# Electromagnetic Theory with Quantum Principal Bundles

Gustavo Amilcar Saldaña Moncada  Gustavo Amilcar Saldaña Moncada
CIMAT, Unidad Guanajuato<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to formulate a non–commutative geometrical version of
the Electromagnetic Theory in a non–commutative space–time by using the theory
of quantum principal bundles and quantum principal connections. To accomplish
this purpose, we will dualize the geometrical formulation of this theory.

MSC 2010: 46L87, 58B99.

Keywords: Quantum principal bundles, non–commutative $U(1)$ gauge theory.

##  1\. Introduction

It is well–known the relationship between Geometry and Physics; particularly
when we deal with the Electromagnetic Theory [1]. Indeed, one of the most
general starting points of this theory in the vacuum is to consider it as a
Yang–Mills theory for the trivial principal $U(1)$–bundle over the Minkowski
space–time: $\mathbb{R}^{4}$ with the metric $\eta=\mathrm{diag}(-1,1,1,1)$
[1].

In Appendix A we show a brief summary of this well–known development. In
particular, we comment how the (second) Bianchi identity (see Equation A.3)
gives rise to the Gauss Law for magnetism (Equation A.8) and the Faraday
equation (see Equation A.9); while critical points of the Yang–Mills
functional (see Equation A.4), a functional that measures the square norm of
the curvature of a principal connection, gives rise to the Gauss Law (Equation
A.10) and the Ampere equation (Equation A.11).

It is worth remarking that in Equation A.4 (or equivalently the Equations
A.10, A.11) we are looking for critical points of the Yang–Mills functional,
so in principle, not every principal connection (or equivalently, not every
$1$–form potential) can satisfy it. Equations A.10, A.11 are usually called
dynamical equations because they rule the dynamics of the electric and the
magnetic field.

On the other side, the Equation A.3 (or equivalently the Equations A.8, A.9)
is always satisfied for every potential $1$–form. This equation can be thought
as a condition impose on the electromagnetic field by the Minkowski space–time
and the group $U(1)$; so Equations A.8, A.9 are usually called geometrical
equations, reflecting the fact that they comes from the Bianchi equation, a
geometrical equation.

This paper aims is recreating the geometrical formulation of the
Electromagnetic Theory showed in the Appendix A by using quantum principal
bundles in a concrete non–commutative space–time: the Moyal–Weyl algebra. Of
course, during the text we will show the link between our formulation with the
common results about this subject in literature, for example [5, 6, 7, 8, 9,
10, 11]. The importance of this paper lies in this link; as well as in the
opportunity of using quantum principal bundles in $SU(2)$ or
$SU(3)$–non–commutative gauge theories.

In particular, we are going to show the correct geometrical equation by using
the non–commutative Bianchi identity, and the correct dynamical equation by
identifying critical points of the non–commutative Yang–Mills functional,
which is now a functional that measures the square norm of the curvature of a
quantum principal connection. With that, we will present the correct
non–commutative Maxwell Equations as well as the correct Lagrangian of the
system and its quantization. An interesting result of this formulation is the
fact that the four–potential produces electric charges and currents in the
vacuum! and magnetic charges and currents in the vacuum too!. This turns the
photon field into a kind of Dyon gauge field.

Moreover, in order to illustrate one more time the powerful tool that the
theory quantum principal bundles is in non–commutative gauge theories (the
reader should remember the link between them in the classical case [1]) and
based on the fact that in non–commutative case the photon field produces
electric and magnetic charges/currents, we are going to develop a toy model of
the Electromagnetic Theory in the Moyal–Weyl algebra with two gauge boson
fields: one associated to electric charges and currents densities, the photon
field, and the other one associated to magnetic charges and currents
densities, a kind of magnetic photon field. In addition, having two gauge
boson fields in the Electromagnetic Theory leads to full symmetry between the
electric field and the magnetic field.

The idea of using two gauge fields in the Electromagnetic Theory comes from
the formalism of Cabibbo–Ferrari [12, 13]. Nevertheless, in the classical
context, these leads a theory with $U(1)\times U(1)$ as gauge group. In the
framework quantum principal bundles it is possible to describe two gauge
fields with only one $U(1)$ (by using a $2D$–dimensional differential calculus
of $U(1)$), which seems to be the correct group of symmetries of the
Electromagnetic Theory.

This paper breaks down into $6$ sections. In the second one we are going to
build the quantum principal $U(1)$–bundle used along the whole work; in
particular, we are going to show the two differential calculus of $U(1)$ used
for our two electromagnetic models. The third section is the heart of this
paper. In this section we will present the non–commutative Maxwell equations
for our two models by using only the geometry of the spaces. It is worth
mentioning that in this section we will show a covariant formulation of both
models, as well as the correct Lagrangian densities. Following the classical
case, in the fourth section we are going to quantize our two models. The last
section is for some concluding comments and just like we have mentioned
before, in Appendix $A$ there is a brief summary of the geometrical
formulation of the Electromagnetic Theory.

To accomplish our purpose, we are going to use M. Durdevich’s theory of
quantum principal bundles and quantum principal connection ([14, 15, 16, 17])
and the general formulation of Yang–Mills equations presented on [18, 19] and
tested in other quantum bundles with several exciting and interesting results
[20, 21]. It is worth remarking that the theory showed on [18, 19] was
formulated in the most general way, it was not created for the particular case
of the quantum bundle used in this paper. Of course, we will continue with the
notation presented on these papers.

Finally it is worth mentioning that we have chosen Durdevich’s framework to
develop this paper instead of [4, 22, 23] because of its purely
geometrical/algebraic formulation and because its generality in terms of
differential calculus and quantum connections: this theory allows to work with
almost every differential calculus on the quantum spaces and with any quantum
principal connections. This generality in terms of differential calculus and
connections makes (at least in the context of Yang–Mills equations in which
connections play the main role) a richer theory.

In Non–Commutative Geometry it is common to use the word quantum as a
synonymous of non–commutative and we will use it in the paper sometimes. On
the other hand, in Physics it is common to use the word non–commutative to
denote theories with non–abelian groups, which is no the case of this paper;
so we expect that the reader do not confuse with all these terms.

##  2\. The Quantum Principal $U(1)$–Bundle

In this section we are going to present the quantum bundle on which we will
work. Since we are not interesting in changing the topology of the spaces nor
the bundle, we have to consider a trivial quantum principal $U(1)$–bundle. The
general theory of this kind of bundles can be checked on [15, 17].

###  2.1. A Non–Commutative Minkowski Space–Time

Let us start by considering the Minkowski space–time
$(\mathbb{R}^{4},\eta=\mathrm{diag}(1,-1,-1,-1))$ and its space of
complex–valued smooth functions $C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})$. By
choosing a $4\times 4$ antisymmetric matrix $(\theta^{\mu\nu})$ $\in$
$M_{4}(\mathbb{R})$, it is possible to take
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$, formal power
series in $\theta^{\mu\nu}$ with coefficients in
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})$. Finally it is possible to apply a
$\theta^{\mu\nu}$–twist on
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ by defining a new
product: for every $f$, $h$ $\in$
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ we define

(2.1) $f\cdot
h:=m\circ\mathrm{exp}\left(\frac{i\,\theta^{\mu\nu}}{2}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}\right)(f\otimes h),$

where $m$ denotes the usual product on
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$. Explicitly we
have

$f\cdot h:=\left(fh+\sum_{\mu,\nu}\frac{i\,\theta^{\mu\nu}}{2}\frac{\partial
f}{\partial x^{\mu}}\frac{\partial h}{\partial
x^{\nu}}+\sum_{\mu,\nu,\alpha,\beta}\frac{i^{2}\,\theta^{\mu\nu}\,\theta^{\alpha\beta}}{8}\frac{\partial
f}{\partial x^{\alpha}\partial x^{\mu}}\frac{\partial h}{\partial
x^{\beta}\partial x^{\nu}}+\cdots\right).$

With this new product,
$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ is a
non–commutative unital $\ast$–algebra ([6]), where the unit is
$\mathbbm{1}(x)=1$ for all $x$ and the $\ast$ operation is the complex
conjugate.

It is worth mentioning that

(2.2) $[x^{\mu},x^{\nu}]:=x^{\mu}\cdot x^{\mu}-x^{\mu}\cdot
x^{\mu}=i\,\theta^{\mu\nu}\,\mathbbm{1}.$

This non–commutative unital $\ast$–algebra receives the name of Moyal–Weyl
algebra ([6]) and it will be our quantum space–time, which we are going to
denote by $M$.

The next step is to extend the Moyal product $\cdot$ to the space of
differential forms [6]. In fact, let us take
$\Omega^{\bullet}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ the space
of formal power series in $\theta^{\mu\nu}$ with coefficient in the algebra of
complex–valued differential forms. Now Equation 2.1 is easily extended to
$\Omega^{\bullet}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ by
considering the action of $\frac{\partial}{\partial x^{\gamma}}$ on forms by
means of the Lie derivative. We will denoted by $\Omega^{\bullet}(M)$ the
space $\Omega^{\bullet}_{\mathbb{C}}(\mathbb{R}^{4})[[\theta^{\mu\nu}]]$ with
the Moyal product $\cdot$ extended. This graded differential $\ast$–algebra is
going to play the role of quantum differential forms on our quantum space–time
$M$.

In accordance with [19], in order to apply its theory there are certain
structures that previously we have to define on $\Omega^{\bullet}(M)$.

Let us define the following left quantum Pseudo–Riemmanian metric on $M$

(2.3)
$\\{\langle-,-\rangle^{k}_{\mathrm{L}}:\Omega^{k}(M)\times\Omega^{k}(M)\longrightarrow
M\mid k=0,1,2,3,4\\}$

such that $\langle f,h\rangle^{0}_{\mathrm{L}}=f\cdot h^{\ast}$ and for
$k=1,2,3,4$ we extend the usual metric of the de Rham differential algebra of
the Minkowski space–time. For example

$\displaystyle\langle\sum^{3}_{\mu=0}f_{\mu}\,dx^{\mu},\sum^{3}_{\nu=0}h_{\nu}\,dx^{\nu}\rangle^{1}_{\mathrm{L}}=\sum^{3}_{\mu,\nu=0}\eta^{\mu\,\nu}f_{\mu}\cdot
h^{\ast}_{\nu}=f_{0}\cdot h^{\ast}_{0}-f_{1}\cdot h^{\ast}_{1}-f_{2}\cdot
h^{\ast}_{2}-f_{3}\cdot h^{\ast}_{3}$

and

$\langle f\,\mathrm{dvol},h\,\mathrm{dvol}\rangle^{4}_{\mathrm{L}}=f\cdot
h^{\ast},$

where

(2.4) $\mathrm{dvol}:=dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}$

is the volume form. Furthermore, by postulating the orthogonality between
quantum forms of different degrees, we can induce Pseudo–Riemannian structures
in the whole graded space; so we will not use superscripts anymore.

By taking the integral operator

$\int_{M}\mathrm{dvol}$

we can define the left quantum Hodge pseudo inner product111Whenever the
integrals converge and taking into a count the corresponding equivalence
classes.

(2.5)
$\langle-|-\rangle_{\mathrm{L}}:=\int_{M}\langle-,-\rangle_{\mathrm{L}}\,\mathrm{dvol}.$

Furthermore, and according to [19], the left quantum Hodge operator is the
antilinear $M$–isomorphism

(2.6) $\star_{\mathrm{L}}:\Omega^{k}(M)\longrightarrow\Omega^{4-k}(M)$

that satisfies

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

Explicitly, for our case in the canonical basis we have

(2.7)
$\star_{\mathrm{L}}\mathbbm{1}=\mathrm{dvol},\;\;\star_{\mathrm{L}}\mathrm{dvol}=-\mathbbm{1},$
(2.8) $\displaystyle\star_{\mathrm{L}}dx^{0}=dx^{1}\wedge dx^{2}\wedge
dx^{3},\quad\star_{\mathrm{L}}dx^{1}=dx^{0}\wedge dx^{2}\wedge dx^{3},$
$\displaystyle\star_{\mathrm{L}}dx^{2}=-dx^{0}\wedge dx^{1}\wedge
dx^{3},\quad\star_{\mathrm{L}}dx^{3}=dx^{0}\wedge dx^{1}\wedge dx^{2},$ (2.9)
$\displaystyle\star_{\mathrm{L}}dx^{0}\wedge dx^{1}=-dx^{2}\wedge
dx^{3},\quad\star_{\mathrm{L}}dx^{0}\wedge dx^{2}=dx^{1}\wedge
dx^{3},\quad\star_{\mathrm{L}}dx^{0}\wedge dx^{3}=-dx^{1}\wedge dx^{2},$
$\displaystyle\star_{\mathrm{L}}dx^{1}\wedge dx^{2}=dx^{0}\wedge
dx^{3},\quad\star_{\mathrm{L}}dx^{1}\wedge dx^{3}=-dx^{0}\wedge
dx^{2},\quad\star_{\mathrm{L}}dx^{2}\wedge dx^{3}=dx^{0}\wedge dx^{1},$ (2.10)
$\displaystyle\star_{\mathrm{L}}dx^{1}\wedge dx^{2}\wedge
dx^{3}=dx^{0},\quad\star_{\mathrm{L}}dx^{0}\wedge dx^{2}\wedge dx^{3}=dx^{1},$
$\displaystyle\star_{\mathrm{L}}dx^{0}\wedge dx^{1}\wedge
dx^{3}=-dx^{2},\quad\star_{\mathrm{L}}dx^{0}\wedge dx^{1}\wedge
dx^{2}=dx^{3}.$

Of course in all cases we have
$\star^{2}_{\mathrm{L}}=(-1)^{k(4-k)+1}\,\mathrm{id}.$ Finally the left
quantum codifferential is defined as the operator

(2.11)
$d^{\star_{\mathrm{L}}}:=(-1)^{k+1}\,\star^{-1}_{\mathrm{L}}\,\circ\,d\,\circ\,\star_{\mathrm{L}}:\Omega^{k+1}(M)\longrightarrow\Omega^{k}(M),$

and it is the formal adjoint operator of $d$ with respect to
$\langle-|-\rangle_{\mathrm{L}}$ [19].

It is worth mentioning that by considering the right structure as

(2.12)
$\langle\hat{\mu},\mu\rangle_{\mathrm{R}}:=\langle\hat{\mu}^{\ast},\mu^{\ast}\rangle_{\mathrm{L}}$

we get $\langle-|-\rangle_{\mathrm{R}}$,
$\star_{\mathrm{R}}:=\ast\circ\star_{\mathrm{L}}\circ\ast$ and
$d^{\star_{\mathrm{R}}}:=(-1)^{k+1}\,\star^{-1}_{\mathrm{R}}\,\circ\,d\,\circ\,\star_{\mathrm{R}}=\ast\circ
d^{\star_{\mathrm{L}}}\circ\ast$, which is the formal adjoint operator of $d$
with respect to $\langle-|-\rangle_{\mathrm{R}}$ [19].

###  2.2. The Quantum Group of $U(1)$ and its Differential Calculus

Let us start this subsection by considering the $\ast$–Hopf algebra of the
polynomial Laurent algebra $(G=\mathbb{C}[z,z^{-1}],\phi,\epsilon,\kappa)$,
where $z^{-1}=z^{\ast}$, $\phi$ is the coproduct, $\epsilon$ is the counity
and $\kappa$ the coinverse. The space $g$ will play the role of the quantum
structure group of our bundle.

The next step is to find differential calculus on $G$ different to the
well–known algebra of classical differential forms in order to create our two
models. The reasons of these changes will be explored in whole text.

####  2.2.1. The Non–Standard $1D$–Differential Calculus

In accordance with [17, 24], a bicovariant $\ast$–First Order Differential
Calculus ($\ast$–FODC) can be defined as an $\mathrm{Ad}$–invariant right
ideal $\mathcal{R}\subseteq\mathrm{Ker}(\epsilon)$ such that
$\kappa(\mathcal{R})^{\ast}\subseteq\mathcal{R}$. In this way, let us consider
any $\ast$–FODC

(2.13) $(\Gamma,d)$

such that the set of invariant elements or the quantum (dual) Lie algebra
${{}_{\mathrm{inv}}}\Gamma$ satisfies

$\mathrm{dim}_{\mathbb{C}}({{}_{\mathrm{inv}}}\Gamma)=1,\qquad
z\,\pi(z)=-\pi(z)\,z,$

where $\pi:\mathrm{Ker}(\epsilon)\longrightarrow{{}_{\mathrm{inv}}}\Gamma$ is
the quantum germs map given by $\pi(g)=\kappa(g^{(1)})dg^{(2)}$ for all $g$
$\in$ $\mathrm{Ker}(\epsilon)$ with $\phi(g)=g^{(1)}\otimes g^{(2)}$ (in
Sweedler’s notation) [17, 24]. Notice

(2.14)
${{}_{\mathrm{inv}}}\Gamma=\mathrm{span}_{\mathbb{C}}\\{\vartheta:=i\,\pi(z)\\}.$

Also we can calculate the adjoint (co)action of $G$ on
${{}_{\mathrm{inv}}}\Gamma$

(2.15) $\displaystyle\mathrm{ad}:{{}_{\mathrm{inv}}}\Gamma$
$\displaystyle\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes G$
$\displaystyle\vartheta$ $\displaystyle\longmapsto\vartheta\otimes\mathbbm{1}$

because of

$\mathrm{ad}(\pi(g))=((\pi\otimes\mathrm{id})\circ\mathrm{Ad})(g)=\pi(g)\otimes\mathbbm{1}$

for all $g$ $\in$ $G$, where $\mathrm{Ad}:G\longrightarrow G\otimes G$ is the
adjoint (right co)action on $G$ [14, 15, 17].

In Durdevich’s framework of quantum principal bundles we have to take the
universal differential envelope $\ast$–calculus [14, 15, 17]

(2.16) $(\Gamma^{\wedge},d,\ast).$

This graded differential $\ast$–algebra will play the role of quantum
differential forms of $U(1)$ in the first model that we will present. This
space has the particularity that

$\Gamma^{\wedge\,2}\cong G\otimes{{}_{\mathrm{inv}}}\Gamma^{\wedge\,2},$

where

(2.17)
${{}_{\mathrm{inv}}}\Gamma^{\wedge\,2}=\mathrm{span}_{\mathbb{C}}\\{\vartheta\,\vartheta\\};$

so there are quantum differential forms of grade $2$. Moreover, there is no a
top grade, which is a big difference with the algebra of classical
differential forms of $U(1)$.

The main reason to use the universal differential envelope $\ast$–calculus
instead of, for example, the universal differential calculus is the fact that
$(\Gamma^{\wedge},d,\ast)$ allows to extend the structure of $\ast$–Hopf
algebra to any grade and it is maximal with this property; reflecting the
classical fact the tangent bundle of a Lie group is also a Lie Group. The
structure of graded differential $\ast$–Hopf algebra will be denoted by the
same symbols.

In this specific case we have ([14, 15, 17])

(2.18) $\vartheta^{\ast}=\vartheta,\qquad d\vartheta=i\vartheta\,\vartheta.$

####  2.2.2. A $2D$–differential calculus

Now let $q$ $\in$ $\mathbb{R}-\\{0,1\\}$,
$\mathcal{L}=\mathrm{span}_{\mathbb{C}}\\{z,z^{-1}\\}$ and its linear dual
space
$\hat{\mathcal{L}}:=\mathrm{span}_{\mathbb{C}}\\{\theta_{-},\theta_{+}\\}$,
where $\theta_{-}(z)=1$, $\theta_{-}(z^{-1})=0$, $\theta_{+}(z)=0$,
$\theta_{+}(z^{-1})=1$. The map

$\varpi:\mathrm{Ker}(\epsilon)\longrightarrow\hat{\mathcal{L}},\qquad
g\longmapsto\varpi(g),$

where $\varpi(g)(x)={\mathcal{Q}}(g\otimes x)$ with ${\mathcal{Q}}$ such that
${\mathcal{Q}}(z^{m}\otimes z^{n})=q^{mn}$ for all $m$, $n$ $\in$
$\mathbb{Z}$; defines

(2.19) $(\Gamma,d),$

a $\ast$–First Order Differential Calculus ($\ast$–FODC) by means of its space
of invariant elements, or equivalently, its (dual) quantum Lie algebra ([4,
24, 17])

(2.20)
${{}_{\mathrm{inv}}}\Gamma:=\frac{\mathrm{Ker}(\epsilon)}{\mathrm{Ker}(\varpi)}.$

It is worth remarking that $\mathrm{dim}({{}_{\mathrm{inv}}}\Gamma)=2$, a big
difference with the classical case in which
$\displaystyle\mathrm{dim}\left(\frac{\mathrm{Ker}(\epsilon)}{\mathrm{Ker}^{2}(\epsilon)}\right)=\mathrm{dim}(\mathfrak{u(1)})=1$
and clearly a linear basis of the quantum Lie algebra is given by

(2.21)
$\beta:=\\{\vartheta^{e}:=-i\,\theta_{-},\,\vartheta^{m}:=-i\,\theta_{+}\\}.$

By considering the quantum germs map [17, 24]

(2.22) $\pi:\mathrm{Ker}(\epsilon)\longrightarrow{{}_{\mathrm{inv}}}\Gamma$

it is easy to look for relations, such as

(2.23) $\displaystyle\pi(z^{n})$
$\displaystyle=i(q^{n}-1)\vartheta^{e}+i(q^{-n}-1)\vartheta^{m},$
$\displaystyle\pi(z^{-n})$
$\displaystyle=i(q^{-n}-1)\vartheta^{e}+i(q^{n}-1)\vartheta^{m}.$

Also we can calculate the adjoint (co)action of $G$ on
${{}_{\mathrm{inv}}}\Gamma$

(2.24) $\displaystyle\mathrm{ad}:{{}_{\mathrm{inv}}}\Gamma$
$\displaystyle\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes G$
$\displaystyle\vartheta$ $\displaystyle\longmapsto\vartheta\otimes\mathbbm{1}$

given by

$\mathrm{ad}(\pi(g))=((\pi\otimes\mathrm{id})\circ\mathrm{Ad})(g)=\pi(g)\otimes\mathbbm{1}$

for all $g$ $\in$ $G$.

Now we have to take the universal differential envelope $\ast$–calculus [14,
15, 17]

(2.25) $(\Gamma^{\wedge},d,\ast).$

This graded differential $\ast$–algebra will play the role of quantum
differential forms of $U(1)$ in the second that we will present.

To conclude this subsection we are to present some relations in this algebra

(2.26) $\vartheta^{e\ast}=\vartheta^{e},\quad\vartheta^{m\ast}=\vartheta^{m}$
(2.27)
$d\pi(z)=d\pi(z^{-1})=-\frac{(q-1)^{2}}{q}(\vartheta^{e}\vartheta^{m}+\vartheta^{m}\vartheta^{e}),$
(2.28)
$d\vartheta^{e}=d\vartheta^{m}=i(\vartheta^{e}\vartheta^{m}+\vartheta^{m}\vartheta^{e}).$

###  2.3. The Quantum Bundle

Finally we have all the ingredients to build the trivial quantum bundle that
we will used in the rest of this paper. Let

(2.29) $\zeta=(GM:=M\otimes G,M,{{}_{GM}}\Phi:=\mathrm{id}\otimes\phi)$

be the trivial quantum principal $U(1)$–bundle over $M$ [15, 17]. Now we can
also take the trivial differential calculus on the bundle ([15, 17]):

(2.30)
$\Omega^{\bullet}(GM)=\Omega^{\bullet}(M)\otimes\Gamma^{\wedge},\;\;{{}_{\Omega}}\Psi:=\mathrm{id}\otimes\phi:\Omega^{\bullet}(GM)\longrightarrow\Omega^{\bullet}(M)\otimes\Gamma^{\wedge}$

(here $\phi$ is the extension of the coproduct in $\Gamma^{\wedge}$). The
graded differential $\ast$–algebra $\Omega^{\bullet}(GM)$ will play de role of
quantum differential forms  on the total quantum space $GM$. It is worth
mentioning that

(2.31) $\mathrm{Hor}^{\bullet}GM:=\Omega^{\bullet}(M)\otimes G$

and of course the graded differential $\ast$–subalgebra of forms on the base
quantum space matches with $\Omega^{\bullet}(M)$. In accordance with the
general theory, the restriction of ${{}_{\Omega}}\Psi$ in
$\mathrm{Hor}^{\bullet}GM$ is a (co)representation of $G$ on
$\mathrm{Hor}^{\bullet}GM$ [15, 17]. This map will be denoted by
${{}_{\mathrm{H}}}\Phi$.

In the light of [15, 17], the set of quantum principal connections (qpcs)

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

on trivial bundles is in biyection with the space of non–commutative gauge
potentials

(2.33)
$\\{A^{\omega}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{1}(M)\mid
A^{\omega}\mbox{ is linear}\\}$

by means of

$\omega(\theta)=A^{\omega}(\theta)\otimes\mathbbm{1}+\mathbbm{1}\otimes\theta$

for all $\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$ (for both cases). For
$A^{\omega}=0$ the corresponding qpc is called the trivial qpc and it is
regular and multiplicative [15, 17].

######  Proposition 2.1.

The trivial qpc is the only regular connection.

###### Proof.

Let us assume that $\omega$ is a regular qpc. Then according to [15, 17] its
corresponding non–commutative gauge potential has to satisfies

$A^{\omega}(\theta\circ g)=\epsilon(g)A^{\omega}(\theta),$

for all $\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$ and all $g$ $\in$ $G$. So
it is enough to evaluate the last equality in $g=z$ to find that it is
satisfied if and only if $A^{\omega}=0$. ∎

Despite the ease of the previous proof, the last proposition is quite
important for our purpose because every (classical) principal connection is
regular [15]; so the last result tells that up to the trivial qpc, there is no
classical counterpart of any qpc, i.e., up to the trivial qpc, all results in
the rest of this paper will be completely quantum, they will not have
classical analogous.

In order to talk about the curvature of a qpc and develop the theory we need
to consider the two different cases given by our two different calculus of
$G$.

####  2.3.1. The Non–Standard $1D$–Differential Calculus

First of all, it is necessary to choose a embedded differential [15, 17]. In
our case, the only embedded differential

(2.34)
$\delta:{{}_{\mathrm{inv}}}\Gamma\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes{{}_{\mathrm{inv}}}\Gamma$

is given by

$\delta(\vartheta)=i\vartheta\otimes\vartheta.$

With this, the space of curvatures of qpcs

(2.35)
$\\{R^{\omega}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{2}(GM)\\}$

is in biyection with the space of non–commutative fields strength

(2.36) $\\{F^{\omega}:=dA^{\omega}-\langle
A^{\omega},A^{\omega}\rangle:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{2}(M)\\},$

where

$\langle A^{\omega},A^{\omega}\rangle=m\circ(A^{\omega}\otimes
A^{\omega})\circ\delta$

and $m$ is the product map. The reader can comparison this relation between
qpcs and their curvatures in terms of potentials
($A^{\omega}\longleftrightarrow F^{\omega}=dA^{\omega}-\langle
A^{\omega},A^{\omega}\rangle$) with Equation A.2
($A^{\omega}\longleftrightarrow F^{\omega}=dA^{\omega}$). The embedded
differential map in the definition of the curvature of a qpc allows to see it
as a gauge field in the general case [15, 17].

