# Quantum Principal Bundles and Yang–Mills–Scalar–Matter Fields

Gustavo Amilcar Saldaña Moncada  Gustavo Amilcar Saldaña Moncada
Instituto de Matemáticas UNAM<EMAIL_ADDRESS>
###### Abstract.

This paper aims to develop a non–commutative geometrical version of the theory
of Yang–Mills and space–time scalar matter fields. To accomplish this purpose,
we will dualize the geometrical formulation of this theory, in which principal
$G$–bundles, principal connections, and linear representations play the most
important role. In addition, we will present the non–commutative geometrical
Lagrangian of the system as well as the non–commutative geometrical associated
field equations. At the end of this work, we show an illustrative example.

MSC 2010: 46L87, 58B99.

Keywords: Quantum bundles, quantum connections, quantum representations,
fields.

##  1\. Introduction

The Standard Model is one of the most successful and important theoretical
achievement in modern physics. From a philosophical/mathematical point of
view, it is another example of the intrinsic relations and interplay between
Physics and Differential Geometry, which in this case, is given by the
geometrical framework of principal bundles, their connections and the
associated structures.

Despite all of this, it presents some basic and fundamental problems that it
cannot solve. For example, a consistant and coherent description of the
space–time at the level of the Plank scale. The need to investigate further is
evident. Non–Commutative Geometry, also known as Quantum Geometry, arises as a
kind of algebraic and physical generalization of geometrical concepts [C],
[Pr], [W]. There are a variety of reasons to believe that this branch of
mathematics could solve some of the Standard Model’s fundamental problems.

The purpose of this paper is to develop a non–commutative geometrical version
of the theory of Yang–Mills–Scalar–Matter fields, following the line of
research of M. Durdevich and also in agreement with [HM], [LRZ], [Z], [L]. To
accomplish this, we are going to dualize the geometrical formulation of the
classical theory, in which principal $G$–bundles, principal connections, and
linear representations play the most important role. In addition, we shall
present a non–commutative geometrical Lagrangian for the system as well as the
non–commutative geometrical associated field equations. At the end of this
work, we are going to show a concrete example. This paper continues of the
theory formulated in [Sa2] and in this way, we shall use its notation and all
the concepts developed.

The importance of this paper lies in its geometrical approach, but also in the
generality of this theory, which could be applied on too many quantum
principal bundle and it works for all quantum principal connection, there is
no need to assume that the connection is strong either regular [BDH], [D2],
[Sa2]. Furthermore, this paper opens the door to get a geometrical formulation
of the Standard Model in the framework of Non–Commutative Geometry and all
that this entails: Spin Geometry, characterization of Yang–Mills connections
by characteristic classes, Higgs mechanism, and Higgs bundles, Etc. as well as
the possibility of researching in Standard Model’s extensions.

Our work is organized into five sections. In the second one, we are going to
present the theory of the left/right quantum star Hodge operator concerning
quantum Riemannian metrics, as well as the left/right quantum codifferential.
Furthermore, by considering associated left/right quantum vector bundles, we
will present the non–commutative geometrical version of the adjoint operators
of exterior covariant derivatives. By using the theory of the second section,
in the third one, we will develop the theory of Yang–Mills and space–time
scalar matter fields, starting with pure Yang–Mills fields, later dealing with
$n$–multiples of space–time scalar matter fields and concluding with
space–time scalar matter fields coupled to Yang–Mills fields.

To keep a correct size of this paper, in the fourth section, we are going to
present just one example by using trivial quantum principal bundles in the
sense of [D2] to show that the theory developed in this paper is non–trivial.
Especifically, we will use a quantum principal $U(1)$–bundle over the space of
$2\times 2$ matrices with complex coefficients. The last section is about some
concluding comments. In Appendix A, the reader can appreciate a little summary
of complementary basic concepts. If the reader is not familiar with [Sa2], we
highly recommend for start the reading of this paper with Appendix A.

In [Sa3] we present another example of our theory using the quantum Hopf
fibration also known as the $q$–Dirac monopole bundle and in [Sa4] we present
another example using a trivial quantum principal bundle with the two–point
space as the base space and the symmetric group of order $2$ as the structure
group.

In the whole work, we will use Sweedler’s notation and we will use the symbols
$\langle-,-\rangle_{\mathrm{L}}$, $\langle-,-\rangle_{\mathrm{R}}$ to denote
hermitian structures, quantum Riemannian metrics and their extensions.

For the aim of this paper, to define the Lagrangian of
Yang–Mills–Scalar–Matter fields in Differential Geometry, it is necessary a
closed Riemannian manifold $(M,g)$, a principal $G$–bundle over $M$, an
$\mathrm{ad}$–invariant inner product of the Lie algebra $\mathfrak{g}$ of
$G$, a unitary finite–dimensional representation $\alpha$ of $G$ on
$V^{\alpha}$, and a smooth function $V:\mathbb{R}\longrightarrow\mathbb{R}$.
By using these elements, we define

(1.1)
$\mathscr{L}_{\mathrm{YMSM}}(\omega,T):=\mathscr{L}_{\mathrm{YM}}(\omega)+\mathscr{L}_{\mathrm{SM}}(\omega,T),$

with

(1.2) $\mathscr{L}_{\mathrm{YM}}(\omega)=-\dfrac{1}{2}\langle
R^{\omega},R^{\omega}\rangle\,,\quad\mathscr{L}_{\mathrm{SM}}(\omega,T):=\dfrac{1}{2}\left(\langle\nabla^{\omega}_{\alpha}T,\nabla^{\omega}_{\alpha}T\rangle-V(T)\right),$

where $R^{\omega}$ is the canonical $\mathfrak{g}M$–valued differential
$2$–form of $M$ associated to the curvature of the principal connection
$\omega$ (by means of the Gauge Principle [KMS], [SW]), $T$ $\in$
$\Gamma(M,V^{\alpha}M)$ is a section of the associated vector bundle with
respect to $\alpha$, $\nabla^{\omega}_{\alpha}$ is the induced linear
connection of $\omega$ in $V^{\alpha}M$ and111Now it should be clear the
definition of the corresponding maps $\langle-,-\rangle$.
$V(\Phi):=V\circ\langle T,T\rangle$. This Lagrangian is gauge–invariant and
the critical points of its associated action

(1.3)
$\mathscr{S}_{\mathrm{YMSM}}(\omega,T)=\int_{M}\mathscr{L}_{\mathrm{YMSM}}(\omega,T)\;d\mathrm{vol}_{g}$

are pairs $(\omega,T)$ that satisfy

(1.4) $\langle
d^{\nabla^{\omega}_{\mathrm{ad}}\star}R^{\omega}\mid\lambda\rangle=\langle\nabla^{\omega}_{\alpha}T\mid\alpha^{\prime}(\lambda)T\rangle,$

for all $\mathfrak{g}M$–valued $1$–form $\lambda$; and

(1.5)
$(\nabla^{\omega\,\star}_{\alpha}\,\nabla^{\omega}_{\alpha}-V^{\prime}(T))\,T=0,$

where $\nabla^{\omega\,\star}_{\alpha}$ is the formal adjoint operator of
$\nabla^{\omega}_{\alpha}$ and ${d^{\nabla^{\omega}_{\mathrm{ad}}}}^{\star}$
is the formal adjoint operator of the exterior covariant derivative associated
to $\nabla^{\omega}_{\mathrm{ad}}$ [Bl]. These equations are called
Yang–Mills–Scalar–Matter equations and they represent the dynamical of
space–time scalar matter particles coupled to gauge boson particles in the
Riemannian space $(M,g)$. In Subsection $3.3$ we show the non–commutative
geometrical version of Equations 1.1–1.5.

Other viewpoints on quantum bundles can be found in the literature, for
example in [BM], [BK], [Pl]. All these formulations are intrinsically related
by the theory of Hopf–Galois extensions [KT]. Moreover, there are other
proposals to bring Yang–Mills theory in Non–Commutative Geometry, for example
[CR], [Dj], [CCM] in which the authors directly used quantum vector bundles,
and the concept of spectral triples.

We have decided to use quantum principal bundles to develop this work because
we believe that a Yang–Mills–Matter theory in Non–Commutative Geometry should
be approached from the respective concepts of principal bundles and
representations, just like in the classical case. In addition, we have decided
to use Durdevich’s formulation of quantum principal bundles because of its
purely geometrical–algebraic framework, which will be evident a long this
work.

##  2\. The Quantum Hodge Operator and Adjoint Operators of Quantum Linear
Connections

In this section, we are going to assume that $(M,\cdot,\mathbbm{1},\ast)$ is a
$\ast$–subalgebra equipped with a $C^{\ast}$–norm (in other words, its
corresponding completion is a $C^{\ast}$–algebra).

######  Definition 2.1.

Given a quantum space $(M,\cdot,\mathbbm{1},\ast)$ and a graded differential
$\ast$–algebra $(\Omega^{\bullet}(M)$ $,d,\ast)$ generated by its degree $0$
elements $\Omega^{0}(M)=M$ (quantum differential forms on $M$), we shall say
that

  1. (1)

$M$ is oriented if for some $n$ $\in$ $\mathbb{N}$,

$\Omega^{k}(M)=0$

for all $k>n$ and

$\Omega^{n}(M)=M\,\mathrm{dvol},$

where $0\not=\mathrm{dvol}$ $\in$ $\Omega^{n}(M)$ satisfies

$p\,\mathrm{dvol}=0\;\Longleftrightarrow\;p=0.$

The element $\mathrm{dvol}$ is called quantum $n$–volume form and if we choose
one, we are going to say that $M$ has an orientation.

  2. (2)

A left quantum Riemannian metric (lqrm) on $M$ is a family of hermitian
structures (antilinear in the second coordinate)

$\\{\langle-,-\rangle^{k}_{\mathrm{L}}:\Omega^{k}(M)\times\Omega^{k}(M)\longrightarrow
M\\}$

where for $k=0$

$\displaystyle\langle-,-\rangle^{0}_{\mathrm{L}}:M\times M$
$\displaystyle\longrightarrow M$ $\displaystyle(\;\hat{p}\;,\;p\;)$
$\displaystyle\longmapsto\hat{p}p^{\ast}$

and such that

$\langle\hat{\mu}p,\mu\rangle^{k}_{\mathrm{L}}=\langle\hat{\mu},\mu
p^{\ast}\rangle^{k}_{\mathrm{L}}\quad\mbox{ and
}\quad\langle\mu,\mu\rangle^{k}_{\mathrm{L}}=0\;\Longleftrightarrow\;\mu=0$

for all $\hat{\mu}$, $\mu$ $\in$ $\Omega^{k}(M)$, $p$ $\in$ $M$ and $k\geq 1$.
If $M$ has an orientation $\mathrm{dvol}$, and

$\displaystyle\langle-,-\rangle^{n}_{\mathrm{L}}:\Omega^{n}(M)\times\Omega^{n}(M)$
$\displaystyle\longrightarrow\quad M$
$\displaystyle(\;\hat{p}\;\,\mathrm{dvol}\;,\;p\,\;\mathrm{dvol}\;)$
$\displaystyle\longmapsto\quad\hat{p}p^{\ast},$

then we will say that $\mathrm{dvol}$ is a left quantum Riemannian $n$–volume
form (lqr $n$–form). Now it should be clear the dual definition of right
quantum Riemannian metric (rqrm) on $M$

$\\{\langle-,-\rangle^{k}_{\mathrm{R}}:\Omega^{k}(M)\times\Omega^{k}(M)\longrightarrow
M\\}$

and the right quantum Riemannian $n$–volume form (rqr $n$–form)

  3. (3)

If $M$ has an orientation $\mathrm{dvol}$ and $s$ is a state of $M$, we define
a quantum integral (qi) on $M$ as

$\displaystyle\int_{M}:\Omega^{n}(M)$ $\displaystyle\longrightarrow\mathbb{C}$
$\displaystyle p\,\mathrm{dvol}$ $\displaystyle\longmapsto s(p).$

We can interpret that a given qi satisfies the Stokes theorem by explicitly
defining

$\displaystyle\int_{\partial M}:\Omega^{n-1}(M)$
$\displaystyle\longrightarrow\mathbb{C}$ $\displaystyle\mu$
$\displaystyle\longmapsto\int_{M}d\mu.$

If $\mathrm{Im}(d)$ $\subseteq$
$\mathrm{Ker}\left(\displaystyle\int_{M}\right)$ we are going to say that
$(M,\cdot,\mathbbm{1},\ast)$ is a quantum space without boundary (with respect
to the given qi).

Better yet, it is easy to see that

(2.1) $\mathrm{dvol}\,p=\varepsilon(p)\,\mathrm{dvol}$

for all $p$ $\in$ $M$, where $\varepsilon$ is a multiplicative unital linear
isomorphism and the composition $\varepsilon\circ\ast$ is an involution.
Notice that if the qi is a closed graded trace, it is possible to establish a
link with the cyclic cohomology [C]. Furthermore, by postulating the
orthogonality between quantum forms of different degrees, we can induce
riemannian structures in the whole graded space $\Omega^{\bullet}(M)$; so we
will not use superscripts anymore.

Given a quantum space $(M,\cdot,\mathbbm{1},\ast)$ with a qi, the maps

(2.2)
$\langle-|-\rangle_{\mathrm{L}}:=\int_{M}\langle-,-\rangle_{\mathrm{L}}\,\mathrm{dvol}\,,\qquad\langle-|-\rangle_{\mathrm{R}}:=\int_{M}\langle-,-\rangle_{\mathrm{R}}\,\mathrm{dvol}$

are an inner products for all $k=0,1,...,n$, and they are called the
left/right quantum Hodge inner products, respectively.

######  Remark 2.2.

Given a lqrm $\\{\langle-,-\rangle_{\mathrm{L}}\\}$ on $M$, we can define a
rqrm on $M$ by means of

$\langle\hat{\mu},\mu\rangle_{\mathrm{R}}:=\langle\hat{\mu}^{\ast},\mu^{\ast}\rangle_{\mathrm{L}}$

and viceversa.

From this moment on, we shall work just with lqrms; however, every single
result presented has a counterpart for rqrms.

In many cases, Non–Commutative Geometry is too general in the sense that we
have a lot of freedom to choose the appropriate structures, which is in a
clear opposition with the classical theory. So in order to develop a
meaningful theory, in many concrete situations we have to impose additional
restrictions in some way. The reader should not worry about this because the
theory keeps being non–trivial: there are still a lot of illustrative and rich
examples, as we shall appreciate in the last section of this work and in
[Sa3], [Sa4].

######  Remark 2.3.

From this point on, we shall assume that $M$ has a fixed qr $n$–form
$\mathrm{dvol}$, and a qi for which $M$ does not have boundary. Furthermore,
we shall assume that for a given $\mu$ $\in$ $\Omega^{n-k}(M)$, the left
$M$–module map

$\displaystyle F_{\mu}:\Omega^{k}(M)$ $\displaystyle\longrightarrow M$
$\displaystyle\hat{\mu}$ $\displaystyle\longmapsto f_{\mu}(\hat{\mu}),$

where $\hat{\mu}\mu=F_{\mu}(\hat{\mu})\,\mathrm{dvol},$ satisfies

$F_{\mu}=\langle-,\star^{-1}_{\mathrm{L}}\mu\rangle_{\mathrm{L}}$

for a unique element $\star^{-1}_{\mathrm{L}}\mu$ $\in$ $\Omega^{k}(M)$. We
will suppose that this identification induces an antilinear isomorphism.

