A zero-curvature representation of electromagnetism and the conservation of
electric charge

G. Luchini1, V. B. Zaché1
1Departamento de Física, Universidade Federal do Espírito Santo (UFES),

CEP 29075-900, Vitória-ES, Brazil

###### Abstract

We show that the laws of electromagnetism in $(D+1)$-dimensional Minkowski
space-time $\mathcal{M}$, explicitly for $D=1$, $2$ and $3$, can be obtained
from an integral representation of the zero-curvature equation in the
corresponding loop space $\mathcal{L}^{(D-1)}(\mathcal{M})$. The conservation
of the electric charge can be seen as the result of a hidden symmetry in this
representation of the dynamical equations.

##  1 Introduction

Hidden symmetries play a special role in the construction of soliton solutions
and their conserved charges in integrable field theories[1]. In
$(1+1)$-dimensional space-time these theories will generally admit what is
called a zero-curvature representation[2, 3] in which their dynamical
equations become equivalent to the flatness of a Lie algebra-valued 1-form
connection whose components are functions of the physical fields and their
derivatives.

This representation lies in the core of the construction of conserved
charges[4] which are obtained from the holonomy operator associated to that
1-form connection: its flatness implies the path-independence of the holonomy
in space-time and leads to an isospectral evolution of the holonomy evaluated
over space; the preserved eigenvalues of this operator are the conserved
charges.

In integrable field theories, some of these charges may coincide with those
obtained from Noether’s theorem but the symmetries here are hidden in the
gauge invariance of the zero-curvature equation, which also defines the ground
for the development of algebraic methods to construct soliton solutions.

The possibility of extending the zero-curvature formulation to field theories
in higher-dimensional space-time as an attempt to understand integrability in
this context was explored in [5]. The crucial step towards this approach was
the generalization of the holonomy operator through a non-abelian Stokes
theorem and the interpretaion of this construction in loop space[6]. In [7, 8]
it was shown how this non-abelian Stokes theorem can be used to define the
integral version of Yang-Mills equations leading to dynamically gauge-
invariant conserved charges. Some consequences of this formulation of gauge
theories in loop space were discussed in [9, 10].

In the present paper we show that the integral equations of electromagnetism
can be represented in loop space by an integral equation for a flat connection
$\mathcal{A}$. In a Minkowski $(D+1)$-dimensional space-time $\mathcal{M}$,
this connection is constructed in the corresponding loop space which is
defined by the maps
$\mathcal{L}^{(D-1)}(\mathcal{M})=\left\\{\Gamma:S^{D-1}\to\mathcal{M}|\Gamma(0)=x_{R}\right\\}$,
taking $(D-1)$-dimensional spheres in space-time, based at $x_{R}$, to points
in this loop space. The connection is defined in terms of an exact $D$-form in
space-time evaluated on the loop $S^{D-1}$. The flatness of the loop space
connection, $\delta\mathcal{A}=0$, will give the local conservation of the
electric charge and the charge itself can be obtained as the eigenvalues of
the generalized $D$-holonomy evaluated over the space as a consequence of the
(hidden) symmetry of the integral equation in loop space: its invariance under
homotopic transformations of the path.

The integral equation in loop space is written in space-time as a Lorentz
scalar integral equation and by choosing appropriately a space-time slicing
with $D$-dimensional hyper-volumes one can recover the usual integral
expressions, namely Gauss’, Faraday and Ampère-Maxwell laws.

##  2 The integral equations of electromagnetism in $1+1$ dimensions

The Maxwell equations in 2-dimensional space-time, in Gaussian coordinates,
are given by111The coordinates of the $(D+1)$-dimensional Minkowski space-time
are $x^{\mu}=(ct,x^{i})$, $\mu=0,1,\dots,D$ and the Minkowski metric has
signature $\eta_{\mu\nu}=\textrm{diag}(1,-1,\dots,-1)$.

$\partial_{\mu}F^{\mu\nu}=\frac{2}{c}J^{\nu}\quad\mu,\nu\ =0,1$ (1)

where $J^{\mu}=(c\rho,j)$ is the covariant electric current density and the
Faraday tensor defined in terms of the gauge potential $a_{\mu}$ is given by
$F_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$ having
$F_{01}=E=-F_{10}$ the only non-vanishing components, with $E$ the electric
field.

We consider the loop space $\mathcal{L}^{(0)}(\mathcal{M})$. This is the space
of maps $\Gamma$ from the sphere $S^{0}$ into the space-time manifold
$\mathcal{M}$ defined as
$\mathcal{L}^{(0)}(\mathcal{M})=\left\\{\Gamma:S^{0}\to\mathcal{M}|\Gamma(0)=x_{R}\right\\}$
where $x_{R}$ is a fixed point in $\mathcal{M}$ which we call the reference
point. The image of the map $\Gamma$ will be, in this case, also points in the
loop space $\mathcal{L}^{(0)}(\mathcal{M})$.

Let us define a $1$-form connection $\mathcal{A}$ in this loop space[6] as

$\displaystyle\mathcal{A}=A_{\mu}\delta x^{\mu}.$ (2)

We want to find an integral representation of the Maxwell equations in
$\mathcal{L}^{(0)}(\mathcal{M})$ based on a flat connection in this space.
This can be done if we write the field in space-time $A_{\mu}$ as the
components of the exact $1$-form $A=df=\partial_{\mu}fdx^{\mu}$, since then
$\delta\mathcal{A}=\frac{1}{2}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\delta
x^{\mu}\wedge\delta x^{\nu}=0$.

Let us write $f$ in terms of the physical fields as

$\displaystyle f=\frac{i\beta}{2}\epsilon^{\mu\nu}F_{\mu\nu},$ (3)

where $\beta$ is an arbitrary constant.

Using the equations of motion we obtain the components $A_{\mu}$ as

$\displaystyle A_{\mu}=\frac{i2\beta}{c}\epsilon_{\mu\nu}J^{\nu}$ (4)

and consequently the loop space connection is given by

$\displaystyle\mathcal{A}=\frac{i2\beta}{c}\epsilon_{\mu\nu}J^{\nu}\delta
x^{\mu}.$ (5)

Given the flatness of the connection we can associate its integral over a path
$\Gamma$ in loop space with the value of a potential $\varphi$ evaluated at
the borders of this path. This is the integral representation of the zero-
curvature equation for the connection in loop space:

$\displaystyle\Delta\varphi=\int_{\Gamma}\mathcal{A}(\sigma)\ d\sigma$ (6)

where $\mathcal{A}(\sigma)\equiv A_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma$, with
$\sigma$ parameterizing the path $\Gamma$. Clearly, by construction we have
that $\varphi=f$ and $A_{\mu}=\partial_{\mu}f$.

From the definitions of $f$ and $A_{\mu}$ in terms of the physical fields and
their local relations given by equation (1) this equation becomes

$\displaystyle
E(x)-E(x_{R})=\frac{2}{c}\int_{0}^{2\pi}\epsilon_{\mu\nu}J^{\nu}\frac{dx^{\mu}}{d\sigma}d\sigma.$
(7)

This is the Lorentz scalar integral equation of electromagnetism in $1+1$
dimensional space-time. In order to obtain the usual version of the integral
equations, here equivalent to the Gauss’ and Ampère’s laws for the electric
field, we need to specify the curve $\gamma$ is space-time where we integrate
the dual of the electric current (see figure 1).

When $\gamma$ is considered to be purely spatial at a constant time $t$ we
have, from (7)

$E(t,x)-E(t,x_{R})=\frac{2}{c}\int_{x_{R}}^{x}\epsilon_{10}J^{0}\frac{dx^{1}}{d\sigma}d\sigma=2\int_{x_{R}}^{x}\rho
dx^{\prime}$ (8)

which can be recognized as the Gauss law for the electric field.