Following the classical case, let us consider the non–commutative gauge
potential $A^{\omega}$ given by

(2.37)
$A^{\omega}(\vartheta)=\phi\,dx^{0}-A_{1}\,dx^{1}-A_{2}\,dx^{2}-A_{3}\,dx^{3}.$

In this way, the non–commutative field strength is defined by

(2.38)
$F^{\omega}(\vartheta)=dA^{\omega}(\vartheta)-iA^{\omega}(\vartheta)\wedge
A^{\omega}(\vartheta)$

and in terms of coordinates we have

(2.39) $F^{\omega}(\vartheta)=\sum_{0\leq\mu<\nu\leq
3}F_{\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\;\;\mathrm{where}\;\;(F_{\mu\nu})=\begin{pmatrix}0&D_{1}&D_{2}&D_{3}\\\
-D_{1}&0&-H_{3}&H_{2}\\\ -D_{2}&H_{3}&0&-H_{1}\\\
-D_{3}&-H_{2}&H_{1}&0\end{pmatrix},$

where

(2.40) ${\bf D}=(D_{1},D_{2},D_{3}):={\bf E}+i[\phi,{\bf A}],\qquad{\bf
E}:=-{\partial{\bf A}\over\partial x^{0}}-\nabla\phi$

and

(2.41) ${\bf H}=(H_{1},H_{2},H_{3}):={\bf B}+i{\bf A}\times{\bf A},\qquad{\bf
B}:=\nabla\times{\bf A},$

with ${\bf A}=(A_{1},A_{2},A_{3})$. The definition of the commutators can be
deduced by the context and we have considered that in the cross product the
multiplication of elements always start from up to down.

The field ${\bf D}$, ${\bf H}$ will be consider as the non–commutative
electric field and the non–commutative magnetic field, respectively. Also
$F_{\mu\nu}$ will recieved the name of the non–commutative electromagnetic
tensor field.

####  2.3.2. The $2D$–Differential Calculus

For this other case the only embedded differential

(2.42)
$\delta:{{}_{\mathrm{inv}}}\Gamma\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes{{}_{\mathrm{inv}}}\Gamma$

is given by

$\delta(\vartheta^{e})=\delta(\vartheta^{m})=i(\vartheta^{e}\otimes\vartheta^{m}+\vartheta^{m}\otimes\vartheta^{e}).$

With this, the space of curvature of qpcs

(2.43)
$\\{R^{\omega}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{2}(GM)\\}$

is in biyection with the space of non–commutative fields strength

(2.44) $\\{F^{\omega}:=dA^{\omega}-\langle
A^{\omega},A^{\omega}\rangle:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{2}(M)\\},$

where

$\langle A^{\omega},A^{\omega}\rangle=m\circ(A^{\omega}\otimes
A^{\omega})\circ\delta$

and $m$ is the product map. The reader can comparison this relation between
qpcs and their curvatures in terms of potentials
($A^{\omega}\longleftrightarrow F^{\omega}=dA^{\omega}-\langle
A^{\omega},A^{\omega}\rangle$) with Equation A.2
($A^{\omega}\longleftrightarrow F^{\omega}=dA^{\omega}$).

In this way, let us calculate the non–commutative field strength $F^{\omega}$
of the non–commutative gauge potential $A^{\omega}$ given by

(2.45)
$A^{\omega}(\vartheta^{e})=\phi^{e}\,dx^{0}-A^{e}_{1}\,dx^{1}-A^{e}_{2}\,dx^{2}-A^{e}_{3}\,dx^{3},$
(2.46)
$A^{\omega}(\vartheta^{m})=\phi^{m}dx^{0}\,-A^{m}_{1}\,dx^{1}-A^{m}_{2}\,dx^{2}-A^{m}_{3}\,dx^{3}$

Thus

(2.47)
$F^{\omega}(\vartheta^{e})=dA^{\omega}(\vartheta^{e})-iA^{\omega}(\vartheta^{e})\wedge
A^{\omega}(\vartheta^{m})-iA^{\omega}(\vartheta^{m})\wedge
A^{\omega}(\vartheta^{e}),$ (2.48)
$F^{\omega}(\vartheta^{m})=dA^{\omega}(\vartheta^{m})-iA^{\omega}(\vartheta^{e})\wedge
A^{\omega}(\vartheta^{m})-iA^{\omega}(\vartheta^{m})\wedge
A^{\omega}(\vartheta^{e}).$

In terms of coordinates we get

(2.49) $F^{\omega}(\vartheta^{e})=\sum_{0\leq\mu<\nu\leq
3}F^{e}_{\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\;\;\mathrm{where}\;\;(F^{e}_{\mu\nu})=\begin{pmatrix}0&D^{e}_{1}&D^{e}_{2}&D^{e}_{3}\\\
-D^{e}_{1}&0&-H^{e}_{3}&H^{e}_{2}\\\ -D^{e}_{2}&H^{e}_{3}&0&-H^{e}_{1}\\\
-D^{e}_{3}&-H^{e}_{2}&H^{e}_{1}&0\end{pmatrix},$ (2.50) ${\bf
D}^{e}=(D^{e}_{1},D^{e}_{2},D^{e}_{3}):={\bf E}^{e}+i[\phi^{m},{\bf
A}^{e}]+i[\phi^{e},{\bf A}^{m}],\;\quad{\bf E}^{e}:=-\frac{\partial{\bf
A}^{e}}{\partial x^{0}}-\nabla\phi^{e}$

and

(2.51) ${\bf H}^{e}=(H^{e}_{1},H^{e}_{2},H^{e}_{3}):={\bf B}^{e}+i{\bf
A}^{e}\times{\bf A}^{m}+i{\bf A}^{m}\times{\bf A}^{e},\;\quad{\bf
B}^{e}:=\nabla\times{\bf A}^{e},$

where ${\bf A}^{e}=(A^{e}_{1},A^{e}_{2},A^{e}_{3})$ and ${\bf
A}^{m}=(A^{m}_{1},A^{m}_{2},A^{m}_{3})$; while

(2.52) $F^{\omega}(\vartheta^{m})=\sum_{0\leq\mu<\nu\leq
3}F^{m}_{\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\;\;\mathrm{where}\;\;(F^{m}_{\mu\nu})=\begin{pmatrix}0&H^{m}_{1}&H^{m}_{2}&H^{m}_{3}\\\
-H^{m}_{1}&0&-D^{m}_{3}&D^{m}_{2}\\\ -H^{m}_{2}&D^{m}_{3}&0&-D^{m}_{1}\\\
-H^{m}_{3}&-D^{+}_{2}&D^{m}_{1}&0\end{pmatrix},$ (2.53) ${\bf
H}^{m}=(H^{m}_{1},H^{m}_{2},H^{m}_{3}):={\bf B}^{m}+i[\phi^{m},{\bf
A}^{e}]+i[\phi^{e},{\bf A}^{m}],\;\quad{\bf B}^{m}:=-\frac{\partial{\bf
A}^{m}}{\partial x^{0}}-\nabla\phi^{m}$

and

(2.54) ${\bf D}^{m}=(D^{m}_{1},D^{m}_{2},D^{m}_{3}):={\bf E}^{m}+i{\bf
A}^{e}\times{\bf A}^{m}+i{\bf A}^{m}\times{\bf A}^{e},\;\quad{\bf
E}^{m}:=\nabla\times{\bf A}^{m}.$

The notation chosen is not a coincidence: we will consider that
$A^{\omega}(\vartheta^{e})$ is the non–commutative electric potential
$1$–form; $(F^{e}_{\mu,\nu}),$ ${\bf D}^{e}$ and ${\bf H}^{e}$ are the
non–commutative electromagnetic tensor field, the non–commutative electric
field and the non–commutative magnetic field generated by
$A^{\omega}(\vartheta^{e})$, respectively; and ${\bf E}^{e}$ and ${\bf B}^{e}$
are their corresponding classical parts. In the same way, we will consider
that $A^{\omega}(\vartheta^{m})$ is the non–commutative magnetic potential
$1$–form; $(F^{m}_{\mu,\nu}),$ ${\bf D}^{m}$ and ${\bf H}^{m}$ are the
non–commutative magnetoelectric tensor field, the non–commutative electric
field and the non–commutative magnetic field generated by
$A^{\omega}(\vartheta^{m})$, respectively; and ${\bf E}^{m}$ and ${\bf B}^{m}$
are their corresponding classical parts.

##  3\. Non–Commutative Maxwell Equations

Just like we have exposed at the beginning of this paper, Maxwell equations in
the vacuum comes from the Bianchi identity and critical points of the
Yang–Mills functional. In this sections we are going to recreate that process
in order to find their quantum counterparts on our bundle.

###  3.1. Non–Commutative Geometrical Equations

In accordance with [15, 19], every qpc satisfies the non–commutative Bianchi
identity, which is

(3.1)
$(D^{\omega}-S^{\omega})R^{\omega}=\langle\omega,\langle\omega,\omega\rangle\rangle-\langle\langle\omega,\omega\rangle,\omega\rangle,$

where $D^{\omega}$ is the covariant derivative and the operator $S^{\omega}$
measures the lack of regularity of the qpc, in the sense that $S^{\omega}=0$
when $\omega$ is regular. Furthermore, when $\omega$ is multiplicative [15,
17], the right–hand side of the last equation is equal to $0$. In summary,
when $\omega$ is regular and multiplicative, for example for classical
principal connections, Equation 3.1 turns into the well–known classical
Bianchi identity $D^{\omega}R^{\omega}=0$.

####  3.1.1. The Non–Standard $1D$–Differential Calculus

For our quantum bundle, Equation 3.1 becomes into

(3.2) $(d-d^{S^{\omega}})F^{\omega}=0,$

where

(3.3)
$d^{S^{\omega}}\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k+1}(M)$

is given by

$d^{S^{\omega}}\tau(\vartheta)=i[A^{\omega}(\vartheta),\tau(\vartheta)]^{\partial}:=i(A^{\omega}(\vartheta)\wedge\tau(\vartheta)-(-1)^{k}\tau(\vartheta)\wedge
A^{\omega}(\vartheta))\;\;\;\in\;\;\;\Omega^{k+1}(M)$

for all linear maps
$\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k}(M).$ In this way,
Equation 3.2 is given by

(3.4)
$dF^{\omega}(\vartheta)=[A^{\omega}(\vartheta),dF^{\omega}(\vartheta)]=i[A^{\omega}(\vartheta),dA^{\omega}(\vartheta)],$

which is the non–commutative geometrical equation for our trivial quantum
bundle with a non–standard $1D$–dimensional differential calculus of $U(1)$.
The reader can comparison the last equation with Equation A.3. It is worth
mentioning that like in the classical case, Equation 3.4 is satisfied by all
qpcs! and in general, $dF^{\omega}(\vartheta)\not=0$.

In the light of Equation 2.37, Equation 3.4 becomes into the non–commutative
Gauss Law for magnetism and the non–commutative Faraday equation

(3.5) $\nabla\cdot{\bf H}=\rho^{m},$ (3.6) $\nabla\times{\bf
D}+\frac{\partial{\bf H}}{\partial x^{0}}=-{\bf j}^{m}.$

For these equations we have defined the magnetic charge density like

(3.7) $\rho^{m}:=i[{\bf B},{\bf A}]$

and the magnetic current density like

(3.8) $-{\bf j}^{m}:=i[\phi,{\bf B}]-i({\bf E}\times{\bf A}+{\bf A}\times{\bf
E})$

Again, the definition of the commutators can be deduced by the context. The
reader can comparison the Equations 3.5, 3.6 with their classical counterparts
in Equations A.8, A.9.

It is worth remarking that Equation 3.5 tells the existence of magnetic
charges for the non–commutative gauge potential $A^{\omega}(\vartheta)$; while
Equation 3.6 tells the existence of magnetic currents, but both of them in the
vacuum! and like in the classical case, this is only a consequence of the
geometry of our spaces. Notice that the magnetic charge and current depend on
the interaction of $A^{\omega}(\vartheta)$ with the classical part of the
non–commutative field strength $F^{\omega}(\vartheta)$.

####  3.1.2. The $2D$–Differential Calculus

Now for this quantum bundle, Equation 3.1 becomes into

(3.9) $(d-d^{S^{\omega}})F^{\omega}=\langle A^{\omega},\langle
A^{\omega},A^{\omega}\rangle\rangle-\langle\langle
A^{\omega},A^{\omega}\rangle,A^{\omega}\rangle,$

where

(3.10)
$d^{S^{\omega}}\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k+1}(M)$

is given by

$d^{S^{\omega}}\tau(\vartheta^{e})=d^{S^{\omega}}\tau(\vartheta^{m})=[A^{\omega}(\vartheta^{m}),\tau(\vartheta^{e})]^{\partial}+[A^{\omega}(\vartheta^{e}),\tau(\vartheta^{m})]^{\partial}\;\;\;\in\;\;\;\Omega^{k+1}(M)$

for all linear maps
$\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k}(M)$, with
$[-,-]^{\partial}$ the graded–commutator. In this way the Equation 3.9 for
$\vartheta^{e}$, $\vartheta^{m}$ is

(3.11) $\displaystyle dF^{\omega}(\vartheta^{e})=dF^{\omega}(\vartheta^{m})$
$\displaystyle=-d(A^{\omega}(\vartheta^{e})\cdot
A^{\omega}(\vartheta^{m}))-d(A^{\omega}(\vartheta^{m})\cdot
A^{\omega}(\vartheta^{e})).$
$\displaystyle=[A^{\omega}(\vartheta^{e}),dA^{\omega}(\vartheta^{m})]+[A^{\omega}(\vartheta^{m}),dA^{\omega}(\vartheta^{e})]$

which are the non–commutative geometrical equations for our trivial quantum
bundle with a $2D$–dimensional differential calculus of $U(1)$. The reader can
comparison the last equation with Equation A.3. It is worth mentioning that
like in the classical case, Equation 3.4 is satisfied by all qpcs!