######  Definition 2.4.

For a given quantum space $(M,\cdot,\mathbbm{1},\ast)$, we define the left
quantum Hodge star operator as

$\displaystyle\star_{\mathrm{L}}:\Omega^{k}(M)$
$\displaystyle\longrightarrow\Omega^{n-k}(M)$ $\displaystyle\mu$
$\displaystyle\longmapsto\star_{\mathrm{L}}\mu.$

By construction, for $k=0,...,n$

(2.3)
$\hat{\mu}\,\mu=\langle\hat{\mu},\star^{-1}_{\mathrm{L}}\mu\rangle_{\mathrm{L}}\,\mathrm{dvol},$

with $\hat{\mu}$ $\in$ $\Omega^{k}(M)$ and $\mu$ $\in$ $\Omega^{n-k}(M)$ and
$\star^{-1}_{\mathrm{L}}$ is uniquely determined by the above equation.

The next result straightforwardly follows.

######  Theorem 2.5.

  1. (1)

For all $\hat{\mu}$, $\mu$ $\in$ $\Omega^{k}(M)$ the following equality holds

$\hat{\mu}\,(\star_{\mathrm{L}}\mu)=\langle\hat{\mu},\mu\rangle_{\mathrm{L}}\,\mathrm{dvol}.$

  2. (2)

For all $p$ $\in$ $M$ and $\mu$ $\in$ $\Omega^{\bullet}(M)$ we get

$\star^{-1}_{\mathrm{L}}(p\,\mu)=(\star^{-1}_{\mathrm{L}}\mu)\,p^{\ast},\qquad\star^{-1}_{\mathrm{L}}(\mu\,p)=\varepsilon(p)^{\ast}(\star_{\mathrm{L}}\mu),$
$\star_{\mathrm{L}}(\varepsilon(p)^{\ast}\,\mu)=(\star_{\mathrm{L}}\mu)\,p,\qquad\star_{\mathrm{L}}(\mu\,p)=p^{\ast}(\star_{\mathrm{L}}\mu).$

  3. (3)

We have

$\star_{\mathrm{L}}\mathbbm{1}=\mathrm{dvol},\quad\star_{\mathrm{L}}\mathrm{dvol}=\mathbbm{1}.$

  4. (4)

For $\widetilde{\mu}$ $\in$ $\Omega^{m}(M)$, $\hat{\mu}$ $\in$
$\Omega^{l}(M)$, $\mu$ $\in$ $\Omega^{k}(M)$ such that $m+l+k=n$

$\langle\hat{\mu},\star^{-1}_{\mathrm{L}}(\widetilde{\mu}\mu)\rangle_{\mathrm{L}}=\langle\hat{\mu}\widetilde{\mu},\star^{-1}_{\mathrm{L}}\mu\rangle_{\mathrm{L}}.$

  5. (5)

The following formula holds

$\displaystyle\langle\hat{\mu}\,|\,\mu\rangle_{\mathrm{L}}=\int_{M}\hat{\mu}\,(\star_{\mathrm{L}}\mu)$

for all $\hat{\mu}$, $\mu$ $\in$ $\Omega^{\bullet}(M)$.

Our next and final step here is to present the construction of the
non–commutative counterparts of the codifferential and the Laplace–de Rham
operators.

######  Definition 2.6.

Let $(M,\cdot,\mathbbm{1},\ast)$ be a quantum space. By considering the left
quantum Hodge star operator $\star_{\mathrm{L}}$, we define the left quantum
codifferential as the linear operator

$\displaystyle
d^{\star_{\mathrm{L}}}:=(-1)^{k+1}\star^{-1}_{\mathrm{L}}\circ\;d\circ\star_{\mathrm{L}}:\Omega^{k+1}(M)$
$\displaystyle\longrightarrow\Omega^{k}(M)$ $\displaystyle\mu$
$\displaystyle\longmapsto d^{\star_{\mathrm{L}}}\mu.$

For $k+1=0$ we take $d^{\star_{\mathrm{L}}}=0.$

Let $\hat{\mu}$ $\in$ $\Omega^{k}(M)$, $\mu$ $\in$ $\Omega^{k+1}(M)$. Then
$\star_{\mathrm{L}}\mu$ $\in$ $\Omega^{n-k-1}(M)$ and
$\hat{\mu}\star_{\mathrm{L}}\mu$ $\in$ $\Omega^{n-1}(M)$; so in the virtue of
Theorem 2.5 point $1$ and since $M$ is a quantum space without boundary

$\displaystyle 0\;=\;\int_{M}d(\hat{\mu}(\star_{\mathrm{L}}\mu))$
$\displaystyle=$
$\displaystyle\int_{M}(d\hat{\mu})\star_{\mathrm{L}}\mu+(-1)^{k}\int_{M}\hat{\mu}(d\star_{\mathrm{L}}\mu)$
$\displaystyle=$
$\displaystyle\int_{M}(d\hat{\mu})\star_{\mathrm{L}}\mu-(-1)^{k+1}\int_{M}\hat{\mu}(\star_{\mathrm{L}}\star^{-1}_{\mathrm{L}}d\star_{\mathrm{L}}\mu)$
$\displaystyle=$ $\displaystyle\int_{M}\langle
d\hat{\mu},\mu\rangle_{\mathrm{L}}\,\mathrm{dvol}-\int_{M}\hat{\mu}(\star_{\mathrm{L}}\,d^{\star_{\mathrm{L}}}\mu)$
$\displaystyle=$ $\displaystyle\int_{M}\langle
d\hat{\mu},\mu\rangle_{\mathrm{L}}\,\mathrm{dvol}-\int_{M}\langle\hat{\mu},d^{\star_{\mathrm{L}}}\mu\rangle_{\mathrm{L}}\,\mathrm{dvol}$

and thus

$\langle
d\hat{\mu}\,|\,\mu\rangle_{\mathrm{L}}=\langle\hat{\mu}\,|\,d^{\star_{\mathrm{L}}}\mu\rangle_{\mathrm{L}}.$

In other words, we have just proved

######  Theorem 2.7.

The map $d^{\star_{\mathrm{L}}}$ is the adjoint operator of $d$ concerning the
left quantum Hodge inner product $\langle-|-\rangle_{\mathrm{L}}$.

Moreover, the following formulas hold

(2.4) $d^{\star_{\mathrm{L}}}\circ d^{\star_{\mathrm{L}}}=0,$ (2.5)
$d^{\star_{\mathrm{L}}}(\varepsilon(p)^{\ast}\mu)=\varepsilon(p)^{\ast}\,d^{\star_{\mathrm{L}}}\mu+(-1)^{n}\star^{-1}_{\mathrm{L}}((\star_{\mathrm{L}}\mu)\,dp),$
(2.6)
$d^{\star_{\mathrm{L}}}(\mu\,p)=(d^{\star_{\mathrm{L}}}\mu)\,p+(-1)^{k+1}\star^{-1}_{\mathrm{L}}(dp^{\ast}(\star_{\mathrm{L}}\mu)),$

for all $p$ $\in$ $M$ and $\mu$ $\in$ $\Omega^{k+1}(M)$. Now we are ready to
define the quantum Laplacian.

######  Definition 2.8.

Given a quantum space $(M,\cdot,\mathbbm{1},\ast)$ and the left quantum Hodge
star operator $\star_{\mathrm{L}}$, the left quantum Laplace–de Rham operator
is defined as

$\vartriangle_{\mathrm{L}}:=d\circ
d^{\star_{\mathrm{L}}}+d^{\star_{\mathrm{L}}}\circ
d=(d+d^{\star_{\mathrm{L}}})^{2}:\Omega^{\bullet}(M)\longrightarrow\Omega^{\bullet}(M).$

Finally, it is easy to see that

######  Proposition 2.9.

The left quantum Laplace–de Rham operator is symmetric and non–negative, i.e.,
$\langle\vartriangle_{\mathrm{L}}\hat{\mu}\,|\,\mu\rangle_{\mathrm{L}}=\langle\hat{\mu}\,|\vartriangle_{\mathrm{L}}\mu\rangle_{\mathrm{L}}$
and $\langle\vartriangle_{\mathrm{L}}\mu\,|\,\mu\rangle_{\mathrm{L}}\geq 0$.

Now it is possible to define left quantum harmonic differential forms, left
quantum de Rham cohomology, and left quantum Hodge theory; but it is not the
main focus of this work.

###  2.1. Adjoint Operators of Quantum Linear Connections

Let $\alpha$ be a finite–dimensional $\mathcal{G}$–representaiton and
$\zeta=(GM,M,{{}_{GM}}\Phi)$ be a quantum principal bundle (qpb) with a
quantum principal connection (qpc) $\omega$ (see Appendix A.2). We define the
hermitian structure for left quantum vector bundle–valued differential forms
(left qvb–valued differential forms, see Appendix A.3)

$\displaystyle\langle-,-\rangle_{\mathrm{L}}:\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$
$\displaystyle\longrightarrow M$

in such a way that
$\langle\mu_{1}\otimes_{M}T_{1},\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}=\langle\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}.$ By using the
previous definition and the qi we can define the map

(2.7)
$\langle-|-\rangle_{\mathrm{L}}:\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\longrightarrow\mathbb{C}$

by

$\displaystyle\langle\mu_{1}\otimes_{M}T_{1}\,|\,\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}=\int_{M}\langle\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}.$

######  Proposition 2.10.

The map $\langle-|-\rangle_{\mathrm{L}}$ is an inner product for left
qvb–valued forms.

###### Proof.

The only part of the statement that it is not trivial is the positive-
definiteness; so let us proceed to prove it. Notice that it is enough to prove
the statement for $\alpha$ $\in$ $\mathcal{T}$ [Sa2]. Let
$\psi=\displaystyle\sum_{k}\mu_{k}\otimes_{M}T_{k}$ such that
$\langle\psi,\psi\rangle^{k}_{\mathrm{L}}=0$. Then
$\tau:=\Upsilon^{-1}_{\alpha}(\psi)=\displaystyle\sum_{k}\mu_{k}\,T_{k}$ $\in$
$\textsc{Mor}(\alpha,{{}_{\mathrm{H}}}\Phi)$ and
$\psi=\displaystyle\sum_{k}\mu_{k}\otimes_{M}T_{k}=\sum^{d_{\alpha}}_{i=1}\mu^{\tau}_{i}\otimes_{M}T^{\mathrm{L}}_{i},$
where
$\mu_{k}=\displaystyle\sum^{n_{\alpha}}_{i=1}\tau(e_{i})\,x^{\alpha\,\ast}_{ki}$
(see Equation A.10). Hence

$\displaystyle
0\;=\;\langle\psi,\psi\rangle_{\mathrm{L}}\;=\;\sum^{d_{\alpha}}_{i,j=1}\langle\mu^{\tau}_{i}\otimes_{M}T^{\mathrm{L}}_{i},\mu^{\tau}_{j}\otimes_{M}T^{\mathrm{L}}_{j}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\sum^{d_{\alpha}}_{i,j=1}\langle\mu^{\tau}_{i}\langle
T^{\mathrm{L}}_{i},T^{\mathrm{L}}_{j}\rangle_{\mathrm{L}},\mu^{\tau}_{j}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\sum^{d_{\alpha},n_{\alpha}}_{i,j,k=1}\langle\mu^{\tau}_{i}\,x^{\alpha}_{ik}x^{\alpha\,\ast}_{jk},\mu^{\tau}_{j}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\sum^{d_{\alpha},n_{\alpha}}_{i,j,k,l=1}\langle\tau(e_{l})\,x^{\alpha\,\ast}_{il}x^{\alpha}_{ik}x^{\alpha\,\ast}_{jk},\mu^{\tau}_{j}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\sum^{d_{\alpha},n_{\alpha}}_{j,k,l=1}\langle\tau(e_{l})\,\delta_{lk}\,x^{\alpha\,\ast}_{jk},\mu^{\tau}_{j}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\sum^{d_{\alpha},n_{\alpha}}_{j,k=1}\langle\tau(e_{k})\,x^{\alpha\,\ast}_{jk},\mu^{\tau}_{j}\rangle_{\mathrm{L}}\;=\;\sum^{d_{\alpha}}_{j=1}\langle\mu^{\tau}_{j},\mu^{\tau}_{j}\rangle_{\mathrm{L}}.$

Since $(M,\cdot,\mathbbm{1},\ast)$ is a $\ast$–subalgebra of a
$C^{\ast}$–algebra

$0\leq\langle\mu^{\tau}_{j},\mu^{\tau}_{j}\rangle_{\mathrm{L}}\leq\sum^{d_{\alpha}}_{j=1}\langle\mu^{\tau}_{j},\mu^{\tau}_{j}\rangle_{\mathrm{L}}=0\;\;\Longrightarrow\;\;\langle\mu^{\tau}_{j},\mu^{\tau}_{j}\rangle_{\mathrm{L}}=0\;\;\Longrightarrow\;\;\mu^{\tau}_{j}=0$

and therefore $\psi=0$. ∎

Also we have

######  Definition 2.11.

Considering the exterior covariant derivative associated to the induced qlc
$\nabla^{\omega}_{\alpha}$, $d^{\nabla^{\omega}_{\alpha}}$ (see Appendix A.3),
and the left quantum Hodge star operator $\star_{\mathrm{L}}$, we define

$d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}:\Omega^{k+1}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\longrightarrow\Omega^{k}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M),$

by

$d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}:=(-1)^{k+1}((\star^{-1}_{\mathrm{L}}\circ\ast)\otimes_{M}\mathrm{id}_{\Gamma^{\mathrm{L}}(M,V^{\alpha}M)})\circ\;d^{\nabla^{\omega}_{\alpha}}\circ((\ast\circ\star_{\mathrm{L}})\otimes_{M}\mathrm{id}_{\Gamma^{\mathrm{L}}(M,V^{\alpha}M)}).$

For $k+1=0$ we take $d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}=0$ and for
$k+1=1$ we are going to write
$d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}:=\nabla^{\omega\,\star_{\mathrm{L}}}_{\alpha}.$

The following statement shows that our definition is in a total agreement with
the classical theory.

######  Theorem 2.12.

The operator $d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}$ is the adjoint
operator of $d^{\nabla^{\omega}_{\alpha}}$ with respect to the inner product
for left qvb–valued forms for any qpc $\omega$.