Next, the Ampère-Maxwell law follows from the integral equation (7) when we
consider the curve $\gamma$ to have constant spatial coordinate $x$:

$E(t,x)-E(0,x)=\frac{2}{c}\int_{0}^{t}\epsilon_{01}J^{1}\frac{dx^{0}}{d\sigma}d\sigma=-2\int_{0}^{t}jdt^{\prime}.$
(9)

We notice that if the curve $\gamma$ is taken to be infinitesimal, i.e.,
$x=x_{R}+\delta x$, then the differential equations (1) can be recovered from
(8) and (9):

$\displaystyle E(t,x)-E(t,x_{R})\approx 2\rho\Delta
x\quad\Longrightarrow\quad\frac{\partial E}{\partial x}=2\rho$ $\displaystyle
E(t,x)-E(0,x)\approx-2j\Delta t\quad\Longrightarrow\quad\frac{\partial
E}{\partial t}=-2j.$

Figure 1: With an appropriate choice for the paths in space-time the Lorentz
scalar integral equations will give the usual integral expressions for the
laws of electrodynamics.

###  2.1 The conservation of the electric charge in 2 dimensions

The loop space representation of the integral dynamical equations defines a
relation between the 0-form $\varphi$ at the borders of the path $\Gamma$ with
the integral of the 1-form connection $\mathcal{A}$ along this path.

Considering an infinitesimal variation of $\Gamma$ keeping its borders fixed,
we find from (6) that

$\displaystyle\delta\left(\Delta\varphi\right)=\int_{\Gamma}(\partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu})\frac{dx^{\mu}}{d\sigma}\delta
x^{\nu}d\sigma.$

Since the border of $\Gamma$ remains fixed, $\Delta\varphi$ should not change
and consequently the l.h.s of the equation above vanishes. On the other hand,
given that $A$ is exact, also the r.h.s above vanishes and we conclude that
equation (6) remains invariant under homotopic deformations of the path
$\Gamma$. In other words, $\varphi$ is path-independent and equivalently
$\mathcal{A}$ is flat. Writing the loop space connection in terms of the
physical fields we have

$\delta\mathcal{A}=\partial_{\nu}A_{\mu}\delta x^{\nu}\land\delta
x^{\mu}=\frac{i2\beta}{c}\epsilon^{\nu\mu}\epsilon_{\mu\lambda}\partial_{\nu}J^{\lambda}\delta
x^{0}\delta x^{1}=\frac{i4\beta}{c}\partial_{\lambda}J^{\lambda}\delta
x^{0}\delta x^{1}$ (10)

and consequently $\delta\mathcal{A}=0$ implies $\partial_{\mu}J^{\mu}=0$, i.e,
the local conservation of the electric charge is obtained as a consequence of
the zero-curvature of the loop space connection.

This path-independence of the integral equation (6) in loop space is the
hidden symmetry behind this conservation law .

The conserved charges associated to this symmetry can be obtained from the
holonomy operator defined by the parallel transport equation along a curve
$\gamma$ in space-time parameterized by $\sigma\in[0,2\pi]$,

$\displaystyle\frac{dW}{d\sigma}+A_{\mu}\frac{dx^{\mu}}{d\sigma}W=0,$ (11)

whose solution can be formally written as

$\displaystyle
W_{\gamma}=e^{-\int_{0}^{2\pi}A_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma}W_{\circ},$
(12)

where $W_{0}$ is obtained from the initial conditions.

We consider the paths given in figure 2 joinning the points $x_{R}$ and
$x=(ct,L)$. We assume that the two paths $\gamma_{L}\circ\gamma_{0}$ and
$\gamma_{t}\circ\gamma_{x_{R}}$ can be deformed into each other by continuous
transformations $x^{\mu}\to x^{\mu}+\delta x^{\mu}$ which make the holonomy
$W$ calculated over $\gamma_{L}\circ\gamma_{0}$, changes as222See appendix A

$\delta
W=\int_{0}^{2\pi}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})\frac{dx^{\mu}}{d\sigma}\delta
x^{\nu}d\sigma.$ (13)

Figure 2: The curve in the $(1+1)$-dimensional space-time is equivalent to the
path defined in the corresponding loop space.

The r.h.s of equation (13) vanishes since $A$ is an exact $1$-form, giving
$\delta W=0$, which means that the holonomy is in fact path-independent. So,
the holonomy calculated over $\gamma_{L}\circ\gamma_{0}$ is identical to that
calculated over $\gamma_{t}\circ\gamma_{x_{R}}$ and we have the identity

$\displaystyle W_{\gamma_{t}}\ \cdot\ W_{\gamma_{x_{R}}}=W_{\gamma_{L}}\
\cdot\ W_{\gamma_{0}}.$ (14)

In terms of the physical fields, using (12) with $W_{0}=1$ for simplicity, the
operator on the r.h.s. is given by

$\displaystyle W_{\gamma_{L}}\ \cdot\
W_{\gamma_{0}}=e^{i2\beta\int_{0}^{t}j\big{|}_{x=L}dt^{\prime}}e^{-i2\beta
Q|_{t=0}}$

and similarly for the l.h.s.

$\displaystyle W_{\gamma_{t}}\ \cdot\ W_{\gamma_{x_{R}}}=e^{-i2\beta
Q|_{t>0}}e^{i2\beta\int_{0}^{t}j|_{x=x_{R}}dt^{\prime}},$

where $Q$ is the electric charge

$Q=\int_{x_{R}}^{x_{L}}\rho\;dx.$ (15)

We can rewrite (14) as

$W_{\gamma_{t}}=W_{\gamma_{L}}\ \cdot\ W_{\gamma_{0}}\ \cdot\
W_{\gamma_{x_{R}}}^{-1}$ (16)

and assuming that $j(t,x)\to 0$ in the limit where $x_{R}$ and $L$ go to
infinity, the operators $W_{\gamma_{L}}$ and $W_{\gamma_{x_{R}}}$ become the
identity and we remain with

$\displaystyle e^{-i2\beta Q(t)}=e^{-i2\beta Q(0)}\Rightarrow
e^{-i2\beta\Delta Q}=1$ (17)

where $\Delta Q\equiv Q|_{t>0}-Q|_{t=0}$. Since $\beta$ is arbitrary this
identity implies

$\displaystyle\Delta Q=0$ (18)

giving us the conservation of electric charge in time. These charges are
defined as the eigenvalues of the holonomy restricted to space

$W_{\gamma_{t}}=e^{-i2\beta Q|_{t}}.$ (19)

This conservation law is a consequence of the path invariance of the holonomy
operator or equivalently, of the flatness of the connection in loop space.

##  3 The integral equations of electromagnetism in 2+1 dimensions

In $3$-dimensional space-time the differential equations of electrodynamics
are given by[11]

$\displaystyle\partial_{\mu}F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{c}J^{\nu}$ (20)
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu}$ $\displaystyle=$
$\displaystyle 0,\qquad\mu,\nu=0,1,2$ (21)

where $J^{\mu}=(c\rho,j^{1},j^{2})$ is the electric $3$-current density,
$F_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$ is the
electromagnetic field whose components are the electric vector field
$\mathbf{E}$ and the magnetic scalar field $B$ given as $F_{0i}=E_{i}$ and
$F_{ij}=-\epsilon_{ij}B$, and
$\widetilde{F}^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\lambda}F_{\nu\lambda}$ is the
Hodge dual of the electromagnetic field333We use $\epsilon^{012}=1$ and
$\epsilon_{ij}\equiv\epsilon^{0ij}$..