In the light of Equation 2.45, Equation 3.11 becomes for
$A^{\omega}(\vartheta^{e})$ into

(3.12) $\nabla\cdot{\bf H}^{e}=\rho,$ (3.13) $\nabla\times{\bf
D}^{e}+\frac{\partial{\bf H}^{e}}{\partial x^{0}}=-{\bf j}.$

In the same way, by Equation 2.46, Equation 3.11 for
$A^{\omega}(\vartheta^{m})$ becomes into

(3.14) $\nabla\cdot{\bf D}^{m}=\rho,$ (3.15) $\nabla\times{\bf
H}^{m}+\frac{\partial{\bf D}^{m}}{\partial x^{0}}=-{\bf j}.$

For these equations we have defined the magnetic and electric charge density
generated by $A^{\omega}(\vartheta^{e})$ and $A^{\omega}(\vartheta^{m})$
respectively, as

(3.16) $\rho=i[{\bf B}^{e},{\bf A}^{m}]+i[{\bf E}^{m},{\bf A}^{e}];$

and the magnetic current density and the electric current density generated by
$A^{\omega}(\vartheta^{e})$ and $A^{\omega}(\vartheta^{m})$ respectively, as

(3.17) $-{\bf j}=i[\phi^{m},{\bf B}^{e}]+i[\phi^{e},{\bf E}^{m}]-i({\bf
E}^{e}\times{\bf A}^{m}+{\bf A}^{m}\times{\bf E}^{e}+{\bf B}^{m}\times{\bf
A}^{e}+{\bf A}^{e}\times{\bf B}^{m}).$

The reader can comparison the set of Equations 3.12–3.15 with their classical
counterparts: Equations A.8, A.9.

At this point there are a couple of comments to do about the last equations.
First of all, it is important to notice that Equation 3.12 tells the existence
of magnetic charges! for the non–commutative electric potential $1$–form;
while Equation 3.14 tells the existence of electric charges for the
non–commutative magnetic potential $1$–form. Like in the classical case, all
of this is a consequence of the geometry of our spaces. Second, $\\{{\bf
H}^{e},{\bf D}^{m}\\}$ and $\\{\\{{\bf D}^{e},{\bf H}^{e}\\},\\{{\bf
D}^{m},{\bf H}^{m}\\}\\}$ satisfied the same equalities! and they depend on a
non–trivial way of both non–commutative potentials and the classical fields:
there are electric/magnetic charges and electric/magnetic current densities
due to self–interaction in the vacuum!.

###  3.2. Non–Commutative Dynamical Equations

In accordance with [19], a qpc $\omega$ is a Yang–Mills qpc, i.e., a critical
point of the non–commutative Yang–Mills functional (a functional that measures
the square norm of the curvature of a qpc) if and only if

(3.18)
$\langle\Upsilon_{\mathrm{ad}}\circ\lambda\,|\,(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})R^{\omega}\rangle_{\mathrm{L}}+\langle\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{\lambda}\,|\,(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}\rangle_{\mathrm{R}}=0$

for all $\lambda$ $\in$ $\overrightarrow{\mathfrak{qpc}(\zeta)}$, where
$d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}$ is the formal adjoint
operator of the exterior derivative of the induced quantum linear connection
on the left associated quantum vector bundle of $\mathrm{ad}$ (just like in
the classical case); while $d^{S^{\omega}\star_{\mathrm{L}}}$ is the formal
adjoint operator of $\Upsilon_{\mathrm{ad}}\circ
S^{\omega}\circ\Upsilon^{-1}_{\mathrm{ad}}$ [19]. The operators
$d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}$ and
$d^{S^{\omega}\star_{\mathrm{R}}}$ are defined in an analogous way for the
right structures. Here we are considering that
$(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})R^{\omega}:=(d^{\nabla^{\omega}_{\mathrm{ad}}\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})\circ\Upsilon_{\mathrm{ad}}\circ
R^{\omega}$,
$(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\widehat{R}^{\omega}:=(d^{\widehat{\nabla}^{\omega}_{\mathrm{ad}}\star_{\mathrm{R}}}-d^{\widehat{S}^{\omega}\star_{\mathrm{R}}})\circ\widetilde{\Upsilon}_{\mathrm{ad}}\circ\widehat{R}^{\omega}$
and $\widehat{\lambda}=\ast\circ\lambda\circ\ast$,
$\widehat{R}^{\omega}=\ast\circ R^{\omega}\circ\ast$.

####  3.2.1. The Non–Standard $1D$–Differential Calculus

For our quantum bundle, Equation 3.18 is reduced into

(3.19)
$(d^{\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})F^{\omega}=0,$

where $d^{\star_{\mathrm{L}}}$ is the left quantum codifferential (see
Equation 2.11) and $d^{S^{\omega}\star_{\mathrm{L}}}$ is the formal adjoint
operator of $d^{S^{\omega}}$ with respect to $\langle-|-\rangle_{\mathrm{L}}$.
In concrete

(3.20)
$d^{S^{\omega}\star_{\mathrm{L}}}\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k}(M)$

is given by

$d^{S^{\omega}\star_{\mathrm{L}}}\tau(\vartheta)=(-1)^{k}\,i\,\star^{-1}_{\mathrm{L}}\left([A^{\omega}(\vartheta),\star_{\mathrm{L}}\tau(\vartheta)]^{\partial}\right),$

for all linear maps
$\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k+1}(M)$. The reader can
comparison Equation 3.19 with Equation A.4. It is worth mentioning that like
in the classical case, not all qpc satisfies Equation 3.19.

In the light of Equation 2.37, Equation 3.19 becomes into the non–commutative
Gauss Law and the non–commutative Ampere equation

(3.21) $\nabla\cdot{\bf D}=\rho^{e},$ (3.22) $\nabla\times{\bf
H}-\frac{\partial{\bf D}}{\partial x^{0}}={\bf j}^{e}.$

Here we have defined the electric charge density and the electric current
density by

(3.23) $\rho^{e}=i[{\bf D},{\bf A}^{\ast}],$

and

(3.24) ${\bf j}^{e}=i[{\bf D},\phi^{\ast}]-i({\bf H}\times{\bf A}^{\ast}+{\bf
A}^{\ast}\times{\bf H}),$

where ${\bf A}^{\ast}=(A^{\ast}_{1},A^{\ast}_{2},A^{\ast}_{3})$.

In summary, Equations 3.5–3.6 and 3.21–3.22 are the non–commutative Maxwell
equations in the vacuum associated to the non–commutative gauge potential
$A^{\omega}$ in the quantum space–time $M$.

It is work remarking again that even in the vacuum there are electric/magnetic
charges as well as magnetic/electric current densities, which are generated by
non–trivial self–interactions on $M$.

Finally, it is possible to proceed exactly like in the classical case to find
conservation laws for all cases

(3.25) $\nabla\cdot{\bf j}^{e}+\frac{\partial\rho^{e}}{\partial
x^{0}}=0,\qquad\nabla\cdot{\bf j}^{m}+\frac{\partial\rho^{m}}{\partial
x^{0}}=0.$

####  3.2.2. The $2D$–Differential Calculus

For our quantum bundle, Equation 3.18 is reduced into

(3.26)
$(d^{\star_{\mathrm{L}}}-d^{S^{\omega}\star_{\mathrm{L}}})F^{\omega}=0,$

where $d^{\star_{\mathrm{L}}}$ is the left quantum codifferential (see
Equation 2.11) and $d^{S^{\omega}\star_{\mathrm{L}}}$ is the formal adjoint
operator of $d^{S^{\omega}}$ with respect to $\langle-|-\rangle_{\mathrm{L}}$.
In concrete

(3.27)
$d^{S^{\omega}\star_{\mathrm{L}}}\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k}(M)$

is given by

$d^{S^{\omega}\star_{\mathrm{L}}}\tau(\vartheta^{e})=(-1)^{k}\,i\,\star^{-1}_{\mathrm{L}}\left([A^{\omega}(\vartheta^{m}),\star_{\mathrm{L}}\tau(\vartheta^{m})]^{\partial}+[A^{\omega}(\vartheta^{m}),\star_{\mathrm{L}}\tau(\vartheta^{e})]^{\partial}\right),$
$d^{S^{\omega}\star_{\mathrm{L}}}\tau(\vartheta^{m})=(-1)^{k}\,i\,\star^{-1}_{\mathrm{L}}\left([A^{\omega}(\vartheta^{e}),\star_{\mathrm{L}}\tau(\vartheta^{m})]^{\partial}+[A^{\omega}(\vartheta^{e}),\star_{\mathrm{L}}\tau(\vartheta^{e})]^{\partial}\right),$

for all linear maps
$\tau:{{}_{\mathrm{inv}}}\Gamma\longrightarrow\Omega^{k+1}(M)$. The reader can
comparison Equation 3.12 with Equation A.4. It is worth mentioning that like
in the classical case, not all qpc satisfies Equation 3.12.

In the light of Equation 2.28, Equation 3.8 becomes into the non–commutative
Gauss Law and the non–commutative Ampere equation for
$A^{\omega}(\vartheta^{e})$

(3.28) $\nabla\cdot{\bf D}^{e}=\rho^{e},$ (3.29) $\nabla\times{\bf
H}^{e}-\frac{\partial{\bf D}^{e}}{\partial x^{0}}={\bf j}^{e}$

and for $A^{\omega}(\vartheta^{m})$

(3.30) $\nabla\cdot{\bf H}^{m}=\rho^{m},$ (3.31) $\nabla\times{\bf
D}^{m}-\frac{\partial{\bf H}^{m}}{\partial x^{0}}={\bf j}^{m},$

where ${\bf A}^{e\ast}=(A^{e\ast}_{1},A^{e\ast}_{2},A^{e\ast}_{3})$, ${\bf
A}^{m\ast}=(A^{m\ast}_{1},A^{m\ast}_{2},A^{m\ast}_{3})$. Here we have defined
the electric and magnetic charge density generated by
$A^{\omega}(\vartheta^{e})$ and $A^{\omega}(\vartheta^{m})$, respectively

(3.32) $\rho^{e}=i[{\bf D}^{e}+{\bf H}^{m},{\bf
A}^{m\ast}],\qquad\rho^{m}=i[{\bf D}^{e}+{\bf H}^{m},{\bf A}^{e\ast}],$

and the electric current density and the magnetic current density generated by
$A^{\omega}(\vartheta^{e})$ and $A^{\omega}(\vartheta^{m})$, respectively

(3.33) ${\bf j}^{e}=i[{\bf D}^{e}+{\bf H}^{m},\phi^{m\ast}]-i(({\bf
H}^{e}+{\bf D}^{m})\times{\bf A}^{m\ast}+{\bf A}^{m\ast}\times({\bf
H}^{e}+{\bf D}^{m})),$ (3.34) ${\bf j}^{m}=i[{\bf D}^{e}+{\bf
H}^{m},\phi^{e\ast}]-i(({\bf H}^{e}+{\bf D}^{m})\times{\bf A}^{e\ast}+{\bf
A}^{e\ast}\times({\bf H}^{e}+{\bf D}^{m})).$

The reader can comparison Equations 3.14–3.17 with their classical
counterparts: Equations A.10, A.11.

It is work remarking again that even in the vacuum there are electric/magnetic
charges as well as magnetic/electric current densities, which are generated by
non–trivial self–interactions on $M$.

In summary, Equations 3.12–3.15 and 3.28–3.31 are the non–commutative Maxwell
equations in the vacuum associated to the non–commutative gauge potential
$A^{\omega}$ in the quantum space–time $M$: there are $4$ for the
non–commutative electric potential $1$–form $A^{\omega}(\vartheta^{e})$, and
other $4$ for the non–commutative magnetic potential $1$–form
$A^{\omega}(\vartheta^{m})$. Since now we have $4$ new Maxwell equations and
all of them are coupled, it is necessary to present a non–trivial solution; in
particular, we are going to present a solution for which the non–commutative
electric potential $1$–form produces a non–zero magnetic charge density (and
hence, the non–commutative magnetic potential $1$–form produces a non–zero
electric charge density).

In fact, by taking $\theta^{\mu\nu}=0$ for all $\mu$, $\nu$ except
$\theta^{23}$; let us consider the non–commutative gauge potential

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

defined by

$A^{\omega}(\vartheta^{e})=x^{3}\,dx^{2}$
$A^{\omega}(\vartheta^{m})=x^{1}x^{2}\,dx^{3}.$

In this way

$E^{e}=(0,0,0),\quad B^{e}=(1,0,0),$ $D^{e}=(0,0,0),\quad
H^{e}=(1+\theta^{23}x^{1},0,0),$

and

$B^{m}=(0,0,0),\quad E^{m}=(-x^{1},x^{2},0),$ $H^{m}=(0,0,0),\quad
D^{m}=(-(1-\theta^{23})x^{1},x^{2},0).$

Thus the non–commutative Maxwell equations are satisfied:

$\nabla\cdot{\bf H}^{e}=\theta^{23},\qquad\nabla\cdot{\bf D}^{e}=0,$
$\nabla\times{\bf D}^{e}+\frac{\partial{\bf H}^{e}}{\partial
x^{0}}=0,\qquad\nabla\times{\bf H}^{e}-\frac{\partial{\bf D}^{e}}{\partial
x^{0}}=0;$

and

$\nabla\cdot{\bf D}^{m}=\theta^{23},\qquad\nabla\cdot{\bf H}^{m}=0,$
$\nabla\times{\bf H}^{m}+\frac{\partial{\bf D}^{m}}{\partial
x^{0}}=0,\qquad\nabla\times{\bf D}^{m}-\frac{\partial{\bf H}^{m}}{\partial
x^{0}}=0.$

This solution is consistence with the zero slope limit of string theory:
$\theta^{0j}=0$ for $j=1,2,3$ [25].