###### Proof.

This proof consists of a large calculation. Let us first assume that $\omega$
is real (see Appendix A.2). Notice that taking
$\nabla^{\omega}_{\alpha}(T_{2})=\displaystyle\sum_{i}\mu^{D^{\omega}(T_{2})}_{i}\otimes_{M}T^{\mathrm{L}}_{i}$
$\in$ $\Omega^{1}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$, one
obtains

$d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}(\mu_{2}\otimes_{M}T_{2})=d^{\star_{\mathrm{L}}}\mu_{2}\otimes_{M}x_{2}+\,(-1)^{k+1}\sum_{i}\star^{-1}_{\mathrm{L}}(\mu^{D^{\omega}(T_{2})\,\ast}_{i}(\star_{\mathrm{L}}\mu_{2}))\otimes_{M}T^{\mathrm{L}}_{i}$

for all $\mu_{2}$ $\in$ $\Omega^{k+1}(M)$, $T_{2}$ $\in$
$\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$. Now for $\mu_{1}$ $\in$ $\Omega^{k}(M)$
and $T_{1}$ $\in$ $\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$ we get

$\displaystyle\langle
d\mu_{1}\otimes_{M}x_{1},\mu_{2}\otimes_{M}T_{1}\rangle_{\mathrm{L}}$
$\displaystyle=$ $\displaystyle\langle d\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}$ $\displaystyle=$
$\displaystyle\langle d(\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}}),\mu_{2}\rangle_{\mathrm{L}}+(-1)^{k+1}\langle\mu_{1}d\langle
T_{1},T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}$ $\displaystyle=$
$\displaystyle\langle d(\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}}),\mu_{2}\rangle_{\mathrm{L}}+(-1)^{k+1}\langle\mu_{1}\langle\nabla^{\omega}_{\alpha}(T_{1}),T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}$
$\displaystyle+$ $\displaystyle(-1)^{k+1}\,\langle\mu_{1}\langle
T_{1},\nabla^{\omega}_{\alpha}(T_{2})\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}},$

since in this case, $\langle-,-\rangle_{\mathrm{L}}$ and
$\nabla^{\omega}_{\alpha}$ are compatible [Sa2] (watch out with our abuse of
notation!). By definition of our hermitian structures

$\langle\mu_{1}\langle\nabla^{\omega}_{\alpha}(T_{1}),T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}=\langle\mu_{1}\,\nabla^{\omega}_{\alpha}(T_{1}),\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}$

and

$\langle\mu_{1}\langle
T_{1},\nabla^{\omega}_{\alpha}(T_{2})\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}=\sum_{i}\langle\mu_{1}\otimes_{M}T_{1},\star^{-1}_{\mathrm{L}}(\mu^{D^{\omega}(T_{2})\,\ast}_{i}(\star_{\mathrm{L}}\mu_{2}))\otimes_{M}T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}}.$

In fact

$\langle\mu_{1}\langle
T_{1},\nabla^{\omega}_{\alpha}(T_{2})\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}=\sum_{i}\langle\mu_{1}\langle
T_{1},T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}}\,\mu^{D^{\omega}(T_{2})\,\ast}_{i},\mu_{2}\rangle_{\mathrm{L}};$

while by Theorem 2.5 point $4$

$\sum_{i}\langle\mu_{1}\otimes_{M}T_{1},\star^{-1}_{\mathrm{L}}(\mu^{D^{\omega}(T_{2})\,\ast}_{i}(\star_{\mathrm{L}}\mu_{2}))\otimes_{M}T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}}=\sum_{i}\langle\mu_{1}\langle
T_{1},T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}},\star^{-1}_{\mathrm{L}}(\mu^{D^{\omega}(T_{2})\,\ast}_{i}(\star_{\mathrm{L}}\mu_{2}))\rangle_{\mathrm{L}}=$
$\sum_{i}\langle\mu_{1}\langle
T_{1},T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}}\,\mu^{D^{\omega}(T_{2})\,\ast}_{i},\star^{-1}_{\mathrm{L}}\star_{\mathrm{L}}\mu_{2}\rangle_{\mathrm{L}}=\sum_{i}\langle\mu_{1}\langle
T_{1},T^{\mathrm{L}}_{i}\rangle_{\mathrm{L}}\,\mu^{D^{\omega}(T_{2})\,\ast}_{i},\mu_{2}\rangle_{\mathrm{L}};$

thus the last assertion holds. Now taking into account these equalities and
Theorem 2.7 we find

$\displaystyle\langle
d^{\nabla^{\omega}_{\alpha}}(\mu_{1}\otimes_{M}T_{1})\,|\,\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}$
$\displaystyle=$ $\displaystyle\langle
d\mu_{1}\otimes_{M}T_{1}\,|\,\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}$
$\displaystyle+$
$\displaystyle(-1)^{k}\langle\mu_{1}\,\nabla^{\omega}_{\alpha}(T_{1})\,|\,\mu_{2}\otimes_{M}T_{2}\rangle_{\mathrm{L}}$
$\displaystyle=$ $\displaystyle\int_{M}\langle d(\mu_{1}\langle
T_{1},T_{2}\rangle_{\mathrm{L}}),\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle+$
$\displaystyle(-1)^{k+1}\int_{M}\langle\mu_{1}\langle\nabla^{\omega}_{\alpha}(T_{1}),T_{2}\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle+$ $\displaystyle(-1)^{k+1}\int_{M}\langle\mu_{1}\langle
T_{1},\nabla^{\omega}_{\alpha}(T_{2})\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle+$
$\displaystyle(-1)^{k}\int_{M}\langle\mu_{1}\,\nabla^{\omega}_{\alpha}(T_{1}),\mu_{2}\otimes_{M}T_{1}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle=$ $\displaystyle\int_{M}\langle\mu_{1}\langle
T_{1},T_{2}\rangle,d^{\star_{\mathrm{L}}}\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle+$ $\displaystyle(-1)^{k+1}\int_{M}\langle\mu_{1}\langle
T_{1},\nabla^{\omega}_{\alpha}(T_{2})\rangle_{\mathrm{L}},\mu_{2}\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle=$
$\displaystyle\int_{M}\langle\mu_{1}\otimes_{M}T_{1},d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}(\mu_{2}\otimes_{M}T_{2})\rangle_{\mathrm{L}}\,\mathrm{dvol}$
$\displaystyle=$
$\displaystyle\langle\mu_{1}\otimes_{M}x_{1}\,|\,d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}(\mu_{2}\otimes_{M}T_{2})\rangle_{\mathrm{L}}$

and the statement in this case follows from linearity.

Since every real quantum connection displacement $\lambda$ (see Equations A.3)
can written as $\lambda=\omega-\omega^{\prime}$, where $\omega$,
$\omega^{\prime}$ are real qpcs, we have that the operator
$\Upsilon_{\alpha}\circ K^{\lambda}\circ\Upsilon^{-1}_{\alpha}$ is
adjointable, where

(2.8)
$K^{\lambda}(\tau):=(D^{\omega}-D^{\omega^{\prime}})(\tau)=-(-1)^{k}\tau^{(0)}\lambda(\pi(\tau^{(1)}))$

with ${{}_{\mathrm{H}}}\Phi(\tau(v))=\tau^{(0)}(v)\otimes\tau^{(1)}(v)$ and
$\mathrm{Im}(\tau)$ $\in$ $\mathrm{Hor}^{k}GM$. This implies that
$\Upsilon_{\alpha}\circ i\,K^{\lambda^{\prime}}\circ\Upsilon^{-1}_{\alpha}$ is
also adjointable. By Equation A.4 we get that
$D^{\omega}=D^{\omega^{\prime}}+i\,K^{\lambda}$ for every qpc $\omega$ and the
theorem follows. ∎

Of course, there is a natural generalization of the left quantum Laplace–de
Rham operator for left qvb–valued forms by means of

(2.9) $\square^{\omega\mathrm{L}}_{\alpha}:=d^{\nabla^{\omega}_{\alpha}}\circ
d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}+d^{\nabla^{\omega}_{\alpha}\star_{\mathrm{L}}}\circ
d^{\nabla^{\omega}_{\alpha}}.$

This operator satisfies

$\langle\square^{\omega\mathrm{L}}_{\alpha}\hat{\psi}\,|\,\psi\rangle_{\mathrm{L}}=\langle\hat{\psi}\,|\,\square^{\omega\mathrm{L}}_{\alpha}\psi\rangle_{\mathrm{L}}\quad\mbox{
and
}\quad\langle\square^{\omega\mathrm{L}}_{\alpha}\psi\,|\,\psi\rangle_{\mathrm{L}}\geq
0$

for all $\hat{\psi}$, $\psi$ $\in$
$\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$.

######  Remark 2.13.

Of course, for rqrms and associated right qvbs, all this formalism still holds
with similar properties. For example the hermitian structure for right
qvb–valued forms

(2.10)
$\displaystyle\langle-,-\rangle_{\mathrm{R}}:\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\otimes_{M}\Omega^{\bullet}(M)\times\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\otimes_{M}\Omega^{\bullet}(M)$
$\displaystyle\longrightarrow M$

is given by $\langle
T_{1}\otimes_{M}\mu_{1},T_{2}\otimes_{M}\mu_{2}\rangle_{\mathrm{R}}=\langle\mu_{1},\langle
T_{1},T_{2}\rangle_{\mathrm{R}}\,\mu_{2}\rangle_{\mathrm{R}}$ and the inner
product

(2.11)
$\langle-|-\rangle_{\mathrm{R}}:\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\otimes_{M}\Omega^{\bullet}(M)\times\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\otimes_{M}\Omega^{\bullet}(M)\longrightarrow\mathbb{C}$

is defined by

$\langle T_{1}\otimes_{M}\mu_{1}\mid
T_{2}\otimes_{M}\mu_{2}\rangle_{\mathrm{R}}=\int_{M}\langle\mu_{1},\langle
T_{1},T_{2}\rangle_{\mathrm{R}}\,\mu_{2}\rangle_{\mathrm{R}}\,\mathrm{dvol}.$

In the context of Remark 2.2, the right quantum Hodge star operator and the
right quantum codifferential are given by

(2.12) $\star_{\mathrm{R}}=\ast\circ\star_{\mathrm{L}}\circ\ast,\qquad
d^{\star_{\mathrm{R}}}=(-1)^{k+1}\star^{-1}_{\mathrm{R}}\circ\;d\circ\star_{\mathrm{R}}=\ast\circ
d^{\star_{\mathrm{L}}}\circ\ast;$

while the adjoint operator of the exterior covariant derivative of
$\widehat{\nabla}^{\omega}_{\alpha}$ is

(2.13)
$d^{\widehat{\nabla}^{\omega}_{\alpha}\star_{\mathrm{R}}}:=(-1)^{k+1}(\mathrm{id}_{\Gamma^{\mathrm{R}}(M,V^{\alpha}M)}\otimes_{M}(\star^{-1}_{\mathrm{R}}\circ\ast))\circ\;d^{\widehat{\nabla}^{\omega}_{\alpha}}\circ(\mathrm{id}_{\Gamma^{\mathrm{R}}(M,V^{\alpha}M)}\otimes_{M}(\ast\circ\star_{\mathrm{R}})).$

For $k+1=1$ we are going to write
$d^{\widehat{\nabla}^{\omega}_{\alpha}\star_{\mathrm{R}}}:=\widehat{\nabla}^{\omega\,\star_{\mathrm{R}}}_{\alpha}.$
For the right structure we will use these relations.

##  3\. Yang–Mills–Scalar–Matter Fields in Noncommutative Geometry

By using the theory developed in the previous section, we can accomplish our
aim: the non–commutative geometrical version of the classical theory of
Yang–Mills–Scalar–Matter fields. Examples will be presented in the next
section.

###  3.1. Non–Commutative Geometrical Yang–Mills Fields

The next definition closely follows the classical formulation.

######  Definition 3.1 (Non–commutative geometrical Yang–Mills model).

In Non–Commutative Geometry a Yang–Mills model (ncg YM model) consists of

  1. (1)

A quantum space $(M,\cdot,\mathbbm{1},\ast)$ such that it is a $\ast$–algebra
completable into $C^{\ast}$–algebra.

  2. (2)

A quantum $\mathcal{G}$–bundle over $M$, $\zeta=(GM,M,{{}_{GM}}\Phi)$, with a
differential calculus such that the left quantum Hodge star operator exists
for the space of base forms.

  3. (3)

The operators $d^{S^{\omega}_{\mathrm{L}}}:=\Upsilon_{\mathrm{ad}}\circ
S^{\omega}\circ\Upsilon^{-1}_{\mathrm{ad}}$ and
$d^{\widehat{S}^{\omega}_{\mathrm{R}}}:=\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{S}^{\omega}\circ\widetilde{\Upsilon}^{-1}_{\mathrm{ad}}$
are assumed to be adjointable for any $\omega$ with respect to the inner
products of qvb–valued forms, where $\widehat{S}^{\omega}=\ast\circ
S^{\omega}\circ\ast$ (see Equation A.7).

The first two points are necessary to guarantee Theorem 2.12. Comments about
the last point will be presented in the final section.

######  Definition 3.2 (Noncommutative Yang–Mills Lagrangian and its action).

Given a ncg YM model, we define the non–commutative geometrial Yang–Mills
Lagrangian (ncg YM Lagrangian) as the association

$\displaystyle\mathscr{L}_{\mathrm{YM}}:\mathfrak{qpc}(\zeta)$
$\displaystyle\longrightarrow M$ $\displaystyle\omega$
$\displaystyle\longmapsto-\dfrac{1}{4}\left(\langle
R^{\omega},R^{\omega}\rangle_{\mathrm{L}}+\langle\widehat{R}^{\omega},\widehat{R}^{\omega}\rangle_{\mathrm{R}}\right),$

where $\langle
R^{\omega},R^{\omega}\rangle_{\mathrm{L}}:=\langle\Upsilon_{\mathrm{ad}}\circ
R^{\omega},\Upsilon_{\mathrm{ad}}\circ R^{\omega}\rangle_{\mathrm{L}}$,
$\langle\widehat{R}^{\omega},\widehat{R}^{\omega}\rangle_{\mathrm{R}}:=\langle\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{R}^{\omega},\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{R}^{\omega}\rangle_{\mathrm{R}}$
(see Appendix A.2). We define its associated action as

$\displaystyle\mathscr{S}_{\mathrm{YM}}:\mathfrak{qpc}(\zeta)$
$\displaystyle\longrightarrow\mathbb{R}$ $\displaystyle\omega$
$\displaystyle\longmapsto\int_{M}\mathscr{L}_{\mathrm{YM}}(\omega)\,\mathrm{dvol}=-\dfrac{1}{4}\left(\langle
R^{\omega}|R^{\omega}\rangle_{\mathrm{L}}+\langle\widehat{R}^{\omega}|\widehat{R}^{\omega}\rangle_{\mathrm{R}}\right)$

and we shall call it the non–commutative geometrical Yang–Mills action (ncg YM
action).

Let us consider the quantum gauge group (qgg) $\mathfrak{qGG}$ [Sa2]. If
$\mathfrak{F}_{\mathfrak{f}}$ is a graded differential $\ast$–algebra
morphism, then
$R^{\mathfrak{f}^{\circledast}\omega}=\mathfrak{F}_{\mathfrak{f}}\circ
R^{\omega}$, and since the maps $\mathbf{A}_{\mathfrak{f}}$ and
$\widehat{\mathbf{A}}_{\mathfrak{f}}$ are unitary, a direct calculation shows
that
$\mathscr{L}_{\mathrm{YM}}(\omega)=\mathscr{L}_{\mathrm{YM}}(\mathfrak{f}^{\circledast}\omega)$
for all $\omega$ $\in$ $\mathfrak{qpc}(\zeta)$. It is important to observe
that in general such relation does not hold for an arbitrary $\mathfrak{f}$
$\in$ $\mathfrak{qGG}$.

######  Definition 3.3.

We define the quantum gauge group of the Yang-Mills model as the group
$\mathfrak{qGG}_{\mathrm{YM}}:=\\{\mathfrak{f}\in\mathfrak{qGG}\mid\mathscr{L}_{\mathrm{YM}}(\omega)=\mathscr{L}_{\mathrm{YM}}(\mathfrak{f}^{\circledast}\omega)\mbox{
for all }\,\omega\in\mathfrak{qpc}(\zeta)\\}\subseteq\mathfrak{qGG}$.

It is worth mentioning that every qgt induced by a graded differential
$\ast$–algebra morphism $\mathfrak{F}_{\mathfrak{f}}$ is an element of
$\mathfrak{qGG}_{\mathrm{YM}}$, so $\mathfrak{qGG}_{\mathrm{YM}}$ always has
at least one element.

Our next step is getting field equations for $\omega$ $\in$
$\mathfrak{qpc}(\zeta)$ by postulating a variational principle for the ncg YM
action, in total agreement with the classical case.

######  Definition 3.4 (Yang–Mills quantum principal connections).

A stationary point of $\mathscr{S}_{\mathrm{YM}}$ is an element $\omega$ $\in$
$\mathfrak{qpc}(\zeta)$ such that for any $\lambda$ $\in$
$\overrightarrow{\mathfrak{qpc}(\zeta)}$ (see Equation LABEL:3.f2.2)

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YM}}(\omega+z\,\lambda)=0.$

Stationary points are also called Yang–Mills qpcs (YM qpcs or non–commutative
geometrical Yang–Mills fields). In terms of a physical interpretation, they
should be considered as gauge boson fields without sources and possessing the
symmetry $\mathfrak{qGG}_{\mathrm{YM}}$.

