf a}\in
\Om (\bM, \bC)^1$, ${\bf b}^.. \in
(\Om (\bM, i \bH)^1$, $H, H' \in C^\infty (\bM, \bH)$: one has for the
bihomogeneous components the following $2 \times 2$ matrices with
entries acting on the fibers of $L^2 (\bS_{\bM}) \otimes \bC^N$:
\beq
i \rho = (\ba, \bb, H, H'): ~\left\{
\begin{array}{l} i \rho_{[1,0]}
= \left(
\begin{array}{cc} \ul{\gamma} (\ba^. . ) \otimes \bone_N & 0
\\ & \\
0 & \ul{\gamma} (\bb^..) \otimes \bone_N
\end{array} \right) \\
{}~~\\
i \rho_{[0,1]} = \left(
\begin{array}{cc} 0 & \bM^* (\gamma^5 \ul H' \otimes
\bone_N) \\
& \\
(\gamma^5 \ul H \otimes \bone_N) \bM & 0
\end{array} \right)
\end{array}
\right.
\eeq
where we used the shorthands
\beq
\left\{
\begin{array}{ll}
\gamma (\ba^. .) = \left(
\begin{array}{cc} - \ul{\gamma} (\ol{\ba}) & 0 \\
~~ & \\
0 & \ul{\gamma} (\ba)
\end{array} \right)~, & \ba \in \Om (\bM, \bC)^1
{}~(\ba^. . = \left(
\begin{array}{cc} - \ol{\ba} & 0 \\
{}~ & ~ \\
0 & \ba
\end{array} \right) \in \Om (\bM, i \bH_{diag})^1 )~, \\
& \\
\ul{\gamma} (\bb^. .) = \left(
\begin{array}{cc} \ul{\gamma} (\bb^1_{~1}) &
\ul{\gamma}(\bb^1_{~2}) \\
& \\
\ul{\gamma} (\bb^2_{~1}) & \ul{\gamma}(\bb^2_{~2})
\end{array} \right)~,
& \bb^. . = \left(
\begin{array}{cc}
\bb^1_{~1} & \bb^1_{~2} = \bb^2_{~1} \\
& \\
\bb^2_{~1} & \bb^2_{~2} = \bb^1_{~1}
\end{array} \right) \in \Om (\bM, \,
i \bH)^1~.
\end{array} \right.
\eeq
{}~~\
The elements of $\pi_{\cD_q} (\delta K^1)$ have the form (statement
synonimous with (E.25)):
\beq
\left\{
\begin{array}{l}
\left(
\begin{array}{cc}
(\ul{\bS}^i_{~k}) \otimes \bone_N & 0 \\
& \\
0 & (\ul{\bf T}^i_{~k} \otimes \bone_N
\end{array} \right) + \left(
\begin{array}{cc}
0 & 0 \\
& \\
0 & i \ul{\bR} \otimes \Delta
\end{array} \right) \\
{}~~ \\
\mbox{where}~~~\left\{
\begin{array}{l}
({\bf S}^i_{~k}) \in C^\infty (\bM, \bH_{diag}) \\
({\bf T}^i_{~k}) \in C^\infty (\bM, \bH) \\
\bR \in C^\infty (\bM, \bH)
\end{array} \right.
\end{array} \right.
\eeq
The vector potentials (antihermitean elements of $\Om_{\cD_q} \cA^1$,
singled out as the antihermitean $V = \rho$ above are accordingly
specified by triples
$(\ba, \, \bb,\, H)$ of a hermitean $U(1)$-connection-one-form $\ba$,
a hermitean $SU (2)$-connection-one-form $\bb^. .$, and a doublet
field $H \in C^\infty (\bM, \bH)$: one has\footnote{
Note that a quaternion is antihermitean
iff it is traceless.
}
\beqn
& & iV = (\ba, \, \bb, H, \, H^*): \\
& & \left\{
\begin{array}{l}
{\dsp iV_{[1,0]} = \left(
\begin{array}{cc} \ul{\gamma} (\ba^. . )
\otimes \bone_N & 0 \\
0 & \ul{\gamma} (\bb^. . ) \otimes \bone_N
\end{array} \right) } \\
{}~~ \\
{}~~~~~~~~~~~~~~~~~{\dsp \left( \left\{
\begin{array}{l} \ba = \ol{\ba} \\
\bb = \bb^* ~~\mbox{i.e.}~ \bb^i_{~k} = \bb^k_{~i}, ~~i,k = 1,2,
\end{array}
\right. \right) } \\
{}~~ \\
iV_{[0,1]} = {\dsp \left(
\begin{array}{cc} 0 & \bM^* (\gamma^5 \ul{H}^*
\otimes \bone_N) \\
& \\
\gamma^5 \ul{H} \otimes \bone_N) \bM & 0
\end{array} \right) }
\end{array}
\right. \nonumber
\eeqn
{}~~\\
The field $H \in C^\infty (\bM, \bH)$ is identified with a doublet
field according to \hfill \break
${\dsp H = \left(
\begin{array}{cc}
\ol{H}^2 & H^1 \\
& \\
- \ol{H}^1 & H^2
\end{array} \right) \Leftrightarrow H^. = (H^1, H^2)
\Leftrightarrow
H_. = (\ol{H}^1, \ol{H}^2) }$.
Under the gauge transformation $\bu = (u,v = (v^i_{~k}))$ the
$\cD_q$-quantum-one-form $-i (\ba, \bb, H, H')$ becomes $-i (\,
^{\bu}\ba,\, ^{\bu}\bb, \, ^{\bu}H, \, ^{\bu}H')$ with:
\beq
\left\{
\begin{array}{l}
^{\bu}\ba = \ba + iu {\bf d} u^* \\
{}~~ \\
^{\bu}\bb = v \bb v^* + iv {\bf d} v^* \\
{}~~ \\
^{\bu}(H + \bone) = u^* v (H + \bone) \\
{}~~ \\
^{\bu}(H' + \bone ) = u^* v (H' + \bone)
\end{array} \right.
\eeq
In the particular case of vector potentials, $\ba$, and
$(\ba^i_{~k})$ thus behave as the respective one-forms of a $U(1)$-
and a $SU(2)$-connection, whilst $\Phi_. = H_. + (0,1)$ {\sl and}
$\Phi^. = H^. + (0,1)$ behave as respective covariant and
contravariant $SU(2)$-doublets.
The elements of $\pi_{\cD_q} (\Om \cA^2)$ are of the type\footnote{
Whilst $\rho_{[1,0]}$ and $\rho_{[0,1]}$ are linearly independent,
this does not hold for $\eta_{[20]}$, $\eta_{[11]}$ and $\eta_{[02]}$
(in the subscript$_{[i,j]}$, $i$ refers to the $\gamma^5-$ and $j$ to
the $\chi_q$-parity).
}:
\beq
\left\{
\begin{array}{l}
\eta_{[2,0]} = {\dsp \left(
\begin{array}{cc}
[ (\ul{\gamma} (\lambda^i_{~k})) -
(\ul{X}^i_{~k} )] \otimes \bone_N & 0 \\
& \\
0 & [ ( \ul{\gamma} (\mu^i_{~k})) - (\ul{Y}^i_{~k} )] \otimes 1_N
\end{array} \right) } \\
{}~~ \\
\eta_{[1,1]} = {\dsp \left(
\begin{array}{cc}
0 & \bM^* \left[ i \ul{\gamma}
(\bq') \gamma^5 \otimes \bone_N \right] \\
& \\
\left[ i \ul{\gamma} (\bq) \gamma^5 \otimes \bone_N \right] \bM & 0
\end{array}
\right) } \\
{}~~ \\
\eta_{[0,2]} = {\dsp \left(
\begin{array}{cc}
\bM^* (\bQ \otimes \bone_N) \bM
& 0 \\
& \\
0 & \bQ' \otimes \Sigma + i \bQ^{''} \otimes \Delta
\end{array} \right) }
\end{array} \right.
\eeq
\bigskip
{}~~\\
where ${\dsp \left\{
\begin{array}{l}
(\lambda^i_{~k}) \in \Om (\bM, \bH_{diag})^2 \\
{}~~ \\
(X^i_{~k}) \in C^\infty (\bM, \bH_{diag} ) \\
{}~~ \\
(\mu^i_{~k}) \in \Om (\bM, \bH)^2 \\
{}~~ \\
(Y^i_{~k}) \in C^\infty (\bM, \bH) \\
{}~~ \\
\bq, \, \bq' \in \Om (\bM, \bH)^1 \\
{}~~ \\
\bQ, \, \bQ',\, \bQ^{''} \in C^\infty (\bM, \bH)
\end{array} \right. }$
{}~~\\
with the following action of $P_2$:
\beq
\left\{
\begin{array}{l}
(P_2 \eta)_{[2,0]} = {\dsp \left(
\begin{array}{l} [ ( \ul{\gamma}
(\lambda^i_{~k})) - (2 N)^{-1}\, Tr (\mu_u + \mu_d) Q_{diag}] \otimes
\bone_N ~~~~~0 \\
{}~~~~\\
0~~~~~ [ ( \ul{\gamma} (\mu^i_{~k})) - (2 N)^{-1}\, Tr (\mu_u + \mu_d)
\cdot Q' ]
\otimes \bone_N
\end{array} \right) } \\
{}~~ \\
(P_2 \eta)_{[1,1]} = {\dsp \left(
\begin{array}{l} 0~~~~~ \bM^* [i \ul{\gamma}
(\bq') \gamma^5 \otimes \bone_N ] \\
{}~~~~\\
\left[ i \ul{\gamma} (\bq) \gamma^5 \otimes \bone_N \right] \bM~~~~~ 0
\end{array} \right)} = \eta_{[1,1]} \\
{}~~ \\
(P_2 \eta)_{[0,2]} = \left(
\begin{array}{cc} \bM^* (\bQ \otimes 1_N) \bM &
0 \\
& \\
0 & \bQ' \otimes \Sigma
\end{array} \right)
\end{array} \right.
\eeq
{}~~
The next step (which we do not describe, referring to [5] III for
details) consists in adapting the results above to a situation
incorporating the leptons, replacing the inner $K$-cycle $(H_q, D_q)$
above by a convex combination $\ap_l (H_l, D_l) + \ap_q (H_q, \,
D_q)$, $\ap_l = (1 + x)/2$, $\ap_q = (1 - x)/2$, $-1
\leq x \leq 1$. Using leptonic reduction, one gets the following
structure: the projection $P_2$ is modified as follows, where we
set \break
$\underline L =Tr [\ap_l \mu_l + \ap_q
(\mu_u + \mu_d)]$, $\mu_l = M_l M_l^*$, $\mu_u = M_u M_u^*$, $\mu_d =
M_d M_d^*$:
\beq
\left\{
\begin{array}{l}
(P_2 \eta)_{[2,0]} = {\dsp \left(
\begin{array}{l}
[ ( \ul{\gamma} (\lambda^i_{~k})) - (\ap_l + 2 \ap_q)^{-1}
\underline LQ_{diag}]
\otimes 1_N ~~~~~~~~~~~~0 \\
{}~~~ \\
0~~~~~~[ ( \ul{\gamma} (\mu^i_{~k})) - (\ap_l + 3 \ap_q )^{-1} (2
\bN)^{-1}
\underline L \cdot Q' ] \otimes \bone_N \end{array} \right) } \\
{}~~ \\
(P_2 \eta)_{[1,1]} = {\dsp \left(
\begin{array}{l} 0~~~~~~~~~ \bM^* \left[ i
\ul{\gamma} (\bq) \gamma^5 \otimes \bone_N \right] \\
{}~~ \\
\left[ i \ul{\gamma} (\bq) \gamma^5 \otimes \bone_N \right] \bM
{}~~~~~~0
\end{array} \right) = \eta_{[1,1]} }\\
{}~~ \\
(P_2 \eta)_{[0,2]} = {\dsp \left(
\begin{array}{l} \bM^* (\bQ \otimes \bone_N)
\bM ~~~~~0 \\
{}~~~~ \\
0~~~~~~~~~~~~~~~ \bQ' \otimes \Sigma
\end{array} \right) }
\end{array} \right.
\eeq
the action of $P_2$ on the leptonic reduction of $\eta$ yielding the
leptonic reduction of $P_2 \eta$.
We now describe the curvature corresponding to the vector-potential
specified in (E.29): it is specified as follows by its
quark-component
$\theta_q$:
\beq\label{E.34}
\left\{
\begin{array}{l}
- \theta_{q [2,0]} = {\dsp \left(
\begin{array}{c}
{\dsp \left[ \frac{i}{2}
\ul{\gamma} (f^..) - (\ap_l + 2 \ap_q)^{-1} N^{-1} \underline L v_\Phi
\bone \right] \otimes \bone_N} \qquad 0 \quad\\
{}~~ \\
\quad 0 \qquad \qquad \qquad {\dsp \left[ \frac{i}{2} \ul{\gamma}
(\bh^. . ) - (2 N)^{-1}
\ul{L} v_\Phi \bone \right] \otimes \bone_N }
\end{array} \right) } \\
{}~~ \\
- \theta_{q [1,1]} = {\dsp \left(
\begin{array}{cc}
0 & \bM^*
\left[ \ul{\gamma} (i \bD \Phi^*) \gamma^5 \otimes \bone_N \right] \\
& \\
\left[ \ul{\gamma} (i \bD \Phi) \gamma^5 \otimes \bone_N \right] \bM
& 0
\end{array} \right) } \\
{}~~ \\
{\dsp - \theta_{q [0,2]} = \left(
\begin{array}{cc} v_\Phi \bM^* \bM & 0 \\
& \\
0 & v_\Phi \bone \otimes \Sigma
\end{array} \right) }
\end{array} \right.
\eeq
the leptonic component $\theta_l$ resulting by leptonic reduction.
Here we use the shorthands\footnote{
We denoted covariant derivatives by bold-faced letters to avoid
confusion with the Dirac operator. We recall that ${\dsp
\Sigma = \frac{1}{2} (\mu_u + \mu_d)}$ with $\mu_u = M_u M_{u^*}$ and
$\mu_d = M_d M_{d^*}$.
}:
\beq
\left\{
\begin{array}{ll}
\ul{\gamma} ({\bf f}^. .) = {\dsp \left(
\begin{array}{cc}
- \ul{\gamma} ({\bf f}) & 0 \\
& \\
0 & \ul{\gamma} ({\bf f})
\end{array} \right) }~, & {\bf f} = {\bf d}
\ba \in C^\infty (\bM, \bC)~, \\
& \\
{\dsp \ul{\gamma} (\bh^. . ) = \left(
\begin{array}{cc} \ul{\gamma}
(\bh^1_{~1})
& \ul{\gamma}(\bh^1_{~2}) \\
& \\
\ul{\gamma}(\bh^2_{~1}) & \ul{\gamma} (\bh^2_{~2})
\end{array} \right)}~, &
\bh^j_{~k} = {\bf d} \bb^j_{~k} - i \bb^j_{~m} \wedge \bb^m_{~k} ~, \\
& \\
\Phi = \bH + \bone\, , ~~~v_\Phi = \Phi_i \Phi^i - 1~, & \bH \in
C^\infty (\bM, \bH)~, \\
& \\
\bD \Phi = {\bf d} \Phi + i (\Phi \ba^. . - \bb^. . \Phi) ~, & \bD
\Phi^* = (\bD \Phi)^* = {\bf d} \Phi^* - i (\ba^. . \Phi^* - \Phi^*
\bb^. . )~.
\end{array} \right.
\footnote{
i.e. $\bh_{\mu \nu} = \partial_\mu \bb_\nu - \partial_\nu \bb_\mu - i
\left[ \bb_\mu, \bb_\nu \right] $.
}
\eeq
{}~~
Here ${\bf f},. \, \bh^. . = (\bh^i_{~j})$, $ i,j =1,2$ are the
respective curvatures of the $U (1)$-connection $\ba$ and the
$SU(2)$-connection $\bb^.. = (\bb^i_{~j}) = \bb^a {\dsp
\frac{\tau_a}{2}}$, $\ap = 1, \, 2,\, 3$:
\beq
\left\{
\begin{array}{l}
{\bf f} = {\bf d} \ba = {\dsp \frac{1}{2} } {\bf f}_{\mu \nu} d x^\mu
\wedge dx^\nu \\
{}~~ \\
\mbox{with}~ {\bf f}_{\mu \nu} = \partial_\mu \ba_\nu - \partial_\nu
\ba_\mu
\end{array} \right. ~,
\eeq
\beq
\left\{
\begin{array}{l}
\bh^. . = \nabla \bb^. . = {\dsp \frac{1}{2} } \bh_{\mu \nu} {\bf d}
\bx^\mu \wedge {\bf d} x^\nu (\bh^i_{~k} ) = \bh^a {\dsp
\frac{\tau_a}{2}} \\
{}~~ \\
\mbox{with} ~ \bh^i_{~k} = {\bf d} \bb^j_{~k} - i \bb^j_{~m} \wedge
\bb^m_{~k}~, ~~~~\mbox{i.e.}~~\bh^j_{~k \mu \nu} = \partial_\mu
\bb^j_{~k\nu} - \partial_\mu
\bb^j_{~km} - i [\bb_\mu, \bb_\nu]^j_{~k} ~, \\
{}~~ \\
\mbox{or}~~\bh_{\mu \nu} = \partial_\mu \bb_\nu - \partial_\nu
\bb_\mu - i [\bb_\mu, \bb_\nu]\, , ~~~~\mbox{i.e.}~\bg^a_{~\mu \nu}
= \partial_\mu
\bb^a_{~\nu} - \partial_\nu \bb^a_{~\mu} + \vr_{abc}
\bb^b~{\mu}
\bb^c_{~\nu}
\end{array} \right.
\eeq
The usual combined $U (1) \times SU (2)$-covariant derivatives
\beq
\left\{
\begin{array}{l}
\bD \Phi^.: \bD \Phi^j = {\bf d} \Phi^j + i (\ba \Phi^j - \bb^j_{~k}
\Phi^k)~, ~~\mbox{or}~~\bD \Phi^. = {\bf d} \Phi + i {\dsp \left(
\ba -
\bb^a
\frac{\tau_a}{2}\right)} \Phi^. \\
{}~~ \\
{\dsp \bD \Phi.: \bD \Phi_j = {\bf d} \Phi_j - i (\ba \Phi_j -
\bb^k_{~j} \Phi_k) =
\bD \ol{\Phi}^j ~, ~~~\mbox{or}~ \bD \Phi. = {\bf d} \Phi. - i \Phi.
\left( \ba -
\bb^a \frac{\tau_a}{2} \right) }
\end{array} \right.
\eeq
has the following relationship with the above quaternion covariant
derivatives: for
{}~~\\
${\dsp \Phi = \left(
\begin{array}{cc}
\ol{\Phi}^2 & \Phi^1 \\
{}~~ & \\
- \ol{\Phi}^1 & \Phi^2
\end{array} \right)
\leftrightarrow \Phi^. = (\Phi^1, \Phi^2) \leftrightarrow \Phi. =
(\ol{\Phi}^1,
\ol{\Phi}^2)}$ one has:
\beq
\bD \Phi = {\bf d} \Phi + i (\Phi (\ba^. .)- (\bb^. .) \Phi = \left(
\begin{array} {cc}
\bD \Phi_2 & \bD \Phi^1 \\
& \\
- \bD \Phi_1 & \bD \Phi^2
\end{array} \right)~.
\eeq
Having at hand the Yang-Mills action of the full electroweak sector
one could think of exploiting it physically (electroweak physics).
But the corresponding theory {\sl leads to wrong hypercharges}, this
being in fact a success of Connes' approach, which thus {\sl entails
the necessity of putting together the electroweak and chromodynamics
sectors}\footnote{
This aspect is in fact enhanced
by interpreting the duality electroweak-strong as a Poincar\'e
duality.
}.
\bigskip
\noindent {\bf \S 4.~~ Appending the chromodynamics sector}
{}~~~~
On the
chromodynamics side the group of inner degrees of freedom is $SU (3)$
and the interaction mediated by the gluons is of the Yang-Mills type
with unbroken symmetry. The relevant ``inner space" algebra is
\beq
B_{chrom} = \bC \oplus M (\bC^3) ~,
\eeq
the best we can do to accomodate $SU(3)$\footnote{
We are not here in the
fortunate situation of the electroweak $U(1) \otimes SU(2)$, unitary
group of $\bC \oplus \bH$. We are compelled to work with $U(1)$ and
the $U(3)$ of $M (\bC^3)$, and later collapse three $U(1)$ group into
one.
}. The overall chromodynamics algebra, counterpart of $\cA$ in
(E.19), is then, with
$A = C^\infty (M)$ as above:
\begin{eqnarray}
\cB & = & A \otimes B_{chrom} = \bC^\infty (M, \bC) \oplus C^\infty
(M, M_3) (\bC)) \\
& = & \{ (g', m); \, g' \in C^\infty (M, \bC), \, m \in C^\infty (M,
M_3 (\bC) \}~. \nonumber
\end{eqnarray}
The relevant $K$-cycle is now, in terms of the previous $H_q$, $H_l$:
\beq
H \otimes ( H_q \otimes \bC_{colour}^3 \oplus H_l \otimes \bC)~,
{}~~~~~H = L^2 (\bS_M)~,
\eeq
in conformity with the fact that the quarks acquire a threefold colour
degree of freedom, the leptons being colourless. On this new Hilbert
space enriched by colour
\begin{itemize}
\item[---] the electroweak algebra acts trivially on the colour
tensorial factors $\bC_{colour}^3$ and $\bC$, acting as above on the
factors $H_q \otimes H$ and $H_l \otimes H$;
\item[---] the generalized Dirac operator also proceeds
in this way from the former $\cD_q \oplus \cD_l$ (indifference to
colour);
\item[---] the algebra $\cB$ acts by its space-time factor on $H$ in
the usual way, and by its $B_{chrom}$-factor, as naturally expected:
the $\bC$-part acting on the $\bC$ factor of $H_l \otimes C$, and the
$M(\bC^3)$-part on the
$\bC_{colour}^3$-factor of $H_q \otimes \bC^3_{colour}$.
\end{itemize}
It is important to note that one has {\sl commutation of the actions
of $\cA$ and $\cB$} (in other terms an action of $\cA \otimes \cB$)
and {\sl commutation of the action of $\cB$ with the discrete part of
the generalized Dirac operator}, this producing the setting of
``non-commutative Poincar\'e duality"~[3a] for which we refer to \S 5.
The quantum one-forms of the chromodynamics sector and their
curvatures have in fact been already computed - modulo tensorization
and non-commutativity of the gauge group - in section~D describing
electrodynamics. The respective lepton and quark contribution are
\beq
iV'_l = \bordermatrix{& e_R &
\nu_L & e_L \cr
e_R & \gamma (\ba') \otimes \bone_N & 0 & 0 \cr
\nu_L & 0 & - & 0 \cr
e_L & 0 & 0 & - \cr}~,
\eeq
\bigskip
\beq
- \theta'_l = \bordermatrix{&e_R & \nu_L & e_L \cr
e_R & {\dsp \frac{i}{2}
\gamma (f') \otimes \bone_N } & 0 & 0 \cr
\nu_L & 0 & - & 0 \cr
e_L & 0 & 0 & - \cr}~,
\eeq
and\footnote{
Here $a'$ and $c^0$ are $U(1)$-potentials with curvatures $f'$,
resp. $g^0$, $c^a$ being a $SU(3)$-potential with curvature $g^a$,
$a= 1, \ldots, 8$.
}
\bigskip
\beqn
iV'_q & = & \bordermatrix{& u_R & d_R& u_L & d_L \cr
u_R & \gamma (c_{~k}^j)
\otimes \bone_N & 0 & 0 & 0 \cr
d_r & 0 & - & 0 & 0 \cr
u_L & 0 & 0 & - & 0 \cr
d_L & 0 & 0 & 0 & - \cr}
\otimes e_{~j}^k \\
& & \nonumber \\
& = & ~~~~~ \left(
\begin{array}{cccc}
\gamma (c^0) \otimes \bone_N & 0 & 0 & 0 \\
0 & - & 0 & 0 \\
0 & 0 & - & 0 \\
0 & 0 & 0 & -
\end{array} \right) \otimes \bone_3 + \left(
\begin{array}{cccc}
\gamma (c^a) \otimes \bone_N & 0 & 0 & 0 \\
0 & - & 0 & 0 \\
0 & 0 & - & 0 \\
0 & 0 & 0 & -
\end{array} \right) \otimes \frac{\lambda_a}{2}
\nonumber
\eeqn
\bigskip
\beqn
- \theta'_q & = & \bordermatrix{&u_R & d_R& u_L& d_L \cr
u_R & {\dsp \frac{i}{2}
\gamma (g_{~k}^i) \otimes \bone_N} & 0 & 0 & 0 \cr
d_R & 0 & - & 0 & 0 \cr
u_L & 0 & 0 & - & 0 \cr
d_L & 0 & 0 & 0 & - \cr}
\otimes e_{~j}^k ~, \\
& & \nonumber \\
& = & ~~~~~\left(
\begin{array}{cccc}
{\dsp \frac{i}{2} \gamma (g^0) \otimes \bone_N} & 0 & 0 & 0 \\
0 & - & 0 & 0 \\
0 & 0 & - & 0 \\
0 & 0 & 0 & -
\end{array} \right) \otimes \bone_3 + \left(
\begin{array}{cccc}
{\dsp \frac{i}{2} \gamma (g^a) \otimes \bone_N} & 0 & 0 & 0 \\
0 & - & 0 & 0 \\
0 & 0 & - & 0 \\
0 & 0 & 0 & -
\end{array} \right) \otimes \frac{\lambda_a}{2} \nonumber
\eeqn
where the second lines correspond to decompositions of $3 \times 3$
matrices as sums
\beq
m =(m_k^i) = m^0 \bone + m^a \frac{\lambda_a}{2} ~,
\eeq
with the $\lambda_a$, $ a=1, \ldots,8$ the $SU (3)$ Gell-Mann
matrices. This alternative corresponds to the isolation of the
$U(1)$-part of the group $U(3)$, necessary in order to remove two
related unwanted features: a plethoral gauge group $U (1) \times S
U(2) \times U(1) \times U(3)$ and an excess by two of the number of
gauge fields. These two drawbacks can be removed by imposing a ``{\sl
modularity condition"} (see [3], [4] and \S 6 for details) leading to
the following coalescence of the gauge fields $a,
\, a'$ and $c^0$:\footnote{
Modular connections
are couples $(V,V')$ of an electroweak and a chromodynamic connection
for which one makes the identifications~\rf{E.48}, \rf{E.49}. The
corresponding curvature is the sum ``$\theta + \theta'$'' with those
identifications. We must in addition take account of the colour -
tripling the quark multiplicity by making the change
$\alpha_q \to 3 \alpha_q$ in $\theta_q$ of~\rf{E.34} as well as in its
leptonic reduction $\theta_l$.
}
\beq\label{E.48}
\left\{
\begin{array}{l}
a' = a \\
{}~~ \\
{\dsp c^0 = - \frac{1}{3} a }
\end{array} \right.
\eeq
with, correspondingly:
\beq\label{E.49}
\left\{
\begin{array}{l}
f' = f \\
{}~~ \\
{\dsp g^0 = - \frac{1}{3} f~. }
\end{array} \right.
\eeq
Once this is done, calculation of the Yang-Mills action corresponding
to the leptonic, resp.\ quark percentages $\ap_l = (1 -x) /2$ and
$\ap_q = (1 - x)/2$ yields the following standard-model bosonic
Lagrangian density
\begin{eqnarray}
\cL_B & = & - \frac{1}{2} (1- x) \bN g_{a \mu \nu} g^{a \mu \nu} -
\frac{1}{3} (10 - x) \bN f_{\mu \nu} f^{\mu \nu} - \frac{1}{4}(2- x)
\bN \bh^s_{~\mu \nu}
\bh^{\mu \nu}_s \nonumber \\
& & + 2 L (\bD \Phi_j)(\bD \Phi^j) + K (\Phi_i \Phi^i
- 1)^2~,
\end{eqnarray}
where
\beq
\left\{
\begin{array}{l}
{\dsp K = \frac{3}{2} F - 6 \ap_q Tr (\mu_u \mu_d) - \bN^{-1} [2^{-1}
(\ap_1 + 3
\ap_q)^{-1} + (\ap_l + 6\ap_q)^{-1} ]L^2 } \\
{}~~ \\
F = Tr [ \ap_l \mu_e^2 + 3 \ap_q (\mu_d + \mu_u)^2] \\
{}~~ \\
L = Tr [\ap_l \mu_e + 3 \ap_q (\mu_u + \mu_d)]
\end{array} \right.
\eeq
\beq
\left\{
\begin{array}{l}
{\dsp D \Phi^. = {\bf d} \Phi + i \left( a - b^a \frac{\tau_a}{2}
\right) \Phi^. } \\
{}~~ \\
{\dsp D \Phi. = {\bf d} \Phi. - i \Phi. \left( a - b^a
\frac{\tau_a}{2} \right)
~~~\mbox{i.e.~~} D \Phi_j = \ol{D \Phi}^j\, , ~~~i = 1,2 }
\end{array} \right.
\eeq
now correspond to correct hypercharge assignements. These satisfactory
covariant derivatives also appear in the fermionic part of the
Lagrangian density (see E \S 7).
We conclude with two remarks:
\begin{itemize}
\item[---] Caveat: this astonishing reinterpretation of the standard
model in non-commuta\-tive geometry (embodying the ``inner degrees of
freedom" as features of a (mildly) non commutative space inseparable
{}from the elementary particle structure) is for the moment {\sl
confined to the classical (Lagrangian) level}. Field quantization and
renormalization studies still lie ahead.
\item[---] A preliminary calculation at tree level (without deep
significance) exhibits for the most symmetric Ansatz $x = 0$ results
with a grand unification flavour, and for lepton-dominance $(x = 1)$
results resembling the realistic situation [7] (details in~[5]). We
reproduce this results borrowed from~[7]:
\end{itemize}
$$
\vbox{\halign{\hfill#\hfill&\quad\hfill#\hfill&
\quad\hfill#\hfill&\quad
\hfill#\hfill&\quad\hfill#\hfill&\quad\hfill#\cr
$x$&-1&0&${1\over2}$&0.99&1\cr
\noalign{\medskip}
$\left({{g_3}/ g_2}\right)^2$&${3\over4}$&1&${3\over2}$&
50.5&$\infty$\cr
\noalign{\medskip}
$\sin^2\theta_W$&${9\over{20}}$&${3\over8}$&${9\over{28}}$&0.252&
${1\over4}$\cr
\noalign{\medskip}
${m_t/ m_W}$&$\sqrt 3$&2&$\sqrt 6$&14.2&$\infty$\cr
\noalign{\medskip}
${m_H/m_W}$&$2.65$&$3.14$&$3.96$&$24.5$&$\infty$\cr}}
$$
\bigskip
\noindent We note that the ratio $m_H/m_t$ shows little variationfrom
1.53 to $\sqrt 3$.
The table suggests the following remarks:
\begin{itemize}
\item[(i)] all tabulated functions are monotonic in $x$.
\item[(ii)] the value $x=0$ seems to correspond to a situation of the
``unification'' type.
\item[(iii)] for the limit value $x=1$, i.e. $\alpha_q=0$, the
Weinberg angle is near its experimental
value, whilst strong interactions prevail. Indication of lepton
dominance at experimental energies? Connected with confinement?
\end{itemize}
In fact the theory presented here is not the least constraining: the
most general choice of coupling constant exhausting the degeneracy of
the $K$-cycle $H_l
\oplus H_q$, plainly restitutes the 17 constants of the usual standard
model~[4].
\bigskip
\noindent {\bf \S 5 ~~ Dual pair of quantum spaces}
{}~~~~
In section~B we defined the ``non-commutative differential geometry''
of a (cohomologically finite-dimensional) algebra by specifying a
$d^+$-summable $K$-cycle, this procedure generalizing to the
non-commutative frame the fact that usual differential geometry is
coded in the Dirac operator. However, the mathematical object
consisting of a couple of an algebra and a
$d^+$-summable $K$-cycle does not yet fully deserve the name of
``non-commutative riemannian manifold'' (we propose below to call it a
``riemannian quantum space''): indeed it still lacks the important
feature of ``Poincar\'e duality''. Broadly speaking Poincar\'e duality
for classical manifolds consists in the isomorphism of homology and
cohomology. And it technically arises, in the appropriate
$KK$-theoretic langage, via the presence of a module over the algebra
of smooth functions. Now a module over an abelian algebra is a
bimodule (in a non-agressive way!): Alain Connes realized\footnote{
in fact whilst looking at the standard model!