######  Remark 3.1.

The general theory in which we are based on ([18], [19]), works with qpcs in
the most general way. However, there is a special kind of qpcs called real
qpcs which satisfy

$\omega(\theta^{\ast})=\omega(\theta)^{\ast}$

for all $\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$. This kind of qpcs are the
only ones that other papers consider, for example [14, 15, 17] and in the
classical case, only real principal connections have physical meaning; so we
could think that only real qpcs have physical meaning as well.

For our quantum bundle this condition becomes into

$A^{\omega}(\theta^{\ast})=A^{\omega}(\theta)^{\ast}$

for all $\theta$ $\in$ ${{}_{\mathrm{inv}}}\Gamma$ and due to Equation 2.26,
real qpcs fulfill

(3.36) $A^{\omega}(\vartheta^{e})^{\ast}=A^{\omega}(\vartheta^{e}),\qquad
A^{\omega}(\vartheta^{m})^{\ast}=A^{\omega}(\vartheta^{m}).$

It is worth mentioning that the explicit solution presented above comes from a
real qpc and the charges densities are real constants. Of course, there are
more solutions, this was only an example.

Finally, it is possible to proceed exactly like in the classical case to find
conservation laws for all cases

(3.37) $\nabla\cdot{\bf j}^{e}+\frac{\partial\rho^{e}}{\partial
x^{0}}=0,\qquad\nabla\cdot{\bf j}^{m}+\frac{\partial\rho^{m}}{\partial
x^{0}}=0,\qquad\nabla\cdot{\bf j}+\frac{\partial\rho}{\partial x^{0}}=0.$

###  3.3. Covariant Formulation

In this section we will pass to the physics common index notation with the
metric $\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$.

####  3.3.1. The Non–Standard $1D$–Differential Calculus

Let us consider

(3.38) $A_{\mu}=(\phi,-{\bf A})$

and then Equation 2.39 is given by

(3.39)
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}].$

The geometrical equation (Equation 3.4) can be written as

(3.40) $\partial_{\mu}\widetilde{F}^{\mu\nu}=J^{m\,\nu}\qquad\mbox{ where
}\qquad J^{m\,\nu}=(\rho^{m},{\bf
j}^{m})=i[A_{\mu},\widetilde{F}^{\mu\nu}]=i[A_{\mu},\widetilde{F}^{\mu\nu}_{\mathrm{classical}}].$

Here, we are consider that

(3.41) $\widetilde{F}^{\mu\nu}={1\over
2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta},$

is the dual electromagnetic tensor field with $\epsilon^{\mu\nu\alpha\beta}$
the Levi–Civita symbol, and $\widetilde{F}^{\mu\nu}_{\mathrm{classical}}$ is
the classical part of $\widetilde{F}^{\mu\nu}$, i.e., the part of
$\widetilde{F}^{\mu\nu}$ only with ${\bf E}$ and ${\bf B}$. The reader can
comparison Equation 3.40 with is classical counterpart, Equation $A.14$.

On the other hand, the dynamical equation (Equation 3.9) can be written as

(3.42) $\partial_{\mu}F^{\mu\nu}=J^{e\,\nu}\qquad\mbox{ where }\qquad
J^{e\,\nu}=(\rho^{e},{\bf j}^{e})=i[A^{\ast}_{\mu},F^{\mu\nu}]$

with $A^{\ast}_{\mu}=(\phi^{\ast},-{\bf A}^{\ast})$. The reader can comparison
the last equation with its classical counterpart, Equation $A.15$.

Notice that the form of both four–currents are like in classical non–abelian
gauge theories.

######  Remark 3.2.

In Equation 3.39, the term

$i[A_{\mu},A_{\nu}]$

comes from the general definition of the curvature for this quantum bundle
with this specific differential calculus. In other papers, like [11, 8], this
term comes form the non–commutativity of $M$ in the definition of the gauge
curvature

$i[D_{\mu},D_{\nu}].$

######  Remark 3.3.

If we consider the algebra of differential forms of $U(1)$, we would get

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$

even in $M$! because of in this case the only embedded differential is
$\delta=0$ and hence the curvature is given by $F^{\omega}=dA^{\omega}$ (see
Equation 2.36). The presence of $\delta$ in the definition of the curvature is
needed to see the curvature as a gauge field in the general theory [14, 15,
17].

So, there is discrepancy between the gauge curvature and the definition of the
curvature of a principal connection in the non–commutative case. This
discrepancy is only solved when we consider the non–standard $1D$ differential
calculus of $U(1)$, so this calculus seems like the correct mathematical
environment in the non–commutative context. In other words, in papers like [8,
11] the authors were working with the non–standard differential calculus
without knowing and hence, with a non–zero $S^{\omega}$.

As another reason to consider our formulation is [10]. In this paper the
authors take a variation on $A_{\mu}$ and get the Equation 3.42 for real qpcs
(see Equation 3.36). We arrived to the same result in [19] but in a general
context: varying the qpc on the non–commutative Yang–Mills functional.
Furthermore, Equation 3.40 can be obtained by considering the Jacobi identity
on $M$

$0={1\over 2}\epsilon^{\mu\nu\alpha\beta}[D_{\nu},[D_{\alpha},D_{\beta}]].$

All of these show that our mathematical formulation is the correct one in the
non–commutative case.

By the last Remark, the correct Yang–Mills Langrangian density of the system
is

(3.43) $\mathcal{L}_{\mathrm{YM}}=-{1\over 2}\left(F^{\mu\nu}\cdot
F^{\ast}_{\mu\nu}\right)-A_{\mu}\cdot J^{e\,\ast\,\mu}-J^{e\;\mu}\cdot
A^{\ast}_{\mu}$

and of course, the Euler–Lagrange equations of the last Lagrangian reduces to
Equation 3.42.

In accordance with [10], twisted gauge transformations are symmetries of the
last Lagrangian. Furthermore, the coviarant formulation allows to identify
that our Equations are covariant under the correct set of Lorentz
transformations, for example, the ones who transform the matrix
$(\theta^{\mu\nu})$ correctly [26].

On the other hand, solutions in terms of the four–potential in Lorentz gauge

(3.44) $\partial_{\mu}A^{\mu}=0$

are given by

(3.45)
$\partial_{\mu}\partial^{\mu}A^{\nu}=i[A_{\mu},\widetilde{F}^{\mu\nu}_{\mathrm{classical}}]+i[A^{\mu},\partial_{\mu}A^{\nu}];$

while the conservation laws are

(3.46) $\partial_{\mu}J^{e\,\mu}=0,\qquad\partial_{\mu}J^{m\,\mu}=0.$

######  Remark 3.4.

The whole classical Electromagnetic Theory (Appendix $A$) can be recovered by
considering

$\theta^{\mu\nu}\longrightarrow 0,$

even with the non–standard $1D$ differential calculus of $U(1)$ because by the
graded commutative of $\Omega^{\bullet}(M)$ we would have
$F^{\omega}=dA^{\omega}$ (see Equation 2.36)

Finally, it is worth mentioning that there is no a complete symmetry between
${\bf D}$, ${\bf H}$ in the sense that the electric four–current depends on
the configuration of the four–potential; while the magnetic four–current does
not: this four–current exists only by the presence of the electromagnetic
field in the quantum space–time $M$. Furthermore, when we quantize the model
(next section), there will be only one gauge field: $A_{\mu}$, which is
typically identifying with the photon field. In other words, the photon field
in the non–commutative case turns into a kind of dyon gauge field in the sense
that it produces electric and magnetic charges and currents by itself in the
vacuum.

####  3.3.2. The $2D$–Differential Calculus

Following the last subsection we get

(3.47) $A^{e}_{\mu}=(\phi^{e},-{\bf A^{e}}),\qquad A^{m}_{\mu}=(\phi^{m},-{\bf
A^{m}}).$

Thus, the non–commutative electromagnetic field is given by

(3.48)
$F^{e}_{\mu\nu}=\partial_{\mu}A^{e}_{\nu}-\partial_{\nu}A^{e}_{\mu}-i[A^{e}_{\mu},A^{m}_{\nu}]-i[A^{m}_{\mu},A^{e}_{\nu}];$

while the non–commutative magnetoelectric field is

(3.49)
$F^{m}_{\mu\nu}=\partial_{\mu}A^{m}_{\nu}-\partial_{\nu}A^{m}_{\mu}-i[A^{e}_{\mu},A^{m}_{\nu}]-i[A^{m}_{\mu},A^{e}_{\nu}].$

The non–commutative geometrical equations (Equation 3.11) can be expressed as

(3.50)
$\partial_{\mu}\widetilde{F}^{e\,\mu\nu}=\partial_{\mu}\widetilde{F}^{m\,\nu\lambda}=J^{\nu},$

where

(3.51) $\widetilde{F}^{e\,\mu\nu}={1\over
2}\epsilon^{\mu\nu\alpha\beta}F^{e}_{\alpha\beta},\qquad\widetilde{F}^{m\,\mu\nu}={1\over
2}\epsilon^{\mu\nu\alpha\beta}F^{m}_{\alpha\beta},$

and

$J^{\nu}=(\rho,{\bf j}).$

Notice that

(3.52)
$J^{\nu}=i[A^{e}_{\mu},\widetilde{F}^{m\,\mu\nu}_{\mathrm{classical}}]+i[A^{m}_{\mu},\widetilde{F}^{e\,\mu\nu}_{\mathrm{classical}}],$

where $\widetilde{F}^{m\,\mu\nu}_{\mathrm{classical}}$,
$\widetilde{F}^{e\,\mu\nu}_{\mathrm{classical}}$ are the classical parts of
the tensors $\widetilde{F}^{m\,\mu\nu}$, $\widetilde{F}^{e\,\mu\nu}$. The
reader can comparison Equation 3.50 with its classical counterpart, Equation
A.14 and it is worth remembering that Equation 3.50 is always satisfied, no
matter the values of $A^{e}_{\mu}$, $A^{m}_{\mu}$.

On the other hand, the non–commutative dynamical equation (Equation 3.26) are

(3.53) $\partial_{\mu}F^{e\;\mu\nu}=J^{e\;\nu},$ (3.54)
$\partial_{\mu}F^{m\;\mu\nu}=J^{m\;\nu},$

where

$J^{e\;\nu}=(\rho^{e},{\bf j}^{e})\quad\mbox{ and }\quad
J^{m\;\nu}=(\rho^{m},{\bf j}^{m}).$

Notice that

(3.55) $J^{e\;\nu}=i[A^{m\,\ast}_{\mu},F^{e\;\mu\nu}+F^{m\;\mu\nu}],\qquad
J^{m\;\nu}=i[A^{e\,\ast}_{\mu},F^{e\;\mu\nu}+F^{m\;\mu\nu}],$

where $A^{e\,\ast}_{\mu}=(\phi^{e\ast},-{\bf A^{e\ast}})$,
$A^{m\,\ast}_{\mu}=(\phi^{m\ast},-{\bf A^{m\ast}})$. The reader can comparison
Equations 3.52 and 3.53 with their classical counterparts, Equation A.15.

On the other hand, conservation laws are given by

(3.56)
$\partial_{\mu}J^{e\;\mu}=0,\qquad\partial_{\mu}J^{m\;\mu}=0,\qquad\partial_{\mu}J^{\mu}=0.$

Solutions in terms of potentials, in the Lorentz gauge

$\partial_{\mu}A^{e\,\mu}=0,\qquad\partial_{\mu}A^{m\,\mu}=0$

are given by

(3.57)
$\partial_{\mu}\partial^{\mu}A^{e\,\nu}=i[A^{m\,\ast}_{\mu},F^{e\;\mu\nu}+F^{m\;\mu\nu}]+i[A^{e\,\mu},\partial_{\mu}A^{m\,\nu}]+i[A^{m\,\mu},\partial_{\mu}A^{e\,\nu}],$
(3.58)
$\partial_{\mu}\partial^{\mu}A^{m\,\nu}=i[A^{e\,\ast}_{\mu},F^{e\;\mu\nu}+F^{m\;\mu\nu}]+i[A^{e\,\mu},\partial_{\mu}A^{m\,\nu}]+i[A^{m\,\mu},\partial_{\mu}A^{e\,\nu}].$

Now we are going to analyze non–commutative symmetries. First of all it is
worth mentioning that non–commutative Maxwell equations are invariant under
the following transformation

(3.59) $A^{e}_{\mu}\;\longleftrightarrow A^{m}_{\mu}.$

In other words

${\bf D}^{e}\longrightarrow{\bf H}^{m},\;\;{\bf E}^{e}\longrightarrow{\bf
B}^{m},\;\;\qquad{\bf H}^{m}\longrightarrow{\bf D}^{e},\;\;{\bf
B}^{m}\longrightarrow{\bf E}^{e}$ ${\bf H}^{e}\longrightarrow{\bf
D}^{m},\;\;{\bf B}^{e}\longrightarrow{\bf E}^{m},\;\;\qquad{\bf
D}^{m}\longrightarrow{\bf H}^{e},\;\;{\bf E}^{m}\longrightarrow{\bf B}^{e},$
$\rho^{e}\longrightarrow\rho^{m},\;\;{\bf J}^{e}\longrightarrow{\bf
J}^{m},\;\;\qquad\;\;\rho^{m}\longrightarrow\rho^{e},\;\;{\bf
J}^{m}\longrightarrow{\bf J}^{e}.$

This symmetry produces a kind of non–commutative dual transformations and it
comes from the interchanging of the order basis
$\beta=\\{\vartheta^{e},\vartheta^{m}\\}\,\longleftrightarrow\,-\beta=\\{\vartheta^{m},\vartheta^{e}\\}$;
so it is a $\mathbb{Z}_{2}$–symmetry (see subsection $2.2$). In other words,
there is full symmetry between the electric field and the magnetic field in
non–commutative Maxwell equations.