Now we will proceed to find YM qpcs.

######  Theorem 3.5.

A qpc $\omega$ is a YM qpc if and only if

(3.1)
$\langle\Upsilon_{\mathrm{ad}}\circ\lambda\,|\,(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})R^{\omega}\rangle_{\mathrm{L}}+\langle\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{\lambda}\,|\,(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}\rangle_{\mathrm{R}}=0$

for all $\lambda$ $\in$ $\overrightarrow{\mathfrak{qpc}(\zeta)}$, where
$(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})R^{\omega}:=(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})\circ\Upsilon_{\mathrm{ad}}\circ
R^{\omega}$,
$(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}:=(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\circ\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{R}^{\omega}$
and $d^{S^{\omega}\star_{\mathrm{L}}}$,
$d^{\widehat{S}^{\omega}\star_{\mathrm{R}}}$ are the adjoint operators of
$d^{S^{\omega}_{\mathrm{L}}}$, $d^{\widehat{S}^{\omega}_{\mathrm{R}}}$
respectively.

###### Proof.

For a given $\lambda$ $\in$ $\overrightarrow{\mathfrak{qpc}(\zeta)}$ we have

$\displaystyle\left.\dfrac{\partial}{\partial z}\right|_{z=0}\langle
R^{\omega+z\,\lambda}\,|\,R^{\omega+z\,\lambda}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\langle\Upsilon_{\mathrm{ad}}\circ(d\circ\lambda-\langle\omega,\lambda\rangle-\langle\lambda,\omega\rangle)\,|\,R^{\omega}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\langle\Upsilon_{\mathrm{ad}}\circ(d\circ\lambda+[\lambda,\omega]-S^{\omega}\circ\lambda)\,|\,R^{\omega}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\langle\Upsilon_{\mathrm{ad}}\circ(D^{\omega}-S^{\omega})\circ\lambda\,|\,R^{\omega}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\langle(d^{\nabla^{\omega}_{\mathrm{ad}}}-d^{S^{\omega}_{\mathrm{L}}})\circ\Upsilon_{\mathrm{ad}}\circ\lambda\,|\,R^{\omega}\rangle_{\mathrm{L}}$
$\displaystyle=$
$\displaystyle\langle\Upsilon_{\mathrm{ad}}\circ\lambda\,|\,(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})\circ
R^{\omega}\rangle_{\mathrm{L}}.$

In the same way we get

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\langle\widehat{R}^{\omega+z\,\lambda}\,|\,\widehat{R}^{\omega+z\,\lambda}\rangle_{\mathrm{R}}=\langle\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{\lambda}\,|\,(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}\rangle_{\mathrm{R}}$

and the theorem follows. ∎

We will refer to Equation 3.1 as the non–commutative geometrical Yang–Mills
field equation. It is worth mentioning that every flat qpc is a YM qpc since
it satisfies trivially the equations. Of course,
$\mathfrak{qGG}_{\mathrm{YM}}$ acts on the space of YM qpcs.

###  3.2. Non–Commutative Geometrical Multiples of Space–Time Scalar Matter
Fields

Like in the classical case, we shall introduce the necessary technical
elements.

######  Definition 3.6 (Non–commutative geometrical $n$–multiples for
space–time scalar matter models).

In Non–Commutative Geometry a $n$–multiples for a given space–time scalar
matter model (ncg $n$–sm model) consists of

  1. (1)

A quantum space $(M,\cdot,\mathbbm{1},\ast)$ closeable into a
$C^{\ast}$–algebra.

  2. (2)

A quantum $\mathcal{G}$–bundle over $M$, $\zeta=(GM,M,{{}_{GM}}\Phi)$, with a
differential calculus such that the left quantum Hodge star operator exists
for the space of base forms.

  3. (3)

The trivial $\mathcal{G}$–representation in $\mathbb{C}^{n}$.

  4. (4)

A Fréchet differentiable $V:M\longrightarrow M$ called the potential.

For the rest of this subsection, we shall consider
$\alpha:=\alpha^{\mathrm{triv}}_{\mathbb{C}^{n}}$. It is worth mentioning that
in this case the induced qlcs $\nabla^{\omega}_{\alpha}$,
$\widehat{\nabla}^{\omega}_{\overline{\alpha}}$ do not depend on $\omega$
(where $\overline{\alpha}$ is the complex conjugate representation of
$\alpha$), they take the same values for every qpc; of course, this is because
the representation is trivial [Sa2].

######  Definition 3.7 (Non–commutative geometrical $n$–space–time scalar
matter Lagrangian and its action).

Given a ncg $n$–sm model, we define its non–commutative geometrical Lagrangian
as the association

$\mathscr{L}_{\mathrm{SM}}:\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)\longrightarrow
M$

given by

$\mathscr{L}_{\mathrm{SM}}(T_{1},T_{2})=\dfrac{1}{4}\left(\langle\nabla^{\omega}_{\alpha}T_{1},\nabla^{\omega}_{\alpha}T_{1}\rangle_{\mathrm{L}}-V_{\mathrm{L}}(T_{1})-\langle\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2},\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\rangle_{\mathrm{R}}+V_{\mathrm{R}}(T_{2})\right)$

where $V_{\mathrm{L}}(T_{1}):=V\circ\langle
T_{1}\,,\,T_{1}\rangle_{\mathrm{L}}$ and $V_{\mathrm{R}}(T_{2}):=V\circ\langle
T_{2}\,,\,T_{2}\rangle_{\mathrm{R}}$. We define its associated action as

$\displaystyle\mathscr{S}_{\mathrm{SM}}:\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)$
$\displaystyle\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$
$\displaystyle\longrightarrow$ $\displaystyle\;\;\mathbb{C}$
$\displaystyle(\;T_{1}$ $\displaystyle,\;T_{2}\;)$ $\displaystyle\longmapsto$
$\displaystyle\int_{M}\mathscr{L}_{\mathrm{SM}}(T_{1},T_{2})\,\mathrm{dvol}.$

A direct calculation shows that

$\displaystyle\langle\nabla^{\omega}_{\alpha}T_{1},\nabla^{\omega}_{\alpha}T_{1}\rangle_{\mathrm{L}}-V_{\mathrm{L}}(T_{1})=\sum^{n}_{i=1}\langle
dp^{T_{1}}_{i},dp^{T_{1}}_{i}\rangle_{\mathrm{L}}-V(p^{T_{1}}_{i}(p^{T_{1}}_{i})^{\ast})$
$\displaystyle\langle\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2},\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\rangle_{\mathrm{R}}-V_{\mathrm{R}}(T_{2})=\sum^{n}_{i=1}\langle
dp^{T_{2}}_{i},dp^{T_{2}}_{i}\rangle_{\mathrm{L}}-V((p^{T_{2}}_{i})^{\ast}p^{T_{2}}_{i})$

where $p^{T_{1}}_{i}=T_{1}(e_{i})$, $p^{T_{2}}_{i}=T_{2}(\overline{e}_{i})$
$\in$ $M$ and $\\{e_{i}\\}^{n}_{i=1}$ is the canonical basis of
$\mathbb{C}^{n}$. Since $\mathrm{Im}(T)\subseteq M$ for all $T$ $\in$
$\textsc{Mor}(\alpha,{{}_{GM}}\Phi)$ and all $T$ $\in$
$\textsc{Mor}(\overline{\alpha},{{}_{GM}}\Phi)$, taking any $\mathfrak{f}$
$\in$ $\mathfrak{qGG}$ we get $\mathfrak{F}_{\mathfrak{f}}\circ T=T$; so

######  Proposition 3.8.

The Lagrangian $\mathscr{L}_{\mathrm{SM}}$ is quantum gauge–invariant.

Like in the previous section, our next step is getting field equations
postulating a variational principle for $\mathscr{S}_{\mathrm{SM}}$, in
agreement with the classical case.

######  Definition 3.9 (Non–commutative geometrical $n$–multiples of scalar
matter fields).

A stationary point of $\mathscr{S}_{\mathrm{SM}}$ is an element
$(T_{1},T_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$
such that for all $(U_{1},U_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{SM}}(T_{1}+z\,U_{1},T_{2}+z\,U_{2})=0.$

In terms of a physical interpretation, stationary points should be considered
as space–time scalar matter and antimatter fields.

As before, we will proceed to find stationary points.

######  Theorem 3.10.

Assume that $(T_{1},T_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$
satisfies

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\int_{M}V_{\mathrm{L}}(T_{1}+z\,U_{1})\,\mathrm{dvol}=\int_{M}\left.\dfrac{\partial}{\partial
z}\right|_{z=0}V_{\mathrm{L}}(T_{1}+z\,U_{1})\,\mathrm{dvol}\,,$

and

$\langle
V^{\prime}_{\mathrm{L}}(T_{1})\,U_{1}\,|\,T_{1}\rangle_{\mathrm{L}}=\langle
U_{1}\,|\,V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}\rangle_{\mathrm{L}}$

for all $(U_{1},U_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$,
where $V^{\prime}_{\mathrm{L}}(T_{1}):=V^{\prime}\circ\langle
T_{1}\,,\,T_{1}\rangle_{\mathrm{L}}$ (and analogous assumptions for
$V^{\prime}_{\mathrm{R}}(T_{2}):=V^{\prime}\circ\langle
T_{2}\,,\,T_{2}\rangle_{\mathrm{R}}$) with $V^{\prime}$ the derivative of $V$.
Then $(T_{1},T_{2})$ is a stationary point if and only if

(3.2)
$\nabla^{\omega\,\star_{\mathrm{L}}}_{\alpha}\left(\nabla^{\omega}_{\alpha}T_{1}\right)-V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}=0\,,\quad\widehat{\nabla}^{\omega\,\star_{\mathrm{R}}}_{\overline{\alpha}}\left(\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\right)-T_{2}\,V^{\prime}_{\mathrm{R}}(T_{2})^{\ast}=0.$

###### Proof.

For a given $(U_{1},U_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$
we have

$\displaystyle\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{SM}}(T_{1}+z\,U_{1},T_{2}+z\,U_{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{4}\left(\langle
U_{1}\,|\,\nabla^{\omega\,\star_{\mathrm{L}}}_{\alpha}\left(\nabla^{\omega}_{\alpha}T_{1}\right)-V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}\rangle_{\mathrm{L}}\right.$
$\displaystyle-$
$\displaystyle\left.\langle\widehat{\nabla}^{\omega\,\star_{\mathrm{R}}}_{\overline{\alpha}}\left(\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\right)-T_{2}\,V^{\prime}_{\mathrm{R}}(T_{2})^{\ast}\,|\,U_{2}\rangle_{\mathrm{R}}\right).$

According to Proposition 2.10, we get $\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{SM}}(T_{1}+z\,U_{1},T_{2}+z\,U_{2})=0$ for
all $(U_{1},U_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,\mathbb{C}^{n}M)\times\Gamma^{\mathrm{R}}(M,\overline{\mathbb{C}}^{n}M)$
if and only if Equation 3.2 holds. ∎

Equation 3.2 turns into

(3.3)
$\sum^{n}_{k=1}d^{\star_{\mathrm{L}}}dp^{T_{1}}_{i}-V^{\prime}(p^{T_{1}}_{i}(p^{T_{1}}_{i})^{\ast})^{\ast}p^{T_{1}}_{i}=0\,,\quad\sum^{n}_{k=1}d^{\star_{\mathrm{L}}}d(p^{T_{2}}_{i})^{\ast}-\,V^{\prime}((p^{T_{2}}_{i})^{\ast}p^{T_{2}}_{i})(p^{T_{2}}_{i})^{\ast}=0$

for all $i=1,...,n$. Of course explicit solutions of the last equation depend
completely on the form of $V$ and the differential structure on the quantum
base space; the quantum total space, the quantum group and their differential
structures do not intervene explicitly.

###  3.3. Non–Commutative Geometrical Yang–Mills–Scalar–Matter Fields

Like in the classical case we will start by presenting the necessary elements
to the theory.

######  Definition 3.11 (Non–commutative geometrical Yang–Mills–Scalar–Matter
model).

In Non–Commutative Geometry a Yang–Mills–Scalar–Matter model (ncg YMSM model)
will consist of

  1. (1)

A quantum space $(M,\cdot,\mathbbm{1},\ast)$ such that it is a
$\ast$–subalgebra of a $C^{\ast}$–algebra.

  2. (2)

A quantum $\mathcal{G}$–bundle over $M$, $\zeta=(GM,M,{{}_{GM}}\Phi)$, with a
differential calculus such that the left quantum Hodge star operator exists
for the space of base forms.

  3. (3)

The operators $d^{S^{\omega}_{\mathrm{L}}}:=\Upsilon_{\mathrm{ad}}\circ
S^{\omega}\circ\Upsilon^{-1}_{\mathrm{ad}}$ and
$d^{\widehat{S}^{\omega}_{\mathrm{R}}}:=\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{S}^{\omega}\circ\widetilde{\Upsilon}^{-1}_{\mathrm{ad}}$
are assumed to be adjointable for any $\omega$ with respect to the inner
products of qvb–valued forms, where $\widehat{S}^{\omega}=\ast\circ
S^{\omega}\circ\ast$.

  4. (4)

A $\mathcal{G}$–representation $\alpha$ in a finite–dimensional
$\mathbb{C}$–vector space $V^{\alpha}$.

  5. (5)

A Fréchet differentiable map $V:M\longrightarrow M$ called the potential.

These conditions establish similar frameworks as the ones discuss in the
previous subsections. Taking into account that the complex conjugate
representation $\overline{\alpha}$ of $\alpha$ acts on $\overline{V}$ we have

######  Definition 3.12 (Non–commutative geometrical Yang–Mills–Scalar–Matter
Lagrangian and its action).

Given a ncg YMSM model, we define the non–commutative geometrical
Yang–Mills–Scalar–Matter Lagrangian (ncg YMSM Lagrangian) as the association

$\mathscr{L}_{\mathrm{YMSM}}:\mathfrak{qpc}(\zeta)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)\longrightarrow
M$

given by

$\mathscr{L}_{\mathrm{YMSM}}(\omega,T_{1},T_{2})=\mathscr{L}_{\mathrm{YM}}(\omega)+\mathscr{L}_{\mathrm{GSM}}(\omega,T_{1},T_{2}),$

where $\mathscr{L}_{\mathrm{YM}}$ is the ncg YM Lagrangian and
$\mathscr{L}_{\mathrm{GSM}}$ is the non–commutative geometrical generalized
space–time scalar matter Lagrangian (ncg GSM Lagrangian) which is given by
(comparing with Definition 3.2)

$\mathscr{L}_{\mathrm{GSM}}(\omega,T_{1},T_{2})=\dfrac{1}{4}\left(\langle\nabla^{\omega}_{\alpha}T_{1},\nabla^{\omega}_{\alpha}T_{1}\rangle_{\mathrm{L}}-V_{\mathrm{L}}(T_{1})-\langle\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2},\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\rangle_{\mathrm{R}}+V_{\mathrm{R}}(T_{2})\right)$

where $V_{\mathrm{L}}(T_{1}):=V\circ\langle
T_{1}\,,\,T_{1}\rangle_{\mathrm{L}}$ and $V_{\mathrm{R}}(T_{2}):=V\circ\langle
T_{2}\,,\,T_{2}\rangle_{\mathrm{R}}$. We define its associated action as

$\displaystyle\mathscr{S}_{\mathrm{YMSM}}:\mathfrak{qpc}(\zeta)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,V^{\alpha}M)$
$\displaystyle\longrightarrow\mathbb{C}$
$\displaystyle(\,\omega\;,\;T_{1}\;,\;T_{2}\,)$
$\displaystyle\longmapsto\int_{M}\mathscr{L}_{\mathrm{YMSM}}(\omega,T_{1},T_{2})\,\mathrm{dvol}$

and we shall call it non–commutative geometrical Yang–Mills–Scalar–Matter
action (ncg YMSM action).