} that the appropriate
non-commutative generalization consists in considering a couple $({\bf
A}',{\bf A}'')$ of algebras and a ${\bf A}'-{\bf A}''$-bimodule, in
other terms a ${\bf A}' \otimes {\bf A}''$-module, with conditions
ensuring the equality of the homology of ${\bf A}'$ and the cohomology
of ${\bf A}''$. It is a fascinating fact that the physical
``duality'' of the electroweak and chromodynamics sectors of
elementary particle physics can be viewed as an example (indeed the
historically first example!) of the non-commutative generalization of
Poincar\'e duality.
\bigskip
We begin with definitions fixing terminology (whereby (i) just renames
the lanscape of section~B not yet deserving the name of quantum
manifolds).
\begin{itemize}
\item[(i)] A {\it riemannian quantum space} is a couple $(A,H)$ of a
unital $^{\star}$-algebra $A$ and a $K$-cyle $H=(H,D,\chi)$ over $A$.
The quantum space $(A,H)$ is $d$-dimensional whenever the $K$-cyle $H$
is $d^+$-summable.
\item[(ii)] A {\it dual riemannian quantum space} is a riemannian
quantum space $(A \otimes A', H)$, where $A' \otimes A''$ is the
algebraic tensor product of the unital$^{\star}$-algebras $A',\ A''$
(with respective units
$1\!\!1'$ and
$1\!\!1''$); and where $H = (H,D,\chi)$ is a $d^+$-summable $K$-cyle
of $A' \otimes A''$ fullfilling the algebraic condition:
\beq
\left[ [ D,\, \underline{a' \otimes 1\!\!1 } ],\ \underline{1\!\!1
\otimes a'' } \right] = 0 \quad,\ a' \in A',\ a'' \in A'',
\eeq
\item[(iii)] A d-dimensional {\it riemannian quantum manifold} is a
dual riemannian quantum space
$(A' \otimes A'',H)$ whose $d^+$-summable $K$-cyle $H = (H,D,\chi)$
has its ``Hochschild obstruction'' vanishing under the operator B,
i.e.
\beq
Tr_{\omega} \left\{
\chi \underline a_0 [ D, \underline a_1 ] \dots [ D, \underline a_k ]
| D |^{-k} \right\} = 0 \quad,\ a_0,a_1, \dots, a_k \in A.
\eeq
\noindent (In order to alleviate notation, we shall consider ${\bf
A}'$ and ${\bf A}''$ as acting on $H$ by commuting representations
$a' \to \underline{a'} = \underline{a' \otimes 1\!\!1 }$ and $a'' \to
\underline{a''} = \underline{1\!\!1 \otimes a'' }$, then
reading
\beq
\left.
\left[ [ D,\, \underline{a'} ],\ \underline{a''} \right] = 0 \quad,
\ a' \in A',\ a'' \in A'' \right).
\eeq
\item[(iv)] The quantum space $(A,H),\ H = (H,D,\chi)$, is {\it
self-dual} whenever the operator algebra $\{ \underline a, a \in A \}$
is commutative, and one has:
\beq
\left[ [ D,\, \underline a ],\ \underline b \right] = 0 \quad,\ a, b
\in A.
\eeq
\end{itemize}
The {\it gauge group of the dual riemannian quantum space} $A'
\otimes A^{''}, H)$ is by definition the product ${\cal G}' \times
{\cal G}''$ of the respective gauge group ${\cal G}',\ {\cal G}''$
(= groups of unitaries of $A'$, resp. $A''$).
\bigskip
\noindent {\bf Remarks:}\footnote{
Note that condition (E.55) is symmetric in
$'$ and $''$.
}
\begin{itemize}
\item[(a)] With $(A' \otimes A'',H)$ a dual riemannian quantum space
we have that:
\beq
\left[ \pi_D ( \Omega A' ),\ \pi_D ( A'' ) \right] =
\left[ \pi_D ( \Omega A'' ),\ \pi_D ( A' ) \right] = 0,
\eeq
\item[(b)] There is a bijection between self-dual quantum spaces
$(A,H)$ and quantum mani\-folds $(A \otimes A, H)$.
\item[(c)] The couple of the algebra of smooth functions over a
compact spin$^c$ riemannian manifold, and its Dirac $K$-cycle is (the
archetype of) a self-dual quantum space.
\end{itemize}
The main mathematical point of the definition (iii) of a
$d$-dimensional riemannian quantum manifold is that it allows to
construct maps between homology and cohomology. We do not discuss
this point in these lectures devoted to physics and refer to [3a] for
it . For our physical applications we shall in fact only use the
structure (ii) of what we call a dual riemannian quantum space. We
shall be concerned with two items: tensor products on the one hand,
(bi)connections on the other.
\bigskip
\noindent {\bf Tensor products of dual riemannian quantum spaces}. We
prove the following result showing that the (algebraic condition of)
Poincar\'e duality met in Section~E~\S~6 boils down to Poincar\'e
duality for the inner space:
{\it The (tensor) product of two dual riemannian quantum spaces is a
dual riemannian quantum space.}
With $(A_1 = A'_1 \otimes A''_1, {\bf H}_1),\ {\bf H}_1 =
(H_1,D_1,\chi_1),\ (A_2 = A'_2 \otimes A''_2, {\bf H}_2)
{\bf H}_2 = (H_2,D_2,\chi_2)$, dual riemannian quantum spaces, we
consider the riemannian quantum space $(A_1 \otimes A_2, H_1 \otimes
H_2)$ tensor product of $(A_1,\ H_1)$ and $(A_2,\ H_2)$ as we defined
it in E~\S~1 (cf.(E.1)). With $a'_1 \in A'_1,\ a''_1 \in
A''_1,\ a'_2 \in A'_2,\ a''_2 \in A''_2$, we want to check that one
has:
\beq\label{E.58}
\left[ \left[ D, \ul{a'_1} \otimes \ul{a'_2} \right],\
\ul{a''_1} \otimes \ul{a''_2} \right] = 0
\eeq
We have, since $\ul{a'_1}$ is even:
\beq
\left( D_1 \otimes id + \chi_1 \otimes D_2 \right)
\left( \ul{a'_1} \otimes \ul{a'_2} \right) =
D_1 \ul{a'_1} \otimes \ul{a'_2} +
\chi_1 \ul{a'_1} \otimes D_2 \ul{a'_2},
\eeq
\beq
\left( \ul{a'_1} \otimes \ul{a'_2} \right)
\left( D_1 \otimes id + \chi_1 \otimes D_2 \right) =
\ul{a'_1} D_1 \otimes \ul{a'_2} +
\chi_1 \ul{a'_1} \otimes \ul{a'_2} D_2,
\eeq
hence
\beq
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right] =
\left[ D_1, \ul{a'_1} \right] \otimes \ul{a'_2} +
\chi_1 \ul{a'_1} \otimes
\left[ D_2, \ul{a'_2} \right],
\eeq
and further
\beq
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right]
\left( \ul{a''_1} \otimes \ul{a''_2} \right) =
\left[ D_1, \ul{a'_1} \right] \ul{a''_1} \otimes \ul{a'_2 a''_2}
+ \chi_1 \ul{a'_1 a''_1} \otimes
\left[ D_2, \ul{a'_2} \right] \ul{a''_2},
\eeq
\beq
\left( \ul{a''_1} \otimes \ul{a''_2} \right)
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right] =
\ul{a''_1} \left[ D_1, \ul{a'_1} \right] \otimes
\ul{a''_2 a'_2} +
\chi_1 \ul{a''_1 a'_1} \otimes \ul{a''_2}
\left[ D_2, \ul{a'_2} \right] ,
\eeq
hence, since $\ul{a'_1}$ and $\ul{a''_1}$
commute:
\beq
\left[ \left[ D, \ul{a'_1} \otimes \ul{a'_2} \right],\
\ul{a''_1} \otimes \ul{a''_2} \right] =
\left[ \left[ D_1, \ul{a'_1} \right],\ \ul{a''_1} \right]
\otimes \ul{a'_2 a''_2}
+ \chi_1 \ul{a'_1 a''_1} \otimes
\left[ \left[ D_2, \ul{a'_2} \right],\ \ul{a''_2} \right],
\eeq
vanishing by the duality property of the systems 1 and 2.
\bigskip
\noindent {\bf Biconnections.} Our problem is the definition of
connections of dual riemannian quantum spaces. Indeed the passage
{}from riemannian quantum spaces involving one algebra to dual quantum
spaces involving a commuting\footnote{
in the representation $\pi_D$.
} couple of algebras $(A',A'')$ (i.e.
their tensor product $A = A' \otimes A''$ together with its tensorial
splitting) raises questions as to the maintenance in this
enlarged\footnote{
enlarged in view of Remark (c).
} frame of the previous philosophy pertaining to
riemannian quantum spaces (and of the latter's implications on
connections, the quantum Yang-Mills algorithm, etc.). Taking the
tensor product $A = A' \otimes A''$ as the new ``basic'' algebra, one
is at first tempted to consider the differential envelope ($ \Omega
A,\delta $) as the ``basic'' differential algebra. However the latter
does not contain the information of the commutativity of $A'$ and
$A''$ and of the vanishing of commutators~\rf{E.58} in the
representation
$\pi_D$. This information is incorporated in each of the following
items:
\begin{itemize}
\item[(1)] Definition. Let $(A, {\bf H}),\ A = A' \otimes A'',\
{\bf H} = (H,D,\chi)$, be a dual riemannian quantum space, with
$(\Omega A, \delta)$ the unital differential envelope of $A$; and
consider the following subsets of $\Omega A^1$:\footnote{
Note that ${\bf S}' = - {\bf S}''$ because $\delta \left[
1\!\!1' \otimes a'', a' \otimes 1\!\!1'' \right] = 0 = \left[ \delta
(1\!\!1' \otimes a''),\ a' \otimes 1\!\!1'' \right] + \left[ 1\!\!1'
\otimes a'', \delta (a' \otimes 1\!\!1'') \right].$
}
\beq
{\bf S}' = \left\{
\left[ \delta (a' \otimes 1\!\!1''),\ 1\!\!1' \otimes a'' \right]
\in \Omega A^1 ; a' \in A', a'' \in A''
\right\} = - {\bf S}''
\eeq
resp.
\beq
{\bf S}'' = \left\{
\left[ \delta (1\!\!1' \otimes a''),\ a' \otimes 1\!\!1'' \right]
\in \Omega A^1 ; a' \in A', a'' \in A''
\right\}.
\eeq
We denote by ${\bf j}_P$ the ideal of $\Omega A$ generated by ${\bf
S}'$ (or, for that matter, by ${\bf S}''$), and define the
{\it Poincar\'e ideal} as:
\beq
{\bf k}_P = {\bf j}_P + \delta {\bf j}_P .
\eeq
\item[(2)] Definition. Let $A'$ and $A''$ be unital (real or
complex) $^{\star}$-algebra, with respective units $1\!\!1'$ and
$1\!\!1''$, and with respective unital differential envelopes
$(\Omega A',\delta')$ and $(\Omega A'',\delta'')$. Let $A = A'
\otimes A''$, with unit $1\!\!1$ and unital differential envelope
$(\Omega A, \delta)$, be the algebraic tensor-product of $A'$ and
$A''$. Set $\Omega A_{\otimes} = \Omega A' \widehat{\otimes} \Omega
A''$, with differential $\delta_{\otimes}$, for the skew tensor
product of $\Omega A'$ and $\Omega A''$ defined as follows: we have,
for $\omega', \psi' \in A', \omega'', \psi'' \in A''$:
\end{itemize}
\beq
\left\{
\begin{array}{l}
\Omega A_{\otimes}^n = \sum_{p + q = n} \Omega A'^p \otimes
\Omega A''^q, \\
\\
(\omega' \otimes \omega'') (\psi' \otimes \psi'') = \omega' \psi'
\otimes \omega'' \psi'' \\
\\
\delta_{\otimes} (\omega' \otimes \omega'') = \delta' \omega' \otimes
\omega'' + \omega' \otimes \delta \omega''
\end{array}
\right. .
\eeq
\noindent We now have the following consequences of Definition (1):
\begin{itemize}
\item[(i):] ${\bf j}_P$ is a graded ideal of $\Omega A$.
\item[(ii):] ${\bf k}_P$ is a graded differential ideal of $\Omega
A$, with ${\bf k}_P^n = {\bf j}_P^n + \delta {\bf j}_P^{n-1}$, hence
$\Omega A/{\bf k}_P$ is a {\bf N}-graded differential algebra, with
differential $d$ obtained from that of $\Omega A$ by passage to the
quotient through the ideal ${\bf k}_P$.
\item[(iii):] The quantum DeRham complex $\Omega A_D$ is a
quotient of $\Omega A/{\bf k}_P$ as a {\bf N}-graded differential
algebra.
\item[(iv):] One has $\left[ \delta (a' \otimes 1\!\!1''),
\delta (1\!\!1' \otimes a'') \right] \in {\bf k}_P^2,\ a' \in A', a''
\in A''$.
\item[(v):] The ideal ${\bf k}_P$ is generated by
the elements $\left[ \delta (a' \otimes 1\!\!1''),
1\!\!1' \otimes a'' \right]$ and $\left[ \delta (a' \otimes 1\!\!1''),
\right.$\break
$\left. \delta (1\!\!1' \otimes a'') \right],\ a' \in A', a'' \in A''$.
\end{itemize}
and the following consequence of Definition (2):
The map $\ol i : \Omega A \to \Omega A_{\otimes}$ specified
as:
\beq
\begin{array}{l}
\ol i \left[ (a'_0 \widehat{\otimes} a''_0) \delta (a'_1
\widehat{\otimes} a''_1) \dots \delta (a'_n \widehat{\otimes} a''_n)
\right] = (a'_0 \widehat{\otimes} a''_0) \delta_{\otimes} (a'_1
\widehat{\otimes} a''_1) \dots \delta_{\otimes} (a'_n
\widehat{\otimes} a''_n) \\
\\
= (a'_0 \widehat{\otimes} a''_0) \left[ \delta' a'_1
\widehat{\otimes} a''_1 + a'_1 \widehat{\otimes} \delta'' a''_1
\right] \dots \left[ \delta' a'_n
\widehat{\otimes} a''_n + a'_n \widehat{\otimes} \delta'' a''_n
\right],
\end{array}
\eeq
is a homomorphism of {\bf N}-graded differential algebras.
Furthermore we have the important fact that {\it the kernel of
$\ol i$
coincides with the Poincar\'e ideal ${\bf k}_P$. Consequently we
have the following isomorphism of {\bf N}-graded differential
algebras:}
\beq
( \Omega {\bf A} / {\bf k},\ \delta ) \cong ( \Omega
{\bf A}_{\otimes},\ \delta_{\otimes} ).
\eeq
{\it and the canonical map $\psi : \Omega {\bf A} \to \Omega {\bf
A}_D$ factors through the canonical map $\phi : \Omega {\bf A} \to
\Omega {\bf A}_{\otimes}$ as $\psi = \xi \circ \phi$, where $\xi :
\Omega {\bf A}_{\otimes} \to \Omega {\bf A}_D$ is a homomorphism of
{\bf N}-graded differential algebras.}
We are now in a position to discuss the issue of ``connections'' of
the dual riemannian quantum space $(A = A' \otimes {\bf A}'', {\bf
H}), {\bf H} = (H, D, \chi)$. The quantum forms of ${\bf A} $, if we
forget about its tensorial decomposition ${\bf A} = {\bf A}' \otimes
{\bf A}''$, are the elements of the quantum DeRham complex $(\Omega
{\bf A}_D, {\bf d})$, its connections are thus the grade-one {\bf
d}-derivations $\nabla$ of
$\Omega {\bf A}_D$. Now, utilizing the homomorphism $\xi : \Omega
{\bf A}_{\otimes} \to \Omega {\bf A}_D$ (cf.[6]), the latter can be
considered as $\xi$-images of $\delta$-derivations of $\Omega
{\bf A}_{\otimes}$ (``preconnexions''), this according to the scheme:
\noindent Let $\phi : (\Omega, \delta) \to (\widetilde{\Omega},
\widetilde{\delta})$ be an epimorphism of {\bf N}-graded differential
algebras, and denote by $Der_{\delta} \Omega$ the set of grade-one
$\delta$-derivations of $\Omega$ considered as a right
$\Omega$-module (resp. by $Der_{ \widetilde{\delta} }
\widetilde{\Omega}$ the set of $\widetilde{\delta}$-derivations of
$\widetilde{\Omega}$ considered as a right
$\widetilde{\Omega}$-module). Then, given
$\widetilde{\nabla} \in Der_{ \widetilde{\delta} }
\widetilde{\Omega}$, there is $\nabla \in Der_{\delta} \Omega $
such that $\phi$ intertwines $\nabla$ and
$\widetilde{\nabla} : \phi \circ \nabla = \widetilde{\nabla} \circ
\phi$. Specifically, for $\widetilde{\nabla} = \widetilde{\delta} +
\widetilde{\rho}, \rho' \in \widetilde{\Omega}^1$, one has $\nabla =
\delta + \rho$, picking $\rho \in \Omega^1$ such that $\phi \rho =
\widetilde{\rho}$.\footnote{
In the expression $\nabla = \delta + \rho$ the symbol $\rho$ has to
be interpreted as usual as denoting multiplication from the left by
$\rho$.