We define the Yang–Mills Lagrangian density of the system as

(3.60) $\mathcal{L}_{\mathrm{YM}}=\mathcal{L}^{e}+\mathcal{L}^{m},$

where

(3.61) $\mathcal{L}^{e}=-{1\over 2}\left(F^{e\;\mu\nu}\cdot
F^{e\,\ast}_{\mu\nu}\right)-A^{e}_{\mu}\cdot J^{e\,\ast\,\mu}-J^{e\;\mu}\cdot
A^{e\,\ast}_{\mu}$

and

(3.62) $\mathcal{L}^{m}={1\over 2}\left(F^{m\;\mu\nu}\cdot
F^{m\,\ast}_{\mu\nu}\right)+A^{m}_{\mu}\cdot J^{m\,\ast\,\mu}+J^{m\;\mu}\cdot
A^{m\,\ast}_{\mu}.$

Of course, the Euler–Lagrange equations for $A^{e}_{\mu}$ and $A^{m}_{\mu}$
(or $A^{e\,\ast}_{\mu}$ and $A^{m\,\ast}_{\mu}$) reduce to Equations 3.53,
3.54. The minus sign in $\mathcal{L}^{m}$ with respect to $\mathcal{L}^{e}$
breaks the symmetry between the electric field and the magnetic field,
however, Euler–Lagrange are still the correct ones and

######  Proposition 3.5.

Let $\omega$ be a real qpc (see Equation 3.36). Then the Yang–Mills action

(3.63)
${\mathcal{S}}_{\mathrm{YM}}:=\int_{M}{\mathcal{L}}_{\mathrm{YM}}\,\mathrm{dvol}$

posses the classical gauge transformation symmetry (see Equation A.16), i.e.,
${\mathcal{S}}_{\mathrm{YM}}$ is invariant under the transformation

(3.64)
$A^{e}_{\mu}\longrightarrow{A^{e}_{\mu}}^{\prime}:=A^{e}_{\mu}+\partial_{\mu}\chi,\qquad
A^{m}_{\mu}\longrightarrow{A^{m}_{\mu}}^{\prime}:=A^{m}_{\mu}+\partial_{\mu}\chi$

for any real–valued $\chi$ $\in$ $M$.

The proof of the previous proposition is only a direct and large calculation
taking into account that under Equation 3.64

${F^{e}_{\mu\nu}}^{\prime}=\partial_{\mu}{A^{e}_{\nu}}^{\prime}-\partial_{\nu}{A^{e}_{\mu}}^{\prime}-i[{A^{e}_{\mu}}^{\prime},{A^{m}_{\nu}}^{\prime}]-i[{A^{m}_{\mu}}^{\prime},{A^{e}_{\nu}}^{\prime}],$
${F^{m}_{\mu\nu}}^{\prime}=\partial_{\mu}{A^{m}_{\nu}}^{\prime}-\partial_{\nu}{A^{m}_{\mu}}^{\prime}-i[{A^{e}_{\mu}}^{\prime},{A^{m}_{\nu}}^{\prime}]-i[{A^{m}_{\mu}}^{\prime},{A^{e}_{\nu}}^{\prime}]$

and the cyclic property of the integral; so we are going to omit it.

The previous covariant formulation allows to conclude that our equations are
covariant under the correct set of Lorentz transformations, for example, the
ones who transform the matrix $(\theta^{\mu\nu})$ correctly [26]. In
accordance with [10], twisted gauge transformations can leave the real action
${\mathcal{S}}_{\mathrm{YM}}$ invariant.

######  Remark 3.6.

In the light of Remark 3.3, only when $A^{e}_{\mu}$, $A^{e}_{\mu}$ could have
any physical sense is when the Yang–Mills action becomes invariant under the
classical gauge transformations of the electromagnetic theory.

######  Remark 3.7.

The whole classical Electromagnetic Theory (Appendix $A$) can be recovered by
considering

$\theta^{\mu\nu}\longrightarrow 0\qquad\mbox{ and }\qquad
A^{m}_{\mu}\longrightarrow 0.$

##  4\. Quantization

####  4.0.1. The Non–Standard $1D$–Differential Calculus

####  4.0.2. The $2D$–Differential Calculus

##  5\. Concluding Comments

In Differential Geometry the most general framework of the Electromagnetic
Theory in the vacuum is given by the theory of principal bundles and principal
connections; so it should be natural to think that in the non–commutative case
the correct mathematical framework of the Electromagnetic Theory is given by
the theory of quantum principal bundles and quantum principal connections, and
showing this is exactly the purpose of this paper.

As the reader should had already notice throughout the entire text, specially
by Remark 3.3, the general theory of quantum principal bundles reproduces
previous results of the non–commutative $U(1)$–gauge theory as well as clarify
the correct equation of motion and the correct geometrical equation.

On the other hand, our toy model with a $2D$–dimensional differential calculus
of $U(1)$ presents the correct mathematical formulation of the Cabibbo–Ferrari
idea because is the geometry of the spaces who is telling us how both
four–potentials are related, not us by imposing it. By the previous fact, this
toy model is a perfect example of the powerful tool that is using quantum
principal bundles to describe non–commutative gauge theory [18, 19]. This
should be not a big surprise because of the intrinsic fruitful relationship
between principal bundles and gauge theory in the classical case

All of these are reasons to keep the research going. Particularly, it is
possible to add matter fields in our framework. In addition, all of theses
motive the research to develop a non–commutative $SU(2)$–gauge theory or a
non–commutative $SU(3)$–gauge theory. Moreover, the general formulation
present in [18, 19] allows to work with other quantum space–time spaces, not
only with the Moyal–Weyl algebra [27].

The presence of the operator $S^{\omega}$ changes completely the mathematical
formulation of these kind of models and the equation of motions and it is
worth mentioning that this operator appears naturally in the general
non–commutative Bianchi identity ([15]) and it also appears naturally when you
varying the qpc in the non–commutative Yang–Mills functional ([19]), so it is
essential to consider it in non–commutative gauge theory, just like this paper
shows.

##  Appendix A Classical Maxwell Equations

Consider a principal $U(1)$–bundle over the Minkowski space–time:
$\mathbb{R}^{4}$ with the metric $\eta=\mathrm{diag}(1,-1,-1,-1)$.

Since $\mathbb{R}^{4}$ is contractible, every bundle over it is trivializable,
so without lose of generality, let us consider the trivial principal bundle

(A.1) $\mathrm{proy}:\mathbb{R}^{4}\times U(1)\longrightarrow\mathbb{R}^{4}.$

In this case, principal connections

$\omega:T(\mathbb{R}^{4}\times U(1))\longrightarrow\mathfrak{u}(1)$

are in biyection with globally defined $\mathfrak{u}(1)$–valued differential
$1$–forms: $i\,A^{\omega}$, where $i=\sqrt{-1}$ and

$A^{\omega}=\displaystyle\sum^{3}_{\mu=0}A_{\mu}\,dx^{\mu}\;\in\;\Omega^{1}(\mathbb{R}^{4})$

is usually called the potential 1–form; and the vector
$(A_{0},A_{1},A_{2},A_{3})$ created by the coefficients of $A^{\omega}$
receives the name of four–vector potential. For this case the curvature form
of $\omega$

$R^{\omega}:\Omega^{2}(\mathbb{R}^{4}\times
U(1))\longrightarrow\mathfrak{u}(1)$

is related with the potential $1$–form by means of the (de Rham) differential,
i.e., curvature forms are in biyection with

(A.2) $dA^{\omega}=:F^{\omega}\;\in\;\Omega^{2}(\mathbb{R}^{4}).$

In general every principal connection has to satisfies the (second) Bianchi
identity; however due to the fact that $U(1)$ is an abelian group, for our
case this identity reduces to ([1])

(A.3) $0=dF^{\omega}=d^{2}A^{\omega},$

which is a trivial relation taking into account the properties of $d$.
Nevertheless it is worth remarking again that the last equation arises from:
the relation between the curvature and the $1$–form potential, the Bianchi
identity and the commutative of $U(1)$; all of this in the context of the
graded–commutative de Rham algebra $\Omega^{\bullet}(\mathbb{R}^{4})$.

On the other hand, the Yang–Mills functional is a functional form the space of
principal connections to $\mathbb{R}$ which measures the squared norm of the
curvature of a principal connection and the Yang–Mills equation comes from a
variational principle in which we are looking for critical points. For our
case and again, by the relation between the curvature and the $1$–form
potential and the commutative of $U(1)$, the Yang–Mills equation is ([1])

(A.4) $0=d^{\star}F^{\omega}=d^{\star}dA^{\omega},$

where $d^{\star}=(-1)^{k}\star^{-1}\circ\,d\,\circ\star$ is the
codifferential, the formally adjoint222Whenever the integrals converge.
operator of $d$, and $\star$ is the star Hodge operator [1].

Now from Equations A.3, A.4 it is possible to obtain Maxwell equations [1]. In
fact, by choosing $(A_{0},A_{1},A_{2},A_{3})=(\phi,-{\bf A})$ with ${\bf A}$ a
vector field in $\mathbb{R}^{3}$, the Equation A.2 becomes into

(A.5) $F^{\omega}=\sum_{0\leq\mu<\nu\leq 3}F_{\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\quad\mathrm{where}\quad(F_{\mu\nu})=\begin{pmatrix}0&E_{1}&E_{2}&E_{3}\\\
-E_{1}&0&-B_{3}&B_{2}\\\ -E_{2}&B_{3}&0&-B_{1}\\\
-E_{3}&-B_{2}&B_{1}&0\end{pmatrix},$

where

(A.6) ${\bf E}=(E_{1},E_{2},E_{3}):=-\frac{\partial{\bf A}}{\partial
x^{0}}-\nabla\phi$

is the electric field,

(A.7) ${\bf B}=(B_{1},B_{2},B_{3}):=\nabla\times\bf A$

is the magnetic field and the $2$–form $(F_{\mu,\nu})$ is called the
electromagnetic field tensor. By substituting the value of $F^{\omega}$ in
Equation A.3 we get the Gauss Law for magnetism and the Faraday equation

(A.8) $\nabla\cdot{\bf B}=0,$ (A.9) $\nabla\times{\bf E}+\frac{\partial{\bf
B}}{\partial x^{0}}=0;$

while by substituting it in Equation A.4 we get the Gauss Law and the Ampere
equation

(A.10) $\nabla\cdot{\bf E}=0$ (A.11) $\nabla\times{\bf B}-\frac{\partial{\bf
E}}{\partial x^{0}}=0.$

Equations A.8–A.11 are the Maxwell equations in the vacuum [1]. $\phi$
receives the name of the electric potential or scalar potential; while ${\bf
A}$ receives the name of the magnetic potential or vector potential.

In index notation we have

(A.12) $A_{\mu}=(\phi,-{\bf A})$

and

(A.13) $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$

The geometrical equation can be written as

(A.14) $\partial_{\mu}\widetilde{F}^{\mu\nu}=0,$

and the dynamical equation can be written as

(A.15) $\partial_{\mu}F^{\mu\nu}=0,$

where

$F^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}F_{\mu\nu},\qquad\widetilde{F}^{\mu\nu}={1\over
2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta},$

with $\epsilon^{\mu\nu\alpha\beta}$ the Levi–Civita symbol and
$\eta_{\mu\nu}=\eta^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$. The tensor
$\widetilde{F}_{\mu\nu}$ receives the name of the dual electromagnetic field
tensor.

It is worth mentioning that Equation A.3 shows explicitly the gauge
transformation symmetry:

(A.16)
$A^{\omega}\longrightarrow{A^{\omega}}^{\prime}:=A^{\omega}+d\chi\quad\mbox{
or in index notation }\quad A^{\prime}_{\mu}:=A_{\mu}+\partial_{\mu}\chi,$

for any $f$ $\in$ $C^{\infty}(\mathbb{R}^{4})$.

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# Nonstable $K$–theory for extension algebras of the simple purely infinite
$C^{*}$–algebra by certain $C^{*}$–algebras

Zhihua Li ∗ and Yifeng Xue ∗∗

###### Abstract.

Let $0\longrightarrow\mathcal{B}\stackrel{{\scriptstyle
j}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathcal{A}\longrightarrow
0$ be an extension of $\mathcal{A}$ by $\mathcal{B}$, where $\mathcal{A}$ is a
unital simple purely infinite $C^{*}$–algebra. When $\mathcal{B}$ is a simple
separable essential ideal of the unital $C^{*}$–algebra $E$ with
$\mathrm{RR}(\mathcal{B})=0$ and (PC), $K_{0}(E)=\\{[p]\mid p$ is a projection
in $E\setminus B\\}$; When $B$ is a stable $C^{*}$–algebra,
$\mathfrak{U}(C(X,E))/\mathfrak{U}_{0}(C(X,E))\cong K_{1}(C(X,E))$ for any
compact Hausdorff space $X$.

Keywords $K$-groups; simple purely infinite $C^{*}$–algebra; real rank zero.