The integral equations of electromagnetism will be represented in the loop
space $\mathcal{L}^{(1)}(\mathcal{M})$ which is defined by the maps
$\mathcal{L}^{(1)}(\mathcal{M})=\left\\{\Gamma:S^{1}\to\mathcal{M}|\Gamma(0)=x_{R}\right\\}$.
We consider a reference point $x_{R}$ in space-time $\mathcal{M}$ and a family
of loops (closed curves) based at this point. Then, each of these loop will
correspond to a point in loop space, which is the image of the map defined
above. So, a collection of homotopic loops scanning a $2$-dimensional surface
will define, in the loop space, a path $\Gamma$.

Figure 3: The scanning of the surface $\Sigma$ with loops based at $x_{R}$.
Each loop in this scanning corresponds to a point in
$\mathcal{L}^{(1)}(\mathcal{M})$.

We define a $1$-form connection in the loop space as

$\displaystyle\mathcal{A}=\oint_{\gamma}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\delta x^{\nu}d\sigma$ (22)

where $G_{\mu\nu}$ is an anti-symmetric tensor which we integrate over each
loop based at $x_{R}$ parameterized by $\sigma\in[0,2\pi]$ and labelled by
$\tau\in[0,2\pi]$. So, this connection is defined, in space-time, on each loop
and thus, it takes values at the points of the loop space
$\mathcal{L}^{(1)}(\mathcal{M})$.

Taking $G_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}$, the
components of the exact $2$-form $G=dC$, this connection becomes flat, i.e,
its curvature vanishes: $\delta\mathcal{A}=0$. So, in order to find an
integral representation of the zero-curvature equation as the integral
equations of the electromagnetism in loop space we write the components of the
$1$-form $C=C_{\mu}dx^{\mu}$ in terms of the physical fields as

$\displaystyle C_{\mu}=i\left(eA_{\mu}+\beta\widetilde{F}_{\mu}\right)$ (23)

where $\beta$ is an arbitrary constant and $e$ the elementary electric charge.

Using the dynamical equations (20) and (21) the components of $G$ can be
written as

$\displaystyle
G_{\mu\nu}=ieF_{\mu\nu}+\frac{i2\pi\beta}{c}\epsilon_{\mu\nu\lambda}J^{\lambda}$
(24)

and the connection in loop space reads

$\displaystyle\mathcal{A}=ie\oint_{\gamma}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\delta
x^{\nu}d\sigma+\frac{i2\pi\beta}{c}\oint_{\gamma}\epsilon_{\mu\nu\lambda}J^{\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\delta x^{\nu}d\sigma.$ (25)

As in the previous case, the representation of the integral equations in loop
space will be defined by an equation like (6),

$\Delta\varphi=\int_{\Gamma}\mathcal{A}(\tau)d\tau$ (26)

where $\Gamma$, the path in loop space, stands now for the scanning of the
$2$-dimensional surface $\Sigma$ in space-time with homotopically equivalent
loops, based at $x_{R}$ which is located at the border $\partial\Sigma$ and

$\mathcal{A}(\tau)\equiv\int_{0}^{2\pi}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma.$ (27)

This integral equation in loop space is in fact a representation of the Stokes
theorem in space-time for the 1-form $C$, which is in this case the potential
$\varphi$:

$\displaystyle\oint_{\partial\Sigma}C_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma=-\int_{\Sigma}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau,$
(28)

where $\Sigma$ is a 2-dimensional surface in spacetime and $\tau\in[0,2\pi]$
parameterizes the loops scanning this surface such that the loop with $\tau=0$
is the infinitesimal one (or point-loop) around the reference point $x_{R}$
and the loop with $\tau=2\pi$ is that which defines the border
$\partial\Sigma$ of the surface $\Sigma$.

The integral equations of electromagnetism are a consequence of the Stokes
theorem and the differential equations of motion, so, writing the fields in
(28) as defined by (23) and (24), given the arbitrariness of $\beta$ we have
the set of equations

$\displaystyle\oint
A_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma=-\int_{\Sigma}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
(29)
$\displaystyle\oint_{\partial\Sigma}\widetilde{F}_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma=-\frac{2\pi}{c}\int_{\Sigma}\epsilon_{\mu\nu\lambda}J^{\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau.$
(30)

Let us show that this set of Lorentz scalar integral equations imply those
usually presented, namely, the Gauss law, the Maxwell-Ampère law and the
Faraday law.

The Gauss law is obtained from (30) when we take the surface $\Sigma$ to be
completely spatial. For the l.h.s of this equation we get

$\displaystyle\oint_{\partial\Sigma}\widetilde{F}_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma=\oint_{\partial\Sigma}\widetilde{F}_{i}\frac{dx^{i}}{d\sigma}d\sigma=\oint_{\partial\Sigma}\epsilon_{ij}E_{j}\frac{dx^{i}}{d\sigma}d\sigma$

and defining the normal vector to the curve $\partial\Sigma$ as
$\hat{\mathbf{n}}_{i}\ dr=\epsilon_{ij}dx^{j}$ we can write the above result
as

$\displaystyle\oint_{\partial\Sigma}\widetilde{F}_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma=-\oint_{\partial\Sigma}\mathbf{E}\cdot\hat{\mathbf{n}}\
dr.$ (31)

For the r.h.s. of (30) we have

$\displaystyle-\frac{2\pi}{c}\int_{\Sigma}\epsilon_{\mu\nu\lambda}J^{\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=-\frac{2\pi}{c}\int_{\Sigma}\epsilon_{ij0}J^{0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma d\tau$
$\displaystyle=-2\pi\int_{\sigma}\rho\epsilon_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau=-2\pi\int_{\sigma}\rho\ dS$ (32)

where $dS=\epsilon_{ij}\frac{\partial x^{i}}{\partial\sigma}\frac{\partial
x^{j}}{\partial\tau}d\sigma d\tau$ is the area element.

Then the integral equation (30) becomes the usual Gauss’ law for the electric
field:

$\displaystyle\oint_{\partial\Sigma}\mathbf{E}\cdot\hat{\mathbf{n}}\
dr=2\pi\int_{\Sigma}\rho\ dS.$ (33)

Now, consider the 2-dimensional surface $\Sigma$ with a component in the
$x^{0}$ direction as depicted in figure 4. In this case, the l.h.s. of (30)
reads

$\displaystyle\oint_{\partial\Sigma}\widetilde{F}_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma$
$\displaystyle=\int_{\gamma_{0}}\widetilde{F}_{i}\big{|}_{t=0}dx^{i}+\int_{\gamma_{b}}\widetilde{F}_{0}\big{|}_{b}dx^{0}-\int_{\gamma_{t}}\widetilde{F}_{i}\big{|}_{t>0}dx^{i}-\int_{\gamma_{a}}\widetilde{F}_{0}\big{|}_{a}dx^{0}$
$\displaystyle=\int_{\gamma_{0}}\epsilon_{ij}E_{j}\big{|}_{t=0}dx^{i}-\int_{\gamma_{b}}cB\big{|}_{b}dt-\int_{\gamma_{t}}\epsilon_{ij}E_{j}\big{|}_{t>0}dx^{i}+\int_{\gamma_{a}}cB|_{a}dt$
$\displaystyle=\int_{\gamma_{t}}\mathbf{E}|_{t>0}\cdot\hat{\mathbf{n}}\
dr-\int_{\gamma_{0}}\mathbf{E}\big{|}_{t=0}\cdot\hat{\mathbf{n}}\
dr-c\left(\int_{\gamma_{b}}B|_{b}dt-\int_{\gamma_{a}}B|_{a}dt\right)$ (34)

and the r.h.s. is

$\displaystyle-\frac{2\pi}{c}\int_{\Sigma}\epsilon_{\mu\nu\lambda}J^{\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=-\frac{2\pi}{c}\int_{0}^{2\pi}\int_{0}^{2\pi}\epsilon_{i0j}J^{j}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{0}}{\partial\tau}d\sigma
d\tau=-2\pi\int_{0}^{t}\int_{a}^{b}\mathbf{j}\cdot\hat{\mathbf{n}}\
drdt^{\prime}.$ (35)

Figure 4: In $2+1$ dimensions we consider a 2-dimensional surface in space-
time in order to obtain the Ampère-Maxwell law from the Lorentz scalar
integral equation.

If we consider an infinitesimal time lapse in the results above then the
equation (30) becomes

$\displaystyle\int_{\gamma_{t}}\mathbf{E}\big{|}_{t>0}\cdot\hat{\mathbf{n}}\
dr-\int_{\gamma_{0}}\mathbf{E}\big{|}_{t=0}\cdot\hat{\mathbf{n}}\ dr-c\Delta
t\left(B\big{|}_{b}-B\big{|}_{a}\right)=-2\pi\Delta
t\int_{a}^{b}\mathbf{j}\cdot\hat{\mathbf{n}}\ dr$ (36)

and in the limit where $\Delta t\to 0$ we have finally the usual Ampère-
Maxwell law

$\displaystyle
B\big{|}_{b}-B\big{|}_{a}=\frac{1}{c}\frac{d}{dt}\int_{a}^{b}\mathbf{E}\cdot\hat{\mathbf{n}}\
dr+2\pi\int_{a}^{b}\mathbf{j}\cdot\hat{\mathbf{n}}\ dr.$ (37)

The Faraday law is in fact a mathematical identity and this is clear from the
integral version given by the Lorentz scalar expression (29). In order to
obtain the usual formula of this law we consider a 2-dimensional
spatiotemporal surface defined by folding the previously used open surface
such that $\gamma_{a}\sim\gamma_{b}^{-1}$.

The l.h.s. of (29) reads

$\displaystyle\oint
A_{\mu}dx^{\mu}=\oint_{\gamma_{0}}A_{i}\big{|}_{t=0}dx^{i}-\oint_{\gamma_{t}}A_{i}\big{|}_{t>0}dx^{i}$

and using (29) again for each of the terms above at constant time we get

$\displaystyle\oint
A_{\mu}dx^{\mu}=\int_{S_{0}}B\big{|}_{t=0}dS-\int_{S_{t}}B\big{|}_{t>0}dS$
(38)

where $S_{0}$ and $S_{t}$ are the areas enclosed by $\gamma_{0}$ and
$\gamma_{t}$, respectively.

Now, for the r.h.s of (29) we have

$\displaystyle-\int_{\Sigma}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=-\int_{0}^{2\pi}\int_{0}^{2\pi}F_{i0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{0}}{\partial\tau}d\sigma d\tau$
$\displaystyle=\oint_{\partial S_{0}}\int_{0}^{t}c\ E_{i}dt^{\prime}dx^{i}.$
(39)

Considering an infinitesimal time lapse in these expressions, equation (29)
becomes

$\displaystyle\int_{S_{0}}B\big{|}_{t=0}dS-\int_{S_{t}}B\big{|}_{t>0}dS=-c\Delta
t\oint_{\partial S_{0}}\mathbf{E}\cdot d\mathbf{r}$

which, in the limit of $\Delta t\to 0$ becomes

$\displaystyle\oint_{\partial S_{0}}\mathbf{E}\cdot
d\mathbf{r}=-\frac{1}{c}\frac{d}{dt}\int_{S_{0}}BdS.$ (40)

Figure 5: The 2-dimensional sheet in space-time from figure 4 is folded to
give a cylinder on which the Lorentz scalar integral equation is evaluated,
resulting in the Faraday law.

###  3.1 The conservation of the electric charge in 3 dimensions

The loop space connection is flat and this result is a direct consequence of
the fact that $G$ is exact so $dG=0$ and

$\displaystyle\delta\mathcal{A}=\oint\left(\partial_{\lambda}G_{\mu\nu}+\partial_{\mu}G_{\nu\lambda}+\partial_{\nu}G_{\lambda\mu}\right)\frac{\partial
x^{\lambda}}{\partial\sigma}\delta x^{\mu}\land\delta x^{\nu}d\sigma=0.$ (41)

In terms of the physical fields, we find that

$\displaystyle\delta\mathcal{A}=\frac{i2\pi\beta}{c}\int\partial_{\mu}J^{\mu}\;d^{3}x=0$
(42)

and consequently the electric charge is conserved.

These conserved charges can be obtained from the generalization of the
holonomy operator defined by the $2$-holonomy $V$: a parallel transport
operator in loop space $\mathcal{L}^{(1)}(\mathcal{M})$ obeying the
equation444The reasoning behind the definition of this equation is explained
in the appendix A

$\displaystyle\frac{dV}{d\tau}-\left(\int_{0}^{2\pi}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial
x^{\nu}}{\partial\tau}d\sigma\right)V=0$ (43)

whose solution can be formally written as

$\displaystyle V_{\Sigma}=e^{\int_{\Sigma}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau}\
V_{0}$ (44)

where $\Sigma$ is a $2$-dimensional surface and $V_{0}$ is obtained from the
initial conditions.

Figure 6: The flatness of the connection $\mathcal{A}$ implies that the
parallel transport along any surface between the point-loop and
$\partial\Sigma_{t}$ will give the same result. This can be defined in loop
space in terms of the path-independence of the $2$-holonomy $V$.

We consider the path $\Gamma_{L}\circ\Gamma_{0}$ in loop space, as in figure
6: the spatial surface $\Sigma_{0}\sim\Gamma_{0}$ at constant time $t=0$ is
scanned with loops starting at the point-loop at $x_{R}$, until the loop which
defines the border $\partial\Sigma_{0}$ and then, moving from this last loop
forward in time up to the loop $\partial\Sigma_{t}$. The $2$-holonomy along
this path (over this surface) is given by (considering, for simplicity,
$V_{0}=1$)

$\displaystyle V_{\Gamma_{L}}\ \cdot\
V_{\Gamma_{0}}=e^{\int_{\partial\Sigma_{0}\times\mathbf{R}}G_{i0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{0}}{\partial\tau}d\sigma
d\tau}e^{\int_{\Sigma_{0}}G_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma d\tau}.$ (45)

From (24) we have that

$\displaystyle\int_{\partial\Sigma_{0}\times\mathbf{R}}G_{i0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{0}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$
$\displaystyle\int_{\partial\Sigma_{0}\times\mathbf{R}}\left(ieF_{i0}+\frac{i2\pi\beta}{c}\epsilon_{i0j}J^{j}\right)\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{0}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$ $\displaystyle-iec\int\oint\mathbf{E}\cdot
d\mathbf{r}dt+i2\pi\beta\int\oint\mathbf{j}\cdot\hat{\mathbf{n}}drdt.$

Assuming that $\|\mathbf{E}\|$ falls off quickly enough and that the electric
current is localized, the quantity above should vanish in the limit where the
radius of $\partial\Sigma_{0}$ goes to infinity. What remains is then

$\displaystyle V_{\Gamma_{L}}\ \cdot\
V_{\Gamma_{0}}=e^{ie\int_{\Sigma_{0}}F_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau}e^{\frac{i2\pi\beta}{c}\int_{\Sigma_{0}}\epsilon_{ij}J^{0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau}=e^{-ie\int_{\Sigma_{0}}B\big{|}_{t=0}dS}e^{i2\pi\beta Q\big{|}_{t=0}}.$
(46)

In the construction of the $2$-holonomy over the path
$\Gamma_{t}\circ\Gamma_{x_{R}}$ we notice that $V_{\Gamma_{x_{R}}}$ becomes
trivial as the radius of the point-loop around $x_{R}$ becomes zero and we get

$\displaystyle V_{\Gamma_{t}}\ \cdot\
V_{\Gamma_{x_{R}}}=e^{-ie\int_{\Sigma_{0}}B\big{|}_{t>0}dS}\ \cdot\
e^{i2\pi\beta Q\big{|}_{t>0}}.$ (47)

The path independence of the $2$-holonomy is a direct consequence of the
flatness of the connection in loop space and it implies that, once
$\Gamma_{t}\circ\Gamma_{x_{R}}$ can be obtained from continous deformations of
$\Gamma_{L}\circ\Gamma_{0}$, we have the relation

$\displaystyle V_{\Gamma_{t}}\ \cdot\ V_{\Gamma_{x_{R}}}=V_{\Gamma_{L}}\
\cdot\ V_{\Gamma_{0}}\qquad\Rightarrow\qquad V_{\Gamma_{t}}=V_{\Gamma_{L}}\
\cdot\ V_{\Gamma_{0}}\ \cdot\ V_{\Gamma_{x_{R}}}^{-1}.$ (48)

Given that the operators $V_{\Gamma_{L}}$ and $V_{\Gamma_{x_{R}}}$, as
discussed above, become unity, the path independence of the $2$-holonomy gives
us that the electric charge $Q$, defined by the eigenvalues of

$V_{\Gamma_{t}}=e^{i2\pi\beta Q|_{t}}$ (49)

and the magnetic flux over the entire space, defined by the eigenvalues of

$V_{\Gamma_{t}}=e^{-ie\int_{\Sigma_{0}}B|_{t}\;dS}$ (50)

are conserved in time.

##  4 The integral equations of electromagnetism in $3+1$ dimensions

In $3+1$ dimensions, the electric and magnetic vector fields $\mathbf{E}$ and
$\mathbf{B}$ can be written as the components of the electromagnetic field
strength $F_{\mu\nu}$, $\mu,\nu=0,\dots,3$ as $E^{i}=F_{0i}=-F_{i0}$ and
$B^{k}=-\frac{1}{2}\epsilon_{kij}F_{ij}$.

Maxwell equations are given by

$\displaystyle\partial_{\mu}F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}J^{\nu}$ (51)
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}$ $\displaystyle=$
$\displaystyle 0$ (52)

where $J^{\mu}=(c\rho,j^{i})$ is the Lorentz covariant current and
$\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\gamma}F_{\lambda\gamma}$
is the Hodge dual of the electromagnetic field.

The construction of the integral representation of the Maxwell equation is
done in the loop space
$\mathcal{L}^{(2)}(\mathcal{M})=\left\\{\Gamma:S^{2}\to\mathcal{M}|\Gamma(0)=x_{R}\right\\}$.
This mapping will relate closed $2$-dimensional surfaces in space-time, which
are all based at the reference point $x_{R}$, with points in the loop space.

We define the $1$-form connection in $\mathcal{L}^{(2)}(\mathcal{M})$ by

$\mathcal{A}=\oint_{\Sigma}H_{\mu\nu\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\delta
x^{\lambda}d\sigma d\tau$ (53)

where $H_{\mu\nu\lambda}$ is a $3$-form in space-time which is integrated over
the $2$-dimensional closed surface $\Sigma$, parameterized by
$\sigma\in[0,2\pi]$ and $\tau\in[0,2\pi]$. By scanning a $3$-dimensional
volume in space-time with these closed surfaces as in figure 7 we define a
path in the loop space.

Figure 7: The scanning of space-time is doing by considering a family of
spheres based at $x_{R}$ which can be continuously deformed into each other.

Let us take $H=dB$, i.e. we consider it to be an exact $3$-form. This is a
sufficient condition for $\mathcal{A}$ to be flat, i.e.,
$\delta\mathcal{A}=0$. We look for an integral representation in loop space of
this zero-curvature equation such that it will be equivalent to Maxwell
integral equations of electromagnetism. Such an integral equation in loop
space is given by

$\Delta\varphi=\int_{\Gamma}\mathcal{A}(\zeta)d\zeta$ (54)

with

$\mathcal{A}(\zeta)=\oint_{\Sigma}H_{\mu\nu\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau$ (55)

and the potential $\varphi$ will be given by

$\varphi=\int_{\Sigma}B_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau.$
(56)

As before, the integral equation in loop space is a representation of the
Stokes theorem for the $2$-form $B=\frac{1}{2}B_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}$:

$\oint_{\partial\Omega}B_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=\int_{\Omega}\left(\partial_{\lambda}B_{\mu\nu}+\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}B_{\mu\nu}\right)\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau d\zeta.$ (57)

We now define the components $B_{\mu\nu}$ in terms of the electromagnetic
field strength and its Hodge dual

$B_{\mu\nu}=i(\alpha F_{\mu\nu}+\beta\widetilde{F}_{\mu\nu})$ (58)

where $\alpha$ and $\beta$ are arbitrary constants. Once the Maxwell equations
(51) are satisfied by these fields, the integrand in the r.h.s of (57) can be
written as

$\partial_{\lambda}B_{\mu\nu}+\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}B_{\mu\nu}=i\alpha\left(\partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\lambda}F_{\mu\nu}\right)+i\beta\epsilon_{\mu\nu\lambda\gamma}\partial_{\rho}F^{\rho\gamma}=\frac{4\pi\beta}{c}\epsilon_{\mu\nu\lambda\gamma}J^{\gamma}.$
(59)

and the mathematical relation (57) defines the Lorentz scalar integral
equations of electrodynamics:

$\oint_{\partial\Omega}\left(\alpha
F_{\mu\nu}+\beta\widetilde{F}_{\mu\nu}\right)\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=\frac{4\pi\beta}{c}\int_{\Omega}J^{\gamma}\epsilon_{\mu\nu\lambda\gamma}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau d\zeta.$ (60)

The arbitrariness of $\alpha$ and $\beta$ implies that the following two
equations hold simultaneously

$\displaystyle\oint_{\partial\Omega}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$ $\displaystyle 0$ (61)
$\displaystyle\oint_{\partial\Omega}\widetilde{F}_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$
$\displaystyle\frac{4\pi}{c}\int_{\Omega}J^{\gamma}\epsilon_{\mu\nu\lambda\gamma}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau d\zeta.$ (62)

These are Lorentz scalar equations for the flux of the electromagnetic field
strength and its Hodge dual through 2-dimensional surfaces in Minkowski space-
time.

We now proceed to show that these equations imply the usual integral laws of
electrodynamics when the 3-dimensional volumes in space-time are appropriately
chosen.

Let us start by considering equation (61), which is the integral version of
the Bianchi identity, when $\Omega$ is a 3-dimensional spatial volume at a
given instant of time. The l.h.s of that equation becomes555Here the Hodge
dual of $dx^{i}\wedge dx^{j}$ in $\mathbb{R}^{3}$ is given by
$dS^{k}=\epsilon_{ijk}\frac{\partial x^{i}}{\partial\sigma}\frac{\partial
x^{j}}{\partial\tau}d\sigma d\tau$.

$\displaystyle\oint_{\partial\Omega}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$ $\displaystyle\oint_{\partial\Omega}F_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau=-\oint_{\partial\Omega}B^{k}\epsilon_{ijk}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$ $\displaystyle-\oint_{\partial\Omega}\mathbf{B}\cdot
d\mathbf{S}$

and the r.h.s being equal to zero, we recover the Gauss law for the magnetic
field.

Next we consider the same equation (61) but now with $\Omega$ as a
3-dimensional volume in space-time which we shall take as the cylinder
$\Omega=\mathbb{D}^{2}\times\mathbb{R}$. The l.h.s of equation (61) is now
decomposed in three parts, corresponding to the flux of the electromagnetic
field strength across the three surfaces which form the border of the
cylinder:
$\partial\Omega=(\mathbb{D}^{2}_{0})^{-1}\cup\mathbb{D}^{2}_{t}\cup(\mathbb{S}^{1}\times\mathbb{R})$.
The time direction is taken at the axis of symmetry of this cylinder and we
reverse the orientation of the bottom disk $\mathbb{D}^{2}_{0}$, at $t=0$, so
that we can consider it as a closed orientable surface.

Figure 8: The cylinder defines a closed surface in space-time with its
symmetry axis along the time dimension. The bottom ($\mathbb{D}^{2}_{0}$) and
top ($\mathbb{D}^{2}_{t}$) disks are 2-dimensional spatial surfaces at
different instants of time while the surface on the side of the cylinder (cyl
$\sim\mathbb{S}^{1}\times\mathbb{R}$) has two perpendicular directions in time
and space.

Then we have, for the flux of the electromagnetic field on the cylinder:

$\displaystyle\oint_{\partial\Omega}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
$\displaystyle=$ $\displaystyle-\oint_{\mathbb{D}^{2}_{0}}F_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau+\oint_{\mathbb{D}^{2}_{t}}F_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau+\oint_{\mathbb{S}^{1}\times\mathbb{R}}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau.$

The two first integrals will give the flux of the magnetic field through each
of the disks, at different instants of time. For the third integral we take
$\sigma$ to parameterize the surface of the cylinder at constant $x^{0}=ct$
coordinate which in turn is parameterized with $\tau$. Then

$\oint_{\mathbb{S}^{1}\times\mathbb{R}}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=-c\int_{0}^{t}\int_{0}^{2\pi}E^{i}\frac{\partial
x^{i}}{\partial\sigma}d\sigma dt^{\prime}$

and if we assume an infinitesimal time lapse, the l.h.s of equation (61) reads

$\oint_{\partial\Omega}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=-\Delta\Phi(\mathbf{B},\mathbb{D}^{2})-c\Delta
t\oint_{\partial\mathbb{D}^{2}}\mathbf{E}\cdot d\mathbf{x}$

where $\Delta\Phi(\mathbf{B},\mathbb{D}^{2})$ stands for the change in the
flux of magnetic field from $\mathbb{D}^{2}_{0}$ to $\mathbb{D}^{2}_{t}$.
Finally, the integral Bianchi identity becomes

$\frac{1}{c}\frac{\Delta\Phi(\mathbf{B},\mathbb{D}^{2})}{\Delta
t}+\oint_{\partial\mathbb{D}^{2}}\mathbf{E}\cdot d\mathbf{x}=0$

which, in the limit $\Delta t\to 0$ gives the Faraday law:

$\frac{1}{c}\frac{d\Phi(\mathbf{B},\mathbb{D}^{2})}{dt}+\oint_{\partial\mathbb{D}^{2}}\mathbf{E}\cdot
d\mathbf{x}=0.$ (63)

Now we consider equation (62), which corresponds to the Maxwell equations with
the matter sources. Taking $\Omega$ to be a 3-dimensional spatial volume at a
given time, its l.h.s becomes

$\displaystyle\oint_{\partial\Omega}\widetilde{F}_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=\oint_{\partial\Omega}\widetilde{F}_{ij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau=-\oint_{\partial\Omega}E^{k}\epsilon_{kij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma
d\tau=-\oint_{\partial\Omega}\mathbf{E}\cdot d\mathbf{S},$

i.e., the flux of the electric field across the border of that spatial volume.
The r.h.s of (62), when evaluated in that spatial volume, reads

$\frac{4\pi}{c}\int_{\Omega}J^{\gamma}\epsilon_{\mu\nu\lambda\gamma}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau
d\zeta=\frac{4\pi}{c}\int_{\Omega}J^{0}\epsilon_{ijk0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}\frac{\partial
x^{k}}{\partial\zeta}d\sigma d\tau d\zeta=-4\pi\int_{\Omega}\rho
d^{3}\mathbf{x}$

and equation (62) gives the Gauss law for the electric field.

Finally, considering $\Omega=\mathbb{D}^{2}\times\mathbb{R}$, that cylindrical
volume in space and time, with the same parameterization as before, the l.h.s
of (62) becomes

$\oint_{\partial\Omega}\widetilde{F}_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma
d\tau=\Phi(\mathbf{E},\mathbb{D}^{2}_{0})-\Phi(\mathbf{E},\mathbb{D}^{2}_{t})+c\int_{0}^{t}\int_{0}^{2\pi}B^{i}\frac{\partial
x^{i}}{\partial\sigma}d\sigma dt^{\prime},$

while its r.h.s reads

$\frac{4\pi}{c}\int_{\Omega}J^{\gamma}\epsilon_{\mu\nu\lambda\gamma}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau
d\zeta=\frac{4\pi}{c}\int_{\Omega}J^{k}\epsilon_{ij0k}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}\frac{\partial
x^{0}}{\partial\zeta}d\sigma d\tau
d\zeta=4\pi\int_{0}^{t}\int_{\mathbb{D}^{2}}j^{k}\epsilon_{kij}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma d\tau
dt^{\prime}.$

For an infinitesimal time lapse, equation (62) gives

$-\frac{1}{c}\frac{\Delta\Phi(\mathbf{E},\mathbb{D}^{2})}{\Delta
t}+\oint_{\partial\mathbb{D}^{2}}\mathbf{B}\cdot
d\mathbf{x}=\frac{4\pi}{c}\Phi(\mathbf{j},\mathbb{D}^{2}),$

which in the limit $\Delta t\to 0$ defines the Ampère-Maxwell law of
induction:

$-\frac{1}{c}\frac{d\Phi(\mathbf{E},\mathbb{D}^{2})}{dt}+\oint_{\partial\mathbb{D}^{2}}\mathbf{B}\cdot
d\mathbf{x}=\frac{4\pi}{c}\Phi(\mathbf{j},\mathbb{D}^{2}).$ (64)

###  4.1 The conservation of the electric charge in 4 dimensions

By construction, the loop space connection is flat, i.e,
$\delta\mathcal{A}=0$, since it is defined in terms of an exact field, $H=dB$
integrated over closed surfaces. This implies locally that the electric charge
is conserved when we consider the definition of $B$ in terms of the physical
fields and take into account the differential Maxwell equations:

$\mathcal{A}=\frac{i4\pi\beta}{c}\oint_{\Sigma}J^{\gamma}\epsilon_{\mu\nu\lambda\gamma}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\delta
x^{\lambda}d\sigma d\tau$ (65)

and

$\delta\int\mathcal{A}d\zeta=-\frac{i4\pi\beta}{c}\int\partial_{\mu}J^{\mu}d^{4}x=0.$
(66)

In order to obtain the conserved charges we consider the generalization of the
holonomy operator given by the $3$-holonomy $U$, satisfying the parallel
transport equation

$\frac{dU}{d\zeta}-\int_{0}^{2\pi}\int_{0}^{2\pi}H_{\mu\nu\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau\;U=0$ (67)

whose solution can be formally written as

$U_{\Omega}=e^{\int_{\Omega}H_{\mu\nu\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau d\zeta}\;U_{0},$ (68)

where $U_{0}$ is defined by the initial conditions.

Clearly666See appendix A., the flatness of the connection implies that the
3-holonomy operator is path-independent in loop space.

Now, we split space-time into space and time and construct a volume whose
border changes in time from $t=0$. Since the 3-holonomy operator is path
independent, let us use the following convenient path: a composition
$\Gamma=\Gamma_{L}\circ\Gamma_{0}$ where $\Gamma_{0}$ starts at $\Sigma_{0}$
and goes to $\partial\Omega_{0}$, the border of the completely spatial volume
$\Omega$ at constant time $t=0$ and $\Gamma_{L}$ a path which starts at
$\partial\Omega_{0}$ and changes only in the time direction ending at
$\partial\Omega_{t}$, the purely spatial volume at time $t$.

Figure 9: The path independence of the 3-holonomy defines the eingenvalues of
this operator over a spatial volume as the conserved charges.

The relevant quantity in the construction of the $3$-holonomy operator
$U_{\Gamma}=U_{\Gamma_{L}}\cdot U_{\Gamma_{0}}$ is the integral of the
connection in $\zeta$. Then we have

$\displaystyle\int_{\Gamma_{0}}\mathcal{A}(\zeta)d\zeta$ $\displaystyle=$
$\displaystyle\frac{i4\pi}{c}\beta\int_{\Omega_{0}}J^{0}\epsilon_{ijk0}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}\frac{\partial
x^{k}}{\partial\zeta}d\sigma d\tau d\zeta=i4\pi\beta Q(0)$ (69)
$\displaystyle\int_{\Gamma_{L}}\mathcal{A}(\zeta)d\zeta$ $\displaystyle=$
$\displaystyle
i4\pi\beta\int_{0}^{t}\int_{\partial\Omega}J^{k}\epsilon_{ij0k}\frac{\partial
x^{i}}{\partial\sigma}\frac{\partial x^{j}}{\partial\tau}d\sigma d\tau
dt=i4\pi\beta\int_{0}^{t}\Phi(\mathbf{j},\partial\Omega)dt,$ (70)

where $Q(0)$ stands for the electric charge at time $t=0$.

Now, assuming the charge distribution to be localized, then at $r\to\infty$ we
have $|\mathbf{j}|\sim r^{-(2+\varepsilon)}$, $\varepsilon>0$ and this means
that $U_{\Gamma_{L}}=1$ if the border $\partial\Omega_{0}$ is far enough from
the charges. So, we end up with (considering $U_{0}=1$ for simplicity)

$U_{\Gamma}=e^{i4\pi\beta Q(0)}.$ (71)

Next we consider a similar path joining the same two points in loop space:
$\Gamma^{\prime}=\Gamma_{t}\circ\Gamma_{x_{R}}$ where now $\Gamma_{t}$ is the
path joining the closed surface $\Sigma_{0}$ at time $t=0$ to the surface
$\Sigma_{t}$ at a later time and $\Gamma_{x_{R}}$ is the purely spatial path
joining the surface $\Sigma_{t}$ to the surface $\partial\Omega_{t}$, at
constant time $t$.

The operator $U_{\Gamma_{x_{R}}}$ will become the unity when the radius of the
sphere at the reference point goes to zero and we remain with

$U_{\Gamma^{\prime}}=e^{i4\pi\beta Q(t)}.$ (72)

Now, the path-independence of $U$ dictates that
$U_{\Gamma^{\prime}}=U_{\Gamma}$, given that the two paths $\Gamma$ and
$\Gamma^{\prime}$ are homotopically equivalent in loop space. With the
considerations above for the behaviour of the fields at spatial infinity, this
relation gives

$e^{i4\pi\beta\Delta Q}=1\qquad\Delta Q\equiv Q(t)-Q(0)$ (73)

defining the conservation of the electric charge, which is given by the
eigenvalues of the operator

$U_{\Gamma_{t}}=e^{i4\pi\beta Q(t)}.$ (74)

##  5 Conclusions

We have shown that it is possible to formulate the integral equations of
electrodynamics in $(D+1)$-dimensional space-time $\mathcal{M}$ as the
integral version of the zero-curvature equation in the loop space
$\mathcal{L}^{(D-1)}(\mathcal{M})$:

$\Delta\varphi=\int_{\Gamma}\mathcal{A}(s)ds$ (75)

where $\varphi$ defines the integral of a $(D-1)$-form over a
$(D-1)$-dimensional hyper-surface in space-time and $\mathcal{A}$ is the
corresponding flat connection given in terms of an exact $D$-form. In
particular, for $D=1,2,3$ we have respectively

$\varphi=f,\qquad\varphi=\int_{\gamma}C_{\mu}\frac{dx^{\mu}}{d\sigma}d\sigma,\qquad\varphi=\int_{\Sigma}B_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma d\tau$
(76)

and

$\mathcal{A}=A_{\mu}\delta
x^{\mu},\qquad\mathcal{A}=\oint_{\gamma}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\delta
x^{\nu}d\sigma,\qquad\mathcal{A}=\oint_{\Sigma}H_{\mu\nu\lambda}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\delta
x^{\lambda}d\sigma d\tau$ (77)

with

$A=df,\qquad G=dC,\qquad H=dB.$ (78)

Generally speaking, the connection defined in
$\mathcal{L}^{(D-1)}(\mathcal{M})$ is given in terms of the components of the
$D$-form in space-time
$\omega=\omega_{\mu_{1}\dots\mu_{D}}dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{D}}$ as

$\mathcal{A}=\oint_{\mathcal{K}}\omega_{\mu_{1}\dots\mu_{D}}\frac{\partial
x^{\mu_{1}}}{\partial s_{1}}\dots\frac{\partial x^{\mu_{D-1}}}{\partial
s_{D-1}}\delta x^{\mu_{D}}ds_{1}\dots ds_{D-1}.$ (79)

This integral representation in loop space is equivalent to the differential
equation

$\mathcal{A}=\delta\varphi.$ (80)

so that $\mathcal{A}$ is a pure gauge connection. The local conservation law
for the electric charge becomes a consequence of the mathematical identity
$\delta\mathcal{A}=\delta^{2}\varphi=0$ in loop space. The flatness of the
connection in loop space implies that the generalized holonomy operator
defined by

$\frac{d\mathcal{W}}{ds}+(-1)^{(D-1)}\mathcal{A}(s)\;\mathcal{W}=0$ (81)

whose solution can be formally written as

$\mathcal{W}=e^{(-1)^{D}\int_{\Gamma}\mathcal{A}(s)ds}\mathcal{W}_{\circ}$
(82)

is path-independent.

A homotopic variation of $\Gamma$ implies a variation of $\mathcal{W}$ which
depends on the curvature of $\mathcal{A}$ and so, $\delta\mathcal{W}=0$.
Therefore, the equations of electrodynamics have a symmetry under the
homotopic transformations of the path in loop space and the conserved charges
are obtained as the eigenvalues of the generalized holonomy, evaluated at
constant time.

Aknowledgments

G. L. would like to thank L.A. Ferreira and H. Malavazzi for valuable
discussions on the subject of the paper.

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##  Appendix A The construction of the generalized holonomies and their path-
independence in loop space

In what follows we present the reasoning behind the construction of the
generalized holonomies which define the parallel transport operators in
$\mathcal{L}^{(1)}(\mathcal{M})$ and $\mathcal{L}^{(2)}(\mathcal{M})$. This is
a review but for the abelian case of what was first introduced in [5] and
later discussed in [6] and [7].

Consider the holonomy $W$ defined by the parallel transport equation

$\frac{dW}{d\sigma}+A_{\mu}\frac{dx^{\mu}}{d\sigma}W=0$ (83)

along a curve $\gamma$. Suppose this curve to be closed and obtained by
continuous deformations $x^{\mu}\to x^{\mu}+\delta x^{\mu}$ from a curve
$\gamma_{0}$ sharing a common point $x_{R}$ with $\gamma$.

We label the set of homotopic closed curves with $\tau\in[0,2\pi]$ such that
$\gamma$ has $\tau=2\pi$ and $\gamma_{0}$, $\tau=0$.

Let us assume that we know $W$ on $\gamma_{0}$. For instance, if $\gamma_{0}$
is the point-loop at $x_{R}$, W can be taken as $W_{0}$, an initial value for
(83).

Now, if we want to get $W$ over $\gamma$, then instead of integrating (83)
directly we can compute the change of $W$ which is calculated over
$\gamma_{0}$ while this path is deformed into $\gamma$.

For an infinitesimal deformation of the path we have $W\to W+\delta W$ and the
variation $\delta W$ can be obtained from equation (83) as follows. We start
by cosidering the variation of the equation as a whole:

$\delta\frac{dW}{d\sigma}+\delta\left(A_{\mu}\frac{dx^{\mu}}{d\sigma}\right)W+A_{\mu}\frac{dx^{\mu}}{d\sigma}\delta
W=0.$

Then we multiply this expression by $W^{-1}$ and rewrite the first term
getting

$\frac{d}{d\sigma}\left(W^{-1}\delta
W\right)+W^{-1}\frac{dW}{d\sigma}W^{-1}\delta
W+\delta\left(A_{\mu}\frac{dx^{\mu}}{d\sigma}\right)+W^{-1}A_{\mu}\frac{dx^{\mu}}{d\sigma}\delta
W=0$

and using equation (83) the second and fourth terms cancel each and what
remains is

$\frac{d}{d\sigma}\left(W^{-1}\delta
W\right)+\delta\left(A_{\mu}\frac{dx^{\mu}}{d\sigma}\right)=0.$

This equation can be integrated in $\sigma\in[0,2\pi]$ giving

$\delta
W=-\int_{0}^{2\pi}\delta\left(A_{\mu}\frac{dx^{\mu}}{d\sigma}\right)d\sigma$

and finally, calculating the variation of the term in the r.h.s:

$\delta
W=-\int_{0}^{2\pi}\left(\partial_{\nu}A_{\mu}\frac{dx^{\mu}}{d\sigma}\delta
x^{\nu}-\partial_{\nu}A_{\mu}\frac{dx^{\nu}}{d\sigma}\delta
x^{\mu}\right)d\sigma-A_{\mu}\delta x^{\mu}\Big{|}_{0}^{2\pi}.$

Considering that for the deformations of the loop, $\delta
x^{\mu}(\sigma=0)=\delta x^{\mu}(\sigma=2\pi)=0$ and defining
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ we get that the
change of the holonomy due to a deformation of the loop is

$\delta W=\left(\int_{0}^{2\pi}F_{\mu\nu}\frac{dx^{\mu}}{d\sigma}\delta
x^{\nu}d\sigma\right)W.$ (84)

This shows that if the connection $A=A_{\mu}dx^{\mu}$ is flat, i.e., if
$F_{\mu\nu}=0$, then the holonomy is independent of the path over which it is
calculated if these paths can be deformed into each other while their end-
points remain fixed.

Now, since the set of loops is parameterized by $\tau\in[0,2\pi]$, we can
write the expression above for the variation of the holonomy into a
differential equation for $W$

$\frac{dW}{d\tau}-\left(\int_{0}^{2\pi}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial
x^{\nu}}{\partial\tau}d\sigma\right)W=0$ (85)

and the holonomy over $\gamma$ can be obtained by integrating this equation
from $\tau=0$ up to $\tau=2\pi$.

Now if we consider the loop space $\mathcal{L}^{(1)}(\mathcal{M})$, the
quantity

$\mathcal{A}(\tau)\equiv\int_{0}^{2\pi}F_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}d\sigma$

defines a connection evaluated at each loop in space-time which correspond to
points in the loop space and by varrying these loops with continuous
transformations we defined a path $\Gamma$ in
$\mathcal{L}^{(1)}(\mathcal{M})$.

The above equation which defines $W$ can be seen as the parallel transport
equation in this loop space and therefore we consider the generalization of
this equations and define the $2$-holonomy $V$ as satisfying

$\frac{dV}{d\tau}-\left(\int_{0}^{2\pi}G_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial
x^{\nu}}{\partial\tau}d\sigma\right)V=0,$ (86)

where $G_{\mu\nu}$ is an anti-symmetric tensor.

Now, let us consider a closed $2$-dimensional surface $\Sigma$ which can be
obtained by continuous deformations from another closed surface $\Sigma_{0}$
sharing a common point $x_{R}$ with each other. The $2$-holonomy $V$ can be
calculated over $\Sigma$ by direct integration of (86) but can also be
obtained as the result of the deformation of the surface $\Sigma_{0}$ into
$\Sigma$, once $V$ over $\Sigma_{0}$ is known.

In order to get $V$ following this second approach we need to find how it
varies when we deform the closed surface. This is done in a similar way as it
was done before to find $\delta W$: we start by considering the variation of
the equation (86), then multiply the result by $V^{-1}$ and finally compute
the variation of the term containing $G_{\mu\nu}$ explicitly and integrate the
expression in $\tau$. This shall leave us with

$\delta
V=\int_{0}^{2\pi}\int_{0}^{2\pi}\left(\partial_{\lambda}G_{\mu\nu}+\partial_{\mu}G_{\nu\lambda}+\partial_{\nu}G_{\lambda\mu}\right)\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\delta
x^{\lambda}d\sigma d\tau.$ (87)

From here we see that if $G$ is an exact $2$-form, then the $2$-holonomy is
surface independent.

Now, parameterizing this variation with $\zeta\in[0,2\pi]$, such that
$\zeta=0$ labels the surface $\Sigma_{0}$ and $\zeta=2\pi$, the surface
$\Sigma$, we get the differential equation

$\frac{dV}{d\zeta}-\int_{0}^{2\pi}\int_{0}^{2\pi}\left(\partial_{\lambda}G_{\mu\nu}+\partial_{\mu}G_{\nu\lambda}+\partial_{\nu}G_{\lambda\mu}\right)\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau=0,$ (88)

and integrating this equation in $\zeta$ will give us the desired $V$ on the
surface $\Sigma$.

In the loop space $\mathcal{L}^{(2)}(\mathcal{M})$ the above equation defines
the parallel transport through a $3$-dimensional volume in space-time, which
corresponds to a path $\Gamma$ in the loop space, parameterized by $\zeta$.
This leads us to introduce the $3$-holonomy $U$ defined by

$\frac{dU}{d\zeta}-\int_{0}^{2\pi}\int_{0}^{2\pi}H_{\lambda\mu\nu}\frac{\partial
x^{\mu}}{\partial\sigma}\frac{\partial x^{\nu}}{\partial\tau}\frac{\partial
x^{\lambda}}{\partial\zeta}d\sigma d\tau=0,$ (89)

with $H_{\lambda\mu\nu}$ a completely anti-symmetric tensor.