} The respective
curvatures $\theta$ and $\widetilde{\theta}$ of $\nabla$ and
$\widetilde{\nabla}$ are related by $\widetilde{\theta} = \phi
\theta$.
Our problem was to select, amongst all connexions of $A$, the ones
matching the tensorial decomposition $A = A' \otimes A''$. Looking at
the preconnections acting on $\Omega A_{\otimes} = \Omega A'
\widehat{\otimes} \Omega A''$, one is now naturally led
to the following notion of ``biconnection''.
Definitions. Let $(A = A' \otimes A'', {\bf H})$, with ${\bf H}
= (H, D, \chi)$, be a dual riemannian quantum space, and consider the
above {\bf N}-graded differential algebras $(\Omega
{\bf A}_{\otimes}, \delta)$ and $(\Omega {\bf A}_D, {\bf d})$.
\begin{itemize}
\item[(i):] a {\it preconnection of} $({\bf A}, {\bf H})$ is a
grade-one graded $\delta$-derivation $\nabla$ of the right $\Omega
{\bf A}_{\otimes}$-module $\Omega {\bf A}_{\otimes}$,
with {\it curvature} is $\theta = \nabla^2$. The {\it
preconnection-one form} $\rho \in \Omega {\bf A}_{\otimes}^1$ arises
by asking $\nabla = \delta + \rho$ (one has thus $\theta = \delta
\rho + \rho^2$). The action $\nabla \to \nabla^{\bf u}$ of the element
{\bf u} of the gauge group ${\cal G}$ of ${\bf A}$ on the
preconnection is by definition:\footnote{
${\cal G}$ is the group of unitaries of ${\bf A} = \Omega
{\bf A}_{\otimes}^0$.
}
\beq
\nabla^{\bf u} \omega = {\bf u} \nabla ({\bf u}^{\star} \omega)
\eeq
(one has then $\nabla^{\bf u} = \delta + \rho^{\bf
u}$ with
\beq
\rho^{\bf u} = {\bf u} \rho {\bf u}^{\star} + {\bf u} \delta {\bf
u}^{\star}).
\eeq
\item[(ii):] a {\it biconnection} is a preconnection of the form
$\nabla = \nabla' \widehat{\otimes} id + id \widehat{\otimes}
\nabla''$, where $\nabla'$ and $\nabla''$ are respective connections
of ${\bf A}'$ and ${\bf A}''$.\footnote{
Thus $\nabla'$ is a grade-one $\delta'$-derivation of $\Omega {\bf
A}'$, with $\nabla' = \delta' + \rho', \rho' \in \Omega
{\bf A}_{\otimes}^{' 1}$, and $\nabla''$ is a grade-one
$\delta''$-derivation of $\Omega {\bf A}''$ with $\nabla'' = \delta''
+ \rho'', \rho'' \in \Omega {\bf A}_{\otimes}^{'' 1}$.
} (one has thus $\rho = \rho' \otimes 1\!\!1'' + 1\!\!1' \otimes
\rho''$ for the preconnection-one form $\rho$ of $\nabla$)
\item[(iii):] a {\it connection of the dual riemannian quantum
space} $({\bf A} = {\bf A}' \otimes {\bf A}'', {\bf H})$ is a
grade-one graded
$\delta$-derivation $\nabla$ of the right $\Omega {\bf A}_D$-module
$\Omega {\bf A}_D$ which is the image of a biconnection in the
homomorphism $\xi : \Omega {\bf A}_{\otimes} \to \Omega {\bf A}_D$.
\end{itemize}
We check that $\nabla = \nabla' \widehat{\otimes} id + id
\widehat{\otimes} \nabla''$ is a preconnection with preconnection-one
form $\rho = \rho' \otimes 1\!\!1'' + 1\!\!1' \otimes
\rho''$: for homogeneous $\omega', \psi' \in \Omega {\bf A}',
\omega'',
\psi'' \in \Omega {\bf A}''$, we have:
\beq
\begin{array}{ll}
\nabla (\omega' \widehat{\otimes} \omega'') & = \nabla' \omega'
\widehat{\otimes} \omega'' + (-1)^{\partial \omega'} \omega'
\widehat{\otimes} \nabla'' \omega'' \\
& = (\delta' + \rho') \omega' \widehat{\otimes} \omega'' +
(-1)^{\partial \omega'} \omega' \widehat{\otimes} (\delta'' + \rho'')
\omega'' \\
& = \delta (\omega' \widehat{\otimes} \omega'') + (\rho' \otimes
1\!\!1'' + 1\!\!1' \otimes \rho'') (\omega' \widehat{\otimes}
\omega'').
\end{array}
\eeq
We then have the following satisfying set of facts:
{\it
Let $\nabla = \nabla' \widehat{\otimes} id + id
\widehat{\otimes} \nabla'' = \delta + \rho,\ \rho = \rho'
\widehat{\otimes} 1\!\!1'' + 1\!\!1' \widehat{\otimes} \rho''$, be a
biconnection of $(A = A' \otimes A'', {\bf H})$. Then }
\begin{itemize}
\item[(i):] {\it The curvature $\theta$ of $\nabla$ is the sum of the
curvatures $\theta'$ and $\theta''$ of $\rho'$ and $\rho''$ in the
following sense: one has: }
\beq
\nabla^2 = \nabla'^2 \widehat{\otimes} id + id \widehat{\otimes}
\nabla''^2,
\eeq
\beq
\theta = \theta' \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes}
\theta'',
\eeq
\beq
\rho^2 = \rho'^2 \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes}
\rho''^2.
\eeq
\item[(ii):] {\it The set of biconnections is stable under the gauge
group ${\cal G}' \times {\cal G}''$ of the dual riemannian quantum
space $(A = A' \otimes A'', {\bf H})$ one has, for ${\bf u}' \in
{\cal G}', {\bf u}'' \in {\cal G}'', {\bf u} = {\bf u}'
\widehat{\otimes} {\bf u}''$: }
\end{itemize}
\beq
\nabla^{\bf u} = \nabla'^{\bf u'} \widehat{\otimes} id + id
\widehat{\otimes} \nabla''^{\bf u''},
\eeq
\beq
\rho^{\bf u} = \rho'^{\bf u'} \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes} \rho''^{\bf u''}.
\eeq
\bigskip
\noindent {\bf \S 6. ~~ The modularity condition}
{}~~~~
We recall that, putting together in Section~E \S 4 the electroweak and
chromodynamics sectors, we had the following commuting representations
of the electroweak algebra ${\cal A}$:
\beq
\pi_l ((f, q)) = \bordermatrix{
& e_R & \nu_L e_L \cr
\cr
& \ul f \otimes \bone_N & 0 \cr
\cr
& 0 & \ul q \otimes \bone_N \cr }
\begin{array}{l}
e_R \\
\\
\nu_L e_L \\
\end{array}
\qquad ,\ (f, q) \in {\cal A},
\eeq
\bigskip
\beq
\pi_q ((f, q)) = \bordermatrix{
& u_R & d_R & u_L d_L \cr
\cr
& \ol{\ul f} \otimes \bone_N & 0 & 0 \cr
\cr
& 0 & \ul f \otimes \bone_N & 0 \cr
\cr
& 0 & 0 & \ul q \otimes \bone \cr }
\begin{array}{l}
u_R \\
\\
d_R \\
\\
{\dsp u_L \atop \dsp d_L} \\
\end{array}
\qquad ,\ (f, q) \in {\cal A},
\eeq
resp. the chromodynamics algebra ${\cal B}$:
\beq
\pi_l ((f', m)) = \pi_l ((f', 0)) = \bordermatrix{
& e_R & \nu_L e_L \cr
\cr
& \ul{f'} \otimes \bone_N & 0 \cr
\cr
& 0 & \ul{f'} \otimes \bone_N \cr }
\begin{array}{l}
e_R \\
\\
{\dsp \nu_L \atop \dsp e_L} \\
\end{array}
\qquad ,\ (f', m) \in {\cal B},
\eeq
\bigskip
\beq
\pi_q ((f', m)) = \pi_q ((0, m)) = \bordermatrix{
& u_R & d_R & u_L d_L \cr
\cr
& m_k^j \otimes \bone_N & 0 & 0 \cr
\cr
& 0 & m_k^j \otimes \bone_N & 0 \cr
\cr
& 0 & 0 & m_k^j \otimes \bone_N \cr }
\otimes e_j^k
\begin{array}{l}
u_R \\
\\
d_R \\
\\
{\dsp u_L \atop \dsp d_L} \\
\end{array}
\ ,\ (f', m) \in {\cal B},
\eeq
with the corresponding respective representations of their gauge
groups ${\cal G}' = U (1) \times SU (2) = \left\{ (u,v) ; u \in
C^{\infty} ({\bf M}, U (1)),\ v \in C^{\infty} ({\bf M}, SU (2))
\right\}$:
\beq
\pi_l ((u, v)) = \bordermatrix{
& e_R & \nu_L e_L \cr
\cr
& \ul u \otimes \bone_N & 0 \cr
\cr
& 0 & \ul q \otimes \bone_N \cr }
\begin{array}{l}
e_R \\
\\
{\dsp \nu_L \atop \dsp e_L} \\
\end{array}
\qquad ,\ (u, v) \in {\cal G}',
\eeq
\bigskip
\beq
\pi_q ((u, v)) = \bordermatrix{
& u_R & d_R & u_L d_L \cr
\cr
& \ol{\ul u} \otimes \bone_N & 0 & 0 \cr
\cr
& 0 & \ul u \otimes \bone_N & 0 \cr
\cr
& 0 & 0 & \ul q \otimes \bone_N \cr }
\otimes \bone_3
\begin{array}{l}
u_R \\
\\
d_R \\
\\
{\dsp u_L \atop \dsp d_L} \\
\end{array}
\qquad,\ (u, v) \in {\cal G}',
\eeq
respectively of ${\cal G}'' = U (1) \times U (1) \times SU (3) =
\left\{ (u', u'', v') ; u', u'' \in C^{\infty} ({\bf M}, U
(1)), v'\in C^{\infty} ({\bf M}, SU (3))
\right\}$:
\beq
\pi_l ((u', u'', v')) = \pi_l ((u', 0)) = \bordermatrix{
& e_R & \nu_L e_L \cr
\cr
& \ul{u'} \otimes \bone_N & 0 \cr
\cr
& 0 & \ul{u'} \otimes \bone_N \cr }
\begin{array}{l}
e_R \\
\\
{\dsp \nu_L \atop \dsp e_L} \\
\end{array}
\ ,\ (u', u'', v') \in {\cal G}'',
\eeq
\bigskip
\beq
\pi_q ((u', u'', v')) = \pi_q ((0, u'', v')) = \bordermatrix{
& u_R & d_R & u_L d_L \cr
\cr
& \ul{u''} \otimes \bone_N & 0 & 0 \cr
\cr
& 0 & \ul{u''} \otimes \bone_N & 0 \cr
\cr
& 0 & 0 & \ul{u'} \otimes \bone_N \cr }
\otimes \ul{v'}
\begin{array}{l}
u_R \\
\\
d_R \\
\\
{\dsp u_L \atop \dsp d_L} \\
\end{array}
\ ,\ (u', u'', v') \in {\cal G}'',
\eeq
We have at this point the plethoral inner symmetry group $U (1) \times
SU (2) \times U (1) \times U (1) \times SU (3)$, which phenomenology
commands us to reduce to the usual $U (1) \times
SU (2) \times SU (3)$: we must thus coalesce the threefold $U (1)
\times U (1) \times U (1)$ to a single $U (1)$, a task which will be
achieved by imposing the ``modularity condition'' below. Before
describing the latter, let us however derive the way in which the
coalescence should arise: we can read this off from the table:
\renewcommand{\arraystretch}{1.7}
\beq
\begin{array}{lcccccccc}
\multicolumn{4}{c}{\mbox{\bf Leptons}} & \qquad
& \multicolumn{4}{c}{\mbox{\bf Quarks}}\\
& e_R & \nu_L & e_L & \qquad & u_R & d_R & u_L & d_L \\
Y & -2 & -1 & -1 & \qquad & 4/3 & -2/3 & 1/3 & 1/3 \\
U & -1 & 0 & 0 & \qquad & 1 & -1 & 0 & 0 \\
U' & -1 & -1 & -1 & \qquad & 0 & 0 & 0 & 0 \\
U'' & 0 & 0 & 0 & \qquad & -1 & -1 & -1 & -1 \\
\end{array}
\eeq
\renewcommand{\arraystretch}{1}
\noindent where we plotted the hypercharge $Y$ and the infinitesimal
generators
$U, U', U''$ defined by $u = e^{i U t},\ u' = e^{i U' t},\ u'' = e^{i
U'' t}$: we see that we need:
\beq
\left\{
\begin{array}{l}
U = U' = -Y \\
\\
U'' = -{1 \over 3} Y \\
\end{array}
\right.
\qquad \mbox{i.e.} \qquad
\left\{
\begin{array}{l}
u = u' = e^{-i Y t} \\
\\
u'' = e^{-{i \over 3} Y t} \\
\end{array}
\right. \cdot
\eeq
We now discuss the modularity condition. For the latter we shall use
the following definition of the {\bf phase} $\phi (u)$ {\bf of the
determinant} of a unitary $u$ belonging to a unital
$C^{\star}$-algebra ${\cal C}$ endowed with a hermitean normalized
trace $\tau$:\footnote{
i.e. $\tau (a^{\star}) = \tau \ol{(a')}$ and $\tau (\bone) = 1$.
}
\beq
\begin{array}{l}
\phi (u) = {1 \over 2 \pi i} \int_0^1 \tau \left[ u (t)' u (t)^{-1}
\right] dt \\
\\
( t \to u (t) \ \mbox{a continuous path of unitaries from
1 to}\ u).
\end{array}
\eeq
This formula defines $\phi (u)$ for $u$ in the connected component
$U_0 ({\cal C})$ of the group of unitaries $U ({\cal C})$ of ${\cal
C}$ (in fact up to the image $\tau (K_0 ({\cal C}))$ of the
$K_0$-group of ${\cal C}$ under the trace $\tau$ - a countable
subgroup of ${\bf R}$). By the unitarity invariance of the trace
$\tau$, one has, for all $s \in U ({\cal C}), \phi (\mbox{sus}^{-1}) =
\phi (u)$, hence $\{ u \in U_0 ({\cal A}) ; \phi (u) = 1 \}$ is a
normal subgroup of $U_0 ({\cal C})$. We set:
\beq
U_{\tau} ({\cal C}) = \ \mbox{component of the unit of} \{ u \in U_0
({\cal C}) ; \phi (u) = 1 \}.
\eeq
The name we gave to $\phi (u)$ is heuristically justified as follows
{}from the familiar formula giving the determinant of the exponential of
a self-adjoint $h$ acting on the Hilbert space ${\bf C}^n$ in terms of
$\tau = {1 \over n} Tr$:
\beq
Det \ e^{2 \pi ih} = e^{2 \pi i \tau (h)}.
\eeq
With $u = e^{2 \pi ih}$, we have indeed $h = {1 \over 2 \pi i} lnu$,
hence
$\phi (u) = \tau (h) = {1 \over 2 \pi i} \tau (lnu)$, leading to
(E.89).\footnote{
For a rigourous exposition of (E.89) we refer to the
paper by P.~De la Harpe and G.~Skandalis, Ann. Inst. Fourier (1986).
}
We now apply the notion (E.89) to the situation of the full standard
model, with ${\cal C} = \Omega_D ({\cal A} \otimes {\cal B})$, and
the following family of traces of ${\cal C}$.
\beq
\tau_{\rho} (a) = Tr_{\omega} \{ D^{-4} \rho x \} \kern 60pt , x \in
{\cal C},
\eeq
where $\rho$ ranges through the self-adjoint elements of the center of
${\cal A}$. Setting:
\beq
U_{\cal A} ({\cal C}) = \build{\cap}_{\rho}^{} U_{\tau_{\rho}} ({\cal
C}),
\eeq
we get a subgroup of the unitarity group of $\Omega_D ({\cal A}
\otimes {\cal B})$, i.e. a subgroup of the gauge group ${\cal G}'
\times {\cal G}''$. Now {\it this subgroup turns out to be the group
$U (1) \times SU (2) \times SU (3)$ obtained through the mechanism}
(E.88) {\it necessary to match the hypercharge.}
\bigskip
\noindent {\bf \S 7. ~~ The fermionic action}
{}~~~~
By definition the fermionic action equals:\footnote{
$F$ in ${\bf L}_F$ stands for
fermionic.
}
\beq
{1\over 8 \pi^2} {\bf L}_F dv
\eeq
with the fermionic Lagrangian:
\beq
{\bf L}_F = {\bf L}_{F l} + {\bf L}_{F q} = \left(
\Psi, {\bf D}_{\Delta l} \Psi \right) + \left(
{\bf Q}, {\bf D}_{\Delta q} {\bf Q} \right),
\eeq
specified as follows:
\begin{itemize}
\item[---] the leptonic field $\Psi \in {\bf H}_l = H_l$, is
\end{itemize}
\beq
\Psi = \Psi = \left(
\Psi^R, \Psi^L \right) \ \mbox{with} \
\left\{
\begin{array}{l}
\Psi^R = \left( \Psi^R_f \right)_{f = 1, \dots , N} \in H^R_l \otimes
{\bf C}^N \\
\\
\Psi^L = \left( \Psi^{L 1}_f , \Psi^{L 2}_f \right)_{f = 1, \dots , N}
\in H^L_l \otimes {\bf C}^2_{iso} \otimes {\bf C}^N
\end{array}
\right. ;
\eeq
\indent whilst the operator ${\bf D}_{\Delta l}$ of ${\bf H}_l = H_l$
is given by:
\beq
{\bf D}_{\Delta l} = {\bf D} + i \left( V_l \build{+}_{}^{\approx}
V'_l \right),
\eeq
\begin{itemize}
\item[---] the quark field ${\bf Q} \in {\bf H}_q = H_q \otimes {\bf
C}^3_{color}$, is:
\end{itemize}
\beq
{\bf Q} = \left( {\bf Q}^R,\ {\bf Q}^L \right) = \left( {\bf
Q}_f^{R u},\ {\bf Q}_f^{R d},\ {\bf Q}_f^{L 1},\ {\bf Q}_f^{L 2}
\right)_{f = 1, \dots , N},
\eeq
\indent with
\beq
\left\{
\begin{array}{l}
{\bf Q}_f^{R u} = {\bf Q}_f^{R u m} \otimes e_m \\
\\
{\bf Q}_f^{R d} = {\bf Q}_f^{R d m} \otimes e_m \\
\\
{\bf Q}_f^{L 1} = {\bf Q}_f^{L 1 m} \otimes e_m \\
\\
{\bf Q}_f^{L 2} = {\bf Q}_f^{L 2 m} \otimes e_m
\end{array}
\right. (\mbox{summation on}\ m\ \mbox{from 1 to 3}),
\eeq
with $\{ e_m \}$ the canonical basis of ${\bf C}^3_{color}$;
whilst the operator ${\bf D}_{\Delta q}$ of ${\bf H}_q = H_q
\otimes {\bf C}^3_{color}$ is given by:
\beq
{\bf D}_{\Delta q} = {\bf D} + i \left( V_q \build{+}_{}^{\approx}
V'_q \right) = D_{\Delta q} \otimes 1\!\!1_3 + i V'_q .
\eeq
The letter $f$ denotes the fermion-family index, $R$ and $L$ stand
respectively for right and left, and the spinor fields correspond to
the following particle-types:
\beq
\left\{
\begin{array}{llll}
\mbox{Leptons:} & \Psi^R : e_R & \Psi_f^{L 1} : \nu_L &
\Psi_f^{L 2} : e_R \\
& & &\\
\mbox{Quarks:} & {\bf Q}_f^{R u} : u_R \quad {\bf Q}_f^{R d} :
d_L
& {\bf Q}_f^{L 1} : u_L & {\bf Q}_f^{L 2} : d_L
\end{array}
\right. .
\eeq
{\it One then has the fermionic Lagrangian ${\bf L}_F = {\bf L}_{Fl}
+ {\bf L}_{Fq}$, with the leptonic term:}
\beq
\begin{array}{ll}
{\bf L}_{Fl} = & \Sigma_{f = 1, \dots , N} \left\{
\overline{\Psi}_f^R i \gamma^{\mu} D_{\mu}^{Rl} \Psi_f^R +
\overline{\Psi}_f^L i \gamma^{\mu} D_{\mu}^{Rl} \Psi_f^L \right\} \\
& \\
& \kern -0,3cm + \Sigma_{f_1 f_2 = 1, \dots , N} \left\{
M_{e f_1 f_2}^{\star}
\overline{\Psi}_{f_1}^R \overline{\Phi} \gamma^5 \Psi_{f_2}^L +
M_{e f_1 f_2} \overline{\Psi}_{f_1}^L \Phi \gamma^5 \Psi^R \right\}
\end{array}
\eeq
{\it with the covariant derivatives}
\beq
\left\{
\begin{array}{l}
D_{\mu}^{Rl} = \nabla_{\mu} - 2 i {\bf a}_{\mu} \\
\\
D_{\mu}^{Ll} = \nabla_{\mu} - i {\bf a}_{\mu}
- i {\bf b}_{\mu} = \nabla_{\mu} - i a_{\mu}
- i {\bf b}_{\mu}^a {\tau_a \over 2}
\end{array}
\right. ,
\eeq
{\it and the quark term:}
\beq
\begin{array}{ll}
{\bf L}_{Fq} = & \Sigma_{f = 1, \dots , N} \left\{
\overline{\bf Q}_{f m}^{R u} i \gamma^{\mu} D_{\mu}^{R u}
{\bf Q}_f^{R u m} + \overline{\bf Q}_{f m}^{R d} i \gamma^{\mu}
D_{\mu}^{R d} {\bf Q}_f^{R d m} \right. \\
& \\
& \left. \! \hphantom{
\Sigma_{f = 1, \dots , N} \left\{
\overline{\bf Q}_{f m}^{R u} i \gamma^{\mu} D_{\mu}^{R u}
{\bf Q}_f^{R u m} \right.
} +
\overline{\bf Q}_{f m}^L i \gamma^{\mu} D_{\mu}^{L q}
{\bf Q}_f^{L m} \right\} \\
& \\
& \! \! \! \! + \Sigma_{f_1 f_2 = 1, \dots , N} \left\{
M_{d f_1 f_2}^{\star}
\overline{\bf Q}_{f_1 m}^{R d} \overline{\Phi}
\gamma^5 {\bf Q}_{f_2}^{L m} + M_{e f_1 f_2} {\bf Q}_{f_1 m}^L \Phi
\gamma^5 {\bf Q}_{f_2}^{R d m} \right\} \\
& \\
& \! \! \! \! + \Sigma_{f_1 f_2 = 1, \dots , N} \left\{
M_{u f_1 f_2}^{\star}
\overline{\bf Q}_{f_1 m}^{R u} \overline{ \widetilde{\Phi} }
\gamma^5 {\bf Q}_{f_2}^{L m} + M_{u f_1 f_2}
\overline{\bf Q}_{f_1 m}^L \widetilde{\Phi}
\gamma^5 {\bf Q}_{f_2}^{R u m} \right\}
\end{array}
\eeq
{\it with the covariant derivatives}
\beq
\left\{
\begin{array}{l}
D_{\mu}^{R u} = \nabla_{\mu} + {4 \over 3} i {\bf a}_{\mu}
- i {\bf c}_{\mu} = \nabla_{\mu} + {4 \over 3} i {\bf a}_{\mu}
- i {\bf c}_{\mu}^a {\lambda_a \over 2} \\
\\
D_{\mu}^{R d} = \nabla_{\mu} - {2 \over 3} i {\bf a}_{\mu}
- i {\bf c}_{\mu} = \nabla_{\mu} - {2 \over 3} i {\bf a}_{\mu}
- i {\bf c}_{\mu}^a {\lambda_a \over 2} \\
\\
D^{L q} = \nabla_{\mu} - i {\bf b}_{\mu} + {1 \over 3}
i {\bf a}_{\mu} - i {\bf c}_{\mu} = \nabla_{\mu}
- i {\bf b}_{\mu}^i {\tau_i \over 2} + {1 \over 3} i {\bf a}_{\mu}
- i {\bf c}_{\mu}^a {\lambda_a \over 2}
\end{array}
\right. .
\eeq
Here both sides of these formulae refer to space-time fields dependent
on a (non specified) space-time point $x \in M$. With ${\bf
C}_{spin}^4,\ {\bf C}_{iso}^2$, and ${\bf C}_{color}^3$ the respective
spaces of $O(4)$-spinors, $SU(2)$-vectors, and $SU(3)$-vectors, we
have $\Psi^R \in {\bf C}_{spin}^4,\ \Psi^L \in {\bf C}_{spin}^4
\otimes {\bf C}_{iso}^2,\ {\bf Q}^{R d},\ {\bf Q}^{R u} \in
{\bf C}_{spin}^4 \otimes {\bf C}_{color}^3,\ {\bf Q}^L
\in {\bf C}_{spin}^4 \otimes {\bf C}_{iso}^2
\otimes {\bf C}_{color}^3,\ \Phi, \widetilde{\Phi} \in
{\bf C}_{iso}^2$. Correspondingly, symbols like
$\overline{\Psi}^R \Psi^R$ (resp. $\overline{\Psi}^L {\Psi}^R,\
\overline{\bf Q}_m^R {\bf Q}^{R m},\ \overline{\bf Q}_m^L {\bf Q}^{L
m}$) denote a scalar product in ${\bf C}_{spin}^4$ (resp. ${\bf
C}_{spin}^4 \otimes {\bf C}_{iso}^2,\ {\bf C}_{spin}^4 \otimes {\bf
C}_{color}^3,\ {\bf C}_{spin}^4 \otimes {\bf C}_{iso}^2 \otimes {\bf
C}_{color}^3$). Symbols like
$\overline{\bf Q}^R \overline{\Phi} {\bf Q}^L$ (resp. $\overline Q^L
\Phi Q^R$) denote a scalar product of elements $\overline{\bf Q}^R
\otimes \overline{\Phi}$ and ${\bf Q}^L$ (resp. $\overline{\bf Q}^L$
and $\Phi \otimes {\bf Q}^R$) in ${\bf C}_{spin}^4 \otimes {\bf
C}_{iso}^2$ (given by ${\bf Q}_{\alpha}^R \Phi_k {\bf Q}^{L \alpha
k}$ (resp. ${\bf Q}_{\alpha k}^L \Phi^k {\bf Q}^{R a}$ ) in terms of
coordinates). The operators included in the above tensor products
(e.g. $\gamma_{\mu},\ \gamma^5,\ {\bf a}_{\mu}$) are understood to
be tensorized by the appropriate unit operators.
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{6}
\noindent {\bf F.~~~ The quantum version of classical conformal
manifolds}
\vglue 0,4truecm
In section~A we showed how the differential geometry of a smooth
spin-manifold can be described via the Dirac operator, thus opening
the way to a non-commutative generalization. An analogous situation
prevails for conformal manifolds: there the conformal structure can
again be described in a Hilbert space setting generalizable to the
non-commutative frame, but now simply in terms of a bounded operator
$F$ of square one, archetype of the following general
notion:\footnote{
simpler
}
\noindent Definition: Let $A$ be a $\hphantom{}^{\star}$-algebra
over {\bf C}.\footnote{
possibly Z/2-graded.
} A {\it Fredholm module over} $A$ is a pair $(H,F)$ of
\begin{itemize}
\item[---] a Hilbert
space $H$ which is a left $A$-module via a bounded
$\hphantom{}^{\star}$-representation $\pi$ of
$A$:
\beq
A \ni a \to \pi (a) = \underline a \in B (H),
\eeq
\item[---] and a bounded linear operator $F \in B (H)$,
\end{itemize}
\noindent with the following properties:
\beq
( F^2 - {\bf 1} ) \underline a \in K (H) \quad ,\ a \in A,
\eeq
\beq
[ F, \underline a ] \in K (H) \quad,\ a \in A.
\eeq
\noindent The Fredholm module $(H, F)$ is:
\begin{itemize}
\item[---] {\it involutive} whenever $F^2 = {\bf 1}$,
\item[---] {\it self-adjoint} whenever $F = F^{\star}$,
\item[---] {\it even} whenever $H$ is $Z/2$-graded, $F$ is odd, and
$\pi$ is of grade zero: $H = H^0 + H^1$ with grading involution
$\varepsilon,\ \underline a H^i \in H^i$, and $F H^i \in H^{i+1},\ i
\in Z/2$.
\item[---] {\it odd} otherwise,
\item[---] {\it $p$-summable}, $p \in [ 1 ; \infty )$, whenever
$[F,a] \in L^p (H), a \in A$.
\end{itemize}
\noindent (here $L^p (H),\ p \in [ 1 ; \infty )$, is the Schatten
ideal of bounded operators $A$ s.t. $|A|^p$ is trace-class, i.e.
belongs to $L^1 (H)$).
\noindent The {\it n-character} $\chi^F_n$ of the $(p+1)$-summable
Fredholm module
$(H,F)$, is defined for $N \ge p+1$ as:
\beq
\chi^F_n (a_0, a_1, \dots , a_n) = cste \cdot Tr \left\{
\underline{a_0} [ F,
\underline{a_1} ] \dots [ F, a_n ] \right\}
\eeq
We shall envisage conformal manifolds as conformal classes of
riemannian manifolds. Our setting and notation will be the following:
${\bf M}$ is an orientable smooth pseudo-riemannian manifold of even
dimension $d = 2 m$ with metric
$g$ of signature\footnote{
number of positive eigen values
} $s$. We set $C^{\infty}
({\bf M}) = A$, and denote the DeRham complex of
${\bf M}$ by $\left( \Omega ({\bf M}) = \oplus_{p = 1,\dots ,d} \Omega
({\bf M})^p,\ {\bf d} \right)$. The components of differential forms
and the scalar product of $\Omega ({\bf M})$ are defined as follows:
with $\varepsilon^i
\in \Omega ({\bf M})^1,\ i=1,\dots,d,$ a orthonormal frame of $T^{{\bf
M}
\star}$, we write:
\beq
\lambda = \lambda_I \varepsilon^I = {1\over p !} \lambda_{i_1 \dots
i_p} \varepsilon^{i_1} \land \dots \land \varepsilon^{i_p} =
\lambda_{i_1 \dots i_p}
\varepsilon^{i_1} \oplus \dots \oplus \varepsilon^{i_p} \quad , \
\lambda \in
\Omega ({\bf M})^p ,
\eeq
and
\beq
(\lambda, \mu) = \Sigma_I \lambda_I \mu_I = {1\over p !} \Sigma_{i_1
\dots i_p}
\lambda_{i_1 \dots i_p} \mu_{i_1 \dots i_p} \quad , \ \lambda,\ \mu
\in
\Omega ({\bf M})^p ,
\eeq
\noindent with the $\lambda_I$ lexicographic and the $\lambda_{i_1
\dots i_p}$ totally antisymmetric. The volume form is $d v =
\varepsilon^1 \land
\varepsilon^2 \land \dots \land \varepsilon^d$ for any direct
orthonormal frame $\varepsilon^1 ,\ \varepsilon^2 ,\dots ,
\varepsilon^d$ of
$T^{{\bf M}\star}$: in the coordinate patch $x^{\mu}$:
\beq
{\bf d} v = \left[ ( (-1)^{d-s} ) \det ( g_{\mu \nu} ) \right]^{1/2}
{\bf d} x^1
\land
\dots
\land {\bf d} x^d .
\eeq
\noindent The {\it Hodge involution} $\gamma : \Omega ({\bf M}) \to
\Omega ({\bf M})$ (such that $\gamma^2 = i d$), defined by:\footnote{
The canonical identification of $C l ({\bf M})$ to $\Omega ({\bf M})$
identifies $\gamma $ with the involution $\gamma^{d + 1}$.
}
\beq
\lambda \land^{\underline{\star}} \mu = (-1)^{ d p + {p (p-1) \over
2} } i^m (\lambda, \mu) d v \quad , \ \lambda,\ \mu \in \Omega ({\bf
M})^p ,
\ 0 \le p \le d ,
\eeq
\noindent is such that
\beq
\gamma {\bf d} \gamma = - {\bf d}^{\star} ,
\eeq
\noindent and undergoes the following change in the change of metric
$ g \to g' = e^{2 \phi} g$:
\beq
\gamma_{p'} = e^{ (d - 2 p) \phi} \gamma_p \quad \left( \gamma_p =
\gamma |_{\Omega ({\bf M})^p} \right).
\eeq
Reminder.
\begin{itemize}
\item[(i)] Hodge theory: With $\Delta = {\bf d d}^{\star} + {\bf
d}^{\star} {\bf d}$ the Laplace-Beltrami operator, and denoting by
$\Delta_i$ and $d_i$ the respective restrictions of $\Delta $ and $d$
to $\Omega ({\bf M})^i$, we have the following location of vector
spaces\footnote{ figurative tridimensional representation.
}:
\beq
\begin{array}{ll}
Ker \Delta_i = & Ker {\bf d}_i \cap Ker {\bf d}_{i-1}^{\star} \\
& \\
\hfill \uparrow & Ker {\bf d}^{\star}_{i - 1}\\
Ker {\bf d}_i & \longrightarrow I m {\bf d}^{\star}_i \\
\hfill \swarrow \kern 0,35 cm & I m \Delta_i\\
\quad I m {\bf d}_{i - 1}
\end{array}
\eeq
with $\dim Ker \Delta_i = \dim ( Ker {\bf d}_i / Im {\bf d}_i ) <
\infty $.
\item[(ii)]: description of $Ker^{\perp} \Delta$: The most general
element
$\omega \in Ker^{\perp} \Delta $ has $\gamma -^{ \mbox{even} }_{
\mbox{odd} }$ components of the form:
\end{itemize}
\beq
\left\{
\begin{array}{l}
\omega^+ = {1 + \gamma \over 2} {\bf d} \alpha^+ \\
\\
\omega^- = {1 - \gamma \over 2} {\bf d} \alpha^-
\end{array}
\right.
\qquad \mbox{with} \quad \alpha^+,\ \alpha^- \in \Omega ({\bf M}).
\eeq
\noindent For $\alpha \in \Omega ({\bf M})$, each of the forms $d
\alpha,{1 + \gamma \over 2} d \alpha $ and ${1 - \gamma \over 2} d
\alpha $ determines the other two: one has:
\beq\label{F.12}
\left\{
\begin{array}{l}
{1 - \gamma \over 2} {\bf d} \alpha = F {1 + \gamma \over 2} {\bf d}
\alpha \\
\\
{1 + \gamma \over 2} {\bf d} \alpha = F {1 - \gamma \over 2} {\bf d}
\alpha
\end{array}
\right. ,
\eeq
and
\beq
{\bf d} \alpha = (1 + F) {1 + \gamma \over 2} {\bf d} \alpha =
(1 + F) {1 - \gamma \over 2} {\bf d} \alpha ,
\eeq
\noindent with $F : Ker^{\perp} \Delta \to Ker^{\perp} \Delta $ the
involution:
\beq\label{F.14}
F = { {\bf d d}^{\star} - {\bf d}^{\star} {\bf d} \over {\bf d
d}^{\star} + {\bf d}^{\star} {\bf d} } \cdotp
\eeq
In what follows we shall be concerned with the differential forms of
${\bf M}$ of order half the dimension: indeed the latter are connected
with the conformal structure of ${\bf M}$ (aimed to be described in
terms of non-commutative concepts) due to the fact that the Hodge
involution commutes with conformal changes of the metric exclusively
in restriction to
$ \Omega ({\bf M})^m,\ d = 2 m$. We now have the
following results due to Alain Connes:
\noindent Let $H_0 = H_m \oplus H^{\perp}$ be the Hilbert completion
of
$\Omega ({\bf M})^m = Ker \Delta_m \oplus Ker^{\perp} \Delta_m,\ d = 2
m$, (thus $H_m$ is the Hilbert space of harmonic $m$-forms with
orthogonal complement
$H^{\perp}$ in the Hilbert space of all $m$-forms). If one defines as
follows the triple $( H,F,\gamma )$:
\begin{itemize}
\item[---] $H$ is the Hilbert space $H = H'_m \oplus H_0 = H'_m \oplus
H_m \oplus H^{\perp},\ H'_m$ a {\it second copy of} $H_m$.
\item[---] $\gamma $ is the involution of $H$ restricting to the
Hodge involution on $H_0$, and to minus the Hodge involution on
$H'_m$.
\item[---] the action of $A = C^{\infty} ({\bf M})$ on $H$ restricts
to the usual action on $H_0$ and to the zero action on $H'_m$.
\item[---] the operator $F$ on $H$ restricts to $F$ as given
by~\rf{F.12} and \rf{F.14} {\it on} $H^{\perp}$, and restricts to the
matrix $\left(
\begin{array}{cc}
{\bf 1} & 0 \\
0 & {\bf 1}
\end{array}
\right) $ on $H'_m \oplus H_m$.
\end{itemize}
\noindent one has that $( H,F,\gamma )$ is an even unital self-adjoint
Fredholm module over $A$, moreover $p$-summable for all $p > 2 m$.
Furthermore the Fredholm module $(H,F,\gamma)$ uniquely determines the
oriented conformal structure of {\bf M}.
Note that in restriction to $H^{\perp} = {1 + \gamma \over 2}
H^{\perp} \oplus {1 - \gamma \over 2} H^{\perp}$ one has $F = P \oplus
P^{\star}$, with
\beq
\left\{
\begin{array}{l}
P {1 - \gamma \over 2} {\bf d} \alpha = {1 + \gamma \over 2} {\bf d}
\alpha \\
\\
P^{\star} {1 + \gamma \over 2} {\bf d} \alpha = {1 - \gamma \over 2}
{\bf d} \alpha
\end{array}
\right. .
\eeq
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{7}
\noindent {\bf G.~~~ Fractals}
\vglue 0,4truecm
We sketch a situation inaugurating the differential geometry of
fractals. Let $\Sigma$ and $\Sigma'$ be two Riemann surfaces of the
same genus $>1$. According to Beers, they can be uniformized jointly
on the Riemann sphere (by means of quasi-fuchsian groups) with a
common limit set forming an irregular equator (called a quasi cercle).
\vspace{0,6cm}
\hspace{-3cm}\epsfbox{Fig.K}
\vspace{0,5cm}
By stereographic projection the quasi cercle projects on a Jordan
curve $\Gamma$ of the complex plane $\bC$. Conferring this with the
usual uniformization of $\Sigma$ within the unit disk $D,$ the Riemann
mapping theorem yields a map $Z$ from the closed disk to $\bC$,
analytic conformal in the interior of the disk and bicontinuous on
the boundary $S^1$, in restriction to which $Z$ is a bicontinuous
(highly non differentiable) complex function with range $\Gamma$. Let
now $F$ be the phase of the Dirac operator ${\dsp i \frac{d}{d
\varphi}}$ of $S^1$: $F$ is the identity on the Hardy space, and
minus the identity on its orthogonal complement, in other words $F$
is the {\sl Hilbert transform}. Alain Connes then defines the quantum
differential as $dZ = [F, Z]$. Taking the power $(dZ)^p$ of the
latter, with $p$ the (non integer) Hausdorff dimension of the fractal
$\Gamma$, $(dZ)^p$ happens to lie in $L^{1+} (S^1)$ and one then has
the following astonishing analytic expression for the Hausdorff
measure $\mu_H$ of the fractal $\Gamma$
\beq
\mu_H (f) = Tr_w \{ f(Z) \, (dZ)^p \}~.
\eeq
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{8}
\noindent {\bf H.~~~ Essays on gravitation}
\vglue 0,4truecm
Recently Connes' non-commutative geometry turned out --- besides
yielding the above reinterpretation of the standard model --- to
be relevant to gravitation.
\bigskip
\noindent {\bf \S 1 ~~ The Dirac operator and gravitation}
{}~~~~
Alain Connes first made the challenging observation, that the Wodzicki
residue of the inverse square of the (Atiyah-Singer-Lichn\'erowicz)
Dirac operator yields the Einstein-Hilbert action of general
relativity, a fact which he left unpublished, but mentioned verbally
in different talks. The Wodzicki residue is a (in fact the unique,
thus canonical) trace on the pseudo-differential operators
(concentrated on pseudo-differential operators of order - the
dimension of the manifold).
In the paper~[9] we computed the Wodzicki residue of $D^{-2}$, first
computing this object for the pure Dirac operator (built with the spin
connection of a riemannian spin manifold).
We thus work in the setting of a 4-dimensional oriented riemannian
spin manifold {\bf M} with riemannian metric $g$. The Dirac operator
$D$ is locally given as follows in terms of an orthonormal section
$e_i$ (with dual section $\theta^k$) of the frame bundle of {\bf M}~:
one has
\beq
\left\{\normalbaselineskip=18pt\matrix{
D=i\gamma^i\widetilde{\nabla}_i=i\gamma^i(e_i+\sigma_i)\hfill&\cr
\hbox{with}\ \sigma_i(x)={1\over
4}\gamma_{ij,k}(x)\gamma^j\gamma^k={1\over 8}\gamma_{ij,k}(x)
\left[\gamma^j\gamma^k-\gamma^k\gamma^j\right]&\cr
},\right.
\eeq
where the $\gamma_{ij,k}$ represent the
Levi-Civita connection $\nabla$ with spin connection
$\widetilde{\nabla}$, specifically~:
\beq
\left\{\normalbaselineskip=18pt\matrix{
\gamma_{ij,k}=-\gamma_{ik,j}={1\over
2}[c_{ij,k}+c_{ki,j}+c_{kj,i}]\qquad,\ i,\ j,\ k=1,\ \dots,\ 4.\cr
\hbox{with}\ c_{ij}^k=\theta^k([e_i,e_j])\hfill&\cr
}\right.
\eeq
(The $\gamma^i$ are constant self-adjoint
Dirac matrices s.t.
$\gamma^i\gamma^j+\gamma^j\gamma^i = 2 \delta^{ij}$). In terms of
local coordinates $x^{\mu}$ inducing the alternative vierbein
$\partial_{\mu}=S_{\mu}^i(x)e_i$ (with dual vierbein $dx^{\mu}$) we
have $\gamma^ie_i=\gamma^{\mu}\partial_{\mu}$, the
$\gamma^{\mu}$ being now $x$-dependent Dirac matrices s.t.
$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=g^{\mu\nu}$
(we use latin sub-(super-)scripts for the basis $e_i$ and greek
sub-(super-)scripts for the basis $\partial_{\mu}$, the type of
sub-(super-)scripts specifying the type of Dirac matrices). The
specification of the Dirac operator in the greek basis is then~:
$$
\left\{\normalbaselineskip=18pt\matrix{
D=i\gamma^{\mu}\widetilde{\nabla}_{\mu}=
i\gamma^{\mu}(\partial_{\mu}+\sigma_{\mu})&\cr
\hbox{with}\ \sigma_{\mu}(x)= S_{\mu}^i(x)\sigma_i(x)\hfill&\cr
}.\right.
\eqno(\mbox{H.1a})
$$
In what follows the notation $D^{-1}$ refers
to an inverse modulo smoothing operators. One then finds that the
value of the Wodzicki residue on the inverse square of the Dirac
operator, namely~:
\beq
I=4Tr_{\omega}\{\sigma_{-4}(x,\xi)\}=4(2\pi)^{-4}\int_{\xi\in
S^3}tr\{\sigma_{-4}(x,\xi)\}d^3\xi dv,
\eeq
($tr$ the normalized Clifford trace) where~:
\beq
\sigma_{-4}(x,\xi)=\hbox{part of order}-4\ \hbox{of the total
symbol}\ \sigma(x,\xi)\ \hbox{of}\ D^{-2},
\eeq
coincides
up to a constant with the Hilbert-Einstein action
$\int{\cal L}_g dv$ of general relativity, where~:
\beq
{\cal L}_g=R_{\mu\nu}\land\star ( {\bf d} x^{\mu}\land
{\bf d} x^{\nu})
\eeq
(specifically
$$
{\cal L}_g={1\over 2}R_{ikmn}( {\bf d} x^m\land {\bf d} x^n,\
{\bf d} x^i\land
{\bf d} x^k)=(g^{im}g^{nk}-g^{in}g^{mk})R_{ikmn}=s,
\eqno(\mbox{H.5a})
$$
$s$ the scalar curvature). One has $I=-{1\over 24\pi}\int{\cal L}_g
dv$.
Our proof is a brute-force computation performed in arbitrary
coordinate patches. We start from the Lichn\'erowicz formula for the
square of the Dirac operator~:
\beq
\begin{array}{ll}
D^2 &=-g^{\mu\nu}\left(\widetilde{\nabla}_{\mu}
\widetilde{\nabla}_{\nu}-\Gamma_{\mu\nu}^{\alpha}
\widetilde{\nabla}_{\alpha}\right)+{1\over 4}s \\
&=-g^{\mu\nu}\left[\partial_{\mu}^x\partial_{\nu}^x+2\sigma_{\mu}
\cdot\partial_{\nu}^x-\Gamma_{\mu\nu}^{\alpha}\partial_{\alpha}+
\partial_{\mu}^x\sigma_{\nu}+\sigma_{\mu}\sigma_{\nu}-
\Gamma_{\mu\nu}^{\alpha}\sigma_{\alpha}\right]+{1\over 4}s.
\end{array}
\eeq
Our computations are based on the algorithm yielding the principal
symbol of a product of pseudo-differential operators in terms of the
principal symbols of the factors, namely, with the shorthands
$\partial_{\xi}^{\alpha}=\partial^{\alpha} /\partial\xi_{\alpha},\
\partial_{\alpha}^x=\partial_{\alpha} /\partial x^{\alpha}$~:
\beq
\sigma^{PQ}(x,\xi)=\Sigma_{\alpha}{(-i)^{|\alpha|}\over
\alpha !}\partial_{\xi}^{\alpha}\sigma^P(x,\xi)
\cdot\partial_{\alpha}^x\sigma^Q(x,\xi).
\eeq
We needed to compute the total symbol $\sigma(x,\xi)$ of $D^{-2}$ up
to order $-4$, the computation is involved because of the need to
``dive by two orders'' in evaluating full symbols.
Since the Einstein-Hilbert action and the action of the standard
model are both obtained by algorithms based on the Dixmier trace, one
naturally wishes to obtain these two actions within a single
procedure. Along this line the first natural object to investigate is
the Wodzicki residue of {\bf D}$^{-2}$, {\bf D} the compound
Dirac operator built with the tensor pro\-duct of the spin
connection
$\sigma_{\mu}$ and the electrodynamics $U$(1)-connection $a_{\mu}$.
But computation of this object yields the same
result as that stated above: the connection $a_{\mu}$ drops
out of the calculation~[9].\footnote{
In fact, since our calculation is based on the
Lichn\'erowicz formula for the square or the Dirac operator
holding in the case of general Dirac operators stemming from Clifford
connections on Clifford bundles, our result naturally generalizes to
this frame
}
We thus conclude that the present algorithms of non-commutative
geometry yielding the respective lagrangians of the microworld and the
cosmos seem (superficially) to tend to repel each other~: whilst
$a_{\mu}$ drops out of the Wodzicki residue of the inverse square of
the compound Dirac operator, $\sigma_{\mu}$ drops out of the
non-commutative Yang-Mills algorithm.\footnote{
Indeed
$\sigma_{\mu}$ drops out of the commutators $[D,\ a],\ a\in
C^{\infty}({\bf M})$.
}
\bigskip
\noindent {\bf \S 2 ~~ The quantum Polyakov action in two dimensions}
{}~~~~
We recall the expression of the (classical) Polyakov action
\beq
I_d (X^\ap) = \sum_{\ap, \beta =1}^m \int \eta_{\ap \beta} {\bf d}
X^\ap \wedge * {\bf d} X^\beta
\eeq
where the $X^\ap$, $\ap = 1, \ldots, m$ are smooth functions on a
Riemann surface $\Sigma$ (with metric $g_{ik}$) and $\eta_{\ap
\beta}$ is an Euclidean metric of $R^m$.
Alain Connes obtains the action (F.1) from the following ``quantum
Polyakov action"
\beq\label{F.2}
I_{quant} (X^\ap) = Tr_w \{\eta_{\ap \beta} [F, X^\ap][F, X^\beta] \}
\eeq
specified as follows. Take as Hilbert space the set $\Om (M)^1$
completed for the scalar product stemming from the Riemannian metric.
And take\footnote{
We may ignore the subset of harmonic forms on which the
denominator vanishes.
}
\beq
F = \frac{{\bf d}^* {\bf d} - {\bf dd}^*}{{\bf d}^* {\bf d} + {\bf
dd}^*}
\eeq
Evaluating the Dixmier trace $Tr_w$ as a Wodzicki residue then
restitutes the classical Polyakov action (H.2) of which (H.3) is thus
a quantum version. We now describe this computation.
Quantum differentials are defined as follows:
\beq
df = [F,f] \hspace{3cm} , \ f \in C^{\infty} (\Sigma).
\eeq
Note that, with
$\Phi = ( {\bf dd}^{\star} - {\bf d}^{\star}{\bf d} ), \
\Delta = {\bf dd}^{\star} + {\bf d}^{\star}{\bf d}, \
q = {\bf d} + {\bf d}^{\star}, \ r = i ( {\bf d} - {\bf d}^{\star} )$,
we have
\beq
\left\{
\begin{array}{l}
\Delta = {\bf dd}^{\star} + {\bf d}^{\star}{\bf d} = q^2 = r^2 , \\
\\
\Phi = {\bf dd}^{\star} - {\bf d}^{\star}{\bf d} = iqr = -irq \\
\\
F = -iqr^{-1} = irq^{-1}.
\end{array}
\right.
\eeq
thus
$F$ is involutive:
$F^2 = 1 \! \! 1 $. (we ignored the fact that inverses are not defined
on $ Ker \Delta = Ker {\bf d} \cap Ker {\bf d}^{\star} = Ker q
\cap Ker r = Ker \Phi \cap Ker \Delta = Ker {\bf d} \cap (Im {\bf
d})^{\perp} = Ker {\bf d}^{\star} \cap ( Im {\bf d}^{\star} )^{\perp} ) $
\noindent Note that definition~\rf{F.2}, is well taken: indeed
$ F, \ X^{\mu}, \ [ F, X^{\mu} ] $, and
$[ F, X^{\mu} ] [ F, X^{\nu} ] $ are pseudodifferential operators of
respective orders 0,0,-1, and
$-2 = - \dim \Sigma$:
$[ F, X^{\mu} ] [ F, X^{\nu} ] $ thus belongs to the definition
ideal
$L^{1+}$ of the Dixmier trace.
We first compute the principal symbol
$\sigma^F (x, \xi)$ of
$F$. We recall that the total symbols
(adjoint of each other) of
${\bf d}$ and
${\bf d}^{\star}$ are given
by:
\beq
\left\{
\begin{array}{l}
\sigma^{\bf d} (x, \xi) = \ \mbox{ie}_{\xi} \\
\\
\sigma^{\bf d^{\star}} (x, \xi) = -i i_{g^{-1 \xi}}
\end{array}
, \right.
\eeq
where
$e_{\xi}$ denotes an exterior, and
$i_u$ an inner product,
$\xi \in T^{\Sigma \star}_x$, and
$u \in T^{\Sigma}_x$. We have thus
the principal symbols:
\beq
\left\{
\begin{array}{l}
\sigma^{\Phi} (x, \xi) = e_{\xi} i_{g^{-1 \xi}} -
i_{g^{-1 \xi}} e_{\xi} \\
\\
\sigma^{\Delta} (x, \xi) = e_{\xi} i_{g^{-1 \xi}} +
i_{g^{-1 \xi}} e_{\xi} = || \xi ||^{-2}
\end{array}
, \right.
\eeq
and
\beq
\sigma^F (x, \xi) = ( e_{\xi} i_{g^{-1 \xi}} -
i_{g^{-1 \xi}} e_{\xi} ) || \xi ||^{2}.
\eeq
these formulae following from the rule for computing the total
symbol of a product of two pseudodifferential operators in terms of
the total symbols of the factors, namely:\footnote{
We recall the notation
$ D^x_{\alpha} = (-i)^{| \alpha |} \partial^{\alpha}_x $.
}
\beq
\sigma^{PQ} = \Sigma_{\alpha} {1\over \alpha !}
\partial^{\alpha}_{\xi} \sigma^P \cdot D^x_{\alpha} \sigma^Q
\eeq
\noindent We now show that the principal symbol
$\sigma^{ [ F,f ] } (x,\xi)$ of
$ [ F,f ] $ is given by:
\beq
\sigma^{ [ F,f ] } (x,\xi) = 2 \left( e_{{\bf d} f^{\perp}} i_{\xi} -
i_{\xi} e_{{\bf d} f^{\perp}} \right) || \xi ||^{-2},
\eeq
where
$ {\bf d} f^{\perp} = {\bf d} f - ( \xi, {\bf d} f ) || \xi ||^{-2}
\cdot \xi $, the projection of
${\bf d} f$ on the plane orthogonal to
$\xi$. Indeed, applying (H.7), keeping the first non vanishing
order, we get, taking account of the fact that
$ O (\xi, \eta) = \left( e_{\xi} i_{g^{-1 \eta}} -
i_{g^{-1 \xi}} e_{\eta} \right) $ is a symmetric bilinear form in
$\xi$ and
$\eta$:
\beq
\begin{array}{ll}
\sigma^{ [ F,f ] } (x,\xi) & = \sigma^F \cdot f +
\Sigma_{| \alpha | = 1} \partial^{\alpha}_{\xi}
\sigma^F D^x_{\alpha} f - f \cdot \sigma^F =
\Sigma_{| \alpha | = 1} \partial^{\alpha}_{\xi}
\sigma^F D^x_{\alpha} f \\
& \\
& = \Sigma_{i = 1,2} {\partial \over \xi_i} \sigma^F \partial_i f =
\Sigma_{i = 1,2} {\partial \over \xi_i} \left[ \xi_k \xi_l ()^{kl}
|| \xi ||^{-2} \right] \partial_i f \\
& \\
& = \Sigma_{i = 1,2} O^{kl} || \xi ||^{-2}
\left\{ \delta^i_k \xi_l + \xi_k \delta^i_l - || \xi ||^{-2}_k
\xi_k \xi^{st}_{lg}
\left[ \delta^i_s \xi_t +\xi_s \delta^i_t \right]
\right\} \partial_i f \\
& \\
& = \Sigma_{i = 1,2} O^{kl} || \xi ||^{-2}
\left\{
\left[ \xi_l \partial_k f + \xi_k \partial_l f \right]
- 2 \xi_k \xi_l || \xi ||^{-2} ( {\bf d} f, \xi )
\right\} \\
& \\
& = \Sigma_{i = 1,2} O^{kl} || \xi ||^{-2}
\left\{
\left[ \xi_k
\left[ \partial_l f - \xi_l || \xi ||^{-2} ( {\bf d} f, \xi ) \right]
\right] \right. \\
& \\
& \left. \hphantom{
\Sigma_{i = 1,2} O^{kl}
|| \xi ||^{-2} \left\{ \left[ \xi_k \left[ \partial_l f
\right. \right. \right. - }
+ \xi_l \left[ \partial_k f - \xi_k || \xi ||^{-2} ( {\bf d} f, \xi )
\right]
\right\} \\
& \\
& = \Sigma_{i = 1,2} O^{kl} || \xi ||^{-2}
\left\{
\xi_l ( {\bf d} f^{\perp} )_k + \xi_k ( {\bf d} f^{\perp} )_l
\right\} \\
& \\
& = 2 \left( e_{ {\bf d} f^{\perp} } i_{\xi} -
i_{\xi} e_{ {\bf d} f^{\perp} }
\right) || \xi ||^{-2}.
\end{array}
\eeq
The principal symbol
$\sigma $ of
$ f_0 [ F, f_1 ] [ F, f_2 ] $ is therefore given by the product:
\beq
\sigma (x, \xi) = 4 \left(
e_{ {\bf d} {\bf f}^{\perp}_1 } i_{\xi} -
i_{\xi} e_{ {\bf d} {\bf f}^{\perp}_1 }
\right)
\left(
e_{ {\bf d} {\bf f}^{\perp}_2 } i_{\xi} -
i_{\xi} e_{ {\bf d} {\bf f}^{\perp}_2 }
\right) || \xi ||^{-4}.
\eeq
We now show that the trace of
$\sigma $ on the fiber is given by:
\beq
tr \sigma (x, \xi) = 8 || \xi ||^{-d} f_0 \left(
{\bf d} f^{\perp}_1, {\bf d} f^{\perp}_2
\right).
\eeq
\noindent Let indeed
$\eta_1 = {\bf d} f^{\perp}_1 = \eta_{1 k} \varepsilon^k, \
\eta_2 = {\bf d} f^{\perp}_2 = \eta_{2 k} \varepsilon^k $, with
$\left( e_k,\ \varepsilon^k \right) $ a dual basis of
$ \left( T^M_x = T^{M \star \star}_x , \ T^{M \star}_x \right) $:
using the fact that we have, for a one-form
$\alpha $, with
$g^{-1} \xi = \overline{\xi}$ and
$g^{-1} \eta = \overline{\eta}$:
\beq
\left( e_{\eta} i_{\xi} - e_{\xi} i_{\eta} \right) \alpha =
\alpha ( \overline{\xi} ) \eta -
\alpha ( \overline{\eta} ) \xi ,
\eeq
we get (summation over
$k$):
\beq
\begin{array}{ll}
{1\over 4} tr \sigma (x, \xi) & = e_k
\left( e_{\eta_1} i_{\xi} - e_{\xi} i_{\eta_1} \right)
\left( e_{\eta_2} i_{\xi} - e_{\xi} i_{\eta_2} \right)
\varepsilon^k = \\
& \\
& = e_k \left\{
\xi^k \left[ \eta_2 ( \overline{\xi} ) \eta_1 -
\eta_2 ( \overline{\eta}_1 ) \xi \right] -
\eta^k_2 \left[ \xi ( \overline{\xi} ) \eta_1 -
\xi ( \overline{\eta}_1 ) \xi \right]
\right\} \\
& \\
& = \left( \xi, \eta_1 \right) \left( \xi, \eta_2 \right) -
\left( \eta_1, \eta_2 \right) || \xi ||^2 -
|| \xi ||^2 \left( \eta_1, \eta_2 \right) +
\left( \xi, \eta_1 \right) \left( \xi, \eta_2 \right) \\
& \\
& = 2 \left[
\left( \xi, \eta_1 \right) \left( \xi, \eta_2 \right) +
|| \xi ||^2 \left( \eta_1, \eta_2 \right)
\right] = - 2 || \xi ||^2
\left( {\bf d} f^{\perp}_1 , {\bf d} f^{\perp}_2 \right)
\end{array}
\eeq
\noindent Integrating this in
$\xi$ over the unit ball yields:
\beq
- 2 \int_{S^1} \left( {\bf d} f^{\perp}_1 , {\bf d} f^{\perp}_2
\right) {\bf d} v_{S^1} = {8 \pi \over 3} \left( {\bf d} f_1 , {\bf
d} f_2 \right).
\eeq
\noindent One has indeed, for
$|| \xi || = 1 $:
\beq
\begin{array}{ll}
\left( {\bf d} f^{\perp}_1 , {\bf d} f^{\perp}_2 \right) & =
( {\bf d} f_1 - (\xi, {\bf d} f_1) \xi ), ( {\bf d} f_2 - (\xi, {\bf
d} f_2) \xi )
\\
& \\
& = ( {\bf d} f_1, {\bf d} f_2 ) - (\xi, {\bf d} f_1) (\xi, {\bf d}
f_2) = ( {\bf d} f_1, {\bf d} f_2 ) - ( \xi^i, {\bf d} f_{1_i} ) (
\xi_j, {\bf d} f_{2^j} ),
\end{array}
\eeq
hence
\beq
\begin{array}{ll}
\int_{S^1} \left( {\bf d} f^{\perp}_1 , {\bf d} f^{\perp}_2 \right)
{\bf d} v_{S^1} & = \int_{S^1}
\left[ ( {\bf d} f_1, {\bf d} f_2 ) - \xi^i \xi_j {\bf d} f_{1_i}
{\bf d} f_{2^j}
\right] {\bf d} v_{S^1} \\
& \\
& = 2 \pi ( {\bf d} f_1, {\bf d} f_2 ) - \delta^i_j \int_{S^1}
\xi^l \xi_l {\bf d} f_{1_i} {\bf d} f_{2^j} \\
& \\
& = 2 \pi \left( l - {1\over 3} \right)
( {\bf d} f_1, {\bf d} f_2 ) = {4 \pi \over 3} ( {\bf d} f_1, {\bf d}
f_2 )
\end{array}
\eeq
\noindent Plugging this result in~(A.7a) then yields:
\beq
Tr_{\omega}
\left\{ f_0 [ F, f_1 ] [ F, f_2 ] \right\} =
- (3 \pi)^{-1} \int_{\Sigma} f_0 ( {\bf d} f_1, {\bf d} f_2 ) dv =
- (3 \pi)^{-1} \int_{\Sigma} f_0 {\bf d} f_1 \wedge^{\star} {\bf d}
f_2
\eeq
our result then following from the replacement
$s$: $f_0 \to l,\ f_1 \to X^{\mu},\ f_2 \to X^{\nu}$.
\bigskip
\noindent {\bf \S 3 ~~ The quantum Polyakov action in four dimensions}
{}~~~~
In two dimensions the quantum Polyakov action turned out to be equal
to the classical action. However the quantum version of the action has
an advantage over the classical version: it is adaptable, mutatis
mutandis, to a four-dimensional conformal manifolds $\Sigma $. The
setting is now the following:
\begin{itemize}
\item[---] The basic algebra is $C^{\infty} (\Sigma),\ \Sigma$ a
conformal 4-dimensional manifold equipped with a riemannian metric
$g^2$.
\item[---] The $( X^{\mu} )_{\mu = 1, \dots , N}, X^{\mu} \in
C^{\infty} (\Sigma)$, take values in the euclidean space ${\bf R}^N$
with me\-tric~$\eta$.
\item[---] The even $K$-cycle is $(H = \Omega (\Sigma)^2,\ F = ({\bf
dd}^{\star} - {\bf d}^{\star} {\bf d}) ({\bf dd}^{\star} + {\bf
d}^{\star} {\bf d})^{-1})$. As above, we take as Hilbert space the
Hilbert completion of the set of differential forms of middle
dimension. The scalar product of $\Omega (\Sigma)^2$ is induced by the
riemannian structure of $\Sigma$. In defining the involutive operator
$F$ we ignore the subspace of harmonic two-forms.
\end{itemize}
The quantum Polyakov action is now as above where Res denotes the
Wodzicki residue (now concentrated on pseudo-differential operators of
order $-4 =-
\dim \Sigma$, where it coincides with $4 Tr_{\omega},\ Tr_{\omega}$
the Dixmier trace).
As in \S 1, the computation is involved because of the need to ``dive
by two orders''. Invariance considerations allow one not to drown in
the calculation, which yields the following esoteric quadratic action:
\beq
\begin{array}{ll}
{\cal L} ( X^{\mu}, X^{\nu} ) = - \eta_{\mu \nu} \Delta
\left[ ( {\bf d} X^{\mu}, {\bf d} X^{\nu} ) \right]
& + \eta_{\mu \nu} ( \Delta {\bf d} X^{\mu}, \Delta {\bf d} X^{\nu} )
- {1\over 2} \eta_{\mu \nu} \Delta {\bf d} X^{\mu} \cdot \Delta {\bf
d} X^{\nu} \\
& \\
& + {1\over 3} {\bf s} \cdot \eta_{\mu \nu}
( {\bf d} X^{\mu}, {\bf d} X^{\nu} ) .
\end{array}
\eeq
\bigskip
\bigskip
\setcounter{equation}{0}
\setcounter{chapter}{9}
\noindent {\bf I.~~~ Conclusion and outlook}
\vglue 0,4truecm
Applications as of today to the fundamental physical interactions
concern the classical (Lagrangian) level (field quantization and
renormalization still lie ahead). They comprise an elaborate classical
version of the full standard model (electroweak and chromodynamics
sectors) and promising essays on gravitation.
Concerning the microworld Connes proposes an astounding
interpretation of the full standard model in terms of a
Poincar\'e-dual pair of (mildly) non-commutative spaces, compounds of
usual space-time and the electroweak, resp. chromodynamical, degrees
of freedom. The latter are interpreted as attributes of a
non-commutative space synthetizing usual space time with the ``inner
symmetries" of elementary particles, a new notion of space combining
inseparably space-time with the elementary particle structure. This
theory presents the Higgs boson as a fith gauge-boson corresponding to
a discrete connection of a two-sheeted space and incorporates the
symmetry-breaking in the definition of the $K$-cycle. All the terms of
the usual Glashow-Salam-Weinberg lagrangian proceed miraculously from
a compact quantum Yang-Mills algorithm. The convincing interpretation
of the electroweak-chromodynamical duality as a case of
non-commutative Poincar\'e duality, with concomitant enrichment of the
mathematical notion of non-commutative space, is a beautiful feature.
The only flaw of the theory is the physically ad hoc (if
mathematically coherent) ``modularity condition''. Since the theory
allows restricted choices of coupling constants displaying various
degrees of additional symmetry and implying constraints, it is
important to know if these potentially predictive constraints can
survive renormalization. In this respect it would be important to
display the symmetry (at the moment not visible) associated to the
Higgs boson as a fifth gauge boson - and more generally to look
systematically for ``quantum symmetries'' - progress is expected on
this point in a near future.
Concerning the cosmos, Connes' non-commutative geometry turned out
lately to be relevant to gravitation. On the one hand there is the
recognition that the Dirac operator embodies the lagrangian of general
relativity. On the other hand Connes developped a quantum version of
the usual Polyakov action (within the conformal variety of his
non-commutative geometry). This quantum version has the virtue of
beeing transcribable to the frame of 4-dimensional conformal
manifolds, yielding a conformally invariant lagrangian possibly tied
up with gravitation. The analysis of the anomalies of this lagrangian
is underway.
Of course one wishes to synthetize the standard model and gravitation
within a non-commutative geometry frame a priori adequate for such a
project... for the moment future music! One could hope from such a
development to come nearer to the explanation of the fermion masses -
a problem in the face of which it is frustrating to see the
formidable apparatus of present-day quantum field theory totally
ineffective.
I want to conclude with the personal statement that, contrary to most
of my colleagues, I do not adhere to the view that renormalization
will remain as a standing feature of the future theory - I can not
help feeling repelled by the ugliness and the incomplete nature of
renormalized field theories (anyway confined to a perturbative
approach unfit to gravitation). My hope is that a further, deeper,
step into non-commutativity of the basic algebra will once wipe out
the divergencies. In respect to gravitation, I sympathize with
Ashtekar's philosophy, hoping for a future jointure of the latter with
Alain Connes' project.
\bigskip
\bigskip
\centerline{\bf Bibliography}
\vglue 0,4truecm
The reader will find under I the
references relevant to these lectures, whose subject-matter
corresponds to the my preference for Connes' approach and reflect my
personal bias when I wrote the paper [5]. I regret not to have had the
leisure of including an account of Connes' non-commited study of the
possible choice of coupling constants: for this I refer to [3], [3a],
[4], where results are stated without details of calculations. The
present lectures utilize refs. [3a], [5], [6], [7], [8], [9], [10].
They do not touch the other references quoted under I. For
introductions we refer to~[3],~[8],~[11].
We attempted in II to list the approaches of physical
interactions by means of non-commutative geometry techniques other
than Connes'. Refs.~[38-46], concern the works of the
Marseille-Mainz group whose theory of the electroweak sector of the
standard model is in spirit a parent of Connes' approach (partly
inspired by it) however more in the mood of Kaluza-Klein theories, and
technically based on the use of a differential algebra. In [38-39],
the basic object is a $Z_2$-graded differential algebra defined as the
graded tensor product of the $Z_2$ graded matrix algebra of $p$-forms
times a matrix algebra of even matrices (with a natural $Z_2$
grading). For a detailed comparison between this approach and Connes'
see [47].
In [30 - 37], the role of the differential algebra is played by
$\Omega_{Der}(A)$ which is defined as the homomorphic image of the
universal differential envelope of $A$ into the algebra of
$A$-valued multilinear forms on the space of derivations of $A$. The
oldest paper of this series is [30], whereas the most complete on this
approach is certainly [37].
\bigskip
\noindent {\bf I ~~ Papers concerning Connes' approach}
{}~~~
\begin{description}
\item{\ [1]} A. Connes, Essay on physics and non-commutative geometry.
The interface of Mathematics and Particle Physics. Clarendon Press,
Oxford (1990).
\item{\ [2]} A. Connes and J. Lott, Particle models and
non-commutative geometry.
{\sl Nucl. Phys.} {\bf B18B} (Proc. Suppl.) (1990).
\item{\ [3]} A. Connes, Non commutative differential geometry. Book to
appear. Cambridge University Press.
\item{[3a]} A. Connes, The metric aspect of non-commutative
geometry. Coll\`ege
de France preprint (1991).
\item{\ [4]} A. Connes, Non-commutative geometry and physics. Les
Houches Lecture
Notes (1992).
\item{\ [5]} D. Kastler, A detailed account of Alain Connes' version
of the standard model in non-commutative geometry. I and II, {\sl
Rev. Math. Phys.}, {\bf 5} no.3 477 (1993). State of the art III
CPT-92P/2824.
\item{\ [6]} D. Kastler and D. Testard, Tensor products of quantum
forms. Comm. Math. Phys. 155, 135 (1993).
\item{\ [7]} D. Kastler and T. Sch\"ucker, Remarks on Alain Connes'
approach to the standard model in non-commutative geometry,
{\sl Theoret. Math. Phys.} {\bf 92}, no.3, 523 (1992).
\item{\ [8]} D. Kastler and M.~Mebkhout, Lectures on non-commutative
geometry and application to fundamental physical interactions. In
preparation.
\item{\ [9]} D. Kastler, The Dirac operator and gravitation, to appear
in {\sl Commun. Math. Phys.}.
\item{[10]} D. Kastler, Dual pairs of Riemannian quantum spaces. CPT
Preprint 1994.
\item{[11]} D. Kastler, Towards extracting physical predictions from
Alain Connes' version of the standard model (the new grand
unification?), July 1991 Istambul Nato workshop: Operator Algebras,
Mathematical Physics and Low Dimensal Topo\-lo\-gy, Richard Herman and
Bet\"ul Tambay eds. Research Notes in Mathematics, A.K.~Peters
Wellesley Mass. USA.
\item{[11a]} D. Kastler, State of the art of Alain Connes'
version of the standard model, July 1992 Kyoto Seminar on Quantum and
Non Commutative Analysis, H. Araki et al eds. Kluwer.
\item{[12]} D. Kastler, An invitation to Alain Connes' cyclic
cohomology, Trends and Deve\-lop\-ments in the eighties, Bielefeld
Encounters in Mathematics Physics IV and V, S. Albeverio and Ph.
Blanchard eds., World Scientific (1985).
\item{[12a]} D. Kastler, Introduction to Alain Connes'
non-commutative differential geometry, Proc. XXII Winter school,
Karpacz, A. Jadczyk ed., World Scientific (1986).
\item{[13]} R. Coquereaux, Non-commutative geometry: a physicist's
brief survey, Proc. XXIIX Karpacz Winter school of Theoretical
Physics; Fields and Geometry J. Geom. Phys. {\bf 11} 307 (1993).
\item{[14]} B. Iochum \& T. Sch\"ucker, A
left-right symmetric model \`a la Connes-Lott,
CPT-93/P.2973 Lett. Math. Phys. in press
\item{[15]} T. Sch\"ucker and J.M. Zylinski Connes' model
building kit, CPT-93/P.2960. J. Geom. and Physics in press
\item{[16]} A. Chamseddine, G. Felder \& J. Fr\"ohlich, Grand
Unification in non-commutative geometry, Nucl. Phys. B395 (1993) 672
\item{[17]} A. Chamseddine, G. Felder \& J. Fr\"ohlich, Gravity in
non-commutative geometry, Comm. Math. Phys., {\bf 155} 205 (1993).
\item{[18]} A. Chamseddine, G. Felder \& J. Fr\"ohlich, Unified gauge
theories in non-commuta\-ti\-ve geometry, Phys. Lett. 296B (1993)
109,.
\item{[19]} A. Chamseddine, G. Felder \& J. Fr\"ohlich, SO(10)
Unification in non-commutative geometry, ETH/TH/93-12.
\item{[20]} A. Chamseddine, G. Felder \& J. Fr\"ohlich, Some elements
of Connes' non-commuta\-ti\-ve geometry, ETH Preprint.
\item{[21]} A. Chamseddine \& J. Fr\"ohlich, Constraints on the Higgs
and top quark masses from effective potential and non-commutative
geometry, Phys. Lett. 314B (1993) 308
\item{[22]} J.C. Varilly and J.M. Gracia-Bondia, ``Connes' non
commutative geometry and the standard model'', J. Geom. and Physics
12 (1993) 223.
\item{[23]} E. Alvarez, J. M. Gracia-Bond\'\i a \& C. P. Mart\'\i n, A
Renormalization Group Ana\-ly\-sis of the NCG constraints
$m_{top}=2m_{W}$,
$m_{Higgs}=3.14m_{W}$, FTUAM-94/2, Phys. Lett. B, in press.
\item{[23a]} E. Alvarez, J. M. Gracia-Bond\'\i a \& C. P. Mart\'\i n,
Parameter restriction in non-commutative geometry model do not
survive standard quantum corrections, Phys. Lett. 306B (1993) 55
\item{[24]} W. Kalau and M. Walze, Gravity, non-commutative geometry
and the Wodzicki Residue, MZ-TH/93-38.
\item{[25]} A. Sitarz, ``Higgs mass and non commutative geometry'',
Phys. Lett. B 308 (1993) 311-314
\item{[26]} A. Sitarz, ``Non-commutative geometry and the Ising
model'' J. Phys A: Math. Gen. 26 (1993) 5305-5312
\item{[27]} G. Kasparov, K-functor and extension of
$C^{\star}$-algebras, Izv. Akad. Nauk. SSSR Ser. Mat. {\bf 44} 571
(1980).
\item{[28]} G. Kasparov, Equivariant KK-theory and the Novikov
conjecture, Invent. Math. {\bf 91} 147
(1988).
\item{[29]} G. Kasparov, Operator K-theory and its application:
elliptic operators, group representation, higher signatures,
$C^{\star}$-extensions. Proceed. Intern. Cong. Math. Warzawa (1983).
\end{description}
\bigskip
\noindent {\bf II ~~ Papers concerning other approaches}
{}~~~~
\begin{description}
\item{[30]} M. Dubois--Violette, ``D\'erivations et calcul
diff\'erentiel non--commutatif". {\sl Preprint Orsay 1987}, {\sl C.R.
Acad. Sci. Paris\/ \bf 307}, I, 403--408 (1988).
\item{[31]} M. Dubois--Violette, R. Kerner, J. Madore,
``Non--commutative differential geo\-metry of matrix
algebras" (Orsay Preprint 1988). SLAC-PPF 88-45, {\sl J. Math.
Phys.\/ \bf 31},
316--322
(1990).
\item{[32]} M. Dubois--Violette, R. Kerner, J. Madore,
``Non--commutative differential geo\-metry and new models
of gauge theory" (Orsay Preprint 1988). SLAC-PPF 88-49, {\sl J.
Math. Phys.\/ \ bf 31},
323--329 (1990).
\item{[33]} M. Dubois--Violette, R. Kerner, J. Madore,
``Gauge bosons in a non--commuta\-ti\-ve geometry", {\sl
Phys. Lett.\/ \bf B217}, 485--488 (1989).
\item{[34]} M. Dubois--Violette, R. Kerner, J. Madore,
``Classical bosons in a non--commuta\-ti\-ve geometry"
(1988). {\sl Class. Quantum Grav.\/ \bf 6}, 1709--1724 (1989).
\item{[35]} R. Kerner, M. Dubois--Violette, J. Madore,
``Mod\`eles des th\'eories de jauge bas\'es sur la g\'eom\'etrie
non--commutative". {\sl Annales de Physique, Colloque
n$^\circ$1, suppl\'e\-ment au n$^\circ$6}, Vol. 14 (1989).
\item{[36]} M. Dubois--Violette, R. Kerner, J. Madore,
``Super Matrix Geometry". {\sl Class. Quantum Grav.\/ \bf 8}
(1991), 1077--1089.
\item{[37]} M. Dubois--Violette, ``Non--commutative
differential geometry, quantum mecha\-nics and gauge
theory" in {\sl Differential Geometric Methods in Theoretical
Physics}, C. Bartocci et al (eds), Proceedings of the Rapallo
Conference (1990), 1991 Springer Verlag.
\item{[38]} R. Coquereaux, G. Esposito-Farese, G. Vaillant,
``Higgs Fields as Yang-Mills Fields and Discrete symmetries.
Preprint Marseille (1990) CPT-90-2407. Nucl Phys B353, 689 (1991).
\item{[39]} R. Coquereaux, ``Higgs fields and superconnections" in
{\sl Differential Geometric Methods in Theoretical Physics}, C.
Bartocci et al (eds), Proceedings of the Rapallo Conference (1990),
1991. Lecture Notes in Physics N375. Springer.
\item{[40]} R. Coquereaux, G. Esposito-Farese, F. Scheck, ``Non
Commutative
Geometry and Graded Algebras in Electroweak Interactions''
IHES-preprint,
CPT-91/P.2464, Int. J. Mod. Phys. A, Vol 7, No 26 (1992)
6555-6593.
\item{[41]} R. Coquereaux, Champs de Yang-Mills et brisures de
symm\'etrie: super-alg\`ebres et
g\'eom\'etrie non commutative. CPT-91/PE.2689 Contribution Rencontre
Strasbourg. RCP No25. IRMA, Vol 43. pp.67-80.
\item{[42]} R. Coquereaux, Yang Mills fields and Symmetry Breaking:
``From Lie Superalgebras to Non Commutative Geometry'' .Proceedings
of the first M. Born German-Polish symposium: ``Groups and Related
Topics", 115-127 (1992) Kluver Acad. Pub. Ed. R. Gielerak et al.
\item{[43]} R. Coquereaux, Contribution to the conference ``Dynamics
of complex and Irre\-gu\-lar systems" CPT-91/PE.2645 Ed. Ph Blanchard.
Word Scientific Pub.
\item{[44]} R. Coquereaux, R. Haussling, N.A. Papadopoulos, F.
Scheck. ``Generalized gauge transformations and hidden
symmetry''. Int. Jour. Modern Physics A7 (1992).2809.
\item{[45]} R. Haussling, N.A. Papadopoulos, F. Scheck, Supersymmetry
in the standard model of electro\-weak interactions
Phys. Lett. B303 (1993) 265.
\item{[46]} R. Coquereaux, R. Haussling, F. Scheck
``Algebraic connections on parallel universes''. CPT-93/PE 2947.
\item{[47]} N.A. Papadopoulos, J. Plass and F. Scheck. ``Models of
electroweak interactions in non-commutative geometry: A comparison''
(1993). MZ-TH/93-26, Phys. Lett. 323B (1994) 380
\item{[48]} B.S. Balkrishna, F. G\"ursey and K.C. Wali,Noncommutative
geometry and Higgs mechanism in the standard model, Phys. Lett. B
254 (1991) 430,
Towards a unified treatment of Yang-Mills and
Higgs fields, Phys. Rev. {\bf D 44} 331 {\bf 308} (1988)
\item{[49]} H. Bacry, Localizability and space of quantum physics.
Springer Lecture notes in Physics {\bf 308} (1988).
\item{[50]} A.Z. Jadzyk and B.Jancewicz, Bull. Acad. Pol. Sc. {\bf
21} (1973)
\end{description}
\end{document}