2000 MR Subject Classification 46L05.

Department of Mathematics and Computer Science, Yichun University, Yichun,
Jiangxi, 336000

Department of Mathematics, East China Normal University, Shanghai 200241.
email<EMAIL_ADDRESS>
Project supported by Natural Science Foundation of China (no.10771069) and
Shanghai Leading Academic Discipline Project(no.B407)

##  1\. Introduction

Let $\mathcal{E}$ be a $C^{*}$–algebra. Denote by
$\mathrm{M}_{n}(\mathcal{E})$ the $C^{*}$–algebra of all $n\times n$ matrices
over $\mathcal{E}$. If $\mathcal{E}$ is unital, write
$\mathfrak{U}(\mathcal{E})$ to denote the unitary group of $\mathcal{E}$ and
$\mathfrak{U}_{0}(\mathcal{E})$ to denote the connected component of the unit
in $\mathfrak{U}(\mathcal{E})$. Put
$U(\mathcal{E})=\mathfrak{U}(\mathcal{E})/\mathfrak{U}_{0}(\mathcal{E})$. If
$\mathcal{E}$ has no unit, we set
$U(\mathcal{E})=\mathfrak{U}(\mathcal{E}^{+})/\mathfrak{U}_{0}(\mathcal{E}^{+})$,
where $\mathcal{E}^{+}$ is the $C^{*}$–algebra obtained by adding a unit to
$\mathcal{E}$. Two projections $p,\ q$ in $\mathcal{E}$ are equivalent,
denoted $p\sim q$, if $p=v^{*}v,q=vv^{*}$ for some $v\in\mathcal{E}$. Let
$[p]$ denote the equivalence of $p$ with respect to “$\sim$”. Let $p,\ r$ be
projections in $\mathcal{E}$. $[p]\leq[r]$ (resp. $[p]<[r]$) means that there
is projection $q\leq r$ (resp. $q<r$) such that $p\sim q$. A projection $p$ in
$\mathcal{E}$ is called to be infinite, if $[p]<[p]$. The simple
$C^{*}$–algebra $\mathcal{E}$ is called to be purely infinite if every nonzero
hereditary subalgebra of $\mathcal{E}$ contains an infinite projection.

Let $K_{0}(\mathcal{E})$ and $K_{1}(\mathcal{E})$ be the $K$–groups of the
$C^{*}$–algebra $\mathcal{E}$ and let $i_{\mathcal{E}}\colon
U(\mathcal{E})\rightarrow K_{1}(\mathcal{E})$ be the canonical homomorphism
(cf. [1]).

The main tasks in non–stable $K$–theory are how to use the projection in
$\mathcal{E}$ to represent $K_{0}(\mathcal{E})$ and how to show
$i_{\mathcal{E}}$ is isomorphic. Cuntz showed in [2] that
$K_{0}(\mathcal{E})\cong\\{[p]|\,p\in\mathcal{E}\ \text{nonzero
projection}\\}$ and $i_{\mathcal{E}}$ is isomorphic, when $\mathcal{E}$ is a
simple unital purely infinite $C^{*}$–algebra. Rieffel and Xue proved that
under some restrictions of stable rank on the $C^{*}$–algebra $\mathcal{E}$,
$i_{\mathcal{E}}$ may be injective, surjective or isomorphic (cf. [6, 7],
[12]).

Let $\mathcal{B}$ be a closed ideal of a unital $C^{*}$–algebra $E$. Let
$\pi\colon E\rightarrow E/\mathcal{B}=\mathcal{A}$ be the quotient map. We
will use these symbols $E$, $\mathcal{B}$, $\mathcal{A}$ and $\pi$ throughout
the paper. Liu and Fang proved in [5] that

  1. (1)

$K_{0}(E)=\\{[p]|\,p\ \text{is a projection in}\ E\backslash\mathcal{B}\\}$
and

  2. (2)

$i_{E}\colon U(E)\rightarrow K_{1}(E)$ is isomorphic.

when $\mathcal{B}=\mathcal{K}$ (the algebra of compact operators on some
separable Hilbert space) and $\mathcal{A}$ is a unital simple purely infinite
$C^{*}$–algebra. Visinescu showed in [10] that the above results are also true
when $\mathcal{B}$ is purely infinite.

In this short note, we show that (1) is true when $\mathcal{B}$ is a separable
simple $C^{*}$–algebra with $\mathrm{RR}(\mathcal{B})=0$ and (PC) (see §2
below) and $\mathcal{A}$ is unital simple purely infinite; We also prove that
$i_{C(X,E)}$ is isomorphic for any compact Hausdorff space $X$ when
$\mathcal{B}$ is stable and $\mathcal{A}$ is unital simple purely infinite.

##  2\. $K_{0}$–group of the extension algebra

Let $\mathcal{E}$ be a $C^{*}$–algebra. $\mathcal{E}$ is of real rank zero,
denoted by $\mathrm{RR}(\mathcal{E})=0$, if every self–adjoint element in
$\mathcal{E}$ can be approximated by an self–adjoint element in $\mathcal{E}$
with finite spectra (cf. [3]). A non–unital, $\sigma$–unital $C^{*}$–algebra
$\mathcal{E}$ with $\mathrm{RR}(\mathcal{E})=0$ is said to have property (PC)
if it $\mathcal{E}$ has finitely many (densely defined) traces, say
$\\{\tau_{1},\cdots,\tau_{k}\\}$ such that following conditions are satisfied:

  1. (1)

there is an approximate unit $\\{e_{n}\\}$ of $\mathcal{E}$ consisting of
projections such that $\lim\limits_{n\to\infty}\tau_{i}(e_{n})=\infty$,
$i=1,\cdots,k;$

  2. (2)

for two projections $p,\,q\in\mathcal{E}$, if $\tau_{i}(p)<\tau_{i}(q)$,
$i=1,\cdots,k$, then $[p]\leq[q]$.

Obviously, stable simple AF–algebras with only finitely many extremal traces
have (PC) and $\mathcal{A}_{\theta}\otimes\mathcal{K}$ also has (PC), where
$\mathcal{A}_{\theta}$ is the irrational rotation algebra and $\mathcal{K}$ is
the algebra of compact operators on some complex separable Hilbert space.

######  Remark 2.1.

Let $\mathcal{E}$ be a non–unital, $\sigma$–unital $C^{*}$–algebra with
$\mathrm{RR}(\mathcal{E})=0$ and (PC). Let $\\{f_{n}\\}$ be an approximate
unit of $\mathcal{E}$ consisting of increased projections. Suppose
$\lim\limits_{n\to\infty}\tau_{i}(e_{n})=\infty$, $i=1,\cdots,k$, for some
approximate unit $\\{e_{n}\\}$ of $\mathcal{E}$ consisting of projections.
Then there $\\{e_{n_{j}}\\}\subset\\{e_{n}\\}$ such that
$\tau_{i}(e_{n_{j}})>j$, $j\geq 1$, $i=1,\cdots,k$. Since
$\lim\limits_{s\to\infty}\|f_{s}e_{n_{j}}f_{s}-e_{n_{j}}\|=0$, $j\geq 1$, we
can find projections $f_{s_{j}}\leq f_{s}$ for $s$ large enough such that
$f_{s_{j}}\sim e_{n_{j}}$, $j\geq 1$. Then

$\tau_{i}(f_{s})\geq\tau_{i}(f_{s_{j}})=\tau_{i}(e_{n_{j}})>j,\quad
i=1,\cdots,k,$

so that $\lim\limits_{n\to\infty}\tau_{i}(f_{n})=\infty$, $i=1,\cdots,k$.

With symbols as above, we can extend $\tau_{i}$ to $M(\mathcal{E})$ by
$\tau_{i}(x)=\sup\limits_{n\geq 1}\tau_{i}(f_{n}xf_{n})$ for positive element
$x\in M(\mathcal{E})$ (cf. [4, P324]), $i=1,\cdots,k$, where $M(\mathcal{E})$
is the multiplier algebra of $\mathcal{E}$.

######  Lemma 2.2.

Suppose that $\mathcal{B}$ is an essential ideal of $E$ and
$\mathcal{A},\,\mathcal{B}$ are simple. Then every positive element in
$E\backslash\mathcal{B}$ is full.

###### Proof.

Let $a\in E\backslash\mathcal{B}$ with $a\geq 0$ and let $I(a)$ be closed
ideal generated by $a$ in $E$. Since $\pi(I(a))$ is a nonzero closed ideal in
$\mathcal{A}$ and $\mathcal{A}$ is simple, we get that
$1_{\mathcal{A}}\in\pi(I(a))$ and hence there is $x\in\mathcal{B}$ such that
$1_{E}+x\in I(a)$. Since $\mathcal{B}$ is an essential ideal, it follows that
$a\mathcal{B}a\not=\\{0\\}$. Choose a nonzero element
$b\in\overline{a\mathcal{B}a}\subset I(a)$. Since $\mathcal{B}$ is simple, $x$
is in the closed ideal of $\mathcal{B}$ generated by $b$. Thus, $x\in I(a)$
and consequently, $1_{E}\in I(a)$. ∎

The following lemma slightly improves Lemma 2.1 of [10], whose proof is
essentially same as it in [11, Lemma 3.2] and [10, Lemma 2.1].

######  Lemma 2.3.

Suppose that $\mathrm{RR}(\mathcal{B})=0$. Let $p,\,q$ be projections in $E$
and assume that there is $v\in\mathcal{A}$ such that $\pi(p)=v^{*}v$ and
$vv^{*}\leq\pi(q)$ in $\mathcal{A}$. Then there is a projection $e\in
p\mathcal{B}p$ and a partial isometry $u\in E$ such that $p-e=u^{*}u$,
$uu^{*}\leq q$ and $\pi(u)=v$.

###### Proof.

Let $v\in\mathcal{A}$ such that $\pi(p)=v^{*}v,\ vv^{*}\leq\pi(q)$. Choose
$u_{0}\in E$ such that $\pi(u_{0})=v$ and set $w=qu_{0}p$. Then
$\pi(w^{*}w)=\pi(p),\ \pi(w)=v$. Thus, $p-w^{*}w\in p\mathcal{B}\,p$. Since
$\mathrm{RR}(\mathcal{B})=0$, $p\mathcal{B}p$ has an approximate unit
consisting of projections. So there is a projection $e\in p\mathcal{B}p$ such
that

$\|(p-e)(p-w^{*}w)(p-e)\|=\|(p-e)-(p-e)w^{*}w(p-e)\|<1.$

Then $z=(p-e)w^{*}w(p-e)$ is invertible in $(p-e)E(p-e)$ and $\pi(z)=\pi(p)$.
Let $s=\big{(}(p-e)w^{*}w(p-e)\big{)}^{-1}$, i.e., $zs=sz=p-e$. Then
$\pi(s)=\pi(p)$. Put $u=ws^{\frac{1}{2}}$. Then $uu^{*}=wsw^{*}\leq q$,
$\pi(u)=v$ and

$\displaystyle u^{*}u=$ $\displaystyle
s^{\frac{1}{2}}w^{*}ws^{\frac{1}{2}}=s^{\frac{1}{2}}(p-e)w^{*}w(p-e)s^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle(p-e)w^{*}w(p-e)s=p-e.$

∎

######  Lemma 2.4.

Suppose that $\mathcal{A}$ is unital simple purely infinite and $\mathcal{B}$
is an essential ideal of a unital $C^{*}$–algebra $E$, moreover $\mathcal{B}$
is separable simple with $\mathrm{RR}(\mathcal{B})=0$ and (PC). Let $p,\,q$ be
projections in $E\backslash\mathcal{B}$ and let $r$ be a nonzero projection in
$p\mathcal{B}p$. Then there is a projection $r^{\prime}$ in $q\mathcal{B}q$
such that $[r]\leq[r^{\prime}]$.

###### Proof.

Since $\mathcal{B}$ has (PC), there are densely defined traces
$\tau_{1},\cdots,\tau_{k}$ on $\mathcal{B}$ and an approximate unit
$\\{f_{n}\\}$ of $\mathcal{B}$ consisting of increased projections such that
$\lim\limits_{n\to\infty}\tau_{i}(f_{n})=\infty$, $i=1,\cdots,k$ and
$\tau_{i}(e)<\tau_{i}(f)$, $i=1,\cdots,k$ implies that $[e]\leq[f]$ for any
two projections $e,\,f$ in $\mathcal{B}$.

By Lemma 2.2, there are $x_{1},\cdots,x_{m}\in\mathcal{B}$ such that
$\sum\limits^{m}_{i=1}x_{i}^{*}qx_{i}=1_{E}$. We regard $E$ as a
$C^{*}$–subalgebra of $M(\mathcal{B})$ for $\mathcal{B}$ is essential. Thus,

$\infty=\tau_{i}(1_{E})=\sum\limits^{m}_{j=1}\tau_{i}(x^{*}_{j}qx_{j})\leq\sum\limits^{m}_{j=1}\tau_{i}(\|x_{j}\|^{2}q),$

i.e., $\tau_{i}(q)=\infty$, $i=1,\cdots,k$. Let $r$ be a nonzero projection in
$p\mathcal{B}p$. Let $\\{g_{n}\\}$ be an approximate unit for $q\mathcal{B}q$
consisting of increased projections. Since $\sup\limits_{n\geq
1}\tau_{i}(g_{n})=\tau_{i}(q)=\infty$, $i=1,\cdots,k$, it follows that there
is $n_{0}$ such that $\tau_{i}(g_{n_{0}})>\tau_{i}(r)$, $i=1,\cdots,k$. Put
$r^{\prime}=g_{n_{0}}$. Then we get $[r]\leq[r^{\prime}]$. ∎

Now we can prove the main result of the section as follows:

######  Theorem 2.5.

Suppose that $\mathcal{A}$ is unital simple purely infinite and $\mathcal{B}$
is an essential ideal of $E$, moreover $\mathcal{B}$ is separable simple with
$\mathrm{RR}(\mathcal{B})=0$ and (PC). Then

$K_{0}(E)=\\{[p]|\,p\ \text{is a projection in}\ E\backslash\mathcal{B}\\}.$

###### Proof.

Set $\mathcal{P}(E)=\\{p\ \text{is a projection in}\
E\backslash\mathcal{B}\\}$. By [2, Theroem 1.4], when $\mathcal{P}(E)$
satisfies following conditions:

  1. $(\Pi_{1})$

If $p,\ q\in\mathcal{P}(E)$ and $pq=0$, then $p+q\in\mathcal{P}(E);$

  2. $(\Pi_{2})$

If $p\in\mathcal{P}(E)$ and $p^{\prime}$ is a projection in $E$ such that
$p\sim p^{\prime}$, then $p^{\prime}\in\mathcal{P}(E);$

  3. $(\Pi_{3})$

For any $p,q\in\mathcal{P}(E)$, there is $p^{\prime}$ such that
$p^{\prime}\sim p,\ p^{\prime}<q$ and $q-p^{\prime}\in\mathcal{P}(E);$

  4. $(\Pi_{4})$

If $q$ is a projection in $E$ and there is $p\in\mathcal{P}(E)$ such that
$p\leq q$, then $p\in\mathcal{P}(E)$,

then $K_{0}(E)=\\{[p]|\,p\in\mathcal{P}(E)\\}$. Therefore, we need only check
that $\mathcal{P}(E)$ satisfies above conditions.

Let $\mathcal{P}(\mathcal{A})$ be the set of all nonzero projections in
$\mathcal{A}$. By [2, Proposition 1.5], $\mathcal{P}(\mathcal{A})$ satisfies
$(\Pi_{1})\sim(\Pi_{4})$. Clearly, $\mathcal{P}(E)$ satisfies $(\Pi_{1})$,
$(\Pi_{2})$ and $(\Pi_{4})$. We now show that $\mathcal{P}(E)$ satisfies
$(\Pi_{3})$.

Let $p,\,q\in\mathcal{P}(E)$. Then there exists a projection
$f\in\mathcal{P}(\mathcal{A})$, such that $f\sim\pi(p)$, $f<\pi(q)$ and
$\pi(q)-f\in\mathcal{P}(\mathcal{A})$, that is, there is a partial isometry
$v\in\mathcal{A}$ such that $f=v{v}^{*}<\pi(q)$ and $\pi(p)={v}^{*}v$. Thus,
there are $u\in E$ and a projection $r\in p\mathcal{B}p$ such that
$p-r=u^{*}u$, $uu^{*}\leq q$ and $\pi(u)=v$ by Lemma 2.3. Note that
$q-uu^{*}\not\in\mathcal{B}$ and $(q-uu^{*})\mathcal{B}(q-uu^{*})\not=\\{0\\}$
($\mathcal{B}$ is an essential ideal). Then by Lemma 2.4, there is
$w_{0}\in\mathcal{B}$ such that $r=w_{0}^{*}w_{0}$,
$w_{0}w_{0}^{*}\in(q-uu^{*})\mathcal{B}(q-uu^{*})$. Put $\hat{u}=u+w_{0}$.
Then $p=\hat{u}^{*}\hat{u}$, $\hat{u}\hat{u}^{*}\leq q$ and
$\pi(q-\hat{u}\hat{u}^{*})=\pi(q)-f\not=0$, i.e.,
$q-\hat{u}\hat{u}^{*}\in\mathcal{P}(E)$. ∎

##  3\. $K_{1}$-group of the extension algebra

Recall from [12] that a unital $C^{*}$–algebra $\mathcal{E}$ has
$1$–cancellation, if a projection $p\in\mathrm{M}_{2}(\mathcal{E})$ satisfies
$\mathrm{diag}(p,1_{k})\sim\mathrm{diag}(p_{1},1_{k})$ for some $k$, then
$p\sim p_{1}$ in $\mathrm{M}_{2}(\mathcal{E})$, where
$p_{1}=\mathrm{diag}(1,0)$. If $\mathcal{E}$ has no unit and $\mathcal{E}^{+}$
has $1$–cancellation, we say $\mathcal{E}$ has $1$–cancellation. It is known
that when $\mathcal{B}$ has $1$–cancellation, we have following exact sequence
of groups:

$U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}K_{0}(\mathcal{B})$
(3.1)

(cf. [12, lemma 2.2]), where $j_{*}$ (resp. $\pi$) is the induced homomorphism
of the inclusion $j\colon\mathcal{B}\rightarrow E$ (resp. $\pi$) on
$U(\mathcal{B})$ (resp. $U(E)$), $\eta=\partial_{0}\circ i_{\mathcal{A}}$ and
$\partial_{0}\colon K_{1}(\mathcal{A})\rightarrow K_{0}(\mathcal{B})$ is the
index map.

Since, in general, we have the exact sequence of groups

$U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A}),$

(for $\pi(\mathfrak{U}_{0}(E))=\mathfrak{U}_{0}(\mathcal{A})$), i.e.,
$U(\cdot)$ is a half–exact and homotopic invariant functor, it follows from
Proposition 21.4.1, Corollary 21.4.2 and Theorem 24.4.3 of [1] that the
sequence of groups

$U(S\mathcal{A})\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A})$
(3.2)

is exact, where $\partial=e_{*}^{-1}\circ i_{*}$ and
$e\colon\mathcal{B}\rightarrow C_{\pi}$ given by $e(b)=(b,0)\in C_{\pi}$,
$e_{*}$ is isomorphic and $i\colon S\mathcal{A}\rightarrow C_{\pi}$ is defined
by $i(g)=(0,g)$, here

$C_{\pi}=\\{(x,f)\in E\oplus C_{0}([0,1),\mathcal{A})|\,\pi(x)=f(0)\\},\quad
S\mathcal{A}=C_{0}((0,1),\mathcal{A}).$

We also have the exact sequence

$K_{1}(S\mathcal{A})\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{1}(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}K_{1}(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}K_{1}(\mathcal{A}).$
(3.3)

######  Proposition 3.1.

Suppose that $i_{\mathcal{A}}$, $i_{\mathcal{B}}$ are isomorphic and
$i_{S\mathcal{A}}$ is surjective. Assume that $\mathcal{B}$ has
$1$–cancellation. Then $i_{E}$ is an isomorphism.

###### Proof.

Combining (3.1), (3.2) with (3.3), we have following diagram

$\begin{array}[]{ccccccccc}U(S\mathcal{A})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}&U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}&U(\mathcal{A})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}&K_{0}(\mathcal{B})\\\
\downarrow i_{S\mathcal{A}}&&\downarrow i_{\mathcal{B}}&\downarrow
i_{E}&\downarrow i_{\mathcal{A}}&\parallel\\\
K_{1}(S\mathcal{A})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&K_{1}(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}&K_{1}(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}&K_{1}(\mathcal{A})\stackrel{{\scriptstyle\partial_{0}}}{{\longrightarrow}}&K_{0}(\mathcal{B})\\\
\end{array},$ (3.4)

in which two rows are exact and

$\eta=\partial_{0}\circ i_{\mathcal{A}},\quad\pi_{*}\circ
i_{E}=i_{\mathcal{A}}\circ\pi_{*},\quad j_{*}\circ i_{\mathcal{B}}=i_{E}\circ
j_{*}.$

Since $e_{*}$ is isomorphic, it follows from the commutative diagram

$\begin{array}[]{ccccc}U(S\mathcal{A})&\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}&U(C_{\pi})\stackrel{{\scriptstyle
e_{*}}}{{\longleftarrow}}&U(\mathcal{B})\\\ \downarrow
i_{S\mathcal{A}}&&\downarrow i_{C_{\pi}}&\downarrow i_{\mathcal{B}}\\\
K_{1}(S\mathcal{A})&\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}&K_{1}(C_{\pi})\stackrel{{\scriptstyle
e_{*}}}{{\longleftarrow}}&K_{1}(\mathcal{B})\\\ \end{array}$

that $\partial\circ i_{S\mathcal{A}}=i_{\mathcal{B}}\circ\partial$. Thus,
(3.4) is a commutative diagram. Using the Five–Lemma to (3.4), we can obtain
the assertion. ∎

For a $C^{*}$–algebra $\mathcal{E}$, let $\mathrm{csr}(\mathcal{E})$ and
$\mathrm{gsr}(\mathcal{E})$ be the connected stable rank and general stable
rank of $\mathcal{E}$, respectively, defined in [6]. We summrize some
properties of these stable ranks as follows:

######  Lemma 3.2.

Let $\mathcal{E}$ be a $C^{*}$–algebra. Then

  1. (1)

$\mathrm{gsr}(\mathcal{E})\leq\mathrm{csr}(\mathcal{E})$ (cf. [6]);

  2. (2)

$\mathrm{csr}(\mathcal{E})\leq 2$ when $\mathcal{E}$ is a stable
$C^{*}$–algebra (cf. [9, Theorem 3.12]);

  3. (3)

$\mathcal{E}$ has $1$–cancellation if $\mathrm{gsr}(\mathcal{E})\leq 2$ (cf.
[12]);

  4. (4)

if $\mathrm{csr}(\mathcal{E})\leq 2$ and
$\mathrm{gsr}(C(\mathbf{S}^{1},\mathcal{E}))\leq 2$, then $i_{\mathcal{E}}$ is
isomorphic (cf. [7, Theorem 2.9] or [12, Corollary 2.2]).

Now we present the main result of this section as follows:

######  Theorem 3.3.

Assume that $\mathcal{A}$ is a unital simple purely infinite $C^{*}$–algebra
and $\mathcal{B}$ is a stable $C^{*}$–algebra. Let $X$ be a compact Hausdorff
space. Then $i_{C(X,E)}$ is an isomorphism.

###### Proof.

If $\mathcal{B}$ is stable, then so is $C(Y,\mathcal{B})$ for any compact
Hausdorff space $Y$. Thus,
$\mathrm{gsr}(C(\mathbf{S}^{1},C(X,\mathcal{B})))\leq 2$ and
$\mathrm{csr}(C(X,\mathcal{B}))\leq 2$ by Lemma 3.2 (1) and (2). So we get
that $i_{C(X,\mathcal{B})}$ is isomorphic by Lemma 3.2 (4).

Since $\mathcal{A}$ is unital simple purely infinite, it follows from [12,
Corollary 3.1] that $i_{C(X,\mathcal{A})}$ and $i_{SC(X,\mathcal{A})}$ are all
surjective. Now we prove $i_{C(X,\mathcal{A})}$ is injective by using some
methods appeared in [8].

Let $f\in\mathfrak{U}(C(X,\mathcal{A}))$ with $i_{C(X,\mathcal{A})}([f])=0$ in
$K_{1}(C(X,\mathcal{A}))$. Let $p$ be a non–trivial projection in
$\mathcal{A}$. Then there exists $g\in\mathfrak{U}(C(X,p\mathcal{A}p))$ such
that $f$ is homotopic to $g+1-p$ by [13, Lemma 2.7]. Thus, there is a
continuous path
$f_{t}\colon[0,1]\rightarrow\mathfrak{U}(\mathrm{M}_{n+1}(C(X,\mathcal{A})))$
such that $f_{0}=1_{n+1}$ and $f_{1}=\mathrm{diag}(g+1-p,1_{n})$ for some
$n\geq 2$. Since $\mathrm{M}_{n+1}(\mathcal{A})$ is purely infinite, we can
find a partial isometry $v=(v_{ij})\in\mathrm{M}_{n+1}(\mathcal{A})$ such that
$\mathrm{diag}(1-p,1_{n})=v^{*}v$, $vv^{*}\leq\mathrm{diag}(1-p,0)$.
Consequently, we get that

$v_{11}^{*}v_{11}=1-p,\ v^{*}_{1j}v_{1,j}=1,\ v^{*}_{1j}v_{1,i}=0,\ i\not=j,\
\sum^{n+1}_{i=1}v_{1i}v^{*}_{1i}\leq 1-p.$

Set $v_{1}=p+v_{11}$, $v_{i}=v_{1i}$, $i=2,\cdots n+2$. Then $v_{1},\cdots
v_{n+1}$ are isometries in $\mathcal{A}$ and $v_{i}^{*}v_{j}=0$, $i\not=j$,
$s=\sum\limits^{n+1}_{i=1}v_{i}v_{i}^{*}$ is a projection. Put

$w_{t}(x)=(v_{1},\cdots,v_{n+1})f_{t}(x)\begin{pmatrix}v_{1}^{*}\\\
v_{2}^{*}\\\ \vdots\\\ v^{*}_{n+1}\end{pmatrix}+1-s,\quad t\in[0,1],\ x\in X.$

It is easy to check that $w_{t}$ is a continuous path in
$\mathfrak{U}(\mathrm{M}_{n}(C(X,\mathcal{A})))$ with $w_{0}=1$ and
$w_{1}=g+1-p$. Thus, $i_{C(X,\mathcal{A})}$ is injective.

The final result follows from Proposition 3.1. ∎

Combining Theorem 3.3 with standard argument in Algebraic Topology, we can get

######  Corollary 3.4.

Let $\mathcal{A}$, $\mathcal{B}$ and $E$ be as in Theorem 3.3. Then

$\pi_{n}(\mathfrak{U}(E))=\begin{cases}K_{0}(E)&\ n\ \text{odd}\\\ K_{1}(E)&\
n\ \text{even}\end{cases}.$

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