According to [Sa2], if $\mathfrak{F}_{\mathfrak{f}}$ is a graded differential
$\ast$–algebra morphism, then
$R^{\mathfrak{f}^{\circledast}\omega}=\mathfrak{F}_{\mathfrak{f}}\circ
R^{\omega}$, and since the maps $\mathbf{A}_{\mathfrak{f}}$ and
$\widehat{\mathbf{A}}_{\mathfrak{f}}$ are unitary, a direct calculation shows
that
$\mathscr{L}_{\mathrm{YMSM}}(\omega,T_{1},T_{2})=\mathscr{L}(\mathfrak{f}^{\circledast}\omega,\mathbf{A}_{\mathfrak{f}}(T_{1}),\widehat{\mathbf{A}}_{\mathfrak{f}}(T_{2}))$
for all $\omega$ $\in$ $\mathfrak{qpc}(\zeta)$ and all $T_{1}$ $\in$
$\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$ and $T_{2}$ $\in$
$\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$. In agreement with the
previous observations, in general it will be not true that any $\mathfrak{f}$
$\in$ $\mathfrak{qGG}$ is a Lagrangian symmetry.

######  Definition 3.13.

We define the quantum gauge group of the Yang-Mills–Scalar–Matter model as the
group
$\mathfrak{qGG}_{\mathrm{YMSM}}:=\\{\mathfrak{f}\in\mathfrak{qGG}\mid\mathscr{L}_{\mathrm{YMSM}}(\omega,T_{1},T_{2})=\mathscr{L}(\mathfrak{f}^{\circledast}\omega,\mathbf{A}_{\mathfrak{f}}(T_{1}),\widehat{\mathbf{A}}_{\mathfrak{f}}(T_{2}))\\}\subseteq\mathfrak{qGG}$.

It is worth mentioning that every qgt induced by a graded differential
$\ast$–algebra morphism $\mathfrak{F}_{\mathfrak{f}}$ is an element of
$\mathfrak{qGG}_{\mathrm{YMSM}}$, so $\mathfrak{qGG}_{\mathrm{YMSM}}$ always
has at least one element. Of course, the group
$\mathfrak{qGG}_{\mathrm{YMSM}}$ depends on the potential $V$.

Like in previous subsections, the next step is getting the non–commutative
geometrical field equations for $(\omega,T_{1},T_{2})$ $\in$
$\mathfrak{qpc}(\zeta)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$
by postulating a variational principle for $\mathscr{S}_{\mathrm{YMSM}}$. All
of this in total agreement with the classical case.

######  Definition 3.14 (Non–commutative geometrical Yang–Mills–Scalar–Matter
field).

A stationary point of $\mathscr{S}_{\mathrm{YMSM}}$ is a triplet
$(\omega,T_{1},T_{2})$ $\in$
$\mathfrak{qpc}(\zeta)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$
such that for any $(\lambda,U_{1},U_{2})$ $\in$
$\overrightarrow{\mathfrak{qpc}(\zeta)}\times\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YMSM}}(\omega+z\,\lambda,T_{1},T_{2})=\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YMSM}}(\omega,T_{1}+z\,U_{1},T_{2}+z\,U_{2})=0.$

Stationary points are also called (non–commutative geometrical)
Yang–Mills–Scalar–Matter fields (YMSM fields) and in terms of a physical
interpretation, they can be interpreted as scalar matter and antimatter fields
coupled to gauge boson fields with symmetry $\mathfrak{qGG}_{\mathrm{YMSM}}$.

Now we are going to find the equations of motion.

######  Theorem 3.15.

Assume that $(T_{1},T_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$
satisfies

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\int_{M}V_{\mathrm{L}}(T_{1}+z\,U_{1})\,\mathrm{dvol}=\int_{M}\left.\dfrac{\partial}{\partial
z}\right|_{z=0}V_{\mathrm{L}}(T_{1}+z\,U_{1})\,\mathrm{dvol}\,,$

and

$\langle
V^{\prime}_{\mathrm{L}}(T_{1})\,U_{1}\,|\,T_{1}\rangle_{\mathrm{L}}=\langle
U_{1}\,|\,V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}\rangle_{\mathrm{L}}$

for all $(U_{1},U_{2})$ $\in$
$\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$,
where $V^{\prime}_{\mathrm{L}}(T_{1}):=V^{\prime}\circ\langle
T_{1}\,,\,T_{1}\rangle_{\mathrm{L}}$ (and analogous assumptions for
$V^{\prime}_{\mathrm{R}}(T_{2}):=V^{\prime}\circ\langle
T_{2}\,,\,T_{2}\rangle_{\mathrm{R}}$) with $V^{\prime}$ the derivative of $V$.
Then $(\omega,T_{1},T_{2})$ $\in$
$\mathfrak{qpc}(\zeta)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,\overline{V^{\alpha}}M)$
is a YMSM field if and only if for all $\lambda$ $\in$
$\overrightarrow{\mathfrak{qpc}(\zeta)}$

(3.4) $\begin{split}\langle\Upsilon_{\alpha}\circ
K^{\lambda}(T_{1})\,|\,\nabla^{\omega}_{\alpha}T_{1}\rangle_{\mathrm{L}}\hskip
17.07182pt-\hskip
17.07182pt\langle\widetilde{\Upsilon}_{\overline{\alpha}}\circ\widehat{K}^{\lambda}(T_{2})\,|\,\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\rangle_{\mathrm{R}}\;\hskip
17.07182pt=\\\
\langle\Upsilon_{\mathrm{ad}}\circ\lambda\,|\,(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})R^{\omega}\rangle_{\mathrm{L}}\;+\;\langle\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{\lambda}\,|\,(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}\rangle_{\mathrm{R}}\end{split}$

and

(3.5)
$\nabla^{\omega\,\star_{\mathrm{L}}}_{\alpha}\left(\nabla^{\omega}_{\alpha}\,T_{1}\right)-V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}=0\,,\qquad\widehat{\nabla}^{\omega\,\star_{\mathrm{R}}}_{\overline{\alpha}}\left(\widehat{\nabla}^{\omega}_{\overline{\alpha}}\,T_{2}\right)-T_{2}\,V^{\prime}_{\mathrm{R}}(T_{2})^{\ast}=0.$

###### Proof.

For a given $\lambda$ $\in$ $\overrightarrow{\mathfrak{qpc}(\zeta)}$ notice
that

$\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{GSM}}(\omega+z\,\lambda,T_{1},T_{2})=\dfrac{1}{4}\left(\langle\Upsilon_{\alpha}\circ
K^{\lambda}(T_{1})\,|\,\nabla^{\omega}_{\alpha}T_{1}\rangle_{\mathrm{L}}-\langle\widetilde{\Upsilon}_{\overline{\alpha}}\circ\widehat{K}^{\lambda}(T_{2})\,|\,\widehat{\nabla}^{\omega}_{\overline{\alpha}}T_{2}\rangle_{\mathrm{R}}\right),$

thus $\displaystyle\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YMSM}}(\omega+z\,\lambda,T_{1},T_{2})=0$
if and only if Equation 3.4 holds. Just like in Theorem 3.15, a direct
calculation shows that $\displaystyle\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YMSM}}(\omega,T_{1}+z\,U_{1},T_{2}+z\,U_{2})=0$
if and only Equation 3.5 holds. ∎

We shall refer to Equations 3.4, 3.5 as (the non–commutative geometrical)
Yang–Mills–Scalar–Matter field equations (YMSM field equations). The reader is
invited to compare these equations with their classical counterparts (see
Equations 1.4, 1.5).

It is worth mentioning that in all cases, the variation of the action with
respect to $z^{\ast}$ produces the same field equations.

##  4\. Example: Trivial Quantum Principal Bundles and Matrices

In the previous section, we had to impose some conditions to develop the
theory. At the first instance, these conditions seem too restrictive, so it is
necessary to present some examples to show that our theory is non–trivial and
there are interesting spaces to study. It is worth mentioning that for the
trivial corepresentation on $\mathbb{C}^{n}$, the first part of Equation 3.4
equals to zero; thus the only way to satisfy Equation 3.4 is when $\omega$ is
a YM qpc. Moreover, Equation 3.5 reduces to Equation 3.3. In summary, for the
trivial quantum representation on $\mathbb{C}^{n}$ in any qpb, YMSM fields are
triplets $(\omega,T_{1},T_{2})$ where $\omega$ is a YM qpc and $(T_{1},T_{2})$
is a critical point of $\mathscr{S}_{\mathrm{SM}}$.

Now let us take a particular and illustrative trivial qpb (in the sense of
[D2] and [Sa2]) by using $U(1)$ as a structure group. In an abuse of notation,
we will identify $U(1)$ with the Laurent polynomial algebra. In [Sa3] the
reader can check the results of this paper in the quantum Hopf fibration and
in [Sa4] the reader can appreciated another example by using another trivial
qpb.

For this example, the $\ast$–FODC on $U(1)$ will be given by the right ideal
$\mathrm{Ker}^{2}(\epsilon)$ and hence the universal differential envelope
$\ast$–calculus $(\Gamma^{\wedge},d,\ast)$ is the algebra of differential
forms of $U(1)$. A Hamel basis of

${{}_{\mathrm{inv}}}\Gamma:=\dfrac{\mathrm{Ker}(\epsilon)}{\mathrm{Ker}^{2}(\epsilon)}$

is given by

$\beta_{U(1)}=\\{\varsigma:=\pi(z)\\},$

where $\pi:U(1)\longrightarrow{{}_{\mathrm{inv}}}\Gamma$ is the quantum germs
map and it has the particularity that: $\varsigma\circ
g=\epsilon(g)\,\varsigma$ for all $g$ $\in$ $U(1)$. Furthermore asking that
$\beta_{U(1)}$ be an orthonormal set, the $\mathrm{ad}$ corpresentation, which
in this case is given by

$\mathrm{ad}(\varsigma)=\varsigma\otimes\mathbbm{1},$

is unitary.

Now let us consider the $C^{\ast}$–algebra given by $2\times 2$ matrices

$(M:=M_{2}(\mathbb{C}),\cdot,\mathrm{Id}_{2},||\,\,||_{\mathrm{op}},\ast),$

where $||\,\,||_{\mathrm{op}}$ is the standard operator norm and $\ast$ is the
complex transpose operation. A particular useful Hamel basis of $M$ is given
by

$\beta_{M}:=\left\\{\mathrm{Id}_{2},\;S_{1}=\frac{1}{2}\sigma_{1},\;S_{2}=\frac{1}{2}\sigma_{2},\;S_{3}=\frac{1}{2}\sigma_{3},\right\\}$

where $\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$ are the Pauli matrices and
$\mathrm{Id}_{2}$ is the identity matrix. Let us construct an appropriate
differential calculus. By considering the $\ast$–Lie algebra
$(\mathfrak{sl}(2,\mathbb{C}),i\,[-,-])$ and the representation

(4.1) $\displaystyle\rho:\mathfrak{sl}(2,\mathbb{C})$
$\displaystyle\longrightarrow\mathrm{Der}(M)$ $\displaystyle B$
$\displaystyle\longrightarrow i\,[B,-],$

where $\mathrm{Der}(M)$ is the space of derivations on $M$ [DV], we can obtain
the Chevalley–Eilenberg complex

(4.2)
$(\Omega^{\bullet}(M):=\Omega^{\bullet}_{\mathrm{Der}}(M_{2}(\mathbb{C})),d,\ast)$

and for comfort, its elements will be considered as $M$–valued alternating
multilinear maps of $\mathfrak{sl}(2,\mathbb{C})$.

Due to $\\{S_{1},S_{2},S_{3}\\}$ is a Hamel basis of
$\mathfrak{sl}(2,\mathbb{C})$, we can take its dual basis
$\\{h^{1},h^{2},h^{3}\\}$ and get a left–right $M$–basis of
$\Omega^{\bullet}(M)$ by means of

(4.3)
$\beta_{\Omega^{\bullet}(M)}:=\\{h^{j_{1},...,j_{k}}:=h^{j_{1}}\wedge...\wedge
h^{j_{k}}\,\mathrm{Id}_{2}\mid 1<j_{1}<...<j_{k}<3\\}.$

######  Proposition 4.1.

The quantum space $(M,\cdot,\mathrm{Id}_{2},\ast)$ satisfies all the
conditions mentioned in Remark 2.3 with respect to this graded differential
$\ast$–algebra.

###### Proof.

  1. (1)

The space $M$ is oriented since for all $k>3$ we have $\Omega^{k}(M)=0$ and

$\mathrm{dvol}:=h^{1,2,3}=(h^{1}\wedge h^{2}\wedge h^{3})\,\mathrm{Id}_{2}$

is a quantum $3$–volume form.

  2. (2)

A direct calculation shows that a lqrm can be defined on $M$ by means of

$\displaystyle\langle-,-\rangle:M\times M$ $\displaystyle\longrightarrow M$
$\displaystyle(\;\,\hat{p}\;\,,\,\;p\,\;)$
$\displaystyle\longmapsto\hat{p}\,p^{\ast};$
$\displaystyle\langle-,-\rangle:\Omega^{1}(M)\times\Omega^{1}(M)$
$\displaystyle\longrightarrow M$
$\displaystyle(\quad\,\hat{\mu}\quad\,,\,\quad\mu\,\quad)$
$\displaystyle\longmapsto\sum^{3}_{k=1}\hat{p}_{k}\,p^{\ast}_{k};$

if $\hat{\mu}=\displaystyle\sum^{3}_{k=1}h^{k}\,\hat{p}_{k},$
$\mu=\displaystyle\sum^{3}_{k=1}h^{k}\,p_{k};$

$\displaystyle\langle-,-\rangle:\Omega^{2}(M)\times\Omega^{2}(M)$
$\displaystyle\longrightarrow M$
$\displaystyle(\quad\,\hat{\mu}\quad\,,\,\quad\mu\,\quad)$
$\displaystyle\longmapsto\sum_{1\leq k<j\leq 3}\hat{p}_{kj}\,p^{\ast}_{kj};$

if $\hat{\mu}=\displaystyle\sum_{1\leq k<j\leq 3}h^{k,j}\,\hat{p}_{kj},$
$\mu=\displaystyle\sum_{1\leq k<j\leq 3}h^{k,j}\,p_{kj}$ and finally

$\displaystyle\langle-,-\rangle:\Omega^{3}(M)\times\Omega^{3}(M)$
$\displaystyle\longrightarrow M$
$\displaystyle(\;\hat{p}\,\mathrm{dvol}\;,\;p\,\mathrm{dvol}\;)$
$\displaystyle\longmapsto\hat{p}\,p^{\ast}.$

We have to remark that with this lqrm, $\mathrm{dvol}$ is actually a lqr
$3$–form. In accordance with Remark 2.2, we get a rqrm with a rqr $3$–form.

  3. (3)

By defining the linear map

$\displaystyle\int_{M}:\Omega^{3}(M)$ $\displaystyle\longrightarrow\mathbb{C}$
$\displaystyle p\,\mathrm{dvol}$
$\displaystyle\longmapsto\dfrac{1}{2}\,\mathrm{tr}(p),$

where $\mathrm{tr}$ denotes the trace operator, it should be clear that it is
a qi. Furthermore, the elements of $\mathrm{Im}(d|_{\Omega^{2}(M)})$ are
trace–zero, so $(M,\cdot,\mathrm{Id}_{2},\ast)$ is a quantum space without
boundary (whit respect to this qi).

  4. (4)

A direct calculation shows

$\star_{\mathrm{L}}\,p=p^{\ast}\,\mathrm{dvol}$

for all $p$ $\in$ $M$;

$\star_{\mathrm{L}}(p\,\mathrm{dvol})=p^{\ast}$

for all $p\,\mathrm{dvol}$ $\in$ $\Omega^{3}(M)$;

$\star_{\mathrm{L}}\,\mu=h^{1,2}\,p^{\ast}_{3}-h^{1,3}\,p^{\ast}_{2}+h^{2,3}\,p^{\ast}_{1}.$

for all $\mu=\displaystyle\sum^{3}_{l=1}h^{l}\,p_{l}$ $\in$ $\Omega^{1}(M)$
and finally

$\star_{\mathrm{L}}\,\mu=h^{1}\,p^{\ast}_{23}-h^{2}\,p^{\ast}_{13}+h^{3}\,p^{\ast}_{12}.$

for all $\mu=\displaystyle\sum_{1\leq l<j\leq 3}h^{l,j}\,p_{lj}$ $\in$
$\Omega^{2}(M).$ To define $\star_{\mathrm{R}}$ it is enough to consider the
Equation 2.12.

∎

It is worth mentioning that $\varepsilon=\mathrm{id}_{M}$ and
$\star_{\mathrm{L}}\circ\star_{\mathrm{L}}=(-1)^{k(n-k)}\mathrm{id}_{\Omega^{k}(M)}$
(Equation 2.1).

A direct calculation shows

######  Proposition 4.2.

The quantum codifferential is given by

$d^{\star_{\mathrm{L}}}\mu=-\sum^{3}_{k=1}i\,[S_{k},p_{k}]$

for $\mu=\displaystyle\sum^{3}_{k=1}h^{k}\,p_{k}$ $\in$ $\Omega^{1}(M)$;

$d^{\star_{\mathrm{L}}}\mu=\sum^{3}_{k=1}h^{k}\,p_{k}$

with $p_{1}=i\,[S_{2},p_{12}]+i\,[S_{3},p_{13}]+\,p_{23},$
$p_{2}=-i\,[S_{1},p_{12}]+i\,[S_{3},p_{23}]-\,p_{13},$
$p_{3}=-i\,[S_{1},p_{13}]-i\,[S_{2},p_{23}]+\,p_{12},$ for
$\mu=\displaystyle\sum^{3}_{1\leq k<j\leq 3}h^{k,j}\,p_{kj}$ $\in$
$\Omega^{2}(M)$ and

$d^{\star_{\mathrm{L}}}\mu=\sum^{3}_{1\leq k<j\leq 3}h^{k,j}\,p_{kj}$

with $p_{12}=-i\,[S_{3},p],$ $p_{13}=i\,[S_{2},p],$ $p_{23}=-i\,[S_{1},p],$ if
$\mu=p\,\mathrm{dvol}$ $\in$ $\Omega^{3}(M)$. To define
$d^{\star_{\mathrm{R}}}$ we can apply the Equation 2.12.

Consider now the trivial quantum principal $U(1)$–bundle
$\zeta^{\mathrm{triv}}$ with the trivial differential calculus formed by all
these spaces ([D2]). Qpcs are characterized by the non–commutative gauge
potentials, linear maps

$A^{\omega}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{1}(M)$

such that

(4.4)
$\omega=\omega^{\mathrm{triv}}+(A^{\omega}\otimes\mathrm{id}_{U(1)})\circ\mathrm{ad},$

where $\omega^{\mathrm{triv}}(\theta)=\mathbbm{1}\otimes\theta$ (for all
$\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$) is the trivial qpc, i.e., every
element of $\overrightarrow{\mathfrak{qpc}(\zeta^{\mathrm{triv}})}$ is of the
form $(A^{\omega}\otimes\mathrm{id}_{U(1)})\circ\mathrm{ad}$ for some
$A^{\omega}$. In particular, it is easy to see that $\omega$ is regular if and
only if $A^{\omega}(\varsigma)$ is a linear combination of
$\\{h^{j}\,\mathrm{Id}_{2}\\}^{3}_{j=1}$.

The only possible embedded differential ([So]) is

(4.5)
$\delta:{{}_{\mathrm{inv}}}\Gamma\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes{{}_{\mathrm{inv}}}\Gamma$

given by $\delta=0$; which implies that
$d^{S^{\omega}_{\mathrm{L}}}=d^{S^{\omega}_{\mathrm{R}}}=0$ and consequently,
its adjoint operators are zero as well.

In this way, the non–commutative field strength $F^{\omega}$ ([D2]) is given
by

$F^{\omega}(\varsigma)=dA^{\omega}(\varsigma).$

###  4.1. Non–commutative geometrical Yang–Mills Fields

We claim that every YM qpc is flat. Indeed, a direct calculation shows that

$\displaystyle\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YM}}(\omega+z\,\lambda^{\prime})$
$\displaystyle=$ $\displaystyle-\dfrac{1}{4}\left(\langle
A^{\prime}(\varsigma)\,|\,d^{\star_{\mathrm{L}}}F^{\omega}(\varsigma)\rangle_{\mathrm{L}}+\langle
A^{\prime}(\varsigma)^{\ast}\,|\,d^{\star_{\mathrm{R}}}F^{\omega}(\varsigma)^{\ast}\rangle_{\mathrm{R}}\right)$
$\displaystyle=$ $\displaystyle-\dfrac{1}{4}\left(\langle
dA^{\prime}(\varsigma)\,|\,F^{\omega}(\varsigma)\rangle_{\mathrm{L}}+\langle
dA^{\prime}(\varsigma)^{\ast}\,|\,F^{\omega}(\varsigma)^{\ast}\rangle_{\mathrm{R}}\right)$
$\displaystyle=$ $\displaystyle-\dfrac{1}{2}\langle
dA^{\prime}(\varsigma)\,|\,dA^{\omega}(\varsigma)\rangle_{\mathrm{L}}$

where $\lambda^{\prime}(\varsigma)=A^{\prime}(\varsigma)\otimes\mathbbm{1}$.
Since $\langle-|-\rangle_{\mathrm{L}}$ is an inner product we conclude that
any YM qpc has to satisfy $dA^{\omega}(\varsigma)=F^{\omega}(\varsigma)=0$.
This result is similar to the one obtained in Differential Geometry for a
trivial $\mathrm{U}(1)$–bundle with a Rimannian metric on the base space.

A direct calculation shows that

(4.6)
$\mathfrak{qGG}_{\mathrm{YM}}=\\{\mathfrak{f}\in\mathfrak{qGG}\mid\mathfrak{f}^{\circledast}\omega^{\mathrm{triv}}\mbox{
is flat }\\}.$

In addition, in accordance with [Sa2]
$\mathrm{U}(1)\subset\mathfrak{qGG}_{\mathrm{YM}}$ and the non–commutative
gauge potential of a YM qpc is always given by

$A^{\omega}(\varsigma)=dp$

for some $p$ $\in$ $M$ because the first cohomology group of the
Chevalley–Eilenberg complex is trivial.

###  4.2. Non–commutative geometrical $n$–multiples of Space–Time Scalar
Matter Fields

By Proposition 4.2, the differential algebra of the Equation LABEL:6.f2.26 can
be used, and for all $p=\begin{pmatrix}p_{1}&p_{2}\\\
p_{3}&p_{4}\end{pmatrix}$ $\in$ $M$ we have

$d^{\star_{\mathrm{L}}}d\,p=\begin{pmatrix}p_{1}-p_{4}&2p_{2}\\\
2p_{3}&-p_{1}+p_{4}\end{pmatrix};$

so taking $V=const$ the pair $(T_{1},T_{2})$ with
$p^{T_{1}}=\lambda_{1}\,\mathrm{Id}_{2}$,
$p^{T_{2}}=\lambda_{2}\,\mathrm{Id}_{2}$, $\lambda_{1}$, $\lambda_{2}$ $\in$
$\mathbb{C}$ is a stationary point. As another example, if $V$ is such that
$V^{\prime}(\mathrm{Id}_{2})=\dfrac{1}{2}\,\mathrm{Id}_{2}$, then the pair
$(T_{1},T_{2})$ with $p^{T_{1}}=p^{T_{2}}=S_{1}$ is a stationary point.

###  4.3. Non–commutative geometrical Yang–Mills–Scalar–Matter Fields

It is well–known that a complete set of mutually inequivalent irreducible
unitary $\mathrm{U}(1)$–representation $\tau$ is in biyection with
$\mathbb{Z}$. The trivial representation on $\mathbb{C}$ is given by $n=0$, so
let us consider $n\not=0$. In all these cases, the left–right $M$ basis given
by Proposition LABEL:6.2.1 has just one element defined by

$\displaystyle T^{n}:\mathbb{C}$ $\displaystyle\longrightarrow
M\otimes\mathrm{U}(1)$ $\displaystyle w$ $\displaystyle\longmapsto
w\,\mathrm{Id}_{2}\otimes z^{n}$

and hence, every $T$ $\in$ $\textsc{Mor}(n,{{}_{GM}}\Phi)$ is of the form
$T=p^{T}T^{n}=T^{n}\,p^{T}$ where $p^{T}=T(1)(\mathrm{Id}_{2}\otimes z^{\ast
n})$.

In general, for a qpc $\omega$ with
$\omega(\varsigma)=A^{\omega}(\varsigma)\otimes\mathbbm{1}+\mathrm{Id}_{2}\otimes\varsigma$
and with $A^{\omega}(\varsigma)=\displaystyle\sum^{3}_{i=1}h^{i}\,p_{i}$ we
get that Equation 3.4 reduces to

(4.7)
$-\dfrac{1}{n}(p^{\ast}_{1}\,dp_{1}-p_{2}\,dp^{\ast}_{2})+p^{\ast}_{1}\,p_{1}A^{\omega}(\varsigma)-p_{2}\,p^{\ast}_{2}A^{\omega}(\varsigma)-2d^{\star_{\mathrm{L}}}dA^{\omega}(\varsigma)=0$

for $T_{1}=\dfrac{1}{n}p_{1}T^{n}$, $T_{2}=-\,\dfrac{1}{n}T^{-n}p_{2}$; while
Equation 3.5 becomes

(4.8)
$\begin{split}\nabla^{\omega\,\star_{\mathrm{L}}}_{n}\left(\nabla^{\omega}_{n}\,T_{1}\right)\,=\,&[\,\dfrac{1}{n}d^{\star_{\mathrm{L}}}dp_{1}\,+\,\star^{-1}_{\mathrm{L}}(d((\star_{\mathrm{L}}A^{\omega}(\varsigma))p^{\ast}_{1}))\\\
&+\star^{-1}_{\mathrm{L}}(A^{\omega}(\varsigma)^{\ast}(\star_{\mathrm{L}}dp_{1}))+n\star^{-1}_{\mathrm{L}}(A^{\omega}(\varsigma)^{\ast}(\star_{\mathrm{L}}A^{\omega}(\varsigma))p^{\ast}_{1})]\;T^{n}\\\
\widehat{\nabla}^{\omega\,\star_{\mathrm{R}}}_{-n}\left(\widehat{\nabla}^{\omega}_{-n}\,T_{2}\right)\,=\,&T^{-n}\,[-\dfrac{1}{n}d^{\star_{\mathrm{R}}}dp_{2}-\star^{-1}_{\mathrm{R}}(d(p^{\ast}_{2}(\star_{\mathrm{R}}A^{\omega}(\varsigma)^{\ast})))\\\
&-\star^{-1}_{\mathrm{R}}((\star_{\mathrm{R}}dp_{2})A^{\omega}(\varsigma))+n\star^{-1}_{\mathrm{R}}(p^{\ast}_{2}(\star_{\mathrm{R}}A^{\omega}(\varsigma)^{\ast})A^{\omega}(\varsigma))].\end{split}$

Now it is possible to look for YMSM fields. For example, for $n=1$ the triplet
$(\omega^{\mathrm{triv}},T_{1},T_{2})$, where
$T_{1}(1)=(S_{1}+S_{2}+S_{3})\otimes z$, $T_{2}(1)=(S_{1}+S_{2}+S_{3})\otimes
z^{\ast}$, is a YMSM field for a potential $V$ such that

$V^{\prime}(\dfrac{3}{4}\,\mathrm{Id}_{2})=2\,\mathrm{Id}_{2},\quad\mbox{ for
example }\quad V(p):=2\,p\;\mbox{ for all }\;p\,\in\,M.$

Also for $n=1$, the triplet $(\omega,\sqrt{3}\,T^{1},T^{-1})$, where
$\omega(\varsigma)=(\displaystyle\sum^{3}_{j=1}S_{j}\,h^{j})\otimes\mathbbm{1}+\mathbbm{1}\otimes\varsigma$,
is again a YMSM field for a potential $V$ such that

$V^{\prime}(3\,\mathrm{Id}_{2})=V^{\prime}(\mathrm{Id}_{2})=-\dfrac{3}{4}\,\mathrm{Id}_{2},\quad\mbox{
for example }\quad V(p):=-\dfrac{3}{4}\,p\;\mbox{ for all }\;p\,\in\,M.$

It is important to mention that in this case $\omega$ is not a YM qpc or a
regular qpc and actually, $\displaystyle\sum^{3}_{j=1}S_{j}\,h^{j}$ is an
eigenvector of $d^{\star_{\mathrm{L}}}\circ d$. Of course, there are more YMSM
fields; however, they all in general depend on the form of $V$.

At least we can ensure that

(4.9)
$\\{\mathfrak{f}\in\mathfrak{qGG}\mid\mathfrak{f}(z^{n})=e^{it}\,\mathrm{Id}_{2}\,,\,\mathfrak{f}(z^{\ast
n})=e^{is}\,\mathrm{Id}_{2}\,,\,\mathfrak{f}(\Omega^{1}(M))=0\;\mbox{ with
}t\,,\,s\,\in\,\mathbb{R}\\}$

is a relative large subgroup of $\mathfrak{qGG}_{\mathrm{YMSM}}$ for any $V$.

We just used $M=M_{2}(\mathbb{C})$ just to develop a concrete example.
However, it is possible to use $M_{n}(\mathbb{C})$ and the corresponding
Chevalley–Eilenberg complex and having different results; although, YM qpcs
are always flat.

##  5\. Concluding Comments

Durdevich’s theory of qpbs is really general in the sense that one has the
freedom to choose so many structures (giving us a much richer theory), and the
theory presented in this paper follows the same line. Despite our classically
motivated notation, it is important to notice the incredible dual similarity
with Differential Geometry since [D1], [D2]. Furthermore [Sa1] presents the
quantum version of the major result for principal $G$–bundles in [SW].
Clearly, due to the generality of the theory, it has a number of essential
differences when we compare this work to its classical counterpart. Moreover,
there are differences with the formulations presented in other papers,
although they maintain a similar research philosophy [HM], [LRZ], [Z], [L].
One of the most important differences with these other approaches is the
absence of the fundamental operator $S^{\omega}$ and a lack of the
systematical use of the left/right associated qvbs.

The operator $S^{\omega}$ is completely quantum in the sense that it does not
have a classical counterpart: in Differential Geometry, every principal
connection is regular and hence $S^{\omega}=0$. It is worth mentioning that in
our theory we just assume the existence of $d^{S^{\omega}\star_{\mathrm{L}}}$,
$d^{\widehat{S}^{\omega}\star_{\mathrm{R}}}$, not a specific form of them. In
Differential Geometry, the element
$d^{\nabla^{\omega}_{\mathrm{ad}}\star}R^{\omega}$ fulfills

$d^{\nabla^{\omega}_{\mathrm{ad}}\star}d^{\nabla^{\omega}_{\mathrm{ad}}\star}R^{\omega}=0.$

This equation is known as the continuity equation. In Non–Commutative Geometry
this equation turns into

$(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})^{2}R^{\omega}=(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})^{2}\widehat{R}^{\omega}=0.$

In our example the above equation holds; however, this is simply because of
$S^{\omega}=0$ (since the only possible embedded differential is $\delta=0$).
In [Sa4], we have
$(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})^{2}=0$,
$(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})^{2}=0$;
nevertheless, the previous equalities do not hold in a trivial qpb with
matrices as the space of base forms and with $S_{2}$ as the structure group.
In terms of a physical interpretation, the continuity equation tells us that a
quantity is conserved. In this sense, the non–commutative geometrical
continuity equation could be used to identify physical fields (together with
the fact that only real connections have physical sense) in more realistic
examples. We consider this quite motivating to keep our the research alive and
going on.

On the other hand, in order to talk about the left/right structures we have to
start with Equations A.5, A.6. These equations allow us to define associated
left/right qvbs as finitely generated projective left/right $M$–modules. To
define the Lagrangians, we used both structures; in addition, we have to
emphasize that in the Lagrangians of Subsections $3.2$, $3.3$, we used a
representation $\alpha$ and its complex conjugate representation
$\overline{\alpha}$, making them a little different that their classical
counterpart: now it looks like if in the quantum case left particles and right
antiparticles cannot be separated; they appear naturally interconnected.

The importance of this change becomes more explicit when we play with the
quantum Hopf fibration [Sa3]. For example, if we do not consider the right
structure, Equation 3.4 becomes

$\langle\Upsilon_{n}\circ
K^{\lambda}(T_{1})\,|\,\nabla^{\omega^{\mathrm{c}}}_{n}T_{1}\rangle_{\mathrm{L}}=0,$

which does not have solutions for an arbitrary $n$. Furthermore, the fact that
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}$
and
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}$
are mutually different (they do not share all their eigenvalues and
eigenspaces) is another strong motivating reason to consider the left/right
structure: it appears that ignoring one of the structures leaves to losing
relevant information about the quantum spaces [Sa3].

It is worth emphasizing that the theory presented here is almost entirely
algebraic: the only assumption about continuity or norms is in the potential
$V$, and when we ask that the quantum space $M$ be $C^{\ast}$–clousable; and
as the reader should have already noticed, we have used this hypothesis just
to guarantee that

$\sum_{i}p_{i}\,p^{\ast}_{i}=0\quad\Longleftrightarrow\quad p_{i}=0.$

This is a clear difference with other non–commutative geometrical Yang–Mills
theories; for example, the reader can check [CCM] in which $C^{\ast}$–algebras
and spectral triples play fundamental roles. In this sense, our theory is
purely geometric–algebraic. Using the spectral triplets can be a way to relate
this theory with Connes’ formulations as well as adding a kind of
non–commutative geometrical spin geometry to our theory. Other lines of
research can be studied from this paper in order to complete the whole
non–commutative geometrical description of the Standard Model and the
mathematics that it involves.

The presented formalism can be easily generalized in order to add quantum
Pseudo–Riemannian closed orientable spaces by weakening Definition 2.1 point
$2$. In fact, one can define a left quantum Pseudo–Riemannian metric (lqprm)
on a quantum space $(M,\cdot,\mathbbm{1},\ast)$ as a family of $M$–valued
symmetric sesquilinear maps

$\\{\langle-,-\rangle^{k}:\Omega^{k}(M)\times\Omega^{k}(M)\longrightarrow
M\\}$

such that for $k=0$

$\displaystyle\langle-,-\rangle^{0}:M\times M$ $\displaystyle\longrightarrow
M$ $\displaystyle(\;\hat{p}\;,\;p\;)$
$\displaystyle\longmapsto\hat{p}p^{\ast}$

and such that for $k\geq 1$

$\langle\hat{\mu}p,\mu\rangle^{k}=\langle\hat{\mu},\mu
p^{\ast}\rangle^{k}\quad\mbox{ and
}\quad\langle\hat{\mu},\mu\rangle^{k}=0\;\;\forall\;\;\hat{\mu}\;\in\;\Omega^{k}(M)\;\Longleftrightarrow\;\mu=0.$

It should be clear how to define the left quantum Pseudo–Riemannian $n$–volume
form (lqpr $n$–form) and the right structure. Of course, we would also have to
impose that with this lqprm, the symmetric sesquilinear map given in Equation
2.2 is non–degenerate, as well as the existence of Hodge operators.

##  Appendix A Notation and Basic Concepts

In this appendix we are going to show a little summary about matrix compact
quantum groups, the universal differential envelope $\ast$–calculus, quantum
principal bundles, and associated quantum vector bundles. The reader always
can consult the original work [W1], [W2], [D1], [D2], [D3], [So], [D5], [Sa1]
[Sa2].

###  A.1. Compact Matrix Quantum Groups

The concept of compact matrix quantum group (cmqg) was developed by S. L.
Woronowicz in [W1], [W2]. A cmqg will be denoted by $\mathcal{G}$; while its
dense $\ast$–Hopf (sub)algebra will be denoted by

$\mathcal{G}^{\infty}:=(G,\cdot,\mathbbm{1},\phi,\epsilon,\kappa,\ast),$

where $\phi$ is the comultiplication, $\epsilon$ is the counity, and $\kappa$
is the coinverse.

A (smooth right) $\mathcal{G}$–representation on a $\mathbb{C}$–vector space
$V$ is a linear map

$\alpha:V\longrightarrow V\otimes G$

such that

$\begin{CD}V@>{\hbox to70.0pt{\hfil$\scriptstyle\alpha$\hfil}}>{}>&V\otimes
G\\\ @V{\mathrm{id}_{V}}V{}V\hbox
to0.0pt{\hskip-44.0pt$\circlearrowleft$\hss}&@V{}V{\mathrm{id}_{V}\,\otimes\,\epsilon}V\\\\[-4.0pt]
V@>{}>{\hbox
to70.0pt{\hfil$\scriptstyle\cong$\hfil}}>&V\otimes\mathbb{C}\end{CD}$

and

$\begin{CD}V@>{\hbox to70.0pt{\hfil$\scriptstyle\alpha$\hfil}}>{}>&V\otimes
G\\\ @V{\alpha}V{}V\hbox
to0.0pt{\hskip-44.0pt$\circlearrowleft$\hss}&@V{}V{\mathrm{id}_{V}\,\otimes\,\phi}V\\\\[-4.0pt]
V\otimes G@>{}>{\hbox
to70.0pt{\hfil$\scriptstyle\alpha\,\otimes\,\mathrm{id}_{G}$\hfil}}>&V\otimes\mathcal{G}\otimes\mathcal{G}.\end{CD}$

We say that the representation is finite–dimensional if
$\dim_{\mathbb{C}}(V)<|\mathbb{N}|$. $\alpha$ usually receives the name of
(right) coaction or (right) corepresentation of $\mathcal{G}$ on $V$.

Given two $\mathcal{G}$–representations $\alpha$, $\beta$ acting on $V$, $W$,
respectively, a representation morphism is a linear map

$T:V\longrightarrow W$

such that the following diagram holds

$\begin{CD}V@>{\hbox to70.0pt{\hfil$\scriptstyle\alpha$\hfil}}>{}>&V\otimes
G\\\ @V{T}V{}V\hbox
to0.0pt{\hskip-44.0pt$\circlearrowleft$\hss}&@V{}V{T\,\otimes\,\mathrm{id}_{G}}V\\\\[-4.0pt]
W@>{}>{\hbox
to70.0pt{\hfil$\scriptstyle\beta$\hfil}}>&W\otimes\mathcal{G}.\end{CD}$

If $\alpha$, $\beta$ are two representations, we define the set of all
representation morphisms between them as

$\textsc{Mor}(\alpha,\beta)$

and the set of all finite–dimensional $\mathcal{G}$–representations will be
denoted by

$\textsc{Obj}(\mathbf{Rep}_{\mathcal{G}}).$

It is important to mention that Woronowicz proved in [W1] the non–commutative
version of Weyl’s representation theory. Another important result is the next
one

######  Theorem A.1.

Let $\mathcal{T}$ be a complete set of mutually non–equivalent irreducible
unitary (necessarily finite–dimensional) $\mathcal{G}$–representations with
$\alpha^{\mathrm{triv}}_{\mathbb{C}}$ $\in$ $\mathcal{T}$ (the trivial
corepresentation on $\mathbb{C}$). For any $\alpha$ $\in$ $\mathcal{T}$ that
acts on $(V^{\alpha},\langle-|-\rangle)$,

$\alpha(e_{i})=\sum^{n_{\alpha}}_{j=1}e_{j}\otimes g^{\alpha}_{ji},$

where $\\{e_{i}\\}^{n_{\alpha}}_{i=1}$ is an ortonormal basis of $V^{\alpha}$
and $\\{g^{\alpha}_{i,j=1}\\}^{n_{\alpha}}_{ij}$ $\subseteq$ $G$. Then
$\\{g^{\alpha}_{ij}\\}_{\alpha,i,j}$ is a Hamel basis of $G$, where the index
$\alpha$ runs on $\mathcal{T}$ and $i$, $j$ run from $1$ to $n_{\alpha}$.

Taking a bicovariant first order differential $\ast$–calculus ($\ast$–FODC
[So]) $(\Gamma,d)$ on $G$, the universal differential envelope $\ast$–calculus
$(\Gamma^{\wedge},d,\ast)$ is given by

$\Gamma^{\wedge}:=\otimes^{\bullet}_{G}\Gamma/\mathcal{Q}\,,\quad\otimes^{\bullet}_{G}\Gamma:=\oplus_{k}(\otimes^{k}_{G}\Gamma)\quad\mbox{
with
}\quad\otimes^{k}_{G}\Gamma:=\underbrace{\Gamma\otimes_{G}\cdots\otimes_{G}\Gamma}_{k\;times}$

where $\mathcal{Q}$ is the bilateral ideal of $\otimes^{\bullet}_{G}\Gamma$
generated by $\displaystyle\sum_{i}dg_{i}\otimes_{G}dh_{i}$ such that
$\displaystyle\sum_{i}g_{i}\,dh_{i}=0$ with $g_{i}$, $h_{i}$ $G$. This space
is interpreted as quantum differential forms on $\mathcal{G}$.

Let us consider
${{}_{\mathrm{inv}}}\Gamma^{\wedge}=\\{\theta\in\Gamma\mid\Phi_{\Gamma^{\wedge}}(\theta)=\mathbbm{1}\otimes\theta\\},$
where $\Phi_{\Gamma^{\wedge}}$ is the extension of the canonical left
representation of $G$ in $\Gamma$. This space is a graded $\mathbb{C}$–vector
space and it is well–known that
${{}_{\mathrm{inv}}}\Gamma:={{}_{\mathrm{inv}}}\Gamma^{\wedge
1}\cong\mathrm{Ker}(\epsilon)/\mathcal{R},$ where
$\mathcal{R}\subseteq\mathrm{Ker}(\epsilon)$ is the canonical right $G$–ideal
of $G$ associated to $(\Gamma,d)$. The canonical right corepresentation of $G$
on $\Gamma$ leaves ${{}_{\mathrm{inv}}}\Gamma$ invariant and denoting it by

(A.1)
$\mathrm{ad}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes\mathcal{G}$

we have

$\mathrm{ad}\circ\pi=(\pi\otimes\mathrm{id}_{G})\circ\mathrm{Ad},$

where $\mathrm{Ad}$ is the (right) adjoint action of $G$ and
$\pi:G\longrightarrow{{}_{\mathrm{inv}}}\Gamma$ is the quantum germs map which
is defined by $\pi(g)=\kappa(g^{(1)})dg^{(2)}$. There is a right $G$–module
structure in ${{}_{\mathrm{inv}}}\Gamma$ given by $\theta\circ
g=\kappa(g^{(1)})\theta g^{(2)}=\pi(hg-\epsilon(h)g)$ if $\theta=\pi(h)$.

###  A.2. Quantum Principal Bundles

Let $(M,\cdot,\mathbbm{1},\ast)$ be a quantum space and let $\mathcal{G}$ be a
cmqg. A quantum principal $\mathcal{G}$–bundle over $M$ (qpb) is a quantum
structure formally represented by the triplet

$\zeta=(GM,M,{{}_{GM}}\Phi),$

where $(GM,\cdot\mathbbm{1},\ast)$ is a quantum space called the quantum total
space with $(M,\cdot,\mathbbm{1},\ast)$ as quantum subspace, which receives
the name of quantum base space, and

${{}_{GM}}\Phi:GM\longrightarrow GM\otimes G$

is a $\ast$–algebra morphism that satisfies

  1. (1)

${{}_{GM}}\Phi$ is a $\mathcal{G}$–representation.

  2. (2)

${{}_{GM}}\Phi(x)=x\otimes\mathbbm{1}$ if and only if $x$ $\in$ $M$.

  3. (3)

The linear map $\beta:GM\otimes GM\longrightarrow GM\otimes\mathcal{G}$ given
by

$\beta(x\otimes
y):=x\cdot{{}_{GM}}\Phi(y)=(x\otimes\mathbbm{1})\cdot{{}_{GM}}\Phi(y)$

is surjective.

Given a qpb $\zeta$ over $M$, a differential calculus on it is

  1. (1)

A graded differential $\ast$–algebra $(\Omega^{\bullet}(GM),d,\ast)$ generated
by $\Omega^{0}(GM)=GM$ (quantum differential forms on $GM$).

  2. (2)

A bicovariant $\ast$–FODC (first order differential $\ast$–calculus) over $G$
$(\Gamma,d)$.

  3. (3)

The map ${{}_{GM}}\Phi$ is extendible to a graded differential $\ast$–algebra
morphism

${{}_{\Omega}}\Psi:\Omega^{\bullet}(GM)\longrightarrow\Omega^{\bullet}(GM)\otimes\Gamma^{\wedge},$

where $(\Gamma^{\wedge},d,\ast)$ is the universal differential envelope
$\ast$–calculus (quantum differential forms on $\mathcal{G}$).

The space of horizontal forms is defined as

$\mathrm{Hor}^{\bullet}GM\,:=\\{\varphi\in\Omega^{\bullet}(GM)\mid{{}_{\Omega}}\Psi(\varphi)\,\in\,\Omega^{\bullet}(GM)\otimes
G\\},$

it is a graded $\ast$–subalgebra of $\Omega^{\bullet}(GM)$ and the map
${{}_{\mathrm{H}}}\Phi:={{}_{\Omega}}\Psi|_{\mathrm{Hor}^{\bullet}GM}$ is a
$\mathcal{G}$–representation on $\mathrm{Hor}^{\bullet}GM$. Also one can
define the space of base forms (quantum differential forms on $M$) as

$\Omega^{\bullet}(M):=\\{\mu\in\Omega^{\bullet}(GM)\mid{{}_{\Omega}}\Psi(\mu)=\mu\otimes\mathbbm{1}\\}.$

In this way, a quantum principal connection (qpc) is a linear map

$\omega:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{1}(GM)$

that satisfies
${{}_{\Omega}}\Psi(\omega(\theta))=(\omega\otimes\mathrm{id}_{G})\mathrm{ad}(\theta)+\mathbbm{1}\otimes\theta.$

For every qpb, there always exist qpcs ([D2]). In analogy with the classical
case, the set

(A.2)
$\mathfrak{qpc}(\zeta):=\\{\omega:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{1}(GM)\mid\omega\mbox{
is a qpc on }\zeta\\}$

is an affine space modeled by the $\mathbb{C}$–vector space of all quantum
connection displacements

$\overrightarrow{\mathfrak{qpc}(\zeta)}:=\\{\lambda:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{1}(GM)\mid\lambda\mbox{
is a linear map such that
}{{}_{\mathrm{H}}}\Phi\circ\lambda=(\lambda\otimes\mathrm{id}_{G})\circ\mathrm{ad}\\}.$

Let us consider the involution

$\displaystyle\wedge=\mathfrak{qpc}(\zeta)$
$\displaystyle\longrightarrow\mathfrak{qpc}(\zeta)$ $\displaystyle\omega$
$\displaystyle\longmapsto\widehat{\omega}:=\ast\circ\omega\circ\ast.$

We define the dual qpc of $\omega$ as $\widehat{\omega}$. A qpc $\omega$ is
real if $\widehat{\omega}=\omega$ and we say that it is imaginary if
$\widehat{\omega}=-\omega$.

Of course, the operation $\wedge$ can be defined in
$\overrightarrow{\mathfrak{qpc}(\zeta)}$. In such a way, every real quantum
connection displacement $\lambda$ can be written as

(A.3) $\lambda=\omega-\omega^{\prime}$

where $\omega$, $\omega^{\prime}$ are real elements; while for any qpc
$\omega$

(A.4) $\omega=\omega^{\prime}+i\,\lambda^{\prime}.$

with $\omega^{\prime}$, $\lambda^{\prime}$ real elements.

A qpc is called regular if for all $\varphi$ $\in$ $\mathrm{Hor}^{k}GM$ and
$\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$ we have

$\omega(\theta)\,\varphi=(-1)^{k}\varphi^{(0)}\omega(\theta\circ\varphi^{(1)}),$

where ${{}_{\mathrm{H}}}\Phi(\varphi)=\varphi^{(0)}\otimes\varphi^{(1)};$ and
it is called multiplicative if

$\omega(\pi(g^{(1)}))\omega(\pi(g^{(2)}))=0$

for all $g$ $\in$ $\mathcal{R}$ with $\phi(g)=g^{(1)}\otimes g^{(2)}$.

For any $\ast$–algebra $(\mathcal{X},m,\mathbbm{1},\ast)$ and linear maps
$T_{1},T_{2}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\mathcal{X}$, let us
define

$\langle T_{1},T_{2}\rangle:=m\circ(T_{1}\otimes
T_{2})\circ\delta:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\mathcal{X}$
$[T_{1},T_{2}]:=m\circ(T_{1}\otimes T_{2})\circ
c^{\mathrm{T}}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\mathcal{X}$

where $\delta$ is an embedded differential and $c^{\mathrm{T}}$ is the
transposed commutator [So]. In this way, the curvature of a qpc is defined as

$R^{\omega}:=d\circ\omega-\langle\omega,\omega\rangle:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{2}(GM)$

If $R^{\omega}=0$, it is common to say that $\omega$ is flat. Finally the
covariant derivative of a qpc $\omega$ is the first–order linear map

$D^{\omega}:\mathrm{Hor}^{\bullet}GM\longrightarrow\mathrm{Hor}^{\bullet}GM$

such that for every $\varphi$ $\in$ $\mathrm{Hor}^{k}GM$

$D^{\omega}(\varphi)=d\varphi-(-1)^{k}\varphi^{(0)}\omega(\pi(\varphi^{(1)}));$

while the dual covariant derivative of $\omega$ is the first–order linear map

$\widehat{D}^{\omega}:=\ast\circ D^{\omega}\circ\ast.$

Let $\mathcal{T}$ a complete set of mutually non–equivalent irreducible
$\mathcal{G}$–representations with $\alpha^{\mathrm{triv}}_{\mathbb{C}}$ $\in$
$\mathcal{T}$. In order to develop the theory of associated qvbs, we have to
assume that for a given $\zeta=(GM,M,$ ${{}_{GM}}\Phi)$ and each $\alpha$
$\in$ $\mathcal{T}$ there exists

$\\{T^{\mathrm{L}}_{k}\\}^{d_{\alpha}}_{k=1}\subseteq\textsc{Mor}(\alpha,{{}_{GM}}\Phi)$

for some $d_{\alpha}$ $\in$ $\mathbb{N}$ such that

(A.5)
$\sum^{d_{\alpha}}_{k=1}x^{\alpha\,\ast}_{ki}x^{\alpha}_{kj}=\delta_{ij}\mathbbm{1},$

with $x^{\alpha}_{ki}:=T^{\mathrm{L}}_{k}(e_{i})$, where
$\\{e_{i}\\}^{n_{\alpha}}_{i=1}$ is an orthonormal basis. Also we are going to
assume that the following relation holds

(A.6)
$W^{\alpha\,\mathrm{T}}X^{\alpha\,\ast}=\mathrm{Id}_{n_{\alpha}}\quad\mbox{
where }\quad W^{\alpha}=(w^{\alpha}_{ij})=Z^{\alpha}X^{\alpha}C^{\alpha\,-1*}$

for each $\alpha$. Here $X^{\alpha}=(x^{\alpha}_{ij})$ $\in$
$M_{d_{\alpha}\times n_{\alpha}}(GM)$,
$X^{\alpha\,\ast}=(x^{\alpha\,\ast}_{ij})$, while $\mathrm{Id}_{n_{\alpha}}$
is the identity element of $M_{n_{\alpha}}(GM)$ and
$Z^{\alpha}=(z^{\alpha}_{ij})$ $\in$ $M_{d_{\alpha}}(\mathbb{C})$ is a
strictly positive element. Finally $C^{\alpha}$ $\in$
$M_{n_{\alpha}}(\mathbb{C})$ is the matrix written in terms of the basis
$\\{e_{i}\\}^{n_{\alpha}}_{i=1}$ of the canonical representation isomorphism
between $\alpha$ and
$\alpha^{cc}:=(\mathrm{id}_{V^{\alpha}}\otimes\kappa^{2})\alpha$, and
$W^{\alpha\,\mathrm{T}}$ is the transpose matrix of $W^{\alpha}$ [D2].

It is worth mentioning that in terms of the theory of Hopf–Galois extensions
([KT]), the first condition guarantees that $GM$ is principal [BDH].
Furthermore, the second condition implies the existence of a right $M$–linear
right $G$–colinear splitting of the multiplication $GM\otimes M\longrightarrow
GM$. However, we have decided to use Equations A.5, A.6 because in this way,
it is possible to do explicit calculations as the reader have already noticed.

Finally, for a qpc $\omega$ and every $\tau$ $\in$
$\textsc{Mor}(\mathrm{ad},{{}_{\mathrm{H}}}\Phi)$ such that
$\mathrm{Im}(\tau)$ $\in$ $\mathrm{Hor}^{k}GM$, let us define

(A.7)
$S^{\omega}(\tau):=\langle\omega,\tau\rangle-(-1)^{k}\langle\tau,\omega\rangle-(-1)^{k}[\tau,\omega]\;\in\;\textsc{Mor}(\mathrm{ad},{{}_{\mathrm{H}}}\Phi)$

There is a non–commutative geometrical version of the Bianchi identity:

(A.8)
$(D^{\omega}-S^{\omega})R^{\omega}=\langle\omega,\langle\omega,\omega\rangle\rangle-\langle\langle\omega,\omega\rangle,\omega\rangle.$

When $\omega$ is regular, $S^{\omega}=0$ and if $\omega$ is multiplicative
$\langle\omega,\langle\omega,\omega\rangle\rangle-\langle\langle\omega,\omega\rangle,\omega\rangle=0$
([D2]); so if $\omega$ is regular and multiplicative (for example, for
classical principal connections) we have $D^{\omega}R^{\omega}=0$.

###  A.3. Associated Quantum Vector Bundles

Let us start by taking a quantum $\mathcal{G}$–bundle $\zeta=(GM,$
$M,{{}_{GM}}\Phi)$ and a $\mathcal{G}$–representation $\alpha$ $\in$
$\mathcal{T}$ acting on $V^{\alpha}$. The $\mathbb{C}$–vector space
$\textsc{Mor}(\alpha,{{}_{GM}}\Phi)$ has a natural $M$–bimodule structure
given by multiplication with elements of $M$ and by Equation A.5, it is a
finitely generated projective left $M$–module; while under the assumption of
Equation A.6, it is a finitely generated projective right $M$–module. We
define the associated left quantum vector bunlde (associated left qvb) to
$\zeta$ with respect to $\alpha$ as the finitely generated projective left
$M$–module

$\zeta^{\mathrm{L}}_{\alpha}:=(\Gamma^{\mathrm{L}}(M,V^{\alpha}M):=\textsc{Mor}(\alpha,{{}_{GM}}\Phi),+,\cdot).$

Let $\omega$ be a qpc. Then the map

(A.9)
$\Upsilon^{-1}_{\alpha}:\Omega^{\bullet}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\longrightarrow\textsc{Mor}(\alpha,{{}_{\mathrm{H}}}\Phi)$

such that

$\Upsilon^{-1}_{\alpha}(\mu\otimes_{M}T)=\mu\,T$

is a graded $M$–bimodule isomorphism, where
$\textsc{Mor}(\alpha,{{}_{\mathrm{H}}}\Phi)$ has the $M$–bimodule structure
similar to the one of $\textsc{Mor}(\alpha,{{}_{GM}}\Phi)$; and its inverse is
given by

(A.10)
$\Upsilon_{\alpha}(\tau)=\sum^{d_{\alpha}}_{k=1}\mu^{\tau}_{k}\otimes_{M}T^{\mathrm{L}}_{k}\quad\mbox{
and
}\quad\mu^{\tau}_{k}=\sum^{n_{\alpha}}_{i=1}\tau(e_{i})\,x^{\alpha\,\ast}_{ki}\;\in\;\Omega^{1}(M).$

Elements of this tensor product can be interpreted as left $qvb$–valued
differential forms. Thus the linear map

(A.11)
$\begin{split}\nabla^{\omega}_{\alpha}:\Gamma^{\mathrm{L}}(M,V^{\alpha}M)&\longrightarrow\Omega^{1}(M)\otimes_{M}\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\\\
T&\longmapsto\Upsilon_{\alpha}\circ D^{\omega}\circ T,\end{split}$

is called the induced quantum linear connection (induced qlc) in
$\zeta^{\mathrm{L}}_{\alpha}$.

Now we define the associated right quantum vector bundle (associated right
qvb) to $\zeta$ with respect to $\alpha$ as as the finitely generated
projective right $M$–module

$\zeta^{\mathrm{R}}_{\alpha}:=(\Gamma^{\mathrm{R}}(M,V^{\alpha}M):=\textsc{Mor}(\alpha,{{}_{GM}}\Phi),+,\cdot)$

The map

(A.12)
$\widetilde{\Upsilon}^{-1}_{\alpha}:\Gamma(M,V^{\alpha}M)\otimes_{M}\Omega^{\bullet}(M)\longrightarrow\textsc{Mor}(\alpha,{{}_{\mathrm{H}}}\Phi)$

such that

$\widetilde{\Upsilon}^{-1}_{\alpha}(T\otimes_{M}\mu)=T\mu$

is a graded $M$–bimodule isomorphism as well, with the inverse given by

(A.13)
$\widetilde{\Upsilon}_{\alpha}(\tau)=\sum^{d_{\alpha}}_{k=1}T^{\mathrm{R}}_{k}\otimes_{M}\widetilde{\mu}^{\tau}_{k}\quad\mbox{
with
}\quad\widetilde{\mu}^{\tau}_{k}=\sum^{d_{\alpha},n_{\alpha}}_{i,j=1}y^{\alpha}_{ik}\,w^{\alpha\,\ast}_{ij}\,\tau(e_{j})\;\;\in\;\;\Omega(M).$

where $Y^{\alpha}=(y^{\alpha}_{ij})$ $\in$ $M_{d_{\alpha}}(\mathbb{C})$ is the
inverse matrix of $Z^{\alpha}$ and
$T^{\mathrm{R}}_{k}=\displaystyle\sum^{d_{\alpha}}_{i=1}z_{ki}T^{\mathrm{L}}_{i}$.
Elements of this tensor product can be interpreted as right $qvb$–valued
differential forms. The linear map

(A.14)
$\begin{split}\widehat{\nabla}^{\omega}_{\alpha}:\Gamma^{\mathrm{R}}(M,V^{\alpha}M)&\longrightarrow\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\otimes_{M}\Omega^{1}(M)\\\
T&\longmapsto\widetilde{\Upsilon}_{\alpha}\circ\ast\circ
D^{\omega}\circ\ast\circ T,\end{split}$

is called the induced quantum linear connection (induced qlc) in
$\zeta^{\mathrm{R}}_{\alpha}$. All these constructions can be extended in a
very natural way by using the direct sum operator for every $\alpha$ $\in$
$\textsc{Obj}(\mathbf{Rep}_{\mathcal{G}})$. The following formulas for
exterior covariant derivatives hold

(A.15) $d^{\nabla^{\omega}_{\alpha}}=\Upsilon_{\alpha}\circ
D^{\omega}\circ\Upsilon^{-1}_{\alpha}\,,\quad
d^{\widehat{\nabla}^{\omega}_{\alpha}}=\widetilde{\Upsilon}_{\alpha}\circ\ast\circ
D^{\omega}\circ\ast\circ\widetilde{\Upsilon}^{-1}_{\alpha}$

The canonical hermitian structure on $\zeta^{\mathrm{L}}_{\alpha}$ is the map
given by

(A.16)
$\displaystyle\langle-,-\rangle_{\mathrm{L}}:\Gamma^{\mathrm{L}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{L}}(M,V^{\alpha}M)$
$\displaystyle\longrightarrow M$ $\displaystyle(\quad T_{1}\quad,\quad
T_{2}\quad)$
$\displaystyle\longmapsto\sum^{n_{\alpha}}_{k=1}T_{1}(e_{k})T_{2}(e_{k})^{\ast};$

while the canonical hermitian structure on $\zeta^{\mathrm{R}}_{\alpha}$ is
the map given by

(A.17)
$\displaystyle\langle-,-\rangle_{\mathrm{R}}:\Gamma^{\mathrm{R}}(M,V^{\alpha}M)\times\Gamma^{\mathrm{R}}(M,V^{\alpha}M)$
$\displaystyle\longrightarrow M$ $\displaystyle(\quad T_{1}\quad,\quad
T_{2}\quad)$
$\displaystyle\longmapsto\sum^{n_{\alpha}}_{k=1}T_{1}(e_{k})^{\ast}T_{2}(e_{k}),$

where $\\{e_{i}\\}^{n_{\alpha}}_{i=1}$ is any orthonormal basis of
$V^{\alpha}$. It is worth mentioning that $\langle-,-\rangle_{\mathrm{L}}$,
$\langle-,-\rangle_{\mathrm{R}}$ are non–singular and
$\nabla^{\omega}_{\alpha}$, $\widehat{\nabla}^{\omega}_{\alpha}$ are hermitian
if $\omega$ is real [Sa2].

In accordance with [Br2],
$GM\,\square^{G}\,V^{\alpha\,\ast}\cong\Gamma(M,V^{\alpha}M)$ (for the natural
left action on the dual space of $V^{\alpha}$, $V^{\alpha\,\ast}$), which is
the commonly accepted construction of the associated qvb. Nevertheless, we
have decided to use $\Gamma(M,V^{\alpha}M)$ because in this way, the
definitions of $\nabla^{\omega}_{\alpha}$,
$\widehat{\nabla}^{\omega}_{\alpha}$ are completely analogous to its classical
counterparts; not to mention that it is easier to work with, since it will
allow us to do explicit calculations. In addition, by using intertwining maps,
the definition of the canonical hermitian structure looks more natural.

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