# New formulation of Galilean relativistic Maxwell theory

Rabin Banerjee 111DAE Raja Ramanna fellow Soumya Bhattacharya  Department of
Astrophysics and High Energy Physics, S.N. Bose National Center for Basic
Sciences, Kolkata 700106, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we discuss Galilean relativistic Maxwell theory in detail. We
first provide a set of mapping relations, derived systematically, that connect
the covariant and contravariant vectors in the Lorentz relativistic and
Galilean relativistic formulations. Exploiting this map, we construct the two
limits of Galilean relativistic Maxwell theory from usual Maxwell’s theory in
the potential formalism for both contravariant and covariant vectors which are
now distinct entities. Field equations are derived and their internal
consistency is shown. The entire analysis is then performed in terms of
electric and magnetic fields for both covariant and contravariant components.
Duality transformations and their connection with boost symmetry are discussed
which reveal a rich structure. The notion of twisted duality is introduced.
Next we consider gauge symmetry, construct Noether currents and show their on-
shell conservation. We also discuss shift symmetry under which the Lagrangian
is invariant, where the corresponding currents are now on-shell conserved. At
the end we analyse the theory by including sources for both contravariant and
covariant sectors. We show that sources are now off-shell conserved

##  I Introduction

The formulation of non-relativistic limit of classical field theories received
considerable attention recently. It has found applications in holography [1],
in studying non-relativistic diffeomorphisms (NRDI) [2, 3, 4, 5, 6], condensed
matter systems [7, 8, 9], fluid dynamics [10, 11], gravitation [12, 13]. This
formulation is tricky and markedly different from the relativistic case.
Covariance in non-relativistic physics is subtle due to the absolute nature of
time. The lack of a single non-degenerate metric in the non-relativistic limit
poses some additional difficulties. Here we are interested in the non-
relativistic limit of Maxwellean electrodynamics which is invariant under
galilean transformations. The basic construction of Galilean electrodynamics
was first given by Le Bellac and Levy-Leblond [14] back in 1970’s. This was
done in the field formulation. A similar field based analysis was done in [15]
using embedding techniques. Further directions in this type of analysis were
provided in [16]. Other references on different aspects of Galilean
electrodynamics and gauge theories are [17, 18, 19, 20, 21, 22, 23].
In this paper we provide a detailed analysis of galilean relativistic Maxwell
theory with and without sources. While earlier findings are reproduced we also
find several new results with new interpretations. We know there are two
distinct non-relativistic limits possible for electrodynamics known as
electric and magnetic limit [14, 16]. We derive these two limits from the
Lorentz transformation of an arbitrary four vector. Our derivation of the non-
relativistic scaling relations are consistent with [14, 16]. All previous
works so far treated only the contravariant components of any vector quantity
but the novelty of our treatment is that we have considered both contravariant
and covariant components separately (which are distinct quantities in the non-
relativistic case) and hence help us to explore the rich symmetries involved
in the theory. We then derive the Lagrangians for both electric and magnetic
limits from which the equations of motion are obtained. We also show that
Maxwell’s equations under non-relativistic limit (electric and magnetic) yield
same equations as those we get from the non-relativistic Lagrangians. This
implies the internal consistency of the limiting process. We observe that at
the equations of motion level if we replace the covariant vector components by
the corresponding contravariant ones then the electric limit and magnetic
limits get interchanged. We then define the galilean electric and magnetic
fields for contravariant and covariant cases. Along the way, we discuss the
dualities and point out some of the subtleties involved in the process. In
this regard we observe one unique aspect of the duality symmetries called
twisted duality which is valid only in the galilean limit. Especially we show
that the transformations of the electric and magnetic fields under galilean
boosts is connected with the familiar duality transformations. Next we move to
discuss gauge symmetry. We have shown that we can choose different gauge
parameters for contravariant and covariant four potentials as they represent
different entities in the galilean limits. We then compute the galilean
version of the Noether currents and explicitly show their on-shell
conservation. We discuss shift symmetries which play an important role in the
study of low-energy effective lagrangians in the context of Goldstone’s
theorem. Recently people have explored shift symmetries from different aspects
[24, 25]. We compute corresponding currents and their conservations in this
limit. In the end we introduce sources and write down the Lagrangians for
appropriate galilean limits (electric and magnetic) and the equations of
motion just like the sourceless case.
The paper is organised as follows: in section II we derive mapping relations
between relativistic and non-relativistic vectors for electric and magnetic
limit for both contravariant and covariant vectors. In section III we derive
the non-relativistic lagrangian for both limits and write down the equations
of motion. We discuss Maxwell’s equations in terms of fields and explore the
duality relations in IV. Gauge symmetry, Noether currents and their
conservations are discussed in V. In section VI we discuss shift symmetry and
its galilean counterpart, corresponding currents and their conservations. In
section VII, discussion has been done by including sources for both
contravariant and covariant sectors. Finally, conclusions have been given in
VIII.

##  II  Mapping relations

Here we derive a certain scaling between special relativistic and Galilean
relativistic quantities. As we know there exists two types of such limits for
the vector quantities namely electric and magnetic limits. So first let us
consider the contravariant vectors. Let us consider a generic Lorentz
transformation with the boost velocity as $u^{i}$:

$x^{\prime 0}=\gamma x^{0}-\frac{\gamma u_{i}}{c}x^{i}$ (1) $x^{\prime
i}=x^{i}-\frac{\gamma u^{i}}{c}x^{0}+(\gamma-1)\frac{u^{i}u_{j}}{u^{2}}x^{j}$
(2)

where $\gamma=\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}$. Under such Lorentz
transformations a contravariant vector changes as

$V^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}V^{\nu}$

We can write them component-wise as (also considering $u<<c$, so $\gamma\to
1$)

$V^{\prime 0}=V^{0}-\frac{u_{j}}{c}V^{j}$ (3) $V^{\prime
i}=V^{i}-\frac{u^{i}}{c}V^{0}$ (4)

We next provide a map that relates the Lorentz vectors with their Galilean
counterparts. 222Notation: Here relativistic vectors are denoted by capital
letters ($V^{0},V^{i}$ etc) and Galilean vectors are denoted by lowercase
letters ($v^{0},v^{i}$ etc).

$V^{0}=cv^{0},\,\,\,\,V^{i}=v^{i}$ (5)

This particular map corresponds to the case
$\frac{V^{0}}{V^{i}}=c~{}\frac{v^{0}}{v^{i}}$ in the $c\to\infty$ limit. This
yields largely timelike vectors and is called ’electric limit’. Now using eqn
5 in eqns 3 and 4 we get

$v^{\prime 0}=v^{0}$ (6) $v^{\prime i}=v^{i}-u^{i}v^{0}$ (7)

The above two equations define the Galilean transformations. We can write them
in a single matrix equation as

$\begin{pmatrix}v^{\prime 0}\\\ v^{\prime
i}\end{pmatrix}=\begin{pmatrix}1&0\\\
-u^{i}&1\end{pmatrix}\begin{pmatrix}v^{0}\\\ v^{i}\end{pmatrix}$ (8)

We now consider the magnetic limit which corresponds to largely spacelike
vectors

$V^{0}=-\frac{v^{0}}{c},\,\,\,\,V^{i}=v^{i}$ (9)

Now using 9 in 3 and 4 we get

$v^{\prime 0}=v^{0}+u_{j}v^{j}$ (10) $v^{\prime i}=v^{i}$ (11)

which is again a galilean transformation. We can write eqn 10 and 11 as a
matrix equation

$\begin{pmatrix}v^{\prime 0}\\\ v^{\prime
i}\end{pmatrix}=\begin{pmatrix}1&u_{j}\\\
0&1\end{pmatrix}\begin{pmatrix}v^{0}\\\ v^{j}\end{pmatrix}$ (12)

We will now consider the covariant vectors. We will write first the reverse
transformations of eqn 1 and 2 which is

$x^{0}=\gamma x^{\prime 0}+\frac{\gamma u_{i}}{c}x^{\prime i}$ (13)
$x^{i}=x^{\prime i}+\frac{\gamma u^{i}}{c}x^{\prime
0}+(\gamma-1)\frac{u^{i}u_{j}}{u^{2}}x^{\prime j}$ (14)

And we know covariant vectors transform as

$V^{\prime}_{\mu}=\frac{\partial x^{\nu}}{\partial x^{\prime\mu}}V_{\nu}$

Componentwise we can again write them as

$V^{\prime}_{0}=V_{0}+\frac{u^{i}}{c}V_{i}$ (15)
$V^{\prime}_{i}=V_{i}+\frac{u_{i}}{c}V_{0}$ (16)

Now here we take the electric limit in the following way, which will soon
become clear

$V_{0}=\frac{v_{0}}{c},\,\,\,\,V_{i}=v_{i}$ (17)

Using 17 in 15 and 16 we get

$\displaystyle v^{\prime}_{0}=v_{0}+u^{i}v_{i}$ (18) $\displaystyle
v^{\prime}_{i}=v_{i}$ (19)

which are again Galilean transformations. We can write 18 and 19 as a matrix
equation as

$\begin{pmatrix}v^{\prime}_{0}\\\
v^{\prime}_{i}\end{pmatrix}=\begin{pmatrix}1&u_{i}\\\
0&1\end{pmatrix}\begin{pmatrix}v_{0}\\\ v_{i}\end{pmatrix}$ (20)

We will now consider the magnetic limit as

$V_{0}=-cv_{0},\,\,\,\,V_{i}=v_{i}$ (21)

Using 21 in 15 and 16 we get

$v_{0}^{\prime}=v_{0}$ (22) $v^{\prime}_{i}=v_{i}-u_{i}v_{0}$ (23)

We can write 22 and 23 as

$\begin{pmatrix}v^{\prime}_{0}\\\
v^{\prime}_{i}\end{pmatrix}=\begin{pmatrix}1&0\\\
-u_{i}&1\end{pmatrix}\begin{pmatrix}v_{0}\\\ v_{i}\end{pmatrix}$ (24)

We can show that the transformation matrix in 8 and the transpose of the
matrix 20 satisfies

$\begin{pmatrix}1&0\\\ -v^{i}&1\end{pmatrix}\begin{pmatrix}1&0\\\
v^{i}&1\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ (25)

Similarly the transformation matrix in equation 12 and the transpose of the
transformation matrix in 24 satisfies

$\begin{pmatrix}1&v_{j}\\\ 0&1\end{pmatrix}\begin{pmatrix}1&-v_{j}\\\
0&1\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ (26)

To justify the limiting prescriptions even further, we consider the norm
preservation for both electric and magnetic limits. Let us first consider the
norm in the electric limit

$V^{0}V_{0}+V^{i}V_{i}\xrightarrow[\text{limit}]{\text{electric}}\Big{(}cv^{0}\Big{)}\Big{(}\frac{v_{0}}{c}\Big{)}+\Big{(}v^{i}\Big{)}\Big{(}v_{i}\Big{)}=v^{0}v_{0}+v^{i}v_{i}$
(27)

which clearly indicates that under the scaling, the norm is preserved.
Likewise, the norm in the magnetic limit is also conserved.

Table 1: Mapping relations ${\rm Limit}$ | ${\rm Contravariant~{}~{}mapping}$ | ${\rm Covariant~{}~{}mapping}$
---|---|---
${\rm Electric~{}~{}limit}$ | $V^{0}\to c~{}v^{0},\,\,V^{i}\to v^{i}$ | $V_{0}\to\frac{v_{0}}{c},\,\,V_{i}\to v_{i}$
${\rm Magnetic~{}~{}limit}$ | $V^{0}\to-\frac{v^{0}}{c},\,\,V^{i}\to v^{i}$ | $V_{0}\to-c~{}v_{0}.\,\,V_{i}\to v_{i}$

The mapping relations, systematically derived here for both covariant and
contravariant components, are essential to the subsequent analysis. Any four
vector in relativistic theory will be replaced by the corresponding structure
for the Galilean theory by adopting this map. These relations are summarised
in table 1.

##  III Lagrangian and field equations

Now let us start from the relativistic Maxwell theory described by the
Lagrangian

$\mathcal{L}=-\frac{1}{4}~{}F_{\mu\nu}~{}F^{\mu\nu}=-\frac{1}{4}\eta_{\mu\alpha}\eta_{\nu\beta}F^{\alpha\beta}F^{\mu\nu}$
(28)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$\eta_{\mu\nu}$ is the flat space metric with signature
$\Big{(}-,+,+,+\Big{)}$.

###  III.1 Electric limit

Now using the relations given in table 1 we can write the two terms in 28 as,

$\displaystyle 2F_{0i}F^{0i}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}-2\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}\hskip
72.26999pt$ (29)

$\displaystyle F_{ij}F^{ij}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}\Big{(}\partial_{i}a_{j}-\partial_{j}a_{i}\Big{)}\Big{(}\partial^{i}a^{j}-\partial^{j}a^{i}\Big{)}\equiv
f_{ij}f^{ij}$ (30)

Here $A^{\mu}$ is the relativistic four potential while $a^{0}$ and $a^{i}$
are it’s galilean counterpart. So in the electric limit the full lagrangian
takes the following form

$\mathcal{L}_{e}=\frac{1}{2}\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}-\frac{1}{4}f_{ij}f^{ij}$
(31)

Now we derive the equations of motion. Varying the Lagrangian 31 with respect
to $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$ we get the corresponding equations of
motion

$\displaystyle\partial_{i}\partial^{i}a^{0}=0$ (32)
$\displaystyle\partial_{t}\partial^{j}a^{0}+\partial_{i}\partial^{j}a^{i}-\partial_{i}\partial^{i}a^{j}=0$
(33)
$\displaystyle\partial^{i}\partial_{t}a_{i}-\partial^{i}\partial_{i}a_{0}=0$
(34)
$\displaystyle\partial^{i}\partial_{i}a_{j}-\partial^{i}\partial_{j}a_{i}=0$
(35)

We now derive these equations directly from the equations of motion. The
relativistic equations, in component form, are given by,

$\displaystyle\partial_{i}F^{i0}=0,\,\,\,\,\,\,~{}~{}~{}~{}~{}~{}\partial_{0}F^{0j}+\partial_{i}F^{ij}=0$
(36)

The Galilean version of these equations is found by using table 1, followed by
taking $c\to\infty$. It reproduces 32 and 33 respectively. To get the
remaining pair of equations we have to interpret the relativistic eqns given
in 36 as

$\displaystyle\partial^{i}F_{i0}=0,\,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial^{0}F_{0j}+\partial^{i}F_{ij}=0$
(37)

Once again the Galilean version is obtained from table 1, followed by taking
$c\to\infty$. Equations 34, 35 are reproduced. This shows the consistency of
the eqn of motion in Galilean electrodynamics.

###  III.2 Magnetic limit

Here again using the relations given in table 1 we can write the two terms in
28 as,

$\displaystyle 2F_{0i}F^{0i}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}-2\partial_{i}a_{0}\Big{(}\partial_{t}a^{i}-\partial^{i}a^{0}\Big{)}\hskip
72.26999pt$ (38) $\displaystyle
F_{ij}F^{ij}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}\Big{(}\partial_{i}a_{j}-\partial_{j}a_{i}\Big{)}\Big{(}\partial^{i}a^{j}-\partial^{j}a^{i}\Big{)}\equiv
f_{ij}f^{ij}$ (39)

So the Lagrangian will take the following form

$\mathcal{L}_{m}=\frac{1}{2}\partial_{i}a_{0}\Big{(}\partial_{t}a^{i}-\partial^{i}a^{0}\Big{)}-\frac{1}{4}f_{ij}f^{ij}$
(40)

Varying 40 wrt $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$ we get,

$\displaystyle\partial_{i}~{}\partial_{t}~{}a^{i}-\partial^{i}~{}\partial_{i}~{}a^{0}=0$
(41)
$\displaystyle\partial_{j}\partial^{i}a^{j}-\partial_{j}\partial^{j}a^{i}=0$
(42) $\displaystyle\partial^{i}\partial_{i}a_{0}=0$ (43)
$\displaystyle\partial_{t}\partial_{j}a_{0}+\partial^{i}\partial_{j}a_{i}-\partial^{i}\partial_{i}a_{j}=0$
(44)

Here also we can show that the above equations agree with those derived
directly from relativistic Maxwell equations 36 and 37 corresponding to
contravariant and covariant sectors respectively, by taking the magnetic
limit.

Table 2: Field equations ${\rm Variables}$ | ${\rm Electric~{}~{}limit}$ | ${\rm Magnetic~{}~{}limit}$
---|---|---
$a^{0}$ | $\partial^{i}\partial_{t}a_{i}-\partial^{i}\partial_{i}a_{0}=0$ | $\partial^{i}\partial_{i}a_{0}=0$
$a^{i}$ | $\partial^{j}\partial_{i}a_{j}-\partial^{j}\partial_{j}a_{i}=0$ | $\partial_{t}\partial_{i}a_{0}+\partial^{j}\partial_{i}a_{j}-\partial^{j}\partial_{j}a_{i}=0$
$a_{0}$ | $\partial^{i}\partial_{i}a^{0}=0$ | $\partial_{i}~{}\partial_{t}~{}a^{i}-\partial^{i}~{}\partial_{i}~{}a^{0}=0$
$a_{i}$ | $\partial_{t}\partial^{i}a^{0}+\partial_{j}\partial^{i}a^{j}-\partial_{j}\partial^{j}a^{i}=0$ | $\partial_{j}\partial^{i}a^{j}-\partial_{j}\partial^{j}a^{i}=0$

The field equations for both the limits of Galilean electrodynamics are shown
in table 2.

##  IV Galilean electric and magnetic fields and dual transformations

Here we introduce the galilean limit of electric and magnetic fields and write
down the Maxwell equations. For this purpose we will discuss Contravariant and
Covariant sector separately.

###  IV.1 Contravariant sector

Relativistic electric and magnetic fields are defined as

$\displaystyle E^{i}=\partial^{0}A^{i}-\partial^{i}A^{0}$ (45) $\displaystyle
B^{i}=\epsilon^{ij}_{~{}k}\partial_{j}A^{k}$ (46)

First, we consider the electric limit.
Electric limit:

Using the mapping relations given in table 1 we can write the electric field
as

$\displaystyle E^{i}=-\frac{1}{c}\partial_{t}a^{i}-c\partial^{i}a^{0}$ (47)

And we can define the Galilean electric and magnetic fields as

$e^{i}=\lim_{c\to\infty}\frac{E^{i}}{c}=-\partial^{i}a^{0}.\,\,\,~{}~{}~{}~{}~{}b^{i}=\lim_{c\to\infty}B^{i}=\epsilon^{ij}_{~{}k}\partial_{j}a^{k}$
(48)

Now we write the field equations that we derived in the previous section in
terms of the Galilean electric and magnetic fields. From 32 we get

$\displaystyle\partial_{i}\partial^{i}a^{0}=0\implies\partial_{i}(-e^{i})=0\implies\vec{\nabla}.\vec{e}=0$
(49)

Similarly eqn 33 implies

$\displaystyle\partial_{t}\partial^{j}a^{0}+\partial_{i}\partial^{j}a^{i}-\partial_{i}\partial^{i}a^{j}=0\implies(\vec{\nabla}\times\vec{b})^{j}=\partial_{t}e^{j}$
(50)

We can see clearly that

$\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=\partial_{i}\epsilon^{ij}_{~{}k}\partial_{j}a^{k}=\epsilon^{ij}_{~{}k}\partial_{i}\partial_{j}a^{k}=0$
(51)

We will now compute $\vec{\nabla}\times\vec{e}$,

$\displaystyle(\nabla\times
e)^{i}=\epsilon^{ij}_{~{}k}\partial_{j}e^{k}=\epsilon^{ij}_{~{}k}\partial_{j}(-\partial^{k}a^{0})=0$
(52)

So in electric limit we get the following set of equations

$\displaystyle\vec{\nabla}.\vec{e}=\partial_{i}e^{i}=0,$ (53)
$\displaystyle\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0,$ (54)
$\displaystyle(\vec{\nabla}\times\vec{e})^{i}=\epsilon^{ij}_{~{}k}\partial_{j}e^{k}=0,$
(55)
$\displaystyle(\vec{\nabla}\times\vec{b})^{i}=\epsilon^{ij}_{~{}k}\partial_{j}b^{k}=\partial_{t}(\vec{e})^{i}$
(56)

It is now possible to obtain the equations (53 \- 56) directly from Maxwell’s
equations,

$\vec{\nabla}.\vec{E}=0,~{}~{}~{}~{}\vec{\nabla}.\vec{B}=0,~{}~{}~{}~{}\vec{\nabla}\times\vec{E}=-\frac{1}{c}\frac{\partial\vec{B}}{\partial
t},~{}~{}~{}~{}\vec{\nabla}\times\vec{B}=\frac{1}{c}\frac{\partial\vec{E}}{\partial
t}$ (57)

by using the identification in eqn 48 and taking $c\to\infty$.
For the electric limit, it is seen from 56, a change in the electric field
influences the magnetic field. But a change in the magnetic field does not
influence the electric field since the R.H.S of 55 vanishes. This implies that
the electric field is considerably greater and dominates over the magnetic
field, justifying the nomenclature electric limit. This is different from the
relativistic case where electric and magnetic fields are treated
symmetrically. This asymmetry in Galilean electromagnetism leads to physical
effects, some of which have been discussed in [29].
We will now consider the magnetic limit.
Magnetic limit:

Electric field can be written in this limit from the mapping relations in 1 as

$E^{i}=-\frac{1}{c}\partial_{t}a^{i}+\frac{1}{c}\partial_{i}a^{0}$ (58)

And we can define the galilean electric and magnetic fields as

$e^{i}=\lim_{c\to\infty}cE^{i}=-(\partial_{t}a^{i}-\partial^{i}a^{0}),\,\,\,~{}~{}~{}~{}~{}b^{i}=\lim_{c\to\infty}B^{i}=\epsilon^{ij}_{~{}k}\partial_{j}a^{k}$
(59)

Using these relations, eqns 41, 42 and two Bianchi identities are expressed in
terms of electric/magnetic fields as

$\displaystyle\vec{\nabla}.\vec{e}=\partial_{i}e^{i}=0,$ (60)
$\displaystyle\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0,$ (61)
$\displaystyle(\vec{\nabla}\times\vec{e})^{i}=\epsilon^{ij}_{~{}k}\partial_{j}e^{k}=-\partial_{t}(\vec{b})^{i},$
(62)
$\displaystyle(\vec{\nabla}\times\vec{b})^{i}=\epsilon^{ij}_{~{}k}\partial_{j}b^{k}=0$
(63)

The above equations (60 \- 63) also follow from Maxwell’s equations 57, using
the map 59 and taking the limit $c\to\infty$. In contrast to the electric
limit, here a change in the magnetic field influences the electric field but
the converse does not hold. In this case the magnetic field dominates over the
electric field.
We can clearly see that equations in electric limit are mapped to those of
magnetic limit and vice versa under the following duality transformations,

$e^{i}\to b^{i},\,\,~{}~{}~{}~{}b^{i}\to-e^{i}$ (64)
$e^{i}\to-b^{i},\,\,~{}~{}~{}~{}b^{i}\to e^{i}$ (65)

This is the analogue of the electromagnetic duality in usual Maxwell’s source
free theory.

Table 3: Fields in galilean limit ${\rm Limits}$ | ${\rm Electric~{}field}$ | ${\rm Magnetic~{}field}$
---|---|---
${\rm Electric~{}limit}$ | $E^{i}\to ce^{i},~{}E_{i}\to\frac{e_{i}}{c}$ | $B^{i}\to b^{i},~{}B_{i}\to b_{i}$
${\rm Magnetic~{}limit}$ | $E^{i}\to\frac{e^{i}}{c},~{}E_{i}\to ce_{i}$ | $B^{i}\to b^{i},~{}B_{i}\to b_{i}$

###  IV.2 Covariant sector

Relativistic electric and the magnetic fields are defined as

$\displaystyle E_{i}=-\Big{(}\partial_{0}A_{i}-\partial_{i}A_{0}\Big{)}$ (66)
$\displaystyle B_{i}=\epsilon_{i}^{~{}jk}\partial_{j}a_{k}$ (67)

Now we consider the electric limit.
Electric limit:
In this limit the electric field looks like

$E_{i}=-\Big{(}\frac{1}{c}\partial_{t}a_{i}+\frac{1}{c}\partial_{i}a_{0}\Big{)}$
(68)

And we can define Galilean electric and magnetic fields as

$e_{i}=\lim_{c\to\infty}cE_{i}=-(\partial_{t}a_{i}-\partial_{i}a_{0}),\,\,\,~{}~{}~{}~{}~{}~{}b_{i}=\lim_{c\to\infty}B_{i}=\epsilon_{i}^{~{}jk}\partial_{j}a_{k}$
(69)

From eqn 34 we get

$\displaystyle\partial^{i}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\Big{)}=\partial^{i}(-e_{i})=\partial^{i}e_{i}=\vec{\nabla}.\vec{e}=0$
(70)

Eqn 35 yields

$\displaystyle\partial^{i}f_{ij}=-\epsilon_{ji}^{~{}~{}k}\partial^{i}b_{k}=0\implies\Big{(}\vec{\nabla}\times\vec{b}\Big{)}_{j}=0$
(71)

And finally calculation of $\vec{\nabla}\times\vec{e}$ yields,

$\displaystyle\Big{(}\vec{\nabla}\times\vec{e}\Big{)}_{i}=\epsilon_{i}^{~{}jk}\partial_{j}e_{k}=-\epsilon_{i}^{jk}\partial_{j}\Big{(}\partial_{t}a_{k}-\partial_{k}a_{0}\Big{)}=-\partial_{t}\epsilon_{i}^{~{}jk}\partial_{j}a_{k}=-\partial_{t}b_{i}$
(72)

So the Maxwell equations in the electric limit are

$\displaystyle\vec{\nabla}.\vec{e}=\partial^{i}e_{i}=0,$ (73)
$\displaystyle\vec{\nabla}.\vec{b}=\partial^{i}b_{i}=0,$ (74)
$\displaystyle\Big{(}\vec{\nabla}\times\vec{e}\Big{)}_{i}=\epsilon_{i}^{~{}jk}\partial_{j}e_{k}=-\partial_{t}(\vec{b})_{i},$
(75)
$\displaystyle\Big{(}\vec{\nabla}\times\vec{b}\Big{)}_{i}=\epsilon_{ij}^{~{}~{}k}\partial^{j}b_{k}=0$
(76)

The above equations (73 \- 76) also follow from Maxwell’s equations 57, using
the map 69 and taking the limit $c\to\infty$. Magnetic limit:

$A_{0}\to-ca_{0}\,\,\,\,A_{i}\to a_{i}$

In this limit electric field is scaled as

$E_{i}=-\Big{(}\frac{1}{c}\partial_{t}a_{i}+c\partial_{i}a_{0}\Big{)}$ (77)

And we can define the Galilean electric and magnetic fields as

$e_{i}=\lim_{c\to\infty}\frac{E_{i}}{c}=-\partial_{i}a_{0},\,\,\,~{}~{}~{}~{}~{}b_{i}=\lim_{c\to\infty}B_{i}=\epsilon_{i}^{~{}jk}\partial_{j}a_{k}$
(78)

The equations 43, 44 and two Bianchi identities are now written as

$\displaystyle\vec{\nabla}.\vec{e}=\partial^{i}e_{i}=0,$ (79)
$\displaystyle\vec{\nabla}.\vec{b}=\partial^{i}b_{i}=0,$ (80)
$\displaystyle\Big{(}\vec{\nabla}\times\vec{e}\Big{)}=\epsilon_{i}^{~{}jk}\partial_{j}e_{k}=0,$
(81)
$\displaystyle\Big{(}\vec{\nabla}\times\vec{b}\Big{)}_{i}=\epsilon_{ij}^{~{}~{}k}\partial^{j}b_{k}=\partial_{t}e_{i}$
(82)

The above equations (79 \- 82) also follow from Maxwell’s equations 57, using
the map 78 and taking the limit $c\to\infty$. Here also we see that electric
and magnetic fields satisfy certain duality relations as follows

$e_{i}\to b_{i},\,\,~{}~{}~{}~{}b_{i}\to-e_{i}$ (83) $e_{i}\to-
b_{i},\,\,~{}~{}~{}~{}b_{i}\to e_{i}$ (84)

The galilean limit scalings for the fields (electric/magnetic) are shown in
Table 3.

###  IV.3 Effect of the dualities at the level of Lagrangian

In the electric limit the Lagrangian is represented by eqn 31. Now in this
limit, the contravariant and covariant electric fields as well as magnetic
fields are represented by 48 and 69 respectively. Using these definitions we
can write the electric limit Lagrangian in the following form

$\mathcal{L}_{e}=\frac{1}{2}\Big{(}e^{i}e_{i}-b_{i}b^{i}\Big{)}$ (85)

Similarly in the magnetic limit the Lagrangian is represented by eqn 40. Here
electric and magnetic fields for contravariant and covaraint cases are given
in equations 59 and 78 respectively. So now the Lagrangian in this limit takes
the following form in terms of the fields

$\mathcal{L}_{m}=\frac{1}{2}\Big{(}e^{i}e_{i}-b_{i}b^{i}\Big{)}$ (86)

We observe that both Lagrangians (eqn 85 and 86) are identical

$\mathcal{L}_{e}=\mathcal{L}_{m}=\mathcal{L}$ (87)

In other words, expressed in terms of the gauge invariant fields (electric and
magnetic), the lagrangians in the two limits are same. This is to be
contrasted with the potential formulation where $\mathcal{L}_{e}$ and
$\mathcal{L}_{m}$ appear to be different. However if we interchange the
covariant and the contravariant indices then $\mathcal{L}_{e}$ and
$\mathcal{L}_{m}$ get interchanged. Similar things happen when the lagrangians
are expressed in terms of the electric and magnetic fields. However, since the
expressions are symmetrical with respect to the covariant and contravariant
indices, $\mathcal{L}_{e}$ and $\mathcal{L}_{m}$ become identical.
We observe there is an overall sign change (i.e $\mathcal{L}\to-\mathcal{L}$)
under the duality transformations ($e^{i}\to
b^{i},~{}b^{i}\to-e^{i},~{}e_{i}\to b_{i},~{}b_{i}\to-e_{i}$ or
$e^{i}\to-b^{i},~{}b^{i}\to e^{i},~{}e_{i}\to-b_{i},~{}b_{i}\to e_{i}$)
however the Lagrangians remain invariant (i.e $\mathcal{L}\to\mathcal{L}$)
under the twisted duality relations (i.e $e^{i}\to-b^{i},~{}b^{i}\to
e^{i},~{}e_{i}\to b_{i},~{}b_{i}\to-e_{i}$ or $e^{i}\to
b^{i},~{}b^{i}\to-e^{i},~{}e_{i}\to-b_{i},~{}b_{i}\to e_{i}$). This has been
shown clearly in table 4. We like to mention that the twisted relations have
not been discussed earlier.

Table 4: Effect of duality on the Lagrangian ${\rm Duality~{}relation}$ | ${\rm Change~{}in~{}the~{}Lagrangian}$
---|---
$e^{i}\to b^{i},~{}b^{i}\to-e^{i},~{}e_{i}\to b_{i},~{}b_{i}\to-e_{i}$ | $\mathcal{L}\to-\mathcal{L}$
$e^{i}\to-b^{i},~{}b^{i}\to e^{i},~{}e_{i}\to-b_{i},~{}b_{i}\to e_{i}$ | $\mathcal{L}\to-\mathcal{L}$
$e^{i}\to b^{i},~{}b^{i}\to-e^{i},~{}e_{i}\to-b_{i},~{}b_{i}\to e_{i}$ | $\mathcal{L}\to\mathcal{L}$
$e^{i}\to-b^{i},~{}b^{i}\to e^{i},~{}e_{i}\to b_{i},~{}b_{i}\to-e_{i}$ | $\mathcal{L}\to\mathcal{L}$

###  IV.4 Dual transformation of electric and magnetic fields under Galilean
boost

Contravariant case:
The Field transforms as

$\displaystyle F^{\prime\mu\nu}(x^{\prime})=\frac{\partial
x^{\prime\mu}}{\partial x^{\lambda}}\frac{\partial x^{\prime\nu}}{\partial
x^{\rho}}F^{\lambda\rho}(x)$ (88)

Boost transformations are written as

$x^{\prime 0}=\gamma x^{0}-\frac{\gamma v_{i}}{c}x^{i}$ (89) $x^{\prime
i}=x^{i}-\frac{\gamma v^{i}}{c}x^{0}+(\gamma-1)\frac{v^{i}v_{j}}{v^{2}}x^{j}$
(90)

From 88 using 89 and 90 we get following relations

$\displaystyle F^{\prime 0i}=E^{\prime i}=\gamma
E^{i}+\frac{\gamma-1}{v^{2}}v^{i}v_{j}E^{j}-\frac{\gamma v_{j}}{c}F^{ji}$ (91)
$\displaystyle F^{\prime ij}=-\frac{\gamma v^{i}}{c}E^{j}+\frac{\gamma
v^{j}}{c}E^{i}+F^{ij}+\frac{\gamma-1}{v^{2}}v_{m}v^{j}F^{im}+\frac{\gamma-1}{v^{2}}v_{l}v^{i}F^{lj}$
(92)

Electric limit
From eqn 91 using the electric limit scaling given in table 1 and keeping in
mind that in this limit $\gamma\to 1$ as $c\to\infty$, we get

$\displaystyle e^{\prime i}=e^{i}$ (93)

Similarly, eqn 92 yields

$\displaystyle f^{\prime ij}=-v^{i}e^{j}+v^{j}e^{i}+f^{ij}\implies b^{\prime
k}=b^{k}-\Big{(}\vec{v}\times\vec{e}\Big{)}^{k}$ (94)

The transformations 93 & 94 manifest the same asymmetry that was observed in
the Maxwell’s equations 55, 56. A change in the electric field induces a
change in the magnetic field but the converse is not true. For the magnetic
limit, discussed right below, it is the other way round.
There is a simple group theoretical argument for the absence of any $b$-term
in 93. For argument’s sake if we retain 94 but include a term in 93 like

$\vec{e}^{\prime}=\vec{e}+\Big{(}\vec{v}\times\vec{b}\Big{)}$ (95)

then the group composition law fails since,

$\vec{e}^{\prime\prime}=\vec{e}^{\prime}+\Big{(}u\times\vec{b}^{\prime}\Big{)}=\vec{e}+\Big{(}(\vec{v}+\vec{u})\times\vec{b}\Big{)}-\vec{u}\times(\vec{v}\times\vec{e})$
(96)

and the last term spoils the transformation 95.
Magnetic limit
From eqn 91 using the magnetic limit scaling from table 1 we get

$\displaystyle e^{\prime i}=e^{i}-v_{j}f^{ji}\implies e^{\prime
i}=e^{i}+\Big{(}\vec{v}\times\vec{b}\Big{)}^{i}$ (97)

Similarly from eqn 92 we get

$\displaystyle f^{\prime ij}=f^{ij}\implies\vec{b}^{\prime}=\vec{b}$ (98)

Adopting the same method the transformations in the covariant sector are
obtained. All these results are summarised in table 5.

Table 5: Transformation of fields under Galilean boost ${\rm Limits}$ | ${\rm Contravariant~{}case}$ | ${\rm Covariant~{}case}$
---|---|---
${\rm Electric~{}limit}$ | $e^{\prime i}=e^{i},~{}~{}b^{\prime k}=b^{k}-\Big{(}\vec{v}\times\vec{e}\Big{)}^{k}$ | $e^{\prime}_{i}=e_{i}+\Big{(}\vec{v}\times\vec{b}\Big{)}_{i},~{}~{}\vec{b}^{\prime}=\vec{b}$
${\rm Magnetic~{}limit}$ | $e^{\prime i}=e^{i}+\Big{(}\vec{v}\times\vec{b}\Big{)}^{i},~{}~{}\vec{b}^{\prime}=\vec{b}$ | $e^{\prime}_{i}=e_{i},~{}~{}b^{\prime}_{k}=b_{k}-\Big{(}\vec{v}\times\vec{e}\Big{)}_{k}$

We can clearly see from this table that under duality transformation
($e^{i}\to b^{i},~{}b^{i}\to-e^{i}$ and $e_{i}\to b_{i},~{}b_{i}\to-e_{i}$)
electric limit reproduces magnetic limit and vice-versa for both covariant and
contravariant cases.

##  V Gauge symmetry

We know in the relativistic case the Maxwell lagrangian,

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (99)

is invariant under the following gauge transformation,

$\delta A_{\mu}=\partial_{\mu}\alpha,\,\,\,\delta
A^{\mu}=\partial^{\mu}\alpha$ (100)

We consider the Galilean version of this gauge invarianvce.

###  V.1 Galilean version

Here we can consider a relatively more general gauge condition,

$\delta A_{\mu}=\partial_{\mu}\alpha,\,\,\,\delta A^{\mu}=\partial^{\mu}\beta$
(101)

In the relativistic theory the covaraint and contravariant vectors are related
by a metric implying $\alpha=\beta$. This is mot true in galilean limit. Hence
we take $\alpha\neq\beta$ when deriving the galilean version of the gauge
transformations. First we consider the electric limit.
Electric limit
From eqn 101 and using the mapping relations given in table 1 we deduce the
following relations,

$\displaystyle\delta A_{0}=\partial_{0}\alpha\implies\frac{1}{c}\delta
a_{0}=\frac{1}{c}\partial_{t}\alpha\implies\delta a_{0}=\partial_{t}\alpha$
(102) $\displaystyle\delta A_{i}=\partial_{i}\alpha\implies\delta
a_{i}=\partial_{i}\alpha$ (103) $\displaystyle\delta
A^{0}=\partial^{0}\beta\implies c\delta
a^{0}=-\frac{1}{c}\partial_{t}\beta\xrightarrow[\text{$c\to\infty$}]{\text{}}\delta
a^{0}=0$ (104) $\displaystyle\delta A^{i}=\partial^{i}\beta\implies\delta
a^{i}=\partial^{i}\beta$ (105)

Taking the variation of 31 in the electric limit,

$\displaystyle\delta\mathcal{L}_{e}=\frac{1}{2}\partial^{i}\delta
a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\Big{)}+\frac{1}{2}\partial^{i}a^{0}\Big{(}\partial_{t}\delta
a_{i}-\partial_{i}\delta a_{0}\Big{)}=0$ (106)

on exploiting eqns 102, 103, 104, 105. This shows the invariance of
$\mathcal{L}_{e}$.
Magnetic limit
From eqn 101 and using the mapping relations given in table 1 and repeating
the steps done for the electric limit, we obtain similar results here also.
These are given in table 6.

Taking the variation of the lagrangian 40 and using the results in table 6 we
get

$\displaystyle\delta\mathcal{L}_{m}=\frac{1}{2}\partial_{i}\delta
a_{0}\Big{(}\partial_{t}a^{i}-\partial^{i}a^{0}\Big{)}+\frac{1}{2}\partial_{i}a_{0}\Big{(}\partial_{t}\delta
a^{i}-\partial^{i}\delta a^{0}\Big{)}=0$ (107)

This shows the invariance of $\mathcal{L}_{m}$.

Table 6: Variations of the Galilean potentials ${\rm Variable}$ | ${\rm Electric~{}~{}limit}$ | ${\rm Magnetic~{}~{}limit}$
---|---|---
$a^{0}$ | $\delta a^{0}=0$ | $\delta a^{0}=\partial_{t}\beta$
$a^{i}$ | $\delta a^{i}=\partial^{i}\beta$ | $\delta a^{i}=\partial^{i}\beta$
$a_{0}$ | $\delta a_{0}=\partial_{t}\alpha$ | $\delta a_{0}=0$
$a_{i}$ | $\delta a_{i}=\partial_{i}\alpha$ | $\delta a_{i}=\partial_{i}\alpha$

###  V.2 Noether current conservation

We know in relativistic classical field theory the Noether current is defined
as

$\displaystyle
J^{\mu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}A_{\nu})}\delta
A_{\nu}$ (108)

which is conserved on-shell i.e $\partial_{\mu}J^{\mu}=0$. Specifically for
the Maxwell Lagrangian,

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

the current in eqn 108 has the form,

$J^{\mu}=-F^{\mu\nu}~{}\partial_{\nu}\alpha$ (109)

and is on-shell conserved i.e

$\displaystyle\partial_{\mu}J^{\mu}=-\Big{(}\partial_{\mu}F^{\mu\nu}\Big{)}\partial_{\nu}\alpha-F^{\mu\nu}\partial_{\mu}\partial_{\nu}\alpha=0$
(110)

The first term is zero because $\partial_{\mu}F^{\mu\nu}=0$ and second term is
zero because of anti-symmetry of $F^{\mu\nu}$.
We now consider here a suitable Galilean version of this conservation. For
this we will directly start from the relativistic definition and substitute
the Galilean results in proper limit (electric or magnetic). 333The details of
the Noether current calculations are provided in appendix A. Analogous
conservation laws, either in electric or magnetic limit are obtained.

##  VI Shift symmetry

We know that Goldstone’s theorem is a crucial input of the study of low-energy
effective lagrangians implying that whenever a global symmetry is
spontaneously broken, a gapless mode will appear. In relativistic theories
this leads to a massless Goldstone particle described by a shift symmetry of
the field

$\phi(x)\to\phi(x)+c$ (111)

where $c$ is constant and is characterised by the scalar field action

$S=\frac{1}{2}\int d^{d}x\partial_{\mu}\phi\partial^{\mu}\phi$ (112)

The above action is invariant under 111. Since 111 is a global transformation,
the conserved currents can be found by exploiting Noether’s first theorem

$\displaystyle
J^{\mu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\delta\phi=c~{}\partial^{\mu}\phi$
(113)

And corresponding conservations are demonstrated as

$\partial_{\mu}J^{\mu}=c\partial_{\mu}\partial^{\mu}\phi=0$ (114)

Consider a constant shift in the four potential,

$A^{\prime}_{\mu}=A_{\mu}+C_{\mu},\,\,\,\,A^{\prime\mu}=A^{\mu}+D^{\mu}$ (115)

that leaves Maxwell lagrangian invariant. We take $C$ and $D$ to be different
for reasons stated in section V.1.
Electric limit
We can define following things

$\displaystyle\delta A_{0}=C_{0}\implies\frac{1}{c}\delta
a_{0}=C_{0}\implies\delta a_{0}=cC_{0}$ (116) $\displaystyle\delta
A_{i}=C_{i}\implies\delta a_{i}=C_{i}$ (117)

Similarly, from expressions for $\delta A^{0},~{}\delta A^{i}$ we find,

$\displaystyle\delta a^{0}=0,\,\,\,~{}~{}~{}\delta a^{i}=D^{i}$ (118)

From 31 Noether currents are found to be

$\displaystyle
j^{t}=\frac{1}{2}(\partial^{i}a^{0})C_{i},\,\,\,~{}~{}~{}~{}~{}j^{i}=-\frac{1}{2}\partial^{i}a^{0}C_{0}-\frac{1}{2}f^{ij}C_{j}$
(119)

And current conservation can be explicitly demonstrated as

$\displaystyle\partial_{\mu}J^{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}\partial_{t}j^{t}+\partial_{i}j^{i}=\frac{1}{2}(\partial_{t}\partial^{i}a^{0})C_{i}-\frac{1}{2}(\partial_{i}\partial^{i}a^{0})C_{0}-\frac{1}{2}(\partial_{i}f^{ij})C_{j}=0$
(120)

The covariant components of the currents are

$\displaystyle j_{t}=0,\,\,\,~{}~{}~{}~{}~{}j_{i}=-\frac{1}{2}f_{ij}D^{j}$
(121)

The current conservation gives us

$\displaystyle\partial^{\mu}J_{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}\partial^{i}j_{i}=-\frac{1}{2}(\partial^{i}f_{ij})D^{j}=0$ (122)

Magnetic limit
We can define following things

$\displaystyle\delta A_{0}=C_{0}\implies-c\delta a_{0}=C_{0}\implies\delta
a_{0}=0$ (123) $\displaystyle\delta A_{i}=C_{i}\implies\delta a_{i}=C_{i}$
(124)

Similarly, from expressions for $\delta A^{0},~{}\delta A^{i}$ we find,

$\displaystyle\delta a^{0}=-cD^{0},\,\,\,~{}~{}~{}~{}~{}~{}\delta a^{i}=D^{i}$
(125)

From 40 the Noether currents are found to be

$\displaystyle j^{t}=0,\,\,\,~{}~{}~{}~{}j^{i}=-\frac{1}{2}f^{ij}C_{j}$ (126)

So the current conseravtaions are demonstrated as

$\displaystyle\partial_{\mu}J^{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}\partial_{i}j^{i}=-\frac{1}{2}\partial_{i}f^{ij}C_{j}=0$ (127)

Similarly the covariant current components are

$\displaystyle
j_{t}=\frac{1}{2}\partial_{i}a_{0}D^{i},\,\,\,~{}~{}~{}~{}~{}j_{i}=-\frac{1}{2}\partial_{i}a_{0}D^{0}-\frac{1}{2}f_{ij}D^{j}$
(128)

The current conservations give

$\displaystyle\partial^{\mu}J_{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}\partial_{t}j_{t}+\partial^{i}j_{i}=\frac{1}{2}(\partial_{t}\partial_{i}a_{0})D^{i}-\frac{1}{2}(\partial^{i}\partial_{i}a_{0})D^{0}-\frac{1}{2}\partial^{i}f_{ij}D^{j}=0$
(129)

##  VII Inclusion of sources

The relativistic Maxwell Lagrangian with source is as follows

$\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}-A_{\alpha}J^{\alpha}$
(130)

We can write the Lagrangian in the following form for convenience

$\mathcal{L}=-\frac{1}{4}\Big{(}2F_{0i}F^{0i}+F_{ij}F^{ij}\Big{)}-\frac{1}{2}A_{\alpha}J^{\alpha}-\frac{1}{2}A^{\alpha}J_{\alpha}$
(131)

We know that the Maxwell theory respects the following gauge transformations

$A_{\mu}\to A_{\mu}+\partial_{\mu}\Lambda$ (132)

The gauge invariance of the action demands the following condition

$A_{\mu}J^{\mu}\to\Big{(}A_{\mu}+\partial_{\mu}\Lambda\Big{)}J^{\mu}=A_{\mu}J^{\mu}-\Lambda\partial_{\mu}J^{\mu}\implies\partial_{\mu}J^{\mu}=0$
(133)

Electric limit:
In the electric limit the scaling of the components of the source $J^{\mu}$
will be as follows

$\displaystyle J^{0}\to cj^{0},\,\,J^{i}\to
j^{i},\,\,J_{0}\to\frac{j_{0}}{c},\,\,J_{i}\to j_{i}$ (134)

In this limit the Lagrangian looks like

$\mathcal{L}_{e}=\frac{1}{2}\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\Big{)}-\frac{1}{4}f_{ij}f^{ij}-\frac{1}{2}a_{0}j^{0}-\frac{1}{2}a_{i}j^{i}-\frac{1}{2}a^{0}j_{0}-\frac{1}{2}a^{i}j_{i}$
(135)

Varying the lagrangian with respect to $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$ will
give following set of equations

$\displaystyle\partial^{i}\partial_{i}a^{0}=j^{0}$ (136)
$\displaystyle\partial_{t}\partial^{j}a^{0}+\partial_{i}\partial^{j}a^{i}-\partial_{i}\partial^{i}a^{j}=-j^{j}$
(137)
$\displaystyle\partial^{i}\partial_{t}a_{i}-\partial^{i}\partial_{i}a_{0}=-j_{0}$
(138)
$\displaystyle\partial^{i}\partial_{i}a_{j}-\partial^{i}\partial_{j}a_{i}=j_{j}$
(139)

We now derive these equations directly from the equations of motion. The
relativistic equations are given by,

$\displaystyle\partial_{i}F^{i0}=J^{0},\,\,\,\,\,~{}~{}~{}~{}~{}\partial_{0}F^{0j}+\partial_{i}F^{ij}=J^{j}$
(140)

Using maps given in table 1 and 134 it is simple to verify that they reproduce
eqns 136 and 137. Using the covariant counterpart of eqns 140 we can get eqns
138 and 139. This shows the consistency of the eqn of motion in Galilean
electrodynamics with source.
Taking the variation of the source part of the lagrangian we get

$\displaystyle\delta\mathcal{L}_{e}=-\frac{1}{2}\partial_{t}\alpha
j^{0}-\frac{1}{2}\partial_{i}\alpha j^{i}-\frac{1}{2}\partial^{i}\beta
j_{i}=\frac{1}{2}\alpha\Big{(}\partial_{t}j^{0}+\partial_{i}j^{i}\Big{)}+\frac{1}{2}\beta\partial^{i}j_{i}=0$
(141)

Since $\alpha,\beta\neq 0$ we have two conditions

$\displaystyle\partial_{t}j^{0}+\partial_{i}j^{i}=0,\,\,\,\,\,\partial^{i}j_{i}=0$
(142)

The sources as given in eqns (136)-(139) satisfy the above conditions. We
observe that sources are conserved off-shell.
Magnetic limit:
Here the scaling relations are as follows

$\displaystyle J^{0}\to-\frac{j^{0}}{c},\,\,J^{i}\to j^{i},\,\,J_{0}\to-
cj_{0},\,\,J_{i}\to j_{i}$ (143)

The Lagrangian in this limit is as follows

$\mathcal{L}_{m}=\frac{1}{2}\partial_{i}a_{0}\Big{(}\partial_{t}a^{i}-\partial^{i}a^{0}\Big{)}-\frac{1}{4}f_{ij}f^{ij}-\frac{1}{2}a_{0}j^{0}-\frac{1}{2}a_{i}j^{i}-\frac{1}{2}a^{0}j_{0}-\frac{1}{2}a^{i}j_{i}$
(144)

Varying the lagrangian wrt $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$ we get the
following set of equations

$\displaystyle\partial_{i}\partial_{t}a^{i}-\partial_{i}\partial^{i}a^{0}=-j^{0}$
(145)
$\displaystyle\partial_{i}\partial^{i}a^{j}-\partial_{i}\partial^{j}a^{i}=j^{j}$
(146) $\displaystyle\partial^{i}\partial_{i}a_{0}=j_{0}$ (147)
$\displaystyle\partial_{t}\partial_{j}a_{0}-\partial^{i}\partial_{i}a_{j}+\partial^{i}\partial_{j}a_{i}=-j_{j}$
(148)

These equations may also be derived directly from the equations of motion
following the same method adopted for the electric limit. The field equations
for both electric and magnetic limit have been shown in table 7.

Table 7: Field equations ${\rm Variables}$ | ${\rm Electric~{}~{}limit}$ | ${\rm Magnetic~{}~{}limit}$
---|---|---
$a^{0}$ | $\partial^{i}\partial_{t}a_{i}-\partial^{i}\partial_{i}a_{0}=-j^{0}$ | $\partial^{i}\partial_{i}a_{0}=j^{0}$
$a^{j}$ | $\partial^{i}\partial_{i}a_{j}-\partial^{i}\partial_{j}a_{i}=j_{j}$ | $\partial_{t}\partial_{j}a_{0}+\partial^{i}\partial_{j}a_{i}-\partial^{i}\partial_{i}a_{j}=-j_{j}$
$a_{0}$ | $\partial^{i}\partial_{i}a^{0}=j^{0}$ | $\partial_{i}\partial_{t}a^{i}-\partial^{i}~{}\partial_{i}~{}a^{0}=-j^{0}$
$a_{j}$ | $\partial_{t}\partial^{j}a^{0}+\partial_{i}\partial^{j}a^{i}-\partial_{i}\partial^{i}a^{j}=-j^{j}$ | $\partial_{i}\partial^{i}a^{j}-\partial_{i}\partial^{j}a^{i}=j^{j}$

Taking the variation of the source part of the lagrangian we get

$\delta\mathcal{L}_{m}=-\frac{1}{2}\partial_{i}\alpha
j^{i}-\frac{1}{2}\partial_{t}\beta j_{0}-\frac{1}{2}\partial^{i}\beta
j_{i}=\frac{1}{2}\alpha\Big{(}\partial_{i}j^{i}\Big{)}+\frac{1}{2}\beta\Big{(}\partial_{t}j_{0}+6^{i}j_{i}\Big{)}=0$
(149)

Since $\alpha,\beta\neq 0$ we have two conditions

$\displaystyle\partial_{i}j^{i}=0,\,\,\,\,\partial_{t}j_{0}+\partial^{i}j_{i}=0$
(150)

Here also sources given by eqns (145) to (148) satisfy the above off-shell
conservation equations.

##  VIII Conclusions

Let us summarise, point by point, the new significant findings of the paper,
comparing with existing results found in the literature.

  * •

An unambiguous construction of the non-relativistic (NR) lagrangian, for both
electric and magnetic limits, was given. We have shown that it correctly
reproduces the equations of motion either in the potential or field
(electric/magnetic) formulation. This lagrangian was deduced from the standard
relativistic lagrangian adopting the dictionary given here. It is expressed
either in terms of potentials or fields. In the later case both electric and
magnetic limit lagrangians become identical having the same functional form as
the usual Maxwell lagrangian.
In ref [21], a NR lagrangian has been given, also derived from the
relativistic Maxwell expression, which has, however, several shortcomings.
444Details are provided in the Appendix B

  * •

It is observed from table 2 that if we replace the covariant components by
contravariant ones in the electric limit case we will end up with the magnetic
limit case and vice-versa. This fact manifests itself only if we consider the
covariant and contravariant sectors separately as we have done here.The
interplay between the covariant and the contravaraint indices that leads to an
interchange of the electric and the magnetic limits of the theory is a new
feature observed here. The reason that it was not noticed earlier stems from
the fact that various applications [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]
only considered the contravariant components. There is a paper [28] that only
gives the Galilean transformation for both covariant and contravariant
components and that too confined to the coordinates and derivatives, and not
for an arbitrary field. Our analysis is much more general where we provide
maps, for both covariant and contravariant sectors, relating arbitrary four
vectors in the Lorentz relativistic and Galilean relativistic formulations.
These maps are the genesis of our analysis where we use them to obtain
galilean relativistic expressions from their corresponding Lorentz
relativistic counterparts. These issues are not even remotely mentioned, much
less discussed, in [28].

  * •

A central point is the formulation of a dictionary that translates four
vectors in the relativistic theory to their corresponding vectors in the non-
relativistic theory. Thus the formalism developed in terms of potentials was
extended to field (electric & magnetic) formulation. In this set up the
duality symmetry was discussed. One can clearly see that in the non-
relativistic limit the duality relations are quite non-trivial. In this limit
we show that apart from the usual duality relations a twisted duality relation
also exists. The feature of twisted duality manifests precisely because the
covariant and contravariant vectors are treated separately. This also shows
that, on the lagrangian level, duality plays quite a subtle role.
Duality symmetries have useful physical applications. For standard Maxwell’s
theory, using duality symmetry we can find new solutions from given original
solutions. Here duality symmetry switches from the electric limit to the
magnetic limit. Thus the solutions of the Rowland-Vasilescu Karpen’s effect
which is an example of the Galilean electric limit, can be exploited, using
the duality relations, to find solutions of Wilson’s effect which corresponds
to the Galilean magnetic limit.555These effects and their implications have
been discussed in [29], but there is no mention of duality symmetry..

  * •

Gauge symmetries play a pivotal role in the understanding of gauge theories.
Since covariant ($a_{\mu}$) and the contravarinat ($a^{\mu}$) vectors are not
connected by any non-degenerate metric, they have separate gauge
transformations. While this was noticed earlier [21], its full implications
were not analysed, and not just because of their problematic lagrangian 161.
We show how gauge symmetries in the relativistic case naturally yield their
non-relativistic counterpart, but with distinct gauge parameters. Both
electric and magnetic limits were analysed. The conservation laws were derived
using Noether’s prescription.

  * •

Shift symmetries, which have an important role to describe Goldstone particles
in relativistic theories, were introduced in the non-relativistic context.
Conservation laws, associated with such symmetries, were derived in both
electric and magnetic limits.

  * •

We have provided a completely holistic approach in terms of both potentials
and fields, clearly showing the connection among them, starting from the
rudimentary structures of usual Maxwell’s theory. Such an analysis is lacking
in the literature.

Future prospects:
This is quite a new research area and has gained attention of late as a part
of the resurgence of non-Lorentzian structures in quantum field theories,
holography and string theory and hence many aspects and directions are yet to
be looked at. There is no consistent Hamiltonian formalism for galilean
electrodynamics for example. Also it will be interesting to study the non-
relativistic limits of other gauge theories for example Proca theory which
describes a massive spin 1 field or Maxwell Chern-Simons theory which is a
$2+1$ dimensional gauge theory, in the same way described here. The analysis
described here for vector field could be extended to include tensor fields
like the Kalb-Ramond fields. Since the connection of these fields with non-
relativistic fluid dynamics is known [26, 27], though relatively less studied,
the present formulation could find application to illuminate this connection.
All these possibilities should be tractable since we have provided independent
maps for both covariant and contravariant sectors. Finally, we hope to
elucidate the nature of Carrollian electrodynamics [16] using the methods
developed here. We expect we can address these issues in the near future.

##  IX Acknowledgements

The authors (RB and SB) acknowledge the support from a DAE Raja Ramanna
Fellowship (grant no:
$1003/(6)/2021/RRF/R\&D-II/4031$, dated: $20/03/2021$). They also acknowledge
Sumit Dey for useful discussions.

##  Appendix A Noether current calculation

Electric limit
The contravariant components of the current for the relativistic case are

$\displaystyle
J^{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}A_{\nu})}\delta A_{\nu}$
(151) $\displaystyle
J^{i}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}A_{0})}\delta
A_{0}+\frac{\partial\mathcal{L}}{\partial(\partial_{i}A_{j})}\delta A_{j}$
(152)

Now using the maps for electric limit given in table 1 we get

$\displaystyle
cj^{0}=c\frac{\partial\mathcal{L}_{e}}{\partial(\partial_{t}a_{0})}\delta
a_{0}+c\frac{\partial\mathcal{L}_{e}}{\partial(\partial_{t}a_{i})}\delta
a_{i}\implies j^{0}=\frac{1}{2}\partial^{i}a^{0}\partial_{i}\alpha$ (153)

Similarly,

$\displaystyle
j^{i}=\frac{\partial\mathcal{L}_{e}}{\partial(\partial_{i}a_{0})}\delta
a_{0}+\frac{\partial\mathcal{L}_{e}}{\partial(\partial_{i}a_{j})}\delta
a_{j}=-\frac{1}{2}\partial^{i}a^{0}\partial_{t}\alpha-\frac{1}{2}\Big{(}\partial^{i}a^{j}-\partial^{j}a^{i}\Big{)}\partial_{j}\alpha$
(154)

So we can show the conservation of the galilean currents as follows

$\displaystyle\partial_{\mu}J^{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}\partial_{t}j^{0}+\partial_{i}j^{i}=\frac{1}{2}\partial_{t}\partial^{i}a^{0}\partial_{i}\alpha+\partial_{i}[-\frac{1}{2}\partial^{i}a^{0}\partial_{t}\alpha-\frac{1}{2}\Big{(}\partial^{i}a^{j}-\partial^{j}a^{i}\Big{)}\partial_{j}\alpha]$
$\displaystyle=-\frac{1}{2}\Big{(}\partial_{i}\partial^{i}a^{0}\Big{)}\partial_{t}\alpha+\frac{1}{2}\Big{(}\partial_{t}\partial^{j}a^{0}-\partial_{i}f^{ij}\Big{)}\partial_{j}\alpha+\frac{1}{2}\partial^{i}a^{0}\partial_{t}\partial_{i}\alpha-\frac{1}{2}\partial^{i}a^{0}\partial_{i}\partial_{t}\alpha-\frac{1}{2}f^{ij}\partial_{i}\partial_{j}\alpha=0$
(155)

In the second line of eqn 155, the first and second term is zero from
equations of motion eqn 32 and 33 respectively, third and fourth terms get
cancelled and fifth term vanishes because of antisymmetry.
Following identical arguments current conservation for covariant components
can be shown as

$\displaystyle\partial^{\mu}J_{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}\partial^{i}j_{i}=0,$ (156)

Magnetic limit
Using the maps for magnetic limit given in table 1 and exploiting eqns 151 and
152 we get

$\displaystyle
j^{0}=-c^{2}\frac{\partial\mathcal{L}_{m}}{(-c)\partial(\partial_{t}a_{0})}\delta
a_{0}-\frac{\partial\mathcal{L}_{m}}{\partial(\partial_{t}a_{i})}\delta
a_{i}=0$ (157) $\displaystyle
j^{i}==\frac{\partial\mathcal{L}_{m}}{(-c)\partial(\partial_{i}a_{0})}(-c)\delta
a_{0}+\frac{\partial\mathcal{L}_{m}}{\partial(\partial_{i}a_{j})}\delta
a_{j}=-\frac{1}{2}f^{ij}\partial_{j}\alpha$ (158)

Using these expressions,

$\displaystyle\partial_{\mu}J^{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}\partial_{i}j^{i}=-\frac{1}{2}\Big{(}\partial_{i}\partial^{i}a^{j}-\partial_{i}\partial^{j}a^{i}\Big{)}\partial_{j}\alpha=0$
(159)

where the second equality vanishes from eqn 42. Likewise for the covariant
case, following identical arguments current conservation can be shown as

$\displaystyle\partial^{\mu}J_{\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Magnetic
limit}}\partial_{t}j_{0}+\partial^{i}j_{i}=0$ (160)

##  Appendix B Problems and inconsistencies of the lagrangian formulation
given in [21]

Any consistent lagrangian formulation of galilean electrodynamics must yield
all the equations of motion, for either contravariant or covariant vectors in
both electric and magnetic limits. Simultaneously, these equations must reduce
to those given in [14] using the field (electric/magnetic) formulation. This
is not merely desirable, but essential, since those equations were obtained
directly [14] using galilean relativistic arguments, bypassing the use of
limiting prescriptions. Since the basic variables in the lagrangian are the
potentials, equations of motion are obtained in the potential formulation. One
has to now express the electric and magnetic fields in terms of potentials and
recast the equations of motion in terms of these fields. Only then a
comparison with [14] is feasible. As we explicitly show, the lagrangian given
in [21] fails on all counts.
The covariant Galilean relativistic largrangian given in [21] is

$\mathcal{L}=-\frac{1}{4}\Big{(}\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}\Big{)}\Big{(}\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}\Big{)}$
(161)

which gives rise to following set of equations

$\partial_{\mu}\Big{(}\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}\Big{)}=0,\,\,\,\,\,\,\,~{}~{}~{}~{}~{}({\rm
Electric~{}~{}limit})$ (162)
$\partial^{\mu}\Big{(}\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}\Big{)}=0,\,\,\,\,\,\,\,~{}~{}~{}~{}~{}({\rm
Magnetic~{}~{}limit})$ (163)

Let us first consider eqn 162. If we re-write this equation component wise it
gives the following set of equations 666According to the convention used in
[21], $\partial^{\mu}=(0,~{}\partial^{i})$ and
$\partial_{\mu}=(\partial_{t},~{}\partial_{i})$.

$\displaystyle\partial_{i}\partial^{i}a^{0}=0,\,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}\partial_{t}\partial^{i}a^{0}+\partial_{j}\partial^{i}a^{j}-\partial_{j}\partial^{j}a^{i}=0$
(164)

which are nothing but equation 32 and 33 respectively. But we cannot get eqns
34 and 35 from the lagrangian 161. In fact we cannot get any equation
involving $a_{0}$ and / or $a_{i}$ simply because there are no covariant
components.
Similarly we can open the magnetic limit equation 163 component wise as
follows

$\displaystyle\partial^{i}\partial_{t}a_{i}-\partial^{i}\partial_{i}a_{0}=0,\,\,\,\,\,~{}~{}~{}~{}~{}\partial^{j}\partial_{i}a_{j}-\partial^{j}\partial_{j}a_{i}=0$
(165)

These two equations do not correspond to any of our equations. On top of that
there are no equations for $a^{0}$ and / or $a^{i}$ simply because
contravariant indices do not arise.
Thus the lagrangian 161 fails to yield, in the electric limit, any equation
involving covariant indices for potentials. Likewise, in the magnetic limit,
there are no equations in the contravariant sector. It is also not possible to
express equations (162, 163) in the field formulation since no map ralating
potentials with fields has been given. Hence the mandatory comparison with
[14] cannot be done. All these issues have been discussed successfully in our
approach.
If we push the analysis of [21] further, serious inconsistencies arise. The
master equations 165, from which the lagrangian 161 was written, was claimed
to be derived by opening the relativistic Maxwell equation 36 in space-time
components and exploiting the map given in [21],

$A^{0}=-\frac{1}{c}a_{0},\,\,\,\,\,\,\,\,A^{i}=a_{i}$ (166)

Doing this, however, instead of 165 we find

$\partial_{i}\partial^{i}a_{0}=0,\,\,\,\,\,\,\,\,\partial_{j}\partial^{i}a_{j}-\partial_{j}\partial^{j}a_{i}=0$
(167)

Surprisingly, there is a mismatch with the first equality in 165. Thus the
very construction of the lagrangian and the associated equations of motion in
[21] are all riddled with inconsistencies.

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# Imitation Learning based Auto-Correction of Extrinsic Parameters for A
Mixed-Reality Setup

Shubham Sonawani, Yifan Zhou and Heni Ben Amor

###### Abstract

In this paper, we discuss an imitation learning based method for reducing the
calibration error for a mixed reality system consisting of a vision sensor and
a projector. Unlike a head mounted display, in this setup, augmented
information is available to a human subject via the projection of a scene into
the real world. Inherently, the camera and projector need to be calibrated as
a stereo setup to project accurate information in 3D space. Previous
calibration processes require multiple recording and parameter tuning steps to
achieve the desired calibration, which is usually time consuming process. In
order to avoid such tedious calibration, we train a CNN model to iteratively
correct the extrinsic offset given a QR code and a projected pattern. We
discuss the overall system setup, data collection for training, and results of
the auto-correction model.

##  I System Overview

Mixed reality is receiving increased interest in robotics and human-robot
interaction. In particular, it can be used to project visual signals to a
human partner that can be easily and quickly understood. To this end, the work
in [1] proposed a camera-projector (stereo) mixed reality setup which provides
explicit visual cues to a subject engaged in a human-robot collaborative task.
Specifically, the approach used a pose information about an object of interest
to generate and overlay visual information, e.g., highlighting an object,
connecting two objects via a virtual line, etc. These signals were then
projected into the real 3D scene to create an immersive environment combining
real objects and artificial objects.

However, such systems require precisely calibrated extrinsic parameters in
order to project visual cues perfectly onto real-world objects. The extrinsic
identifies the transformation between the reference frame of camera and the
projection device. Incorrect extrinsic parameters, as shown in an overview
figure, can substantially affect the 3D effect and results in wrongly placed
visual cues. In other words, it is not enough to identify the location of an
object in the camera space alone, but rather its location in the projector
frame is also critical. Given this, Deriving the extrinsic transformation
between the camera and the projector is a non-trivial and delicate endeavor.

Figure 1: Projection of 3D cube’s wire-frame after auto-calibrated extrinsic
parameters

For traditional camera calibration, checkerboard-based calibration procedures
are widely used and produce reasonably robust results. However, in the case of
a project, the calibration procedure changes since a projector is an active
light emitter and can be regarded as an inverse camera. As shown in [2], the
calibration sequence for a camera-projector setup requires tuning parameters
such as global and projector-light components to generate a reasonable
calibration. However, this adds an additional burden to the human expert
performing the procedure and typically requires more than three image
sequences to be recorded. Accordingly, the process is accurate but involves
human intervention and can be time-consuming. Another element that is required
is a physically large calibration pattern, i.e., a checkerboard that is big
enough to capture a large section of the physical environment. Most
importantly, any small change in the setup requires a complete recalibration –
a frustrating experience that can hamper the adoption of mixed reality.

To remove the requirement for such tedious processes, we propose a novel
method that automatically uses a Neural Network model to automatically correct
the extrinsic offset. In this method, we place an April tag (QR code) on the
table and try to project a highlighted pattern onto the tag. Due to the
inaccurate extrinsic parameters, the projected pattern will have an offset
from the ground truth pose. By observing this offset visually, our model
directly predicts the underlying extrinsic error to correct the extrinsic
parameters. Gradually, the highlighted pattern shifts on top of the QR code.
The overall result is that visual cues and patterns can now be accurately
projected into space and on top of 3D objects to create complex and compelling
mixed-reality environments, see Fig. 1.

##  II Training and Testing of the Neural Network

The goal of the model is to find a policy
$\pi_{\boldsymbol{\theta}}(\boldsymbol{e}|\boldsymbol{I})$ which is
parameterized by $\boldsymbol{\theta}$, takes in an image $\boldsymbol{I}$
from the camera and estimates the current error $\boldsymbol{e}$ in the
projector extrinsic parameters. In this image $\boldsymbol{I}$, there is a
highlighted area and an April tag, where the highlight is supposed to cover
the April tag perfectly. Since some error $\boldsymbol{e}$ exists in the
extrinsic parameters, the highlighted area is offset from the intended
projection area. Given this image, we train a neural network as the policy
$\pi_{\boldsymbol{\theta}}$ to directly predict the extrinsic parameters’
offset. This is, in essence, an imitation learning process. The model is
trained on demonstrations where experts adjust the offsets, and the model
imitates these actions.

###  II-A The Neural Network Model

We adopt a ResNet-50 [3] neural network pre-trained on ImageNet-1k [4] dataset
as our backbone. Since we are performing a regression task instead of a
classification task as ResNet-50 originally does, we modify the last layer of
the network to be a fully-connected layer that outputs the estimated offset of
the extrinsic parameters directly.

###  II-B Data Collection

In order to train the network, we need a set of expert demonstrations
$\mathcal{D}=\\{\boldsymbol{d}_{0},...,\boldsymbol{d}_{n}\\}$ where each
demonstration $\boldsymbol{d}_{i}$ contains a current image $\boldsymbol{I}$
and the expert’s estimation of correction $\boldsymbol{e}$. In order to
collect the data, firstly, a camera projector system is calibrated tediously
using [2] method with around eight recordings of calibration sequences. In
addition, manual tuning of extrinsic parameters is required to obtain near
perfect calibration parameters. We project 0.10 meters $\times$ 0.10 meters of
a red colored plane onto the April tag tracked in the camera frame using [5].
In order to mimic the offset in extrinsic parameters, we inject random offset
in the x and y position of the projector w.r.t. camera. Moreover, iteratively,
in a sequence of images, we reduce the offset and capture an image at that
offset showing the April tag and projected plane. We collect around 100 such
sequences, of which we use 70 for training and 30 for testing.

In order to enable high precision in the real world, we use high resolution
images ($700\times 700$ px), so that the images contain clear edges of the
April tag and highlighted area.

###  II-C Testing in Real World

For testing, we calibrate the system before starting the testing process to
get the ground truth extrinsic parameters for comparison. Afterward, we
initialize a test trial by randomly placing the April tag on the table. While
testing, we disturb the ground truth extrinsic parameters and use them as the
noisy extrinsic parameters, which causes an offset on the highlighted pattern
(see overview figure with the convergence of noisy to ground truth sequence
1-4). The trained neural network takes in the image captured by the camera and
spits out the estimated offset of the extrinsic parameters, with which the
correctional update is made, and the pattern is projected back onto the table
with lower error. Repeating this step iteratively, we can see the highlighted
pattern is moving closer to the April tag (overview figure parts 2 and 3).
When the predicted offsets are smaller than a pre-defined small number
($\epsilon$), we mark the convergence of the trial. Empirically, we choose a
small step size for updates, which smoothes out the potential prediction
noises and increases the stability of the convergence.

We measure the l-2 norm between the ground truth and the converged prediction
as the error. During test time, the error is around 5e-4. Corresponding to the
real world set up, we can see the highlight pattern is projected onto the
April tag precisely (checkout the overview figure-part 4).

##  III Conclusion

This paper demonstrates work in progress auto-calibration method for a camera-
projector based mixed reality setup. Importantly, with 100 data sequences for
training and testing, we show that the CNN model can be used to calibrate
extrinsic parameters of the camera-projection stereo system. However, this
paper has not addressed convergence guarantees of CNN model to ground truth
parameters. Given this, future work will involve analysis of convergence
guarantees and robustness of the discussed method.

## References

  * [1] R. K. Ganesan, Y. K. Rathore, H. M. Ross, and H. B. Amor, “Better teaming through visual cues: How projecting imagery in a workspace can improve human-robot collaboration,” IEEE Robotics and Automation Magazine, vol. 25, pp. 59–71, 6 2018.
  * [2] D. Moreno and G. Taubin, “Simple, accurate, and robust projector-camera calibration,” Proceedings - 2nd Joint 3DIM/3DPVT Conference: 3D Imaging, Modeling, Processing, Visualization and Transmission, 3DIMPVT 2012, pp. 464–471, 2012.
  * [3] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016.
  * [4] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255, 2009.
  * [5] E. Olson, “Apriltag: A robust and flexible visual fiducial system,” in 2011 IEEE international conference on robotics and automation, pp. 3400–3407, IEEE, 2011.


# Can Feature Engineering Help Quantum Machine Learning for Malware Detection?

Ran Liu
Univ. of Maryland, Baltimore County
<EMAIL_ADDRESS>
Maksim Eren
Univ. of Maryland, Baltimore County
<EMAIL_ADDRESS>
Charles Nicholas
Univ. of Maryland, Baltimore County
<EMAIL_ADDRESS>

###### Abstract

With the increasing number and sophistication of malware attacks, malware
detection systems based on machine learning (ML) grow in importance. At the
same time, many popular ML models used in malware classification are
supervised solutions. These supervised classifiers often do not generalize
well to novel malware. Therefore, they need to be re-trained frequently to
detect new malware specimens, which can be time-consuming. Our work addresses
this problem in a hybrid framework of theoretical Quantum ML, combined with
feature selection strategies to reduce the data size and malware classifier
training time.

##  1 Introduction

With the increasing number and sophistication of malware attacks, malware
detection systems based on machine learning (ML) grow in importance. At the
same time, many popular ML models used in malware classification are
supervised solutions. These supervised classifiers often do not generalize
well to novel malware. Therefore, they need to be re-trained frequently to
detect new malware specimens, which can be time-consuming. Our work addresses
this problem in a hybrid framework of theoretical Quantum ML, combined with
feature selection strategies to reduce the data size and malware classifier
training time.

Theoretically, Quantum Parallelism can provide a potential speed-up in a
number of ML algorithms. Prior research suggests that an N-dimensional dataset
can be mapped onto quantum states with logN qubits. Thus, classical data takes
O(logN) steps for the mapping process[1]. In addition, quantum operation shows
advantages on some specific steps, which include O(poly(log N)) time
complexity for Quantum Fourier Transform and matrix inversion, and O(log N)
for distance estimation and inner product[2]. Prior work has also shown the
advantages of using quantum algorithms in clustering problems. Harrow et al
achieved runtime complexity of O(log(MN)) with a quantum algorithm for
clustering, while in comparison, the best classical clustering algorithm takes
O(poly(MN))[3]. More specifically, current research on Quantum Annealing shows
advantages on some specific ML algorithms with partial noise robustness ,which
limits its application on some practical problems when the dataset is large.
In this research, we explored a feature selection strategy based on classical
ML methods to help train classifiers on quantum gate model machines (e.g.
IBM’s Quantum Processor) when a machine has very limited qubits capacity. We
will examine our approach on some state-of-the-art quantum algorithms to be
run on gate model machines.

##  2 Qubit

‘Qubit’ or ‘Quantum bit’ is the fundamental analogous concept in Quantum
computation, which is defined in the Hilbert Space. Using Dirac notation,
given an orthonormal basis in Hilbert Space, qubits can be measured to a basis
state with probability. We can express a quantum system using state vector
language. For example, we can generate superpositions by form linear
combinations of states with computational basis $|0\rangle$ and $|1\rangle$:

$|\varphi\rangle=\alpha|0\rangle+\beta|1\rangle$ (1)

where $\alpha,\beta\in\mathbb{C}$, and $|\alpha|^{2}+|\beta|^{2}=1$. Unlike
classical bits which can only be in state 0 or 1, qubit can be in the
continuum of basis until it has been observed. For example, 1 means qubit can
be in state the $|0\rangle$ with probability $|\alpha|^{2}$ or in the state
$|1\rangle$ with probability $|\beta|^{2}$.

##  3 Variational Quantum Classifier

In our research, we use the Variational Quantum Classifier proposed by
Havlíček, V., Córcoles, A.D., Temme, K. et al. [4]. A classical data point in
the interval $\left(0,2\pi\right]$ is first mapped onto the Bloch sphere using
the non-linear circuits constructed as:

$\mathcal{U}_{\Phi}(\vec{x})=\mathit{U}_{\Phi(\vec{x})}\mathit{H}^{\otimes
n}\mathit{U}_{\Phi(\vec{x})}\mathit{H}^{\otimes n}$

H is the Hadamard gate and

$\mathit{U}_{\Phi(\vec{x})}=\exp\left(i\sum_{S\subseteq[n]}\phi_{S}\left(\vec{x}\right)\prod_{\mathit{i}\in
S}Z_{i}\right)$

Then variational circuits are constructed for the optimization purpose, which
is called as the l-layer circuits. The l means repeat constructed variational
circuits l times. Next the Z-basis measurements
$f:\\{0,1\\}^{n}\longrightarrow\\{+1,-1\\}$are applied to the output bit
strings. In the end, measurements are repeated n times to give the probability
distribution. And a label as assigned to the input vectors.

##  4 Related Work

The Support Vector Machine (SVM) algorithm is a kind of supervised learning
algorithm which is used to solve binary classification problems. SVM works on
finding a hyperplane to distinguish instances from two different classes. As a
hyperplane be defined as:

$\vec{w}\cdot\vec{x}+b=0$

where $\vec{x}$ is the normal vector and $b$ is the offset. Thus, for class A
and B, the well trained SVM hyperplanes can give:

$\vec{w}\cdot\vec{x}+b\geq 1if\vec{x}\in A$
$\vec{w}\cdot\vec{x}+b\leq-1if\vec{x}\in B$

Thus, the SVM problem is to maximize the margin of two classes A, B s.t.

$y\left(\vec{w}\cdot\vec{x}+b\right)\geq 1$

The classification result of a new vector $\vec{x_{0}}$ is given by:

$sign\left(\vec{w}\cdot\vec{x_{0}}+b\right)$

Havlíček, V., Córcoles, A.D., Temme, K. et al. [4] proposed two SVM type
classifiers to use the Hilbert Space as the feature space to achieve quantum
advantages. The first classifier is to construct the variational circuits that
are discussed in [5] [6] to generate the hyperplane in the Hilbert Space. The
other classifier is to use the quantum computer to directly estimate the
kernel function of the Hilbert Space. The quantum mechanism can give quantum
advantages if circuits are hard to be simulated by a conventional computer.
Their features mapping strategy can be visualized by Figure 1.

Figure 1: A classical data point in the interval $\left(0,2\pi\right]$ in A is
mapped onto the Bloch sphere using the non-linear circuits in B [7]

A classical data point in the interval $\left(0,2\pi\right]$ is first mapped
onto the Bloch sphere using the non-linear circuits constructed as:

$\mathcal{U}_{\Phi}(\vec{x})=\mathit{U}_{\Phi(\vec{x})}\mathit{H}^{\otimes
n}\mathit{U}_{\Phi(\vec{x})}\mathit{H}^{\otimes n}$

H is the Hadamard gate and

$\mathit{U}_{\Phi(\vec{x})}=\exp\left(i\sum_{S\subseteq[n]}\phi_{S}\left(\vec{x}\right)\prod_{\mathit{i}\in
S}Z_{i}\right)$

In their first approach, the classical data is first mapped onto the Hilbert
Space by the feature mapping strategy described above. Then variational
circuits are constructed for the optimization purpose, which they called it as
the l-layer circuits. The l means repeat constructed variational circuits l
times. Next the Z-basis measurements
$f:\\{0,1\\}^{n}\longrightarrow\\{+1,-1\\}$are applied to the output bit
strings. In the end, measurements are repeated n times to give the probability
distribution. And a label as assigned to the input vectors. In their second
approach, the quantum computer is directly used for the estimated purposes.
The quantum computer is first involved in the training process to estimate the
kernel of all data pairs in the training dataset. Then after the optimization
problem of SVM has been organized, the quantum computer is involved again to
estimate the kernel for a new datum of all support vectors. They experiment
both approaches using generated customized data. Their results show a 100%
success rate even in the presence of noise.

Patrick Rebentrost, Masoud Mohseni et al. [8] proposed a quantum approach to
analyze the least - squares approximation of SVM. On a high level, they tried
to get the optimization parameters by solving the linear equation:

$F\pmatrix{b}\\\ \vec{a}\equiv\pmatrix{0}\&1\\\
I\&K+\gamma^{-1}I_{N}\pmatrix{b}\\\ \vec{a}=\pmatrix{0}\\\ \vec{y}$

where K is the kernel of paris in the training dataset, a is the weight, b is
the basis and $\vec{y}=(y_{1},\dots,y_{M})^{T}$ are labels. So Matrix F is
$(M+1)\times(M+1)$ Their approach can be summarized as follows: First the
classical data points are mapped to the Hilbert space:

$|\vec{x_{i}}\rangle=\frac{1}{|\vec{x_{i}}|}\sum_{i=1}^{M}(\vec{x_{i}})_{j}|j\rangle$

Second, the HHL algorithm [9] is applied to get the optimized hyperplane
parameter $\vec{a}$ and $b$. Then the quantum register is transformed to:

$|b,\vec{a}\rangle=\frac{1}{\sqrt{C}}(b|0\rangle+\sum_{k=1}^{M}(a_{k}|k\rangle)$

where $C=b^{2}+\sum_{k=1}^{M}a_{k}^{2}$. The classification result is given
by:

$y(\vec{x_{0}})=(\sum_{i=1}^{M}a_{i}(\vec{x_{i}}\cdot\vec{x_{0}})+b)$

##  5 Experiment

In our experiments, we use the Drebin dataset consisting of +15k benign and
malware samples with 215 dynamic and static analysis based features[10]. We
utilize XGBoost and Decision Tree (DT) to identify the top 20 most important
features. Using these features, we next conducted our experiments on the IBM
Qiskit Simulator with Variational Quantum Classifier (VQC). The preliminary
results show that VQC with XGBoost selected features can get a 78.91% test
accuracy score on 10,000 samples with a 50% test set split. Differently, VQC
with DT feature selection got 62.41% test accuracy. We also conducted tests
using IBM 5 qubits machines. Our experiments in this round were limited by the
number of qubits in IBM Quantum Machine. Therefore we ran our experiment ten
times with 20 randomly selected samples on each run with a 50% test set split.
To show that our results are statistically significant, we report our final
accuracy score with a 95% Confidence Interval (CI), using QSVM. The final
average accuracy for the model trained using the features selected with
XGBoost was 74% (+- 11.35%).

We next explored oversampling and under-sampling strategies based on classical
ML methods to help train classifiers on quantum gate model machines for
malware detection. Our preliminary experimental results show:

  * •

In the case of using oversampling techniques, the accuracy increased up to
80.15% from 74% with SMOTE, and up to 78.06% with Adaptive Synthetic (ADASYN).

  * •

In the case of using under-sampling techniques, the accuracy increased up to
78.62% from 74% by removing samples using K-means, and up to 77.26% by
removing samples which do not agree “enough” with their neighborhood.

  * •

To overcome noisy samples introduced by SMOTE, we combined the use of
oversampling and under-sampling, the accuracy increased up to 83.78%.

  * •

We conducted our experiments on the IBM 5 qubits Quantum Machine. Due to the
limited number of qubits available on the Quantum Machine, we use 20 samples
with a 50% test set split. The Quantum SVM got 56% accuracy and VQC achieved
80% success for malware detection.

##  6 Conclusion

In this study, we explored the use of feature engineering and the quantum
machine for malware detection tasks. The practical use of a quantum machine
for malware detection is still in its early stages due to the physical
limitations.We hope that our work will inspire other researchers to explore
this direction.

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