# Shift symmetries and duality web in gauge theories

###### Abstract

Using a generalised Noether prescription we are able to extract all the
currents and their conservation laws in space dependent shift symmetric
theories. Various identities among the currents in the matter sector are found
that form the basis for revealing a dual picture when the full interacting
theory is considered by coupling to gauge fields. The coupling is achieved in
terms of vector fields by adhering to a modified minimal prescription which is
also supported by an iterative Noether scheme. Further, this scheme shows that
couplings can also be introduced using higher rank tensor gauge fields that
have appeared in recent discussions on fractons. We reveal a connection among
these descriptions (using vector or tensor fields) through certain duality
maps that relate the various fields (gauge, electric and magnetic) in the two
cases. A correspondence is established among the Gauss’ law, Faraday’s law and
Ampere’s law. Explicit calculations are provided for linear and quadratic
shift symmetric lagrangians.

Rabin Banerjee111Raja Ramanna Fellow<EMAIL_ADDRESS>and Anwesha
<EMAIL_ADDRESS>
S.N. Bose National Centre for Basic Sciences, Kolkata- 700106, India

###### Contents

  1. 1 Introduction
  2. 2 Shift Symmetries and conservation laws
    1. 2.1 Nonrelativistic Maxwell theory
    2. 2.2 Higher derivative scalar field theories
    3. 2.3 The multipole algebra
    4. 2.4 Alternative forms for currents
  3. 3 From shift symmetries to gauge symmetries
  4. 4 Dual description of theory with space dependent linear shift symmetry
    1. 4.1 Gauge coupling from iterative Noether prescription
    2. 4.2 Full interacting theory in dual description
      1. 4.2.1 Lagrangian and hamiltonian analysis in (2+1) dimensional space-time
      2. 4.2.2 Lagrangian and hamiltonian analysis in (3+1) dimensional space-time
  5. 5 Dual description of theory with space-dependent quadratic shift symmetry
    1. 5.1 Gauge coupling from iterative Noether prescription
    2. 5.2 Full interacting theory and equations of motion in (3+1) dimension
  6. 6 Conclusion

##  1 Introduction

Recent studies on quantum phases of matter have indicated the existence of
peculiar quasiparticles, called fractons, with the characteristic feature of
restricted mobility (for reviews, see [1, 2, 3, 4]). Since this is a novel
(theoretical) phase of matter, it has triggered intense activity covering a
wide gamut of topics like field theories with higher rank symmetric tensors
[2, 5, 6, 7, 8, 9, 10, 11] gravitational physics [10, 12, 13, 14, 15],
holography [16] and even quantum information storage [17, 18, 19, 20].

The preeminent defining property of fractons, namely, their partial or
complete immobility [21, 22, 23, 24] is attained by invoking a rank-2 spatial
symmetric tensor gauge field $A_{ij}$ instead of the usual vector field
$A_{i}$. To visualise this it may be recalled that the vector theory with
standard gauge symmetry,

$\delta A_{i}=\partial_{i}\Lambda$ (1.1)

and Gauss constraint,

$\partial_{i}E_{i}=\rho$ (1.2)

where $E_{i}$, the electric field, is the momentum conjugate to $A_{i}$ and
$\rho$ is the charge density, admits two possibilities for extending to
symmetric tensor fields,
(a) as a vector charge theory with,

$\displaystyle\delta A_{ij}$
$\displaystyle=\partial_{i}\Lambda_{j}+\partial_{j}\Lambda_{i}$
$\displaystyle\partial_{i}E^{ij}$ $\displaystyle=\rho^{j}$

where $E_{ij}$, the generalised electric field, is the momentum conjugate to
$A_{ij}$ and $\rho^{j}$ is the vector charge density, or, alternatively,
(b) as a scalar charge theory with,

$\displaystyle\delta A_{ij}$ $\displaystyle=\partial_{i}\partial_{j}\Lambda$
$\displaystyle\partial_{i}\partial_{j}E^{ij}$ $\displaystyle=\rho$ (1.3)

Both possibilities impose restrictions on the fracton mobility which is
partial for case (a) but complete for case (b) [25]. This is obvious from
(1.3) since it implies simultaneous conservation of charge as well as dipole
moment,

$\int_{space}\,\rho=\int_{space}\,x^{k}\rho=0$ (1.4)

Also, comparing gauge transformation properties given in (1.1) and (1.3)
suggests that the tensor and vector fields are related by,

$A_{ij}=\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$ (1.5)

In this paper we focus on the scalar charge theory defined in case (b) and we
show that, using (1.5) it is possible to have another description in terms of
a vector field $A_{i}$ with conventional transformation property (1.1). The
full interaction theory is considered where the starting point is the matter
sector defined in terms of higher derivatives of complex scalars. While the
explicit form of the matter sector is known [11, 26, 27, 28, 29] it is derived
here from a Lifschitz type action involving real scalars with polynomial shift
symmetries. Gauge couplings are now introduced in the matter sector by
following the iterative Noether’s prescription of Deser [30]. Two variants are
revealed. Interactions may be introduced either by a vector field $A_{i}$ or a
symmetric rank-2 field $A_{ij}$, which are related by (1.5). The results for
the vector coupling may, alternatively, be understood as following from a
generalised minimal prescription applied to the matter sector. The tensor
coupling, on the other hand, exactly corresponds to the gauging advocated by
the fracton gauge principle [11, 26]. Once the matter sector and interactions
have been obtained, it remains to define the pure gauge sector. For the
fractonic theory this is known in terms of $A_{ij}$. Its alternative
description is obtained by using the map (1.5). We thus have two clear choices
for the fractonic theory expressed either in terms of $A_{ij}$ or $A_{i}$. A
full dynamical treatment, either in the lagrangian or hamiltonian
formulations, reveals a correspondence among the equations of motion, Gauss
constraint, Faraday’s law and Ampere’s law. We also discuss some contrasting
features of the two descriptions. We extend this correspondence further by
considering conservation of the quadrupole moment, besides charge and dipole
moment. Now there is a duality web involving $A_{i},A_{ij}$ and $A_{ijk}$
where the rank-3 tensor appears due to the new conservation of quadrupole
moment.

The paper is structured as follows: in section 2 we consider arbitrary shift
symmetric real scalar and vector theories and derive identities among various
conserved currents. Alternative forms for these currents are obtained that
differ by local counterterms. By suitable maps, shift symmetries in real
scalar theories are identified with gauge symmetries in complex scalar
theories in section 3. These maps enable us to determine the matter sector as
well as expressions for conserved currents found in the literature [9, 11, 26,
27] on fractons. Sections 4 and 5, respectively, deal with dual formulations
in linear and quadratic shift symmetric theories in 2+1 and 3+1 dimensions,
employing both lagrangian and hamiltonian techniques. Finally, conclusions are
given in section 6.

##  2 Shift Symmetries and conservation laws

An important input in the study of low-energy effective lagrangians is
Goldstone’s theorem which implies that whenever a global symmetry is
spontaneously broken, a gapless mode will appear. For relativistic theories,
this leads to a massless Goldstone particle. Such a particle is described by a
shift symmetry of the field, in the example of a Goldstone scalar,

$\phi(x)\,\to\,\phi(x)+c$ (2.1)

where $c$ is a constant, and is characterised by the usual action of a scalar
field,

$S=\frac{1}{2}\int\,\,d^{d}x\,\,\partial_{\mu}\phi\,\partial^{\mu}\phi$ (2.2)

Since the action is invariant under (2.1), which is a global transformation,
we may obtain the corresponding conserved current by using Noether’s first
theorem,

$j^{\mu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\delta\phi=c\,\partial^{\mu}\phi$
(2.3)

which satisfies,

$\partial_{\mu}j^{\mu}=c\,\,\Box\phi=0$ (2.4)

on using the equation of motion. Further, it is found that the shift symmetry,
on exploiting canonical quantisation of $\phi$, connects the vacuum
$|0\rangle$ with a single particle state $|1\rangle$,

$\langle 0|j^{\mu}|1\rangle=c\langle
0|\partial^{\mu}\phi|1\rangle=cp^{\mu}e^{ip.x}$ (2.5)

and the matrix element vanishes in the strict soft limit, as required by
Goldstone’s theorem.

We now generalise the above analysis by infusing two modifications. First, a
non-relativistic version is taken and, secondly, the constant shift symmetry
is enlarged to a linear spatial shift,

$\phi\,\to\,\phi+c+c_{i}x_{i}=\phi+w$ (2.6)

The non-relativistic free scalar theory is defined as,

$\mathcal{L}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}(\partial_{i}\phi)(\partial_{i}\phi)=\frac{1}{2}\partial_{0}\phi\partial_{0}\phi-\frac{1}{2}\partial_{i}\phi\partial_{i}\phi$
(2.7)

which transforms in the following way,

$\delta\mathcal{L}=-(\partial_{i}\phi)(\partial_{i}w)=-\partial_{i}(\phi\partial_{i}w)$
(2.8)

Thus the Lagrangian is quasi-invariant.

Applying Noether’s theorem, the conserved currents are given by,

$\displaystyle j^{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\phi)}\delta\phi=(\partial_{0}\phi)\,w=\dot{\phi}\,w$
(2.9) $\displaystyle j^{i}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\phi)}\delta\phi+\phi\partial^{i}w=-(\partial^{i}\phi)\,w+\phi(\partial^{i}\,w)$
(2.10)

where the extra piece in (2.10) is the boundary term in (2.8). Current
conservation now follows on shell,

$\partial_{0}j_{0}-\partial_{i}j_{i}=(-\ddot{\phi}+\partial^{2}\phi)w=0$
(2.11)

where $\partial^{2}\,w=0$ has been used since $w$ is linear in $x$ (2.6).

However, there are other possibilities for defining the currents. A more
conventional definition is,

$\displaystyle j_{0}$ $\displaystyle=-j^{0}=-\dot{\phi}$ (2.12) $\displaystyle
j_{i}$ $\displaystyle=j^{i}=-\partial_{i}\phi$ (2.13)

On shell current conservation follows immediately,

$\partial_{0}j_{0}-\partial_{i}j_{i}=-\ddot{\phi}+\partial^{2}\phi=0$ (2.14)

It is now possible to introduce a tensor current $j_{ki}$ whose space
derivative yields the vector current,

$j_{ki}=-\phi\delta_{ki},\qquad j_{i}=\partial_{k}j_{ki}$ (2.15)

so that,

$\partial_{0}j_{0}-\partial_{i}\partial_{k}j_{ki}=0$ (2.16)

The above relation shows the existence of two conserved charges, the usual
charge and the dipole charge,

$Q=\int_{space}\,\,j_{0},\qquad Q^{i}=\int_{space}\,\,x^{i}j_{0}$ (2.17)

This is reminiscent of fractonic theories, the characteristic feature of which
is the presence of higher derivatives. In the present case such terms are
absent.
It is tempting to ask whether a general formulation is possible to generate
the currents (2.12,2.13). Indeed a viable definition is given by,

$\displaystyle J_{i}$
$\displaystyle=wj_{i}-(\partial_{k}w)\,j_{ki}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\phi)}\delta\phi+\phi(\partial_{i}\,w)$
(2.18) $\displaystyle J^{0}$
$\displaystyle=w\,j^{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\phi)}\delta\phi$
(2.19)

while there is no change in $J^{0}$ defined as in (2.9), the vector current
yields,

$wj_{i}-\partial_{k}w\,j_{ki}=-(\partial_{i}\phi)\,w+\phi\,(\partial_{i}w)$
(2.20)

leading to,

$j_{i}=-\partial_{i}\phi,\qquad j_{ki}=-\phi\delta_{ki}$ (2.21)

reproducing (2.13) and (2.15).
The equation for current conservation of $J_{0},J_{i}$ obtained from the
defining equation (2.18,2.19) is given by,

$\partial_{0}J_{0}-\partial_{i}J_{i}=-\delta\mathcal{L}-\partial_{i}(\phi\partial_{i}\,w)=0$
(2.22)

that follows from (2.8). The standard vector current conservation is deduced
by substituting the explicit values of $J_{0},J_{i}$ in (2.22),

$\partial_{0}(wj_{0})-\partial_{i}(wj_{i}-(\partial_{k}w)\,j_{ki})=0$ (2.23)

Extracting $w$ out from the parenthesis, followed by a little algebra
immediately yields (2.14).

This derivation, however, conceals the fact that there are two conservation
laws related to the charges $Q,Q^{i}$ (2.17). To reveal this, we open $w$ in
favour of $c,c_{i}$ using (2.6). Then,

$\partial_{0}[(c+c_{j}x_{j})j_{0}]-\partial_{i}[(c+c_{j}x_{j})j_{i}-\partial_{k}(c+c_{j}x_{j})j_{ki}]=0$
(2.24)

Equating terms proportional to $c$ yields the standard conservation law (2.14)
while equating those of $c_{j}$ yields,

$\partial_{0}(\tilde{j}_{oj})-\partial_{i}(\tilde{j}_{ij}-j_{ij})=0$ (2.25)

where,

$\tilde{j}_{oj}=x_{j}j_{0},\qquad\tilde{j}_{ij}=x_{j}j_{i}$ (2.26)

and leads to the conservation of the dipole charge $Q^{j}$, defined in (2.17).

###  2.1 Nonrelativistic Maxwell theory

It is feasible to develop a similar algorithm for the nonrelativistic Maxwell
field. The new point here is the interplay between shift and gauge symmetries.
The dynamics is governed by the Lagrangian333$C_{1}$ and $C_{2}$ are, in
general, different,

$\mathcal{L}=\frac{C_{1}}{2}F_{oi}^{2}-\frac{C_{2}}{4}F_{ij}^{2}$ (2.27)

where

$F_{0i}=\partial_{0}A_{i}-\partial_{i}A_{0},\quad
F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$ (2.28)

The equations of motion are given by,

$C_{1}\partial_{0}F_{0i}-C_{2}\partial_{j}F_{ji}=0,\quad\partial_{i}F_{0i}=0$
(2.29)

where the second one is the Gauss constraint.
There is a gauge symmetry,

$\delta A_{0}=\partial_{0}w,\quad\delta A_{i}=\partial_{i}w$ (2.30)

that results in a conserved Noether current,

$\displaystyle j^{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{0}A_{i})}\delta
A_{i}=C_{1}F_{0i}(\partial_{i}w)$ (2.31) $\displaystyle j_{i}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{i}A_{j})}\delta
A_{j}+\frac{\partial\mathcal{L}}{\partial(\partial_{i}A_{0})}\delta
A_{0}=-C_{2}F_{ij}(\partial_{j}w)-C_{1}F_{0i}(\partial_{0}w)$ (2.32)

which is easily verified,

$\partial_{0}j_{0}-\partial_{i}j_{i}=(C_{2}\partial_{i}F_{ij}-C_{1}\partial_{0}F_{0j})(\partial_{j}w)+C_{1}\partial_{i}F_{0i}\partial_{0}w=0$
(2.33)

by using the equations of motion.
Gauge symmetry, regarded as an origin of the masslessness of the photon, is
translatable to the linear shift symmetry by choosing,

$w=c_{i}x_{i}+\frac{1}{2}c_{ij}x_{i}x_{j}$ (2.34)

so that,

$\delta A_{i}=c_{i}+c_{ij}x_{j},\quad\delta A_{0}=0$ (2.35)

analogous to (2.6). Thus the gapless photon mode is interpreted as a
consequence of gauge symmetry or shift symmetry similar to Goldstone bosons.
Note that although, in general, $c_{ij}$ has both a symmetric part and an
antisymmetric component, only the former contributes to (2.34).
As we found for the scalar field there are other ways of defining currents. We
adopt the same algorithm as before. It is useful to express $c_{ij}$ in terms
of its symmetric $(s_{ij})$ and antisymmetric $(a_{ij})$ parts,

$s_{ij}=\frac{1}{2}(c_{ij}+c_{ji}),\qquad a_{ij}=\frac{1}{2}(c_{ij}-c_{ji})$
(2.36)

which implies,

$\delta A_{i}=c_{i}+s_{ij}x_{j}+a_{ij}x_{j}=w_{i}(x)$ (2.37)

For the above shift the lagrangian (2.27) changes as,

$\delta\mathcal{L}=2C_{2}\partial_{i}(a_{ij}A_{j})=C_{2}a_{ij}F_{ij}$ (2.38)

and hence is quasi-invariant as well as gauge invariant.

It is pertinent to observe that, retaining only the symmetric part $s_{ij}$ in
(2.38), as dictated by the structure (2.34), would yield $\delta\mathcal{L}=0$
and there would be no boundary term. Exact gauge invariance follows which
shows its direct correspondence with shift invariance. This is to be expected
if one works with standard theories without higher derivative terms in the
action.

Since our motivation is to go beyond usual invariances by involving higher
derivative terms, we envisage a wider possibility here itself by demanding a
general form for $\delta A_{i}$ given by (2.37) that includes both $s_{ij}$
and $a_{ij}$. The actual implementation of these ideas in the context of
scalar field theories with higher derivatives is done in next (sub) section.

Following our general algorithm the currents are defined analogous to
(2.18,2.19),

$\displaystyle J^{0}$
$\displaystyle=w_{j}j^{0j}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}A_{i})}\delta
A_{i}=C_{1}F_{0i}w_{i}$ (2.39) $\displaystyle J_{i}$
$\displaystyle=w_{j}j_{ij}-(\partial_{k}w_{j})j_{ijk}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}A_{j})}\delta
A_{j}-2C_{2}a_{ij}A_{j}$ (2.40)

where the boundary term (2.38) has been included in the expression for
$J_{i}$. While (2.39) yields,

$j^{0j}=-j^{j}_{0}=C_{1}F_{0j}$ (2.41)

simplifying (2.40) leads to,

$w_{j}j_{ij}-(\partial_{k}w_{j})j_{ijk}=-C_{2}F_{ij}w_{j}-2C_{2}a_{ij}A_{j}$
(2.42)

which yields,

$\displaystyle j_{ij}=-C_{2}F_{ij}$ (2.43) $\displaystyle
j_{ijk}=C_{2}(-\delta_{ik}A_{j}+\delta_{ij}A_{k})$ (2.44)

From the above relations we get a condition,

$\partial_{i}j_{ijk}=C_{2}F_{jk}=-j_{jk}$ (2.45)

analogous to (2.15).
The conservation laws are given by,

$\partial_{0}J_{0}-\partial_{i}J_{i}=\partial_{0}(w_{j}j_{0j})-\partial_{i}\Big{(}w_{j}j_{ij}-(\partial_{k}w_{j})j_{ijk}\Big{)}$
(2.46)

Abstracting out $w_{j}$ we find,

$w_{j}(\partial_{0}j_{0j}-\partial_{i}j_{ij})=0$ (2.47)

on exploiting (2.45) and the linearity of $w_{j}$ (2.37). A direct way to
verify (2.47) is to use (2.41) and (2.45) which just yields the onshell
condition (2.29).
As before, opening up $w_{j}$ yields another conservation law,

$\partial_{0}[(c_{j}+s_{jk}x_{k}+a_{jk}x_{k})j_{0j}]-\partial_{i}[(c_{j}+s_{jk}x_{k}+a_{jk}x_{k})j_{ij}]+\partial_{i}[(s_{jk}+a_{jk})j_{ijk}]=0$
(2.48)

Equating coefficients of $c_{j}$ reproduces (2.47). Likewise, equating
coefficients of $(s_{jk}+a_{jk})$ yields,

$\partial_{0}(x_{k}j_{0j})-\partial_{i}[x_{k}j_{ij}-j_{ijk}]=0$ (2.49)

and leads to the conservation of a dipole like (vector) charge,

$Q_{kj}=\int_{space}\,\,x_{k}j_{0j}$ (2.50)

The field momentum and dipole charge satisfy a generalised Heisenberg algebra,

$[P_{i},Q_{kj}]=\Big{[}\int\,dx\,\pi_{l}\partial_{i}A_{l}(x),\int\,dy\,y_{k}j_{0j}(y)\Big{]}=-C_{1}\Big{[}\int\,dx\,dy\pi_{l}\partial_{i}A_{l}(x),y_{k}F_{0j}(y)\Big{]}$
(2.51)

Recalling that $F_{0j}$ is the momentum canonically conjugate to $A_{j}$,

$\pi_{j}=\frac{\partial\mathcal{L}}{\partial\dot{A}_{j}}=C_{1}F_{0j}$ (2.52)

we obtain,

$[P_{i},Q_{kj}]=\delta_{ik}Q_{j}$ (2.53)

where $Q_{j}$ is the (vector) charge,

$Q_{j}=\int_{space}\,\,j_{0j}$ (2.54)

Appearance of higher rank tensor currents in standard Maxwell’s theory is a
precursor to similar currents in higher derivative scalar field theories, both
real and complex. Since it forms a central part of our analysis, we discuss
this in some details.

###  2.2 Higher derivative scalar field theories

The simplest higher derivative extension of the usual theory (2.7) is given
by,

$\mathcal{L}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}(\partial_{i}\partial_{j}\phi)(\partial_{i}\partial_{j}\phi)$
(2.55)

which is a typical example of a Lifshitz type model with a scaling symmetry,

$\delta x^{i}=\lambda x^{i}\,;\,\,\,\delta t=z\lambda t\,\,\,(z=\lambda)$
(2.56)

where $z$ characterises the degree of anisotropy between space and time.

A cubic shift in the coordinates,

$\phi\,\to\,\phi+w,\quad
w=c+c_{i}x_{i}+\frac{1}{2}c_{ij}x_{i}x_{j}+\frac{1}{3}c_{ijk}x_{i}x_{j}x_{k}$
(2.57)

leaves the lagrangian quasi-invariant, since,

$\delta\mathcal{L}=-\partial_{i}[\partial_{j}\phi\partial_{i}\partial_{j}w]+\partial_{j}[\phi\partial^{2}\partial_{j}w]-\phi(\partial^{2})^{2}w$
(2.58)

where the last term drops out since $w=O(x^{3})$.
Since the lagrangian involves double derivatives, the Noether currents are
defined as,

$\displaystyle J^{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\phi)}\delta\phi=\dot{\phi}w$
$\displaystyle J^{i}$
$\displaystyle=J_{i}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\phi)}\delta\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\phi)}\partial_{j}(\delta\phi)-\Big{(}\partial_{j}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\phi)}\Big{)}\delta\phi+\partial_{j}\phi\partial_{i}\partial_{j}w-\phi\partial^{2}\partial_{i}w$
$\displaystyle=-(\partial_{i}\partial_{j}\phi)(\partial_{j}w)+\partial_{j}(\partial_{i}\partial_{j}\phi)w+(\partial_{j}\phi)(\partial_{i}\partial_{j}w)-\phi(\partial^{2}\partial_{i}w)$
(2.59)

where the boundary terms have been inserted. Current conservation now follows,

$\partial_{0}J_{0}-\partial_{i}J_{i}=-w(\ddot{\phi}+(\partial^{2})^{2}\phi)+\phi(\partial^{2})^{2}w$
(2.60)

which vanishes on-shell.

The other (higher order) currents and their associated conservation laws
follow on using our definition,

$J^{0}=wj^{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\phi)}\delta\phi=\dot{\phi}w$
(2.61)

which yields,

$j^{0}=\dot{\phi}$ (2.62)

The space current is given by,

$\displaystyle J_{i}$
$\displaystyle=wj_{i}-(\partial_{j}w)j_{ji}+(\partial_{j}\partial_{k}w)j_{jki}-(\partial_{j}\partial_{k}\partial_{l}w)j_{jkli}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\phi)}\partial_{j}(\delta\phi)-\Big{(}\partial_{j}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\phi)}\Big{)}\delta\phi+(\partial_{j}\phi)\partial_{i}\partial_{j}w-\phi\partial^{2}\partial_{i}w$
$\displaystyle=-(\partial_{i}\partial_{j}\phi)\partial_{j}w+\partial_{j}(\partial_{i}\partial_{j}\phi)w+(\partial_{j}\phi)\partial_{i}\partial_{j}w-\phi\partial^{2}\partial_{i}w$
(2.63)

Comparing coefficients of $w$ and its derivatives yields,

$\displaystyle j_{i}$ $\displaystyle=\partial_{i}\partial^{2}\phi$ (2.64)
$\displaystyle j_{ji}$ $\displaystyle=\partial_{i}\partial_{j}\phi=j_{ij}$
(2.65) $\displaystyle j_{jki}$
$\displaystyle=\frac{1}{2}(\delta_{ki}\partial_{j}\phi+\delta_{ij}\partial_{k}\phi)=j_{kji}$
(2.66) $\displaystyle j_{jkli}$
$\displaystyle=\frac{1}{3}(\delta_{jk}\delta_{li}+\delta_{jl}\delta_{ki}+\delta_{kl}\delta_{ji})\phi$
(2.67)

The following set of identities are obtained from the above relations,

$j_{i}=\partial_{j}j_{ij}=\partial_{j}\partial_{k}j_{ijk}=\partial_{j}\partial_{k}\partial_{l}j_{ijkl}$
(2.68)

Besides this chain, there exists another identity,

$j_{jk}=\partial_{i}j_{jki}$ (2.69)

The above identities are useful for proving conservation laws, as shown below.
It is important to point out that the first relation in (2.68), which connects
a vector current to a derivative of a rank two current, and the identity
(2.69) that connects a rank two current to a derivative of rank three current
contain the genesis of the duality among different gauge theories, as
demonstrated later. Current conservation is now verified as,

$\partial_{0}J_{0}-\partial_{i}J_{i}=\partial_{0}(wj_{0})-\partial_{i}J_{i}=\partial_{0}(-\dot{\phi}w)-\partial_{i}\Big{[}wj_{i}-(\partial_{j}w)j_{ji}+(\partial_{j}\partial_{k}w)j_{jki}-(\partial_{j}\partial_{k}\partial_{l}w)j_{jkli}\Big{]}=0$
(2.70)

Factoring out $w$ and exploiting the identities (2.68) yields,

$\partial_{0}J_{0}-\partial_{i}J_{i}=-w(\ddot{\phi}+(\partial^{2})^{2}\phi)+(\partial_{i}\partial_{j}\partial_{k}\partial_{l}w)j_{jkli}$
(2.71)

which vanishes on shell and the fact that $w$ is cubic in the coordinates.
This derivation hides the whole tower of conservation laws that may be
revealed by opening up $w$ using (2.57). Then equating one by one the factors
multiplying parameters $c,c_{i}$ etc, we have, from (2.70),

$\displaystyle\partial_{0}j_{0}-\partial_{i}j_{i}=$ $\displaystyle 0$
$\displaystyle\partial_{0}(x_{i}j_{0})-\partial_{j}(x_{i}j_{j}-j_{ij})=0$
(2.72)

While the first leads to the conservation of the usual charge
$Q=\int_{space}j_{0}$, the second yields the conservation of the dipole
charge,

$Q_{i}=\int_{space}\,\,x_{i}j_{0}$ (2.73)

Similarly, equating coefficients of $c_{ij}$ yields,

$\partial_{0}\Big{(}\frac{1}{2}x_{i}x_{j}j_{0}\Big{)}-\partial_{k}\Big{[}\frac{1}{2}x_{i}x_{j}j_{k}-\frac{1}{2}(x_{i}j_{jk}+x_{j}j_{ik})+j_{ijk}\Big{]}=0$
(2.74)

which leads to the conservation of the quadrupole charge,

$Q_{ij}=\int_{space}x_{i}x_{j}j_{0}$ (2.75)

A direct check of (2.74) can be done by first simplifying the left side

$\frac{1}{2}x_{i}x_{j}(\partial_{0}j_{0}-\partial_{k}j_{k})-\frac{1}{2}(x_{j}j_{i}+x_{i}j_{j})+\frac{1}{2}(x_{i}\partial_{k}j_{jk}+x_{j}\partial_{k}j_{ik})+\frac{1}{2}(j_{ji}+j_{ij})-\partial_{k}j_{ijk}=0$
(2.76)

followed by exploiting the identities (2.64,2.68). The various terms cancel
out while the first bracket vanishes by usual vector current conservation.
Finally, equating coefficients of $c_{ijk}$ in (2.70) yields the conservation
law,

$\partial_{0}(x_{i}x_{j}x_{k}j_{0})-\partial_{l}\Big{[}x_{i}x_{j}x_{k}j_{l}-\partial_{m}(x_{i}x_{j}x_{k})j_{ml}+\partial_{m}\partial_{n}(x_{i}x_{j}x_{k})j_{mnl}-\partial_{m}\partial_{n}\partial_{p}(x_{i}x_{j}x_{k})j_{mnpl}\Big{]}=0$
(2.77)

The octupole charge is now conserved,

$Q_{ijk}=\int_{space}\,x_{i}x_{j}x_{k}j_{0},\quad\dot{Q}_{ijk}=0$ (2.78)

The various multipole charges form a closed algebra with the the symmetry
generators which will be discussed now.

###  2.3 The multipole algebra

The symmetry algebra of charges is obtained here for different cases like
dipole, quadrupole etc.
Let us first begin with the dipole charge $Q^{i}$. The field momentum
(translation generator) $P_{i}$ and dipole charge $Q^{j}$ satisfy the Lie
algebra of the Heisenberg group. To see this we compute,

$[P_{i},Q^{j}]=\Big{[}\int\,dx\,\pi\partial_{i}\phi(x),\int\,dy\,y^{j}j_{0}(y)\big{]}$
(2.79)

Recalling that $j^{0}$ (2.12) is the canonical momenta $\pi$ and using the
canonical bracket among $\phi-\pi$, we obtain,

$[P_{i},Q^{j}]=-i\delta_{i}\,^{j}\,Q$ (2.80)

which yields the cherished form.
Similarly the angular momentum-dipole charge satisfies the algebra,

$[M_{ij},Q_{k}]=\Big{[}\int\,\,dx(x_{i}\pi\partial_{j}\phi-
x_{j}\pi\partial_{i}\phi),\int\,\,dy\,y^{k}j_{0}(y)\Big{]}=-i(\delta_{jk}Q_{i}-\delta_{ik}Q_{j})$
(2.81)

The monopole charge $Q$ in (2.80) and the dipole charge in (2.81) appear as
central extensions of the algebra. Similar results follow for higher pole
charges. Specifically, for the quadrupole charge,

$\displaystyle[P_{i},Q_{kj}]$
$\displaystyle=-i(\delta_{ik}Q_{j}+\delta_{ij}Q_{k})$ (2.82)
$\displaystyle[M_{ij},Q_{kl}]$
$\displaystyle=-i(\delta_{ik}Q_{jl}+\delta_{il}Q_{jk}-\delta_{jk}Q_{il}-\delta_{jl}Q_{ik})$
(2.83)

and for the octupole charge,

$\displaystyle[P_{i},Q_{jkl}]$
$\displaystyle=-i(\delta_{ik}Q_{jl}+\delta_{ij}Q_{kl}+\delta_{il}Q_{jk})$
(2.84) $\displaystyle[M_{ij},Q_{klm}]$
$\displaystyle=-i(\delta_{jk}Q_{ilm}+\delta_{jl}Q_{ikm}+\delta_{jm}Q_{ikl}-\delta_{ik}Q_{jlm}-\delta_{il}Q_{jkm}-\delta_{im}Q_{jkl})$
(2.85)

The entire algebra is similar to the Heisenberg algebra and its extensions
like, for instance,

$\displaystyle[p_{i},q_{j}]$ $\displaystyle=-i\delta_{ij}$
$\displaystyle[p_{i},q_{j}q_{k}]$
$\displaystyle=-i(\delta_{ij}q_{k}+\delta_{ik}q_{j})$
$\displaystyle[p_{i},q_{j}q_{k}q_{l}]$
$\displaystyle=-i(\delta_{ij}q_{k}q_{l}+\delta_{ik}q_{j}q_{l}+\delta_{il}q_{j}q_{k})$
(2.86)

which are manifested in (2.80), (2.82) and (2.84).
Likewise the angular momentum algebra is another such manifestation,

$[q_{i}p_{j}-q_{j}p_{i},q_{k}]=-i(\delta_{jk}q_{i}-\delta_{ik}q_{j})$ (2.87)

whose analougue can be found in (2.81) and which can be extended to the higher
pole charges.

###  2.4 Alternative forms for currents

As is known, expressions for currents are not unique and may differ by local
counterterms. Indeed this happens here also if we consider an equivalent form
of the lagrangian (2.55),

$\mathcal{L}_{eq}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}(\partial^{2}\phi)^{2}$
(2.88)

Under the cubic shift (2.57), the lagrangian changes by a total derivative,

$\delta\mathcal{L}_{eq}=\partial_{i}[\phi\partial_{i}\partial^{2}w-\partial_{i}\phi\partial^{2}w]$
(2.89)

The Noether currents (2.59) are now defined as,

$\displaystyle\tilde{J}^{0}$
$\displaystyle=\frac{\partial\mathcal{L}_{eq}}{\partial(\partial_{0}\phi)}\delta\phi=\dot{\phi}w$
$\displaystyle\tilde{J}^{i}$
$\displaystyle=\tilde{J}_{i}=\frac{\partial\mathcal{L}_{eq}}{\partial(\partial_{i}\phi)}\delta\phi+\frac{\partial\mathcal{L}_{eq}}{\partial(\partial_{i}\partial_{j}\phi)}\partial_{j}(\delta\phi)-\Big{(}\partial_{j}\frac{\partial\mathcal{L}_{eq}}{\partial(\partial_{i}\partial_{j}\phi)}\Big{)}\delta\phi-\phi\partial_{i}\partial^{2}\omega+\partial_{i}\phi\partial^{2}w$
$\displaystyle=\partial_{i}\partial^{2}\phi-(\partial^{2}\phi)\partial_{i}w+(\partial_{i}\phi)\partial^{2}w-\phi(\partial_{i}\partial^{2}w)$
(2.90)

While $J^{0}$ in (2.59) and $\tilde{J}^{0}$ are identical, the spatial
currents differ by local counterterms,

$\tilde{J}^{i}-J^{i}=\partial_{j}\Lambda_{ji}$ (2.91)

where,

$\Lambda_{ji}=-\Lambda_{ij}=\partial_{j}w\partial_{i}\phi-\partial_{i}w\partial_{j}\phi$
(2.92)

which ensures conservation of $\tilde{J}$ current since,

$\partial_{i}\tilde{J}^{i}=\partial_{i}J^{i}$ (2.93)

The tensor currents, analogous to (2.63), are now obtained by adopting the
definition (2.90), so that,

$\displaystyle
w\tilde{j}_{i}-\partial_{j}w\tilde{j}_{ji}+\partial_{j}\partial_{k}w\tilde{j}_{jki}-\partial_{j}\partial_{k}\partial_{l}wj_{jkli}$
$\displaystyle=(\partial_{i}\partial^{2}\phi)w-(\partial^{2}\phi)\partial_{i}w+(\partial_{i}\phi)\partial^{2}w-\phi(\partial_{i}\partial^{2}w)$
(2.94)

This yields,

$\displaystyle\tilde{j}_{i}$ $\displaystyle=\partial_{i}\partial^{2}\phi$
$\displaystyle\tilde{j}_{ji}$
$\displaystyle=\delta_{ji}\partial^{2}\phi=\tilde{j}_{ij}$
$\displaystyle\tilde{j}_{jki}$
$\displaystyle=\delta_{kj}\partial_{i}\phi=\tilde{j}_{kji}$
$\displaystyle\tilde{j}_{jkli}$
$\displaystyle=\frac{1}{3}(\delta_{jk}\delta_{li}+\delta_{jl}\delta_{ki}+\delta_{kl}\delta_{ji})\phi$
(2.95)

While $j_{i}$ and $\tilde{j}_{i}$, $j_{jkil}$ and $\tilde{j}_{jkil}$ exactly
match, the other two tensor currents differ in a way such that conservation is
retained.

$\displaystyle\partial_{j}j_{ji}=\partial_{j}\tilde{j}_{ji}=\partial_{i}\partial^{2}\phi$
$\displaystyle\partial_{j}\partial_{k}j_{jki}=\partial_{j}\partial_{k}\tilde{j}_{jki}=\partial_{i}\partial^{2}\phi$
(2.96)

Finally, the identities (2.68) and (2.69) are also preserved,

$\displaystyle\tilde{j}_{i}$
$\displaystyle=\partial_{j}\tilde{j}_{ji}=\partial_{j}\partial_{k}\tilde{j}_{ijk}=\partial_{j}\partial_{k}\partial_{l}\tilde{j}_{ijkl}$
$\displaystyle\tilde{j}_{jk}$ $\displaystyle=\partial_{i}\tilde{j}_{jki}$
(2.97)

which is necessary to establish current conservation.
It may be mentioned that the structure (2.64-2.67) or (2.95) is generic while
the identities (2.68) and (2.69) are exactly retained for any higher
derivative theory that is a generalisation of (2.55) or (2.88). For instance
if we consider the lagrangian,

$\mathcal{L}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}(\partial_{i}\partial_{j}\partial_{k}\phi)(\partial_{i}\partial_{j}\partial_{k}\phi)$
(2.98)

then, under a quintic shift of coordinates,

$\phi\to\phi+w,\,\,w=c+c_{i}x_{i}+\frac{1}{2}c_{ij}x_{i}x_{j}+\frac{1}{3}c_{ijk}x_{i}x_{j}x_{k}+\frac{1}{4}c_{ijkl}x_{i}x_{j}x_{k}x_{l}+\frac{1}{5}c_{ijklm}x_{i}x_{j}x_{k}x_{l}x_{m}$
(2.99)

it is left quasi invariant,

$\delta\mathcal{L}=\partial_{i}\Lambda_{i}-\phi(\partial^{2})^{3}w$ (2.100)

where

$\Lambda_{i}=(\partial_{j}\partial_{k}\phi)(\partial_{i}\partial_{j}\partial_{k}w)-(\partial_{k}\phi)(\partial^{2}\partial_{i}\partial_{k}w)+\phi(\partial^{2})^{2}\partial_{i}w$

Following the algorithm developed here the basic conservation law is expressed
as,

$\partial_{0}J_{0}-\partial_{i}J_{i}=\partial_{0}(wj_{0})-\partial_{i}J_{i}=0$
(2.101)

where

$J_{i}=wj_{i}-(\partial_{j}w)j_{ji}+(\partial_{j}\partial_{k}w)j_{jki}-(\partial_{j}\partial_{k}\partial_{l}w)j_{jkli}+(\partial_{j}\partial_{k}\partial_{l}\partial_{m}w)j_{jklmi}-(\partial_{j}\partial_{k}\partial_{l}\partial_{m}\partial_{n}w)j_{jklmni}$
(2.102)

and the individual expressions are given by,

$\displaystyle j_{i}=$ $\displaystyle-\partial_{i}\partial^{4}\phi,\quad
j_{ji}=-\partial_{i}\partial_{j}\partial^{2}\phi,\quad
j_{jki}=-\partial_{i}\partial_{j}\partial_{k}\phi$ $\displaystyle j_{jkli}=$
$\displaystyle-\frac{1}{3}(\delta_{il}\partial_{j}\partial_{k}\phi+\delta_{ik}\partial_{j}\partial_{l}\phi+\delta_{ij}\partial_{k}\partial_{l}\phi)$
$\displaystyle j_{jklmi}=$ $\displaystyle....$ (2.103)

The tower of conservation laws corresponding to dipole and higher pole
symmetries can be calculated as shown earlier and are based on identities
similar to (2.68).

$j_{i}=\partial_{j}j_{ji}=\partial_{j}\partial_{k}j_{jki}=\partial_{j}\partial_{k}\partial_{l}j_{jkli}=\partial_{j}\partial_{k}\partial_{l}\partial_{m}j_{jklmi}=\partial_{j}\partial_{k}\partial_{l}\partial_{m}\partial_{n}j_{jklmni}$
(2.104)

Also, there exist intermediate identities like,

$j_{jk}=\partial_{i}j_{ijk}$ (2.105)

as happened for the theory defined by (2.55).

##  3 From shift symmetries to gauge symmetries

The analysis for polynomial shift symmetries in scalar field theories done in
the previous section is now generalised to complex scalars. This will allow
interactions to be introduced by coupling with gauge fields-either vector or
higher rank tensors. Such theories have recently been profusely discussed in
the literature.
A complex scalar $\Phi$ can be polar decomposed as,

$\Phi=\rho e^{i\phi}$

where $\rho,\phi$ are two real scalars. Then, under the shift,

$\phi\to\phi+w$ (3.1)

where $w$ is an arbitrary space dependent polynomial,

$\Phi\to e^{iw}\Phi$ (3.2)

which is the usual abelian gauge transformation on the complex scalar $\Phi$.
Then,

$ln\Phi\,\to\,ln\Phi+iw$ (3.3)

Comparison with (3.1) shows that, as far as transformations are concerned,
there exists the map,

$i\phi\to ln\Phi$ (3.4)

With this map, we can construct suitable actions of complex scalars begining
from real scalars such that the symmetries are preserved. This is illustrated
by examples.

As a first case we construct an action for complex scalars that is exactly
invariant under a linear shift like (2.6). From (2.55) and (2.58) it is clear
that (2.55) is exactly invariant ($\delta\mathcal{L}=0$) under a linear shift.
Thus our cherished goal will be achieved by starting from (2.55) and using
(3.4) to get the corresponding theory. We find

$i\partial_{i}\partial_{j}\phi\,\,\to\,\,\frac{\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi}{\Phi^{2}}$
(3.5)

and hence,

$\mathcal{L}=(\partial_{i}\partial_{j}\phi)^{2}=|i\partial_{i}\partial_{j}\phi|^{2}\,\to\,\frac{|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}}{|\Phi^{2}|^{2}}$
(3.6)

This implies that the expression on the right will be invariant under the
gauge transformation (3.2) since the corresponding expression on the left side
is invariant under the linear shift (3.1). In either case $w$ is given by
(2.6). Since $|\Phi^{2}|^{2}$ is invariant under a general U(1) gauge
transformation, we just take the numerator in (3.6) as the desired lagrangian.
This reproduces the expression for the lagrangian invariant under a linearly
shifted U(1) gauge transformation, as found in the literature.
The conserved currents may likewise be obtained by a direct application of the
equality connecting the $\phi$ and $\Phi$ fields,

$\phi=-\frac{i}{2}ln\Phi+c.c$ (3.7)

Then it follows from (2.65),

$j_{ji}=j_{ij}=\partial_{i}\partial_{j}\phi=\frac{-i[{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)-\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})]}{2|\Phi|^{4}}$
(3.8)

while the vector current follows by simply using,

$j_{i}=\partial_{j}j_{ji}$ (3.9)

Now we shall similarly construct an action for complex scalar theory which is
invariant under a quadratic shift. From (2.98) and (2.100) it can be
understood that (2.98) is invariant under a quadratic shift. So we shall now
use the map (3.4) in (2.98) to achieve the corresponding complex scalar
theory.

$i\partial_{i}\partial_{j}\partial_{k}\phi\,\to\,\frac{\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)}{\Phi^{3}}$
(3.10)

and hence,

$\mathcal{L}=(\partial_{i}\partial_{j}\partial_{k}\phi)^{2}=|i\partial_{i}\partial_{j}\partial_{k}\phi|^{2}\,\to\,\frac{\Big{|}\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)\Big{|}^{2}}{|\Phi|^{6}}$
(3.11)

As before, one may drop the $|\Phi|^{6}$ in the denominator, retaining only
the numerator. It reproduces the well known expression for the lagrangian
invariant under a quadratically shifted U(1) gauge transformation.
The conserved current can now be derived from (2.103) using the map between
$\phi$ and $\Phi$ (3.7) as

$j_{ijk}=\partial_{i}\partial_{j}\partial_{k}\phi=\frac{-i{\Phi^{*}}^{3}\Big{[}\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)\Big{]}+c.c}{2|\Phi|^{6}}$
(3.12)

while the lower rank currents follow by simplifying

$j_{i}=\partial_{j}\partial_{k}j_{ijk},\quad j_{ij}=\partial_{k}j_{ijk}$

Here we derived the lagrangians and corresponding tensor currents for the
matter sector in higher derivative complex scalar theories invariant under a
gauge transformation, where the gauge parameter is a space dependent linear or
quadratic shift. This was achieved by exploiting a map from real to complex
scalar field using the structure of lagrangians and currents derived for real
scalar fields in the previous section. We shall use these lagrangians in our
upcoming sections to construct a full interacting gauge theory.

##  4 Dual description of theory with space dependent linear shift symmetry

Here we briefly review the coupling prescription for a lagrangian involving
complex scalar fields invariant under space dependent linear shift. It is also
shown that, interactions can be introduced either using usual gauge (vector)
fields or, alternatively, using second rank symmetric tensor gauge fields. In
the literature [2, 4, 9] only the second possibility has been discussed at
length.
The gauge transformation on a scalar field is given by

$\Phi(x)\,\,\to\,\,\Phi^{\prime}(x)=e^{iw(x)}\Phi(x),\quad\Phi^{*}(x)\,\,\to\,\,\Phi^{\prime*}(x)=e^{-iw(x)}\Phi^{*}(x)$
(4.1)

where the transformation parameter $w$ depends linearly on the space
coordinate $x_{i}$ (2.6)444Rotational invariance is assumed.. It has been
shown that [9] the following scalar field lagrangian is invariant under the
above transformation (4.1);

$\mathcal{L}=\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial_{0}}\Phi-s\Big{(}\partial_{i}(\Phi^{*}\Phi)\Big{)}^{2}+it({\Phi^{*}}^{2}\partial_{i}\Phi\partial_{i}\Phi-\Phi^{2}\partial_{i}\Phi^{*}\partial_{i}\Phi^{*})+u|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-\lambda|\Phi|^{4}$
(4.2)

where $s,t$ and $u$ are some normalization factors. The first two terms and
the last term in the above lagrangian are trivially invariant under (4.1),
even when $w(x)$ is an arbitrary local parameter. Let us therefore concentrate
on the $u$ term. This particular term was also obtained here in (3.6), apart
from the $|\Phi^{2}|^{2}$ term which, incidentally, is trivially invariant
under an arbitrary local transformation. The term with coefficient $u$ changes
under a general transformation as

$|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}\to|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi+i\Phi^{2}\partial_{i}\partial_{j}w|^{2}$
(4.3)

Invariance is achieved provided $w$ is linearly dependent on $x_{i}$, as given
in (2.6). This is the reason behind taking the particular form of the
lagrangian (4.2), which is invariant under (4.1) for $w$ having the form
(2.6).
The discussion so far has been confined to taking a specific form for $w(x)$,
appearing in (4.1), that is either constant or linear in $x$. In either case,
(4.2) is invariant under the global transformation (4.1) without the need for
introducing any gauge field. We now wish to lift this global symmetry to a
local symmetry. Thus let us take a general space-dependent parameter, which
will obviously not keep the lagrangian invariant. So we shall have to add a
compensating term in the lagrangian, preferably a gauge field using some
prescription, which will render the invariance of the lagrangian. In this way
gauge couplings can be introduced. As we shall show, this may be achieved
using tensor gauge fields or vector gauge fields [11]. The complete theory
comprising of the matter and gauge (given by either vector or tensor) fields
is now invariant under an arbitrary space-dependent transformation. The issue
of global symmetries, for either matter or the gauge fields now becomes
inconsequential.

Using (4.3) it is easy to find a covariantization prescription involving a
tensorial gauge field, for this part of the lagrangian. It is given by

$\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi\quad\to\quad\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi-
iA_{ij}\Phi^{2}$ (4.4)

so that the covariance can be achieved, under (4.1) and

$A_{ij}\to A_{ij}+\partial_{i}\partial_{j}w$ (4.5)

where $A_{ij}$ is a second rank symmetric tensor field. It is seen that (4.4)
transforms under (4.1) and (4.5) covariantly as

$\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi-
iA_{ij}\Phi^{2}\quad\to\quad
e^{2iw(x)}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi-
iA_{ij}\Phi^{2})$ (4.6)

where the non covariant piece $\partial_{i}\partial_{j}w$ (see (4.3)) is
cancelled by a contribution from the transformation of $A_{ij}$. So the total
lagrangian is given by

$\mathcal{L}=\Big{|}\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi-
iA_{ij}\Phi^{2}\Big{|}^{2}$ (4.7)

At this juncture we make an important observation. It may be shown that the
non-standard gauging prescription (4.4) can be mapped to the standard minimal
prescription using the usual vector gauge field $A_{i}$. The origin of this
lies in the presence of conservation laws other than usual charge
conservation. It was shown in the previous section that, for linear shift
symmetry, there are two currents $j^{i}$ and $j^{ij}$ related by (2.15) This
kind of relation enables one to question whether the vector and the tensor
fields are also related. Infact coupling the vector current with usual vector
gauge field gives,

$A_{i}j^{i}=A_{i}\partial_{j}j^{ij}=-\frac{1}{2}(\partial_{j}A_{i}+\partial_{i}A_{j})j^{ij}=A_{ij}j^{ij}$
(4.8)

where we have dropped a boundary term. So the above relation enables us to
identify immediately

$A_{ij}:=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i}).$ (4.9)

This identification is also compatible with transformations (4.5) and $\delta
A_{i}=-\partial_{i}w$ 555 Note the difference in sign from (2.30).. After
obtaining the relation between a tensor and a vector gauge field we shall now
discuss how one can use the minimal coupling prescription with $A_{i}$ instead
of the non-standard prescription shown in (4.4) and can give the same
interacting theory for linear shift symmetric lagrangian.
For that we first replace the ordinary derivatives $\partial_{i}$ with the
covariant derivative $\partial_{i}+iA_{i}$, where $A_{i}$ is the usual vector
field transforming as (2.30)(see footnote 5),

$(\partial_{i}+iA_{i})\Phi\,\,\to\,\,e^{iw}(\partial_{i}+iA_{i})\Phi$ (4.10)

Now one has to symmetrize the double derivative term of the lagrangian,
followed by covariantization of the derivative, demonstrated below:

$\displaystyle\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi\,\,\to\,\,\frac{1}{2}(\Phi\partial_{i}\partial_{j}\Phi+\Phi\partial_{j}\partial_{i}\Phi)-\partial_{i}\Phi\partial_{j}\Phi$
$\displaystyle\to\,\,\frac{1}{2}\Big{[}\Phi(\partial_{i}+iA_{i})(\partial_{j}+iA_{j})\Phi+\Phi(\partial_{j}+iA_{j})(\partial_{i}+iA_{i})\Phi\Big{]}-(\partial_{i}+iA_{i})\Phi(\partial_{j}+iA_{j})\Phi$
(4.11)

On simplification the above expression yields

$\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi\,\,\to\,\,\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi+\frac{i}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})\Phi^{2}$
(4.12)

This expression exactly matches with the one in (4.4), provided we identify
$A_{ij}$ as gien in (4.9) and the total lagrangian is now given by

$\mathcal{L}=\Big{|}\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi+\frac{i}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})\Phi^{2}\Big{|}^{2}$
(4.13)

So (4.11) gives a minimal prescription using usual vector field that gives the
same result as that of (4.4), which is obtained in terms of a rank-2 symmetric
tensor. The preceding analysis revealed the intriguing possibility of working
with either vector or symmetric higher rank tensor gauge fields when gauging
higher derivative complex scalar theories.

###  4.1 Gauge coupling from iterative Noether prescription

Here we first obtain the currents directly, adopting a definition analogous to
(2.63). Then, following the iