# Determination of the dipole polarizability of the alkali-metal negative ions

B. K. Sahoo<EMAIL_ADDRESS>Atomic, Molecular and Optical Physics Division,
Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India

(Received date; Accepted date)

###### Abstract

We present electric dipole polarizabilities ($\alpha_{d}$) of the alkali-metal
negative ions, from H- to Fr-, by employing four-component relativistic many-
body methods. Differences in the results are shown by considering Dirac-
Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DC
Hamiltonian with the lower-order quantum electrodynamics interactions. At
first, these interactions are included self-consistently in the Dirac-Hartree-
Fock (DHF) method, and then electron correlation effects are incorporated over
the DHF wave functions in the second-order many-body perturbation theory,
random phase approximation and coupled-cluster (CC) theory. Roles of electron
correlation effects and relativistic corrections are analyzed using the above
many-body methods with size of the ions. We finally quote precise values of
$\alpha_{d}$ of the above negative ions by estimating uncertainties to the CC
results, and compare them with other calculations wherever available.

There are several experimental techniques available to produce negative alkali
ions in the laboratory. The techniques to produce these ions have been
engineered time to time over the several decades vermeer ; andersen1 .
Electron affinities (EAs) of these systems have been measured very precisely
hotop ; patterson ; kasdan ; andersen2 ; andersson . Photoabsorption spectra
of these ions have also been extensively investigated both theoretically and
experimentally, in order to understand their structures watanabe ; eiles ;
berzinsh ; lindhal . Starting from seventies, a series of studies on the
photodetachment of negative lithium (Li-), sodium (Na-) and potassium (K-)
ions have been conducted by several groups kasdan ; haeffler ; moores ; slater
; taylor . Theoretical results from these lighter ions were in excellent
agreement with the corresponding experimental values. By solving a set of
coupled equations, Norcross had predicted the existence of bound excited
states in negative alkali ions norcross . This was later conformed by Greene
greene , by applying a combined approach of $jj$-coupling in the $R$-matrix
method and generalized quantum-defect theory while attempting to describe
photodetachment spectra of negative rubidium (Rb-), cesium (Cs-) and francium
(Fr-) ions. Later, another study bahrim  disproved the existence of such
states in these ions by reanalyzing the calculations using the Dirac
$R$-matrix method. Instead, the authors of the work suggested that the lowest
excited state of the above alkali negative ions is a multiplet of
${}^{3}P_{J}^{o}$-shape resonance. This work clearly demonstrated the
importance of relativistic effects for accurate calculations of atomic
properties in these ions. Similarly, several studies on negative hydrogen (H-)
ion have been carried out balling ; wildt , and its applications millar ; rau
and production sources are well-known to the physicists dudinikov ; faircloth
.

Apart from EAs and photodetachment cross-sections, there is scarcity in the
atomic data of the negative alkali ions. Alkali atoms have a closed-shell and
a valence orbital in their electronic configurations. Owing to this, it is
relatively simpler to calculate atomic wave functions of these atoms. However,
it is challenging to determine atomic wave functions of the alkaline earth-
metal atoms due to strong electron correlations among the valence electrons in
such systems. The negative alkali ions are isoelectronic to the alkaline
earth-metal atoms.

Electric dipole polarizability ($\alpha_{d}$) is a very useful property of any
atomic system. This quantity has been measured very precisely schwerdtfeger2
in the alkali atoms as well as in the singly charged positive alkaline earth-
metal ions, and with reasonable accuracy in the alkaline earth-metal atoms.
However, there has not been a single measurement of $\alpha_{d}$ carried out
thus far in any of the negative alkali ions due to difficulties in setting up
their experiments. There are no full relativistic calculation of $\alpha_{d}$
available in these ions, and only a few non-relativistic calculations have
been reported in the lighter H- pipin ; kar ; bhatia , Li- moores ; pouchan ;
agren , Na- moores  and K-moores  ions. Except the high-precision calculations
in H-, the reported values of other ions are not very reliable.

Calculations of $\alpha_{d}$ in the alkali atoms are in very good agreement
with the experiments arora1 ; arora2 . The reason for this is that one can
easily use their experimental energies and electric dipole (E1) matrix
elements inferring from the lifetime measurements of their atomic states in
the evaluation of $\alpha_{d}$ values using the sum-over-states approach. It
is possible to adopt the sum-over-states approach in these atoms because they
possess a large number of bound states in contrast to the negative alkali
ions. Ab initio procedures demonstrate that the electron correlation and the
relativistic effects are pronounced, and they need to be accounted for, in
order to determine $\alpha_{d}$ values of alkaline earth-metal atoms lim ;
bijaya1 ; yashpal1 , which are isoelectronic systems to the negative alkali
ions. A few calculations of the $\alpha_{d}$ values of some heavier negative
ions of the coinage metal atoms have been reported by Sadlej and coworkers
diercksen ; kello , by employing a variety of methods including the coupled-
cluster (CC) theory. In another study by Schwerdtfeger and Bowmaker
schwerdtfeger , these quantities were also evaluated by using total angular
momentum $j$-averaged relativistic pseudo-potentials in the configuration
interaction (CI) method. These works highlight about the unusually large
electron correlation and relativistic effects in the determination of
$\alpha_{d}$ values in the negative ions compared to their counter
isoelectronic neutral atoms. But the relativistic effects were estimated only
approximately in these calculations. Recently, we had evaluated $\alpha_{d}$
values of Cl- and Au- by applying a number of relativistic many-body methods
at different levels of approximation to demonstrate the roles of electron
correlations for their accurate determination. Here, we intend to determine
$\alpha_{d}$ values of all the negative alkali ions very accurately.

It is not possible to adopt a finite-field (FF) approach to determine
$\alpha_{d}$ of atomic states by preserving spherical symmetry property and
treating parity as a good quantum number. Thus, the spherical symmetry of the
systems is exploited in Refs. diercksen ; kello ; schwerdtfeger  in order to
adopt the FF procedure for the determination of $\alpha_{d}$ of negative ions.
Also, the FF approach introduces large uncertainty to the calculation of
$\alpha_{d}$, which stems from both numerical differentiation as well as
neglecting higher-order perturbation corrections in the evaluation of the
second-order perturbed energy due to external electric-field, in a brute-force
manner. To overcome these problems in the determination of $\alpha_{d}$ while
retaining spherical symmetry behavior of atomic orbitals, we follow a
perturbative approach in which the total Hamiltonian of the system is defined
in the presence of a weak external electric field $\vec{\mathcal{E}}$ as
$H=H_{at}+\vec{D}\cdot\vec{\mathcal{E}}$ with the atomic Hamiltonian $H_{at}$
and electric dipole operator $D$ in a similar framework as Dalgarno and Lewis
dalgarno . In such case, the wave function and energy of an atomic state can
be expressed as

$\displaystyle|\Psi_{0}\rangle=|\Psi_{0}^{(0)}\rangle+|\vec{\mathcal{E}}||\Psi_{0}^{(1)}\rangle+\cdots$
(1) and (2) $\displaystyle
E_{0}=E_{0}^{(0)}+|\vec{\mathcal{E}}|E_{0}^{(1)}+\frac{1}{2}|\vec{\mathcal{E}}|^{2}E_{0}^{(2)}\cdots,$

respectively, where superscripts (0), (1), etc. denote order of
$\vec{\mathcal{E}}$ in the expansion. Since $D$ is an odd-parity operator,
$E_{0}^{(1)}=0$ and the second-order energy is traditionally given by
$E_{0}^{(2)}\equiv\alpha_{d}$. It follows that $\alpha_{d}$ can be evaluated
in the perturbative approach as bijaya1

$\displaystyle\alpha_{d}$ $\displaystyle=$ $\displaystyle
2\frac{\langle\Psi_{0}^{(0)}|D|\Psi_{0}^{(1)}\rangle}{\langle\Psi_{0}^{(0)}|\Psi_{0}^{(0)}\rangle}.$
(3)

Thus, it is imperative to determine both the unperturbed wave function
$|\Psi_{0}^{(0)}\rangle$ of $H_{at}$ and the first-order perturbed wave
function $|\Psi_{0}^{(1)}\rangle$ due to $D$ very reliably for an accurate
evaluation of $\alpha_{d}$. Instead of using the sum-over-states approach to
determine $|\Psi_{0}^{(1)}\rangle$, we would like to solve it as the solution
to the first-order inhomogeneous perturbed equation given by

$\displaystyle(H_{at}-E_{0}^{(0)})|\Psi_{0}^{(1)}\rangle=-D|\Psi_{0}^{(0)}\rangle.$
(4)

Though the solution of this equation appears to be similar to the procedure
adopted by Dalgarno and Lewis dalgarno , but it can be kept in mind that we
only obtain the first-order wave function $|\Psi_{0}^{(1)}\rangle$ for Eq. (3)
instead of determining $E_{0}^{(2)}$ directly.

The many-electron atomic wave function can be obtained by

$\displaystyle|\Psi_{0}\rangle=\Omega_{0}|\Phi_{0}\rangle,$ (5)

where $|\Phi_{0}\rangle$ is a mean-field wave function, which is obtained here
by the Dirac-Hartree-Fock (DHF) method, and $\Omega_{0}$ is known as the wave
operator that is responsible for accounting for electron correlation effects
due to the interactions that are neglected in the determination of
$|\Phi_{0}\rangle$. Likewise, for the wave function, we can expand
$\Omega_{0}$ in the presence of weak electric field $\vec{\mathcal{E}}$ as

$\displaystyle\Omega_{0}=\Omega_{0}^{(0)}+|\vec{\mathcal{E}}|\Omega_{0}^{(1)}+\cdots.$
(6)

Using this, we can write the unperturbed and the first-order perturbed wave
function as

$\displaystyle|\Psi_{0}^{(0)}\rangle=\Omega_{0}^{(0)}|\Phi_{0}\rangle\ \ \
\text{and}\ \ \ |\Psi_{0}^{(1)}\rangle=\Omega_{0}^{(1)}|\Phi_{0}\rangle.$ (7)

In the $n^{th}$-order perturbation theory, we express yashpal1

$\displaystyle\Omega^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\Omega^{(k,0)}\ \ \ \text{and}\ \ \
\Omega^{(1)}=\sum_{k=0}^{n-1}\Omega^{(k,1)}$ (8)

with $\Omega^{(0,0)}=1$, $\Omega^{(1,0)}=0$ and
$\Omega^{(0,1)}=\sum_{p,a}\frac{\langle\Phi_{a}^{p}|D|\Phi_{0}\rangle}{\epsilon_{a}^{(0)}-\epsilon_{p}^{(0)}}$
for all the occupied orbitals denoted by the index $a$ and unoccupied orbitals
denoted by the index $p$. In the second-order relativistic perturbation theory
(RMBPT(2) method) that accounts for the lowest-order electron correlation
effects yashpal1  in the many-body theory, it corresponds to $n=1$ in the
above summations.

We present results from two all-order many-body methods: relativistic random-
phase approximation (RRPA) and relativistic CC (RCC) theory. The RCC theory
incorporates electron correlation effects more rigorously, while RRPA has
traditionally been employed to capture these effects due to the core-
polarization only, which can be done to all-orders in a computationally much
less expensive way. The correlation effects arising through RRPA also
represent the orbital relaxation effects that arise naturally in the mixed-
parity orbitals of the DHF method in the FF procedure. In our RRPA
implementation yashpal2 ; bijaya2 , they are contained in $\Omega_{0}^{(0)}=1$
and $\Omega^{(1)}=\sum_{k=0}^{\infty}\sum_{p,a}\Omega_{a\rightarrow
p}^{(k,1)}$. Here, $a\rightarrow p$ means replacement of an occupied orbital
$a$ from $|\Phi_{0}\rangle$ by a virtual orbital $p$, which alternatively
refers to a singly excited state with respect to $|\Phi_{0}\rangle$. The RCC
theory implicitly includes correlation effects arising through RRPA along with
other correlation effects such as pair-correlation effects to all-orders and
is known as the gold standard method of many-body theory for its capabilities
of producing accurate results in multi-electron systems. In this theory, the
wave operators are given by yashpal2 ; bijaya3

$\displaystyle\Omega^{(0)}$ $\displaystyle=$ $\displaystyle e^{T^{(0)}}\ \ \ \
\text{and}\ \ \ \ \Omega^{(1)}=e^{T^{(0)}}T^{(1)},$ (9)

respectively. We consider only singles and doubles excitations in the RCC
calculations (RCCSD method) by expressing

$\displaystyle T^{(0)}=T_{1}^{(0)}+T_{2}^{(0)}\ \ \ \ \text{and}\ \ \ \
T^{(1)}=T_{1}^{(1)}+T_{2}^{(1)},$ (10)

where subscripts (1) and (2) denote the level of excitation. In this method,
$\alpha_{d}$ determined as

$\displaystyle\alpha_{d}$ $\displaystyle=$ $\displaystyle
2\frac{\langle\Phi_{0}|\Omega^{(0)\dagger}D\Omega^{(1)}|\Phi_{0}\rangle}{\langle\Phi_{0}|\Omega^{(0)\dagger}\Omega^{(0)}|\Phi_{0}\rangle}=2\langle\Phi_{0}|(\overbrace{D^{(0)}}T^{(1)})_{c}|\Phi_{0}\rangle,\
\ \ \ \ $ (11)

where $\overbrace{D^{(0)}}=e^{T^{\dagger{(0)}}}De^{T^{(0)}}$ is a non-
truncating series. We have adopted an iterative procedure to take into account
contributions from this non-terminating series self-consistently, as described
in our earlier works on $\alpha_{d}$ calculations in the closed-shell atoms
yashpal3 ; bijaya3 .

Table 1: Calculated $\alpha_{d}$ values (in $ea_{0}^{3}$) of the negative alkali-metal ions from the DHF, RMBPT(2), RRPA and RCCSD methods. Results from the DC, DCB and DCQ Hamiltonians are listed separately to highlight the roles of Breit and QED interactions in the determination of $\alpha_{d}$ of the above ions. Method | DC | DCB | DCQ
---|---|---|---
H- ion
DHF | 44.41 | 44.61 | 44.41
RMBPT(2) | 66.35 | 66.35 | 66.35
RRPA | 91.13 | 91.13 | 91.13
RCCSD | 206.14 | 206.16 | 206.15
Li- ion
DHF | 500.90 | 500.94 | 500.91
RMBPT(2) | 764.68 | 764.74 | 764.70
RRPA | 1176.68 | 1176.76 | 1176.70
RCCSD | 794.06 | 794.08 | 794.07
Na- ion
DHF | 605.91 | 606.00 606.03 |
RMBPT(2) | 923.52 | 923.66 | 923.72
RRPA | 1447.69 | 1447.92 | 1448.01
RCCSD | 952.54 | 952.63 | 952.67
K- ion
DHF | 1053.39 | 1053.58 | 1053.89
RMBPT(2) | 1586.91 | 1587.18 | 1587.68
RRPA | 2565.41 | 2565.88 | 2566.66
RCCSD | 1353.69 | 1353.84 | 1354.19
Rb- ion
DHF | 1214.16 | 1214.43 | 1215.40
RMBPT(2) | 1816.88 | 1817.22 | 1818.75
RRPA | 2968.55 | 2969.21 | 2971.66
RCCSD | 1506.57 | 1506.75 | 1507.88
Cs- ion
DHF | 1534.15 | 1534.48 | 1537.06
RMBPT(2) | 2271.42 | 2271.78 | 2275.81
RRPA | 3770.88 | 3771.71 | 3778.25
RCCSD | 1800.42 | 1800.54 | 1803.59
Fr- ion
DHF | 1357.23 | 1357.78 | 1357.33
RMBPT(2) | 1990.66 | 1991.21 | 1990.77
RRPA | 3308.49 | 3309.76 | 3308.69
RCCSD | 1619.04 | 1619.16 | 1619.03

For the evaluation of $|\Psi_{0}\rangle$, we consider first the Dirac-Coulomb
(DC) Hamiltonian, given by

$\displaystyle H^{DC}$ $\displaystyle=$
$\displaystyle\sum_{i}\left[c\mbox{\boldmath$\alpha$}_{i}\cdot\textbf{p}_{i}+(\beta_{i}-1)c^{2}+V_{n}(r_{i})\right]+\sum_{i,j>i}\frac{1}{r_{ij}},\
\ \ \ \ \ $ (12)

where $c$ is the speed of light, $\alpha$ and $\beta$ are the usual Dirac
matrices, $\textbf{p}_{i}$ is the single particle momentum operator,
$V_{n}(r_{i})$ denotes the nuclear potential, and $\frac{1}{r_{ij}}$
represents the Coulomb potential between two electrons located at the $i^{th}$
and $j^{th}$ positions. We estimate the Breit interaction by using the Dirac-
Coulomb-Breit (DCB) Hamiltonian ($H^{DCB}=H^{DC}+V^{B}$) by defining the
potential

$\displaystyle V^{B}$ $\displaystyle=$
$\displaystyle-\sum_{j>i}\frac{[\mbox{\boldmath$\alpha$}_{i}\cdot\mbox{\boldmath$\alpha$}_{j}+(\mbox{\boldmath$\alpha$}_{i}\cdot\mathbf{\hat{r}_{ij}})(\mbox{\boldmath$\alpha$}_{j}\cdot\mathbf{\hat{r}_{ij}})]}{2r_{ij}},$
(13)

where $\mathbf{\hat{r}_{ij}}$ is the unit vector along $\mathbf{r_{ij}}$.
Similarly, contributions from the quantum electrodynamics (QED) effects are
estimated using the Dirac-Coulomb-QED (DCQ) Hamiltonian
($H^{DCQ}=H^{DC}+V^{Q}$) by considering $V^{Q}=V_{VP}+V_{SE}$ with the vacuum
polarization interaction potential $V_{VP}$ and the self-energy interaction
potential $V_{SE}$. We use the model potentials for $V_{VP}$ and $V_{SE}$ as
defined in Refs. ginges ; ymyu .

We use Gaussian type orbitals (GTOs), as defined in Ref. mohanty , to obtain
the single particle orbitals. We have considered orbitals up to $h$-angular
momentum symmetry (orbital angular momentum $l=5$) to carry out all the
calculations. We have used 40 GTOs for each symmetry to obtain the DHF wave
function. However, we have frozen high-lying orbitals beyond energy 3000
atomic units (a.u.) to account for electron correlation effects through the
employed many-body methods. We have verified contributions from these
neglected orbitals using RRPA and they are found to be extremely small. These
contributions are included in the uncertainty estimation later.

Figure 1:  (X-DHF)/DHF values, where X represents contributions from the DHF,
RMBPT(2), RRPA and RCCSD methods, against the atomic number ($Z$) of the
negative alkali ions. Values from the DHF method act as reference (zero line
on x-axis) for comparison of correlation contributions incorporated at
different levels of many-body methods.

In Table 1, we present $\alpha_{d}$ values of all the negative alkali ions in
a.u., $ea_{0}^{3}$, from the DHF, RMBPT(2), RRPA and RCCSD methods using the
DC, DCB and DCQ Hamiltonians. It can be seen from this table that the DHF
method gives lower values in all the ions, and the electron correlation
contributions enhance their magnitudes. Except in the H- ion, the RRPA yields
the largest value for each ion. The results from the RMBPT(2) method are also
larger than the values obtained using the RCCSD method, except in the lighter
H- and Li- ions. It can be recalled that the results from the RMBPT(2) method
are actually the lowest-order core-polarization terms and results from RRPA
are the contributions from the all-order core-polarization effects including
the DHF value. Thus, the huge differences between the results from the
RMBPT(2) and RRPA methods imply that the core-polarization effects arising
through the higher-order perturbation theory are quite strong in the negative
alkali ions. Since RRPA values are the mean-field contributions (i.e. DHF
values) in the FF procedure, it would require immense efforts to attain
convergence in the values by evaluating the property in the FF framework than
in the perturbation approach through a many-body method as adopted here.
Nonetheless, the RCCSD method implicitly incorporates the RRPA contributions
in addition to correlation effects due to non-RRPA effects such as those from
the pair-correlations. Large differences between the RRPA and RCCSD results
indicate that the non-RRPA contributions are also substantially large, but
with opposite sign than that of the RRPA contributions. As a result, the final
results in the RCCSD method come out to be smaller than the RMBPT(2) values in
the heavier ions.

The electron correlation effects in the simplest two electron H- ion, which is
analogous to He atom, shows a unique trend than the rest of the ions. In this
system, the inclusion of electron correlation effects increases the
$\alpha_{d}$ values gradually through the RMBPT(2), RRPA and RCCSD methods.
Recently, we have evaluated this property for the negative chlorine (Cl-) and
gold (Au-) ions by employing the above many-body methods bijaya2 . We find
similar trends in the electron correlation effects in H- and Cl-, but the
correlation contributions at different levels of approximations in the many-
body theory are found to be quite large in H- compared to Cl-. The DHF value
of $\alpha_{d}$ was larger than the RCCSD value in Au-, in contrast to the
trend seen in the negative alkali ions. In Fig. 1, we plot the fractional
differences of the results from the DC Hamiltonian from different methods
against the atomic number ($Z$) of the negative alkali ions. This shows that
scaling of correlation contributions through RMBPT(2) and RRPA varies linearly
with the size of the considered systems, but no particular trend is followed
by the contributions from the RCCSD method.

Table 2: Contributions from the DC Hamiltonian through different RCC terms in the determination of $\alpha_{d}$ (in $ea_{0}^{3}$) of the negative alkali ions. The differences between the sum of the contributions from the mentioned terms and the final values from the RCCSD method given using the DC Hamiltonian in Table 1 correspond to the contributions from the remaining RCC terms that are not shown explicitly here. Ion | $DT_{1}^{(1)}$ | $T_{1}^{(0)\dagger}DT_{1}^{(1)}$ | $T_{2}^{(0)\dagger}DT_{1}^{(1)}$ | $T_{1}^{(0)\dagger}DT_{2}^{(1)}$ | $T_{2}^{(0)\dagger}DT_{2}^{(1)}$
---|---|---|---|---|---
H- | 149.59 | 30.47 | $-3.88$ | $-1.99$ | 32.25
Li- | 968.86 | $-146.32$ | $-108.09$ | 25.45 | 127.48
Na- | 1160.42 | $-193.82$ | $-118.33$ | 34.24 | 133.65
K- | 1741.75 | $-358.37$ | $-204.67$ | 69.89 | 197.74
Rb- | 1948.21 | $-414.85$ | $-224.50$ | 82.39 | 209.90
Cs- | 2351.13 | $-516.10$ | $-278.24$ | 105.36 | 250.74
Fr- | 2076.99 | $-448.47$ | $-214.02$ | 87.29 | 190.49

We notice from Table 1 that both the higher-order relativistic effects due to
the Breit and QED interactions do not contribute significantly to $\alpha_{d}$
of the considered ions. However, the trends in the results from different
approximations in many-body theories demonstrate that the estimated
corrections from the Breit and QED interactions vary in different methods. It
is, therefore, not prudent to estimate corrections due to these relativistic
effects by applying lower-order many-body methods as done in literature quite
often.

To gain a deeper insight into the behavior of electron correlation effects in
the determination of $\alpha_{d}$ of the negative alkali ions, we present
contributions from different RCC terms in Table 2 by using the DC Hamiltonian.
Contributions from the higher-order non-linear RCC terms are not given
explicitly, but their importance can be found from the differences in the
results after summing contributions from the shown terms and the final RCCSD
values from the DC Hamiltonian given in Table 1. Contrasting behaviors of
correlation effects between H- and other ions can be visibly noticeable. The
first term, $DT_{1}^{(1)}$, includes the DHF value, and the leading-order RRPA
and non-RRPA correlation contributions bijaya1 . This is the reason for the
seemingly dominant contribution of the term over all others. Though the
overall trend of correlation effects between H- and Cl- was earlier found to
be similar, comparison of individual contributions from various RCC terms of
the above table and that given in Ref. bijaya2  for Cl- does not suggest the
same. In Cl- ion, the $DT_{1}^{(1)}$ term accounts for almost all of the
$\alpha_{d}$ value in the RCCSD method, whereas in H-, almost all the RCC
terms are found to be significant. Also, it can be found from Ref. bijaya2
that only the first three terms contribute mostly to the determination of
$\alpha_{d}$ in the heavier Au- ion, where contributions from the
$T_{2}^{(0)\dagger}DT_{1}^{(1)}$ RCC term are found to be quite large in the
alkali negative ions. We plot contributions from the above RCC terms in Fig. 2
to demonstrate their roles quantitatively in different ions. For this purpose,
we have plotted $DT_{1}^{(1)}$ value after subtracting the DHF result. As
seen, the magnitudes of the correlation effects are slowly increasing with the
size of the system through each RCC term, with the only exception being the
trend from Cs- to Fr-.

Figure 2:  Plot demonstrating correlation contributions arising through the
leading order RCC terms against $Z$ of the negative alkali ions. The DHF value
is subtracted from the first term to show only its correlation contribution.

In Table 3, we quote the final values along with uncertainties by considering
the RCCSD results from the DC, DCB and DCQ Hamiltonians of our calculations.
The uncertainties are estimated by extrapolating contributions from the
finite-size basis functions that were used and the neglected higher-level
excitations (especially from the triple excitations). Our final values are
compared with the available literature results in the same table. It is worth
noting again that due to complication in measuring $\alpha_{d}$ of the
negative ions, there are no experimental values available till date and
calculations are reported only for a few lighter negative ions by employing
lower-order non-relativistic many-body methods. We find that among the
negative alkali ions, precise results are available only for H-, by applying
few-body methods. Though the RCC theory is not apt to investigate electron
correlation effects in H-, the applied RCCSD method in this work is the
complete ansatz of the RCC theory for this ion. Thus, its result can serve as
the benchmark to our calculations for other ions. A very good agreement of our
result with the reported precise value bhatia  implies reliability in the
calculation using the RCCSD method. In this context, we would like to
emphasize here that the DHF, RMBPT(2) and RRPA results are far-off from the
RCCSD result in this ion. The entire uncertainty to its $\alpha_{d}$ value
arises due to construction of basis functions owing to its point-size nucleus
in contrast to other ions. An old theoretical study presents $\alpha_{d}$
values of Li-, Na- and K- by analyzing photodetachment cross-sections in both
the length and velocity gauge approximations moores . Values from both the
gauge expressions differ widely, raising questions about their accuracies. Two
more calculations for $\alpha_{d}$ of Li- using CI methods are found in
literature pouchan ; agren . However, results from both the CI calculations
are far apart and much outside of the quoted error bars. Thus, these results
cannot be used assuredly for any application of this ion. Our $\alpha_{d}$
value of Li- matches well with the velocity gauge result of Ref. moores  and
calculation by Ågren et al agren , but this agreement could be coincidental as
large disagreements between our calculations, performed in length gauge, with
the results from gauges of Ref. moores  are seen in the Na- and K- ions. To
our knowledge, there are no theoretical studies on $\alpha_{d}$ of Rb-, Cs-
and Fr- available in the literature to make comparative analysis with our
calculations.

Table 3: The recommended $\alpha_{d}$ values (in $ea_{0}^{3}$) of negative
alkali ions from this work. Previous calculations using non-relativistic
methods are also given for comparison. The estimated uncertainties are quoted
in parentheses.

Ion | This work | Others
---|---|---
H- | 206.2(5) | 206.165 pipin , 206.37683 kar
|  | 206.1487618(37) bhatia
Li- | 794(2) | 832a moores , 798b moores , 650(50) pouchan
|  | 798(5) agren
Na- | 953(5) | 989a moores , 1058b moores
K- | 1354(7) | 1805a moores , 1757b moores
Rb- | 1508(8) |
Cs- | 1804(10) |
Fr- | 1620(10) |

a From the length gauge calculation.
b From the velocity gauge calculation.

We would also like to make an analogy among the roles of electron correlation
effects played in the determination of $\alpha_{d}$ values of the positively
charged ions and neutral atoms belonging to the isoelectronic sequence of the
considered negative ions. Comparing the calculations in the positively charged
ions and alkaline earth-metal atoms from our earlier work yashpal1 , and that
with the results for the undertaken negative ions reported in this work, we
find that the Breit and QED contributions are negligibly small in all these
three types of systems belonging to the same isoelectronic sequence. The
correlation trends are found to be almost similar among these systems, but the
final results are found to be at least an order bigger in the negative ions
than their positively charged counterparts and neutral atoms bijaya1 ;
yashpal1 ; schwerdtfeger2 . This implies that the negative ions are highly
polarized and very sensitive to an applied electric field.

In summary, we have evaluated electric dipole polarizabilities of all the
negative alkali ions very accurately. Propagations of electron correlation
effects are investigated from lower-order to all-order in perturbation in the
determination of these quantities by employing a relativistic second-order
perturbation theory, random phase approximation and coupled-cluster method on
the Dirac-Hartree-Fock calculation. Trends of correlation effects from these
methods and through different terms of the coupled-cluster approach are
demonstrated quantitatively. It shows that the roles of electron correlation
effects follow almost similar trends in all ions except in H-, on which they
bestow a completely different trend. Random-phase approximation predicts the
values by huge magnitudes, while the coupled-cluster method gave moderate
values. This clearly suggests that there are huge cancellations among the
electron correlation effects arising through the pair-correlation and core-
polarization effects. It also demonstrates that magnitudes of the correlation
effects increase with the size of the system with an exception in the trend
from Cs- to Fr-. We have also compared electron correlation trends in the
determination of the above-mentioned quantities among the negative alkali
ions, and their isoelectronic positively charged ions and neutral atoms. This
comparison shows that electric dipole polarizabilities in the above ions are
about an order larger than the values of their respective positively charged
ions and neutral atoms indicating that negative alkali ions are highly
polarized. Higher-order relativistic corrections due to the Breit and quantum
electrodynamics interactions are found to be negligibly small, but their
magnitudes are observed to be modified with the electron correlation effects.
There are only a few existing calculations of the investigated property
available for the lighter ions. Since it is extremely difficult to measure the
electric dipole polarizabilities of the negative alkali ions, our estimated
precise values of these quantities will be very useful to the applications of
these ions. Our results can be further improved after including contributions
from the triple excitations.

We acknowledge use of Vikram-100 HPC cluster of Physical Research Laboratory,
Ahmedabad, India, which we used to carry out computations for this work.

## References

  * (1) A. Vermeer and N. A. van Zwol, Nucl. Instr. Meth. 175, 393 (1980).
  * (2) T. Andersen, Phys. Rep. 394, 157 (2004).
  * (3) K. T. Andersson, J. Sandström, I. Yu. Kiyan, D. Hanstorp and D. J. Pegg, Phys. Rev. A 62, 022503 (2000).
  * (4) H. Hotop and W. C. Lineberger, J. Phys. Chern. Ref. Data 4, 539 (1975).
  * (5) A. Kasdan and W. C. Lineberger, Phys. Rev. A 10, 1658 (1974).
  * (6) T. Andersen, H. K. Haugen and H. Hotop, J. Phys. Chem. Ref. Data 28, 1511 (1999).
  * (7) T. A. Patterson, H. Hotop, A. Kasdan and D. W. Norcross, Phys. Rev. Lett. 32, 189 (1974).
  * (8) U. Berzinsh, G. Haeffler, D. Hanstorp, A. Klinkmüller, E. Lindroth, U. Ljungblad, and D. J. Pegg, Phys. Rev. Lett. 74, 4795 (1995).
  * (9) A. O. Lindahl, J. Rohlén, H. Hultgren, D. J. Pegg, C. W. Walter and D. Hanstorp, Phys. Rev. A 88, 053410 (2013).
  * (10) S. Watanabe and C. H. Greene, Phys. Rev. A 22, 158 (1980).
  * (11) M. T. Eiles and C. H. Greene, Phys. Rev. Lett. 121, 133401 (2018).
  * (12) G. Haeffler, I. Yu. Kiyan, D. Hanstorp, B. J. Davies and D. J. Pegg, Phys. Rev. A 59, 3655 (1999).
  * (13) D. L. Moores and D. W. Norcross, Phys. Rev. A 10, 1646 (1974).
  * (14) J. Slater, F. H. Read, S. E. Novick and W. C. Lineberger, Phys. Rev. A 17, 201 (1978).
  * (15) K. T. Taylor and D. W. Norcross, Phys. Rev. A 34, 3878 (1986).
  * (16) D. W. Norcross, Phys. Rev. Lett. 32, 192 (1974).
  * (17) C. H. Greene, Phys. Rev. A 42, 1405 (1990).
  * (18) C. Bahrim and U. Thumm, Phys. Rev. A 61, 022722 (2000).
  * (19) P. Balling, H. H. Andersen, C. A. Brodie, U. V. Pedersen, V. V. Petrunin, M. K. Raarup, P. Steiner, and T. Andersen, Phys. Rev. A 61, 022702 (2000).
  * (20) R. Wlidt, Astrophys. J. 93, 47 (1941).
  * (21) A. R. P. Rau, J. Astrophys. Astr. 17, 113 (1996).
  * (22) T. J. Millar, C. Walsh and T. A. Field, Chem. Rev. 117, 1765 (2017).
  * (23) V. Dudinikov, Development and Applications of Negative Ion Sources, Springer Series on Atomic, Optical, and Plasma Physics, v. 110, Switzerland (2019).
  * (24) D. Faircloth and S. Lawrie, New J. Phys. 20, 025007 (2018).
  * (25) P. Schwerdtfeger and J. K. Nagle, Mol. Phys. 117, 1200 (2019).
  * (26) J. Pipin and D. M. Bishop, J. Phys. B 25, 17 (1992).
  * (27) S. Kar, Y.-S. Wang, Y. Wang and Y. K. Ho, Int. J Quantum Chem. e25515 (2017).
  * (28) A. K. Bhatia and R. J. Drachman, J. Phys. B: At. Mol. Opt. Phys. 27, 1299 (1994).
  * (29) C. Pouchan and D. M. Bishop, Phys. Rev. A 29, 1 (1984).
  * (30) H. Ågren, J. Olsen, H. J. Aa. Jensen and P. Jo/rgensen, Phys. Rev. A 40, 2265 (1989).
  * (31) B. Arora and B. K. Sahoo, Phys. Rev. A 86, 033416 (2012).
  * (32) B. K. Sahoo and B. Arora, Phys. Rev. A 87, 023402 (2013).
  * (33) I. S. Lim and P. Schwerdtfeger, Phys. Rev. A 70, 062501 (2004) .
  * (34) B. K. Sahoo and B. P. Das, Phys. Rev. A 77, 062516 (2008).
  * (35) Y. Singh, B. K. Sahoo and B. P. Das, Phys. Rev. A 88, 062504 (2013).
  * (36) G. H. F. Diercksen and A. Sadlej, Chem. Phys. Lett. 84, 390 (1981).
  * (37) V. Kellö, M. Urban and A. J. Sadlej, Chem. Phys. Lett. 253, 383 (1996).
  * (38) P. Schwerdtfeger and G.A. Bowmaker, J. Chem. Phys. 100, 4487 (1994).
  * (39) A. Dalgarno and J. T. Lewis, Proc. Roy. Sot. Aa33, 70 (1956).
  * (40) Y. Singh and B. K. Sahoo, Phys. Rev. A 90, 022511 (2014).
  * (41) B. K. Sahoo, Submitted to Phys. Rev. Research.
  * (42) B. K. Sahoo and B. P. Das, Phys. Rev. Letts. 120, 203001 (2018).
  * (43) Y. Singh and B. K. Sahoo, Phys. Rev. A 91, 030501(R) (2015).
  * (44) J. S. M. Ginges and J. C. Berengut, Phys. Rev. A 93, 052509 (2016).
  * (45) Y.-M. Yu and B. K. Sahoo, Phys. Rev. A 99, 022513 (2019).
  * (46) A. Mohanty and E. Clementi, Modern techniques in Computational Chemistry: MOTECC-89, Ch. 4, pg. 169 (1989).


# Electric-Magnetic duality invariance of Galilean relativistic Born-Infeld
theory

Rabin Banerjee1 111DAE Raja Ramanna fellow Soumya Bhattacharya1 Bibhas Ranjan
Majhi2 1Department of Astrophysics and High Energy Physics, S.N. Bose National
Center for Basic Sciences, Kolkata 700106, India<EMAIL_ADDRESS><EMAIL_ADDRESS>2 Department of Physics, Indian Institute of Technology
Guwahati, Guwahati 781039, Assam, India<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce for the first time, in a systematic manner,
Galilean relativistic Born-Infeld action in detail. Exploiting maps connecting
Lorentz relativistic and Galilean relativistic vectors, we construct the two
limits (electric and magnetic) of Galilean relativistic Born-Infeld action
from usual Born-Infeld theory. An action formalism is thereby derived. From
this action, equations of motion are obtained either in the potential or field
formulation. Duality transformations involving the electric and magnetic
fields are defined that map the electric limit relations to the magnetic ones
and vice-versa, exactly as happens for Galilean relativistic Maxwell theory.
We also explicitly show the Galilean boost and gauge invariances of the
theory.

##  I Introduction

The low energy effective description of a system is usually explained by the
study of a non-relativistic (NR) field theory. Non-relativistic limit of
classical field theories received considerable attention recently. It has
found applications in various diverse branches of theoretical physics like
holography [1], non-relativistic diffeomorphism invariant theories [2, 3, 4,
5, 6, 7], condensed matter physics [8, 9], fluid dynamics [10, 11] and
gravitation [12, 13] to name a few. This formulation is tricky and altogether
different from the relativistic case. The absolute nature of time in the non-
relativistic limit leads to the lack of a single non-degenerate metric which
poses some additional difficulties. It is very interesting to understand
Galilean invariance and its role in field theories, especially in gauge
theories [14]. The basic construction of Galilean electrodynamics was first
given by Le Bellac and Levy-Leblond [15] back in 1970’s. This was done in the
field formulation. A similar field based analysis was done in [16] using
embedding techniques. Further directions in this type of analysis were
provided in [17]. A systematic covariant construction of the action principle
of Galilean relativistic Maxwell theory was given in [18] and a covariant
action formulation of Galilean relativistic Proca theory was provided in [19].

The principal motivation of the present paper is to investigate Galilean
versions of non-linear theories of electromagnetism. Among various
possibilities, the Born-Infeld (BI) theory [20] is of particular importance.
Ths is because the inclusion of electron’s self energy demands non-linearity
in the electromagnetic theory. In that case the effective action includes non-
linear terms apart from the standard Maxwell term. Such an action is well
known as Born–Infeld (BI) action [20] and gave birth to the field of nonlinear
electrodynamics. Also, such a theory is a low-energy effective action for the
electromagnetism of $D$-brane. In fact we can use the effective action of an
open string ending on $D$-brane to identify the electromagnetism of $D$-brane
[21]. One indeed can show that a $n$-dimensional BI action, leads to Monge-
gauge-fixed $(p+1)$ dimensional Dirac-Born-Infeld (DBI) action under
dimensional reduction, which for $p=1$ yields the string action [22]. On the
other hand BI action is also derivable from that of the open string action
[23]. In recent times non-relativistic string theory attracted considerable
attention [24]. Therefore understanding the non-relativistic regime of BI
theory serves a two-fold purpose. In one direction we will be able to know
about the non-relativistic nature of non-linear electrodynamics and on the
other side, various non-relativistic features of open-string can be
illuminated.
Given the importance of the BI theory, we are going to construct a Galilean
relativistic version of the leading order BI theory. The duality symmetry of
the leading order relativistic BI theory has been discussed in [26]. We
systematically construct the Galilean relativistic action for both electric
and magnetic limits, respectively. We derived the equations of motion in terms
of Galilean potentials. We then move to the field formulation and recast the
equations of motion in terms of Galilean electric and magnetic fields. We
write down a Galilean version of the duality relations. We have found that
under these duality relations the equations of motion in the electric limit go
to the Bianchi identities of the magnetic limit and the Bianchi identities of
the electric limit map to the equations of motion in the magnetic limit.
Similarly magnetic limit equations map to that of the electric limit case. We
also show the inavariance of the lagrangian under the Galilean boost and gauge
transformations.
The paper is organised as follows: the mapping relations between Lorentz and
galilean vectors are briefly discussed in section II. The relativistic BI
theory and duality symmetry, in leading order, has been discussed in terms of
electric and magnetic fields in section III. In section IV we give a Galilean
covariant action principle for the Born-Infeld theory. The equations of motion
in terms of fields are shown in section V. We discuss the Galilean version of
the duality symmetry in section VI. We show that two Galilean limits (electric
and magnetic) are swapped under these duality transformations. In section VII
the invariance of the Galilean lagrangian under Galilean boost has been shown.
The gauge invariance of the lagrangian is shown in section VIII. Finally
conclusions have been given in section IX.

##  II Maps relating Lorentz and Galilean vectors

Here we briefly review the basic construction of the scaling relations between
special relativistic and Galilean relativistic quantities. We first consider
the two Galilean limits (electric and magnetic) for vector quantities. So
first let us take the contravariant vectors. Under Lorentz transformations,
with boost velocity $u^{i}$, the time-like and the space-like components of a
vector $V^{\mu}$ transforms as (also considering $u<<c$, so $\gamma\to 1$)

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

Galilean vectors $v^{0},~{}v^{i}$ are introduced by the scaling,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)

$V^{0}=cv^{0},\,\,\,\,V^{i}=v^{i}$ (3)

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
3 in eqns 1 and 2 we get

$v^{\prime 0}=v^{0}$ (4) $v^{\prime i}=v^{i}-u^{i}v^{0}$ (5)

These equations define the usual galilean transformations.
We next consider the magnetic limit which corresponds to largely spacelike
vectors

$V^{0}=-\frac{v^{0}}{c},\,\,\,\,V^{i}=v^{i}$ (6)

Now using 6 in 1 and 2 we get

$v^{\prime 0}=v^{0}+u_{j}v^{j}$ (7) $v^{\prime i}=v^{i}$ (8)

which is again a Galilean transformation, although the corresponding group is
not an invariance group of classical mechanics. Expectedly, the role of time-
like and space-like vectors has been reversed from the earlier case.
Similarly one can show that the covariant vectors transform under Lorentz
transformations in the low velocity limit as

$V^{\prime}_{0}=V_{0}+\frac{u^{i}}{c}V_{i}$ (9)
$V^{\prime}_{i}=V_{i}+\frac{u_{i}}{c}V_{0}$ (10)

Now we take the electric limit in the following way [18],

$V_{0}=\frac{v_{0}}{c},\,\,\,\,V_{i}=v_{i}$ (11)

Using 11 in 9 and 10 we get

$\displaystyle v^{\prime}_{0}=v_{0}+u^{i}v_{i}$ (12) $\displaystyle
v^{\prime}_{i}=v_{i}$ (13)

which are again Galilean transformations.
Likewise, in the magnetic limit the covariant vectors scale as

$V_{0}=-cv_{0},\,\,\,\,V_{i}=v_{i}$ (14)

Using 14 in 9 and 10 we get

$v_{0}^{\prime}=v_{0}$ (15) $v^{\prime}_{i}=v_{i}-u_{i}v_{0}$ (16)

The complete mapping relations are summarised in the table 1.

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

##  III Born-Infeld theory in the leading order

The Born-Infeld lagrangian is expressed in terms of the square root of a
determinant as

$\mathcal{L}=-\frac{1}{g^{2}}\Big{(}\sqrt{-{\rm
det}(\eta_{\mu\nu}+gF_{\mu\nu})}-1\Big{)}~{},$ (17)

which in the leading order reduces to the following form [26]

$\mathcal{L}=-\frac{1}{4}F^{2}-g^{2}(F^{2})^{2}+4g^{2}(FFFF)^{\mu}_{~{}\mu}~{}.$
(18)

In the above we denoted $F^{2}=F^{\mu\nu}F_{\mu\nu}$ and
$(AB)^{\mu}_{~{}\nu}=A^{\mu\lambda}B_{\lambda\nu}$. Using the leading order
lagrangian (18) we can write the equations of motion and the Bianchi
identities respectively in a covariant form as

$\partial_{\mu}{}^{*}G^{\mu\nu}=0,~{}~{}~{}~{}~{}~{}\partial_{\mu}{}^{*}F^{\mu\nu}=0~{},$
(19)

where the dual tensor is defined as,

${}^{*}G^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\lambda}G_{\rho\lambda}~{};$
(20)
$G^{\mu\nu}={}^{*}F^{\mu\nu}+8g^{2}F^{2}({}^{*}F^{\mu\nu})+32g^{2}{}^{*}(FFF)^{\mu\nu}~{}.$
(21)

The set of equations in (19) is preserved under the following discrete duality
transformations

$F\to G,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G\to-F~{}.$ (22)

Since we are eventually going to discuss Galilean relativistic Born-Infeld
theory, it is useful to recast the above results in their noncovariant forms
using electric and magnetic fields.
Equations of motion:
The first relation in (19) yields

$\displaystyle\vec{\nabla}.\vec{E}+16g^{2}\vec{\nabla}.\Big{(}\vec{E}(E^{2}-B^{2})\Big{)}+32g^{2}\vec{B}.\vec{\nabla}(\vec{E}.\vec{B})=0~{};$
(23)

and

$\begin{split}-\partial_{t}\vec{E}+(\vec{\nabla}\times\vec{B})-16g^{2}\partial_{t}\Big{(}\vec{E}(E^{2}-B^{2})\Big{)}-32g^{2}\partial_{t}\Big{(}\vec{B}(\vec{E}.\vec{B})\Big{)}&\\\
-16g^{2}\vec{\nabla}\times\Big{(}\vec{B}(E^{2}-B^{2})\Big{)}-32g^{2}\vec{\nabla}\times\Big{(}\vec{E}\times(\vec{E}\times\vec{B})\Big{)}-32g^{2}\vec{\nabla}\times(B^{2}\vec{B})=0~{}.\end{split}$
(24)

Here the electric and magnetic fields are defined as,

$\displaystyle E^{i}=F^{0i}=\partial^{0}A^{i}-\partial^{i}A^{0},$
$\displaystyle
B^{i}=\frac{1}{2}\epsilon^{ijk}F_{jk}=\epsilon^{ijk}\partial_{j}A_{k}~{}.$
(25)

Note that these incorporate nonlinear terms apart from the standard Maxwell
ones.

Bianchi identities:
The other relation in (19) leads to two of the standard Maxwell’s four
equations:

$\displaystyle\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}~{};$ (26) $\displaystyle\vec{\nabla}.\vec{B}=0~{}.$ (27)

Expectedly, the Bianchi identities retain their usual Maxwell form. The
equations of motion in the limit $g\to 0$ also reproduce the desired Maxwell
equations. The duality relations 22 can be expressed explicitly in the
component form as well.
Duality relations:
The first one in (22) implies the following two mappings:

$\displaystyle\vec{E}\to\vec{B}-16g^{2}(E^{2}-B^{2})\vec{B}-32g^{2}\vec{E}\times(\vec{E}\times\vec{B})-32g^{2}B^{2}\vec{B}~{};$
(28)
$\displaystyle\vec{B}\to-\vec{E}-16g^{2}(E^{2}-B^{2})\vec{E}-32g^{2}(\vec{E}.\vec{B})\vec{B}~{}.$
(29)

Whereas the last one signifies

$\displaystyle\vec{B}-16g^{2}(E^{2}-B^{2})\vec{B}-32g^{2}\vec{E}\times(\vec{E}\times\vec{B})-32g^{2}B^{2}\vec{B}\to-\vec{E}~{};$
(30)
$\displaystyle\vec{E}+16g^{2}(E^{2}-B^{2})\vec{E}+32g^{2}(\vec{E}.\vec{B})\vec{B}\to\vec{B}~{}.$
(31)

It can be shown that the two sets of dualities given above i.e. (28), (29) and
(30), (31) are consistent among each other. Also in the limit $g\to 0$
corresponding to Maxwell theory, the results yield the well known duality
maps,

$\vec{E}\to\vec{B},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vec{B}\to-\vec{E}$ (32)

using either set. Under these duality relations the equations of motion i.e.
(23) and (24) will go to the Bianchi identities (27) and (26) respectively,
and vice-versa.

##  IV Galilean relativistic Born-Infeld theory: action principle

Here we wish to systematically build up a Galilean relativistic version of the
Born-Infeld theory. Let us now start from the relativistic Born-Infeld theory
described by the Lagrangian given in (18). Below we consider two Galilean
limits separately.

###  IV.1 Electric limit

Using the relations given in table 1 we can express the terms in (18) as

$\displaystyle F^{2}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}-2\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}+f^{2},\hskip
72.26999pt$ (33)

and

$\displaystyle(FFFF)^{\mu}_{~{}\mu}\xrightarrow[\text{$c\to\infty$}]{\text{Electric
limit}}2\partial^{l}a^{0}\partial^{k}a^{0}(\partial_{l}a_{0}-\partial_{t}a_{l})(\partial_{k}a_{0}-\partial_{t}a_{k})+4\partial^{i}a^{0}(\partial_{t}a_{m}-\partial_{m}a_{0})(ff)^{m}_{~{}i}$
$\displaystyle+(ffff)^{i}_{~{}i}~{};$ (34)

where we denoted $f^{2}=f^{ij}f_{ij}$. So in the electric limit we can write
down the Galilean relativistic action as

$S_{e}=\int\mathcal{L}_{e}dtd^{3}x~{},$ (35)

where the Lagrangian takes the following form

$\displaystyle\mathcal{L}_{e}=\frac{1}{2}\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}-\frac{1}{4}f^{2}-g^{2}\Big{(}-2\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}+f^{2}\Big{)}^{2}+\hskip
57.81621pt$ $\displaystyle
4g^{2}\Big{(}2\partial^{l}a^{0}\partial^{k}a^{0}(\partial_{l}a_{0}-\partial_{t}a_{l})(\partial_{k}a_{0}-\partial_{t}a_{k})+4\partial^{i}a^{0}(\partial_{t}a_{m}-\partial_{m}a_{0})(ff)^{m}_{~{}i}+(ffff)^{i}_{~{}i}\Big{)}~{}.$
(36)

Now varying the action (35) with respect to $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$
we get the following equations respectively,

$\displaystyle\partial_{i}\partial^{i}a^{0}+8g^{2}\partial_{i}(\alpha\partial^{i}a^{0})-32g^{2}\partial_{i}\Big{(}\partial^{i}a^{0}\partial^{k}a^{0}(\partial_{k}a_{0}-\partial_{t}a_{k})-\partial^{j}a^{0}(ff)^{i}_{~{}j}\Big{)}=0~{};$
(37)
$\displaystyle\partial_{t}\partial^{j}a^{0}-\partial_{i}f^{ij}+8g^{2}\partial_{t}(\alpha\partial^{j}a^{0})-8g^{2}\partial_{i}(\alpha
f^{ij})+32g^{2}\partial_{t}\Big{(}\partial^{j}a^{0}\partial^{l}a^{0}(\partial_{l}a_{0}-\partial_{t}a_{l})+\partial^{l}a^{0}(ff)_{l}^{~{}j}\Big{)}+$
$\displaystyle
8g^{2}\partial_{i}\Big{(}4\partial^{i}a^{0}(\partial_{t}a_{k}-\partial_{k}a_{0})f^{jk}-4\partial^{j}a^{0}(\partial_{t}a_{k}-\partial_{k}a_{0})f^{ik}+2(fff)^{ji}-2(fff)^{ij}\Big{)}=0~{};$
(38)
$\displaystyle\partial^{i}(\partial_{t}a_{i}-\partial_{i}a_{0})+8g^{2}\partial^{i}\Big{(}\alpha(\partial_{t}a_{i}-\partial_{i}a_{0})\Big{)}+32g^{2}\partial^{i}\Big{(}\partial^{j}a^{0}(\partial_{i}a_{0}-\partial_{t}a_{i})(\partial_{j}a_{0}-\partial_{t}a_{j})+(\partial_{t}a_{k}-\partial_{k}a_{0})(ff)_{i}^{~{}k}\Big{)}=0~{};$
(39)

and

$\displaystyle\partial^{i}f_{ij}+8g^{2}\partial^{i}(\alpha
f_{ij})-8g^{2}\partial^{i}\Big{(}4\partial^{k}a^{0}(\partial_{t}a_{i}-\partial_{i}a_{0})f_{jk}-4\partial^{k}a^{0}(\partial_{t}a_{j}-\partial_{j}a_{0})f_{ik}+2(fff)_{ji}-2(fff)_{ij}\Big{)}=0~{}.$
(40)

In the above $\alpha$ represents

$\alpha=-2\partial^{i}a^{0}\Big{(}\partial_{t}a_{i}-\partial_{i}a_{0}\big{)}+f^{2}~{}.$
(41)

###  IV.2 Magnetic limit

In the magnetic limit, using table 1, we can write down the Galilean
relativistic action as

$S_{m}=\int\mathcal{L}_{m}dtd^{3}x~{},$ (42)

where the Lagrangian takes the following form

$\displaystyle\mathcal{L}_{m}=\frac{1}{2}\partial_{i}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})-\frac{1}{4}f^{2}-g^{2}\Big{(}-2\partial_{i}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})+f^{2}\Big{)}+\hskip
36.135pt$ $\displaystyle
4g^{2}\Big{(}2\partial_{i}a_{0}\partial_{j}a_{0}(\partial_{t}a^{i}-\partial^{i}a_{0})(\partial_{t}a^{j}-\partial^{j}a^{0})+4\partial_{k}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})(ff)_{i}^{~{}k}+(ffff)^{i}_{~{}i}\Big{)}~{}.$
(43)

Now varying the action (42) with respect to $a_{0},~{}a_{j},~{}a^{0},~{}a^{j}$
we have the following set of equations respectively,

$\displaystyle\partial_{i}(\partial_{t}a^{i}-\partial^{i}a^{0})+8g^{2}\partial_{i}\Big{(}\alpha(\partial_{t}a^{i}-\partial^{i}a^{0})\Big{)}+32g^{2}\partial_{i}\Big{(}\partial_{j}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})(\partial_{t}a^{j}-\partial^{j}a^{0})+(\partial_{t}a^{l}-\partial^{l}a^{0})(ff)_{l}^{~{}i}\Big{)}=0~{};$
(44) $\displaystyle\partial_{i}f^{ij}+8g^{2}\partial_{i}(\alpha
f^{ij})-8g^{2}\partial_{i}\Big{(}4\partial_{k}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})f^{jk}-4\partial_{k}a_{0}(\partial_{t}a^{j}-\partial^{j}a^{0})f^{ik}+2(fff)^{ji}-2(fff)^{ij}\Big{)}=0~{};$
(45)
$\displaystyle\partial^{i}\partial_{i}a_{0}+8g^{2}\partial^{i}(\alpha\partial_{i}a_{0})-32g^{2}\partial^{i}\Big{(}\partial_{i}a_{0}\partial_{k}a_{0}(\partial^{k}a^{0}-\partial_{t}a^{k})-\partial_{j}a_{0}(ff)^{i}_{~{}j}\Big{)}=0~{};$
(46)

and

$\displaystyle\partial_{t}\partial_{j}a_{0}-\partial^{i}f_{ij}+8g^{2}\partial_{t}(\alpha\partial_{j}a_{0})-8g^{2}\partial^{i}(\alpha
f_{ij})+32g^{2}\partial_{t}\Big{(}\partial_{j}a_{0}\partial_{i}a_{0}(\partial_{t}a^{i}-\partial^{i}a^{0})+\partial_{k}a_{0}f_{jm}f^{mk}\Big{)}+$
$\displaystyle
8g^{2}\partial^{i}\Big{(}4\partial_{i}a_{0}(\partial_{t}a^{k}-\partial^{k}a^{0})f_{jk}-4\partial_{j}a_{0}(\partial_{t}a^{k}-\partial^{k}a^{0})f_{ik}+2(fff)_{ji}-2(fff)_{ij}\Big{)}=0~{}.$
(47)

We observe that the electric limit equations of motion ((37) - (40)) and
magnetic limit equations of motion ((44) - (47)) satisfy a symmetry property.
Under an interchange of covariant and contravariant indices, the results for
electric limit go to magnetic limit and vice versa.

${\rm{\bf Contravariant\longleftrightarrow
Covariant}}\Longleftrightarrow{\rm{\bf Electric\longleftrightarrow Magnetic}}$

##  V Galilean electric and magnetic fields

Here we introduce the Galilean limit of electric and magnetic fields and write
down the equations of motion for Galilean relativistic Born-Infeld theory,
obtained in the previous section, in terms of these fields. Note that
relativistic electric and magnetic fields are defined in (25).

###  V.1 Electric limit

Using the maps given in table 1 we can write the contravariant and covariant
components of the electric field as

$\displaystyle
E^{i}=-\frac{1}{c}\partial_{t}a^{i}-c\partial^{i}a^{0},~{}~{}~{}~{}E_{i}=-\Big{(}\frac{1}{c}\partial_{t}a_{i}+\frac{1}{c}\partial_{i}a_{0}\Big{)}~{}.$
(48)

Also we define the Galilean electric and magnetic fields as

$e^{i}=\lim_{c\to\infty}\frac{E^{i}}{c}=-\partial^{i}a^{0},\,\,\,~{}b^{i}=\lim_{c\to\infty}B^{i}=\epsilon^{ij}_{~{}k}\partial_{j}a^{k},~{}~{}~{}~{}e_{i}=\lim_{c\to\infty}cE_{i}=-(\partial_{t}a_{i}-\partial_{i}a_{0}),\,\,\,~{}~{}b_{i}=\lim_{c\to\infty}B_{i}=\epsilon_{i}^{~{}jk}\partial_{j}a_{k}~{}.$
(49)

Now we write the field equations that we derived in the previous section in
terms of the Galilean electric and magnetic fields. Using the maps given in
Eq. (49) we express (37) as

$\partial_{i}e^{i}+16g^{2}\partial_{i}\Big{(}(e^{2}-b^{2})e^{i}\Big{)}+32g^{2}b^{i}\partial_{i}(e^{j}b_{j})=0~{},$
(50)

where $e^{2}=e^{i}e_{i}$, $b^{2}=b^{i}b_{i}$. Similarly (38) takes the form as
follows:

$\displaystyle-\partial_{t}e^{i}+(\vec{\nabla}\times\vec{b})^{i}-16g^{2}\partial_{t}\Big{(}(e^{2}-b^{2})e^{i}\Big{)}-32g^{2}\partial_{t}\Big{(}b^{i}(e^{k}b_{k})\Big{)}-16g^{2}[\vec{\nabla}\times\Big{(}(e^{2}-b^{2})\vec{b}\Big{)}]^{i}$
$\displaystyle-32g^{2}[\vec{\nabla}\times\Big{(}\vec{e}\times(\vec{e}\times\vec{b})\Big{)}]^{i}-32g^{2}[\vec{\nabla}\times(b^{2}\vec{b})]^{i}=0~{}.$
(51)

In an identical manner we can recast equations (39) and (40) in terms of
Galilean covariant electric and magnetic fields, defined in Eq. (49). We are
not presenting them here as one can can easily check that they are of same
form as those in magnetic limit, with the covariant fields replaced by
contravariant fields (as defined in (58)). This we leave for the next
subsection. This fact verifies our observation which is mentioned at the end
of IV, that under the interchange of covariant and contravariant indices
electric limit goes to magnetic limit and vice-versa.

We next write the Bianchi identities using fields. We can see from
relativistic Born-Infeld theory that Bianchi identities ((26) and (27)) retain
the same form as that of Maxwell case. So in the Galilean limit these Bianchi
identities reduce to those of Galilean relativistic Maxwell theory, shown in
[18]. First, using the field definitions given in (49) we can write

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

The same result follow by starting from (26), after an appropriate insertion
of $c$ (the velocity of light in free space),

$\vec{\nabla}\times\vec{E}=-\frac{1}{c}\frac{\partial\vec{B}}{\partial t}$
(53)

and then using (49) followed by $c\to\infty$,

$\vec{\nabla}\times\vec{e}=-\frac{1}{c^{2}}\frac{\partial\vec{b}}{\partial
t}\to 0~{}.$ (54)

Similarly one finds,

$\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0~{}.$ (55)

On the other hand if we start with the relativistic equations directly, then
using the mapping relation summarised in table 2 in (23) we have

$\displaystyle
c~{}\partial_{i}e^{i}+16g^{2}c~{}\partial_{i}(e^{i}(e^{2}-b^{2})\Big{)}+32g^{2}c~{}b^{i}\partial_{i}(e^{k}b_{k})=0~{},$
(56)

which in the $c\to\infty$ will give rise to (50). Similarly from (24) we can
reproduce (51) in the $c\to\infty$ limit. This shows the internal consistency
of the results.

Table 2: Fields in galilean limit ${\rm Limits}$ | ${\rm Electric~{}field}$ | ${\rm Magnetic~{}field}$
---|---|---
${\rm Electric~{}limit}$ | $E^{i}\to ce^{i},~{}E_{i}\to\frac{e_{i}}{c}$ | $B^{i}\to b^{i},~{}B_{i}\to b_{i}$
${\rm Magnetic~{}limit}$ | $E^{i}\to\frac{e^{i}}{c},~{}E_{i}\to ce_{i}$ | $B^{i}\to b^{i},~{}B_{i}\to b_{i}$

###  V.2 Magnetic limit

The contravariant and covariant components of the electric field can be
written in this limit from the mapping relations in table 1 as

$E^{i}=-\frac{1}{c}\partial_{t}a^{i}+\frac{1}{c}\partial_{i}a^{0},~{}~{}~{}~{}~{}~{}~{}E_{i}=-\Big{(}\frac{1}{c}\partial_{t}a_{i}+c\partial_{i}a_{0}\Big{)}~{}.$
(57)

Correspondingly, the Galilean electric and magnetic fields are defined as,

$e^{i}=\lim_{c\to\infty}cE^{i}=-(\partial_{t}a^{i}-\partial^{i}a^{0}),\,\,\,~{}~{}b^{i}=\lim_{c\to\infty}B^{i}=\epsilon^{ij}_{~{}k}\partial_{j}a^{k}~{}~{}~{}~{}~{}~{}e_{i}=\lim_{c\to\infty}\frac{E_{i}}{c}=-\partial_{i}a_{0},\,\,\,~{}~{}b_{i}=\lim_{c\to\infty}B_{i}=\epsilon_{i}^{~{}jk}\partial_{j}a_{k}~{}.$
(58)

Now we write the field equations that we derived earlier in terms of the
Galilean electric and magnetic fields. From equation (44) using the mapping
relation given in (58) we have

$\partial_{i}e^{i}+16g^{2}\partial_{i}(e^{i}(e^{2}-b^{2})\Big{)}+32g^{2}b^{i}\partial_{i}(e^{k}b_{k})=0~{}.$
(59)

Similarly (45) yields

$(\vec{\nabla}\times\vec{b})+16g^{2}\vec{\nabla}\times\Big{(}(e^{2}-b^{2})\vec{b}\Big{)}-32g^{2}\vec{\nabla}\times\Big{(}\vec{e}(\vec{e}.\vec{b})\Big{)}=0~{}.$
(60)

From the Bianchi identities we have the following two equations

$(\vec{\nabla}\times\vec{e})^{i}=\epsilon^{ij}_{~{}~{}k}\partial_{j}e^{k}=-\partial_{t}b^{i}~{};$
(61)

and

$\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0~{}.$ (62)

These equations can also be derived directly from the relativistic Maxwell
equations, as we did for the electric case.

##  VI Duality transformations

The duality transformations given in (28) and (29) will take the following
forms in the non-relativistic limit

$\displaystyle e^{i}\to
b^{i}+16g^{2}(e^{2}-b^{2})b^{i}-32g^{2}e^{i}(e_{k}b^{k})$ (63) $\displaystyle
b^{i}\to-e^{i}-16g^{2}(e^{2}-b^{2})e^{i}-32g^{2}(e^{k}b_{k})b^{i}~{}.$ (64)

Now we use these duality relations in the electric and magnetic limit
equations, obtained in the last section, separately and observe what these
transformations correspond.

Let us start with the equations in electric limit. Using the non-relativistic
duality relations (63) and (64) in (50) and keeping upto $O(g^{2})$ we get

$\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0~{}.$ (65)

Similarly use of them in (51) gives rise to

$(\vec{\nabla}\times\vec{e})^{i}=\epsilon^{ij}_{k}\partial_{j}e^{k}=-\partial_{t}b^{i}~{}.$
(66)

Whereas under these duality transformations the first Bianchi identity (52)
leads to

$(\vec{\nabla}\times\vec{b})+16g^{2}\vec{\nabla}\times\Big{(}(e^{2}-b^{2})\vec{b}\Big{)}-32g^{2}\vec{\nabla}\times\Big{(}\vec{e}(\vec{e}.\vec{b})\Big{)}=0~{},$
(67)

while the second Bianchi identity (55) yields

$\partial_{i}e^{i}+16g^{2}\partial_{i}(e^{i}(e^{2}-b^{2})\Big{)}+32g^{2}b^{i}\partial_{i}(e^{k}b_{k})=0~{}.$
(68)

Interestingly, these relations ((65) - (68)) exactly correspond to those in
magnetic limit (see, (59) - (62)).

Now we will concentrate on the equations in magnetic limit. Using the non-
relativistic duality relations (63) and (64) in (59) and keeping upto
$O(g^{2})$ we get

$\vec{\nabla}.\vec{b}=\partial_{i}b^{i}=0~{}.$ (69)

Again using of the duality relations in (60) will give rise to

$(\vec{\nabla}\times\vec{e})^{i}=\epsilon^{ij}_{k}\partial_{j}e^{k}=0~{}.$
(70)

Under these the first Bianchi identity (61) reduces to

$\displaystyle-\partial_{t}\vec{e}+(\vec{\nabla}\times\vec{b})-16g^{2}\partial_{t}\Big{(}(e^{2}-b^{2})\vec{e}\Big{)}-32g^{2}\partial_{t}\Big{(}\vec{b}(\vec{e}.\vec{b})\Big{)}-16g^{2}\vec{\nabla}\times\Big{(}(e^{2}-b^{2})\vec{b}\Big{)}$
$\displaystyle-32g^{2}\vec{\nabla}\times\Big{(}\vec{e}\times(\vec{e}\times\vec{b})\Big{)}-32g^{2}\vec{\nabla}\times(b^{2}\vec{b})=0~{},$
(71)

while the second Bianchi identity (62) leads to

$\partial_{i}e^{i}+16g^{2}\partial_{i}\Big{(}(e^{2}-b^{2})e^{i}\Big{)}+32g^{2}b^{i}\partial_{i}(e^{j}b_{j})=0~{}.$
(72)

Here we see that these relations ((69)-(72)) exactly correspond to the results
(50-55) found for the electric limit.

This shows that the results for the electric and magnetic limits are swapped
under the non-relativistic duality transformations (63) and (64).

##  VII Invariance under Galilean boost transformations

Here we discuss the invariance of the Galilean Lagrangians under the Galilean
boost transformation. We will consider both electric and magnetic limit cases
separately.

Using the electric limit scaling given in table 1, we find that the Galilean
electric and magnetic fields transform under the Galilean boost as follows

$\displaystyle e^{\prime i}=e^{i},~{}~{}~{}b^{\prime
i}=b^{i}-\Big{(}\vec{v}\times\vec{e}\Big{)}^{i},~{}~{}~{}~{}e^{\prime}_{i}=e_{i}+\Big{(}\vec{v}\times\vec{b}\Big{)}_{i},~{}~{}~{}b^{\prime}_{i}=b_{i}~{}.$
(73)

Now considering the variation we get

$\delta e^{i}=0,~{}~{}\delta
b^{i}=-\Big{(}\vec{v}\times\vec{e}\Big{)}^{i},~{}~{}~{}~{}~{}\delta
e_{i}=\Big{(}\vec{v}\times\vec{b}\Big{)}_{i},~{}~{}\delta b_{i}=0~{}.$ (74)

We can write down the Lagrangian in terms of fields as

$\mathcal{L}_{e}=\frac{1}{2}(e^{i}e_{i}-b^{i}b_{i})+4g^{2}(e^{i}e_{i}-b^{i}b_{i})^{2}+4g^{2}\Big{(}2(e^{i}e_{i})(e^{j}e_{j})-4(e^{i}e_{i})(b^{j}b_{j})+4(e^{i}b_{i})(e_{j}b^{j})+2(b^{i}b_{i})(b^{j}b_{j})\Big{)}~{}.$
(75)

Now computation of the variation of this Lagrangian by using (74) one can
easily verify that

$\delta\mathcal{L}_{e}=0~{}.$ (76)

In a similar manner using the magnetic limit scaling given in table 1, we find
that the

$\delta\mathcal{L}_{m}=0~{}.$ (77)

##  VIII Gauge invariance

It is well known that the relativistic Born-Infeld Lagrangian 18 is invariant
under the following gauge transformation,

$\delta A_{\mu}=\partial_{\mu}\alpha,\,\,\,\delta
A^{\mu}=\partial^{\mu}\alpha~{},$ (78)

where $\alpha$ is an arbitrary function of spacetime. Below we will explore
how the same is maintained at Galilean level. Here we can consider a
relatively more general gauge transformation of the following form:

$\delta A_{\mu}=\partial_{\mu}\alpha,\,\,\,\delta
A^{\mu}=\partial^{\mu}\beta~{}.$ (79)

Note that in the relativistic theory the covaraint and contravariant vectors
are related by a metric implying $\alpha=\beta$. However, this is not true in
Galilean limit due to the lack of a non-degenerate metric. Hence we are free
to choose $\alpha\neq\beta$. We now discuss the electric and magnetic limit
cases separately.

Electric limit
From (79) and using the mapping relations given in table 1 we deduce the
following relations,

$\displaystyle\delta A_{0}=\partial_{0}\alpha\implies\delta
a_{0}=\partial_{t}\alpha~{};$ (80) $\displaystyle\delta
A_{i}=\partial_{i}\alpha\implies\delta a_{i}=\partial_{i}\alpha~{};$ (81)
$\displaystyle\delta A^{0}=\partial^{0}\beta\implies c\delta
a^{0}=-\frac{1}{c}\partial_{t}\beta\xrightarrow[\text{$c\to\infty$}]{\text{}}\delta
a^{0}=0~{};$ (82)

and

$\displaystyle\delta A^{i}=\partial^{i}\beta\implies\delta
a^{i}=\partial^{i}\beta~{}.$ (83)

These results are summarised in table 3. Varying the Lagrangian and using the
results summarised in table 3 we can show that

$\delta\mathcal{L}_{e}=0~{}.$ (84)

Table 3: Variations of the Galilean potentials ${\rm Variable}$ | ${\rm Electric~{}~{}limit}$ | ${\rm Magnetic~{}~{}limit}$
---|---|---
$a^{0}$ | $\delta a^{0}=0$ | $\delta a^{0}=\partial_{t}\beta$
$a^{i}$ | $\delta a^{i}=\partial^{i}\beta$ | $\delta a^{i}=\partial^{i}\beta$
$a_{0}$ | $\delta a_{0}=\partial_{t}\alpha$ | $\delta a_{0}=0$
$a_{i}$ | $\delta a_{i}=\partial_{i}\alpha$ | $\delta a_{i}=\partial_{i}\alpha$

Magnetic limit
Proceeding in a similar way as in the electric limit we can derive the
variations of the Galilean potentials in the magnetic limit which are
summarised in table 3. Using these results and considering the variation of
the Lagrangian 43 we can show that

$\delta\mathcal{L}_{m}=0~{}.$ (85)

This shows that the Galilean version of BI theory, both in electric and
magnetic limits, is gauge invariant.

##  IX Conclusions

Galilean relativistic electrodynamics always play an important role to
describe electrodynamics of moving bodies with low velocities. This
construction helps us to understand a wide range of physics phenomena that we
experience in our day to day life [25]. So in that sense it is a more
practical approach than relativistic electrodynamics which is not used in the
low velocity limit. In the same vein the study of Galilean relativistic
nonlinear electrodynamics is also useful. This is further enhanced by its
connection with Galilean relativistic string theory.
A systematic construction of the Galilean relativistic Born-Infeld (BI) theory
using an action principle has been provided here. BI theory which is a non-
linear theory of electromagnetism, plays an important role in the relativistic
string theory as the electrodynamics on the D-brane is explained by the BI
theory. The Galilean relativistic BI theory emerges in the studies of non-
relativistic open strings. There is no systematic study of Galilean
relativistic BI theory available in the literature. Here in this paper we have
studied quite extensively the Galilean relativistic BI theory in the leading
order and various symmetries involved with it.
We know that BI theory satisfies a duality symmetry. We have considered the
leading order expressions. The conventional covariant expressions have been
expressed in component forms using electric and magnetic fields. The duality
symmetry was discussed in terms of these fields. We then discuss two Galilean
limits (electric and magnetic) separately and write down the equations of
motion in terms of potentials. We consider both contravariant and covariant
potentials as they represent different quantities in the Galilean limit. We
observe that under the interchange of covariant and contravariant indices the
equations of motion in the electric limit go to that of magnetic limit and
vice-versa. We then recast these equations of motion in terms of Galilean
electric and magnetic fields. This is necessary for discussing the electric-
magnetic duality. We express the duality relations in terms of Galilean
electric and magnetic fields up to leading order. We show explicitly that
under these duality transformations the electric and magnetic limit equations
are swapped; i.e. electric limit equations of motion go to magnetic limit
Bianchi identities and electric limit Bianchi identities go to magnetic limit
equations of motion and vice-versa. This is an important feature and can be
used to find new solutions in the magnetic limit from known solutions in the
electric limit and vice-versa. Next we show that both electric and magnetic
limit Lagrangians are invariant under Galilean boost transformations. In the
end we show that these Galilean relativistic Lagrangians are gauge invariant.
We believe our result is important and will shed some light in various aspects
of non-relativistic string theory.
There have been studies on nonlinear electromagnetism defined on
noncommutative spaces [26]. It will be quite interesting to study the Galilean
relativistic aspects of these nonlinear electromagnetism defined on
noncommutative spaces using the method developed here. It is also interesting
to study the effects of this electric-magnetic duality symmetry in the non-
relativistic string theory. We hope we will address these issues in near
future.

##  X Acknowledgements

Two of 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: $20/03/2021$). The other author (BRM)
is supported by Science and Engineering Research Board (SERB), Department of
Science $\&$ Technology (DST), Government of India, under the scheme Core
Research Grant (File no. CRG/2020/000616).

## References

  * [1] M. Taylor, Non-relativistic holography, 0812.0530.
  * [2] O. Andreev, M. Haack and S. Hofmann, On nonrelativistic diffeomorphism invariance, Phys. Rev. D 89, 064012 (2014) doi:10.1103/PhysRevD.89.064012 [arXiv:1309.7231 [hep-th]].
  * [3] R. Banerjee, A. Mitra, P. Mukherjee, A new formulation of non-relativistic diffeomorphism invariance, Phys. Lett. B 737 (2014), [arXiv:1404.4491].
  * [4] R. Banerjee, A. Mitra, P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 4, 045010, [arXiv:1407.3617].
  * [5] K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 1504, 155 (2015). doi:10.1007/JHEP04(2015)155 [arXiv:1412.2738 [hep-th]].
  * [6] R. Banerjee and P. Mukherjee, New approach to non- relativistic diffeomorphism invariance and its applications, Phys. Rev. D 93, no. 8, 085020 (2016) doi:10.1103/PhysRevD.93.085020 [arXiv:1509.05622 [gr-qc]].
  * [7] R. Banerjee, S. Gangopadhyay and P. Mukherjee, On the question of symmetries in nonrelativistic diffeomorphism- invariant theories, Int. J. Mod. Phys. A 32, no. 19n20, 1750115 (2017) doi:10.1142/S0217751X17501159 [arXiv:1604.08711 [hep-th]].
  * [8] D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas, Annals. of. Physics. 321, 197-224 (2006).
  * [9] B. Grinstein and S. Pal, Existence and construction of Galilean invariant $z\neq 2$ theories, Phys. Rev. D 97, no. 12, 125006 (2018).
  * [10] A. Jain, Galilean anomalies and their effect on hydrodynamics Phys. Rev. D 93, no. 6, 065007 (2016).
  * [11] R. Banerjee and P. Mukherjee, Subtleties of nonrelativistic reduction and applications, doi: 10.1016/j.nuclphysb.2018.11.002, [arXiv: 1801.08373].
  * [12] K. Morand, Embedding Galilean and Carrollian geometries. I. Gravitational waves, J. Math. Phys. (N.Y.) 61, 082502 (2020), arXiv:1811.12681 [hep-th].
  * [13] J. Read and N. J. Teh, The teleparallel equivalent of Newton–Cartan gravity, Class. Quant. Grav. 35, no. 18, 18LT01 (2018).
  * [14] N. D. Sengupta, On an Analogue of the Galilei Group, Nuovo Cim. 54 (1966) 512, DOI: 10.1007/BF02740871.
  * [15] M. L. Bellac and J.-M. Levy-Leblond, Galilean Electromagnetism, Nuovo Cimento 14B, 217 (1973) .
  * [16] E. S. Santos, M. de Montigny, F. C. Khanna and A. E. Santana, Galilean covariant Lagrangian models, J. Phys. A 37 (2004) 9771.
  * [17] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [1402.0657].
  * [18] R. Banerjee and S. Bhattacharya, New formulation of Galilean relativistic Maxwell theory, Phys. Rev. D. 107, 105022 (2023), [arXiv: hep-th/2211.12023].
  * [19] R. Banerjee and S. Bhattacharya, A new formulation of Galilean relativistic Proca theory, arXiv: hep-th/2303.13066.
  * [20] M. Born, L. Infeld, ”Foundations of the New Field Theory,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 144 (1934) 425–451.
  * [21] C. Wang, “Born-Infeld Action and Its Applications,” available on: “https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2013/Cong-Wang-Dissertation.pdf”
  * [22] G. W. Gibbons, “Aspects of Born-Infeld theory and string / M theory,” AIP Conf. Proc. 589, no.1, 324-350 (2001) [arXiv:hep-th/0106059 [hep-th]].
  * [23] A. A. Tseytlin, “Born-Infeld action, supersymmetry and string theory,” [arXiv:hep-th/9908105 [hep-th]].
  * [24] J. Gomis, Z. Yan and M. Yu, “Nonrelativistic Open String and Yang-Mills Theory,” JHEP 03, 269 (2021) [arXiv:2007.01886 [hep-th]].
  * [25] G. Rousseaux, Forty years of Galilean Electromagnetism (1973–2013), Eur. Phys. J. Plus (2013) 128: 81.
  * [26] R. Banerjee, A Note on Duality Symmetry in Nonlinear Gauge Theories, Phys. Lett. B 576 (2003) 237-242, [arXiv:hep-th/0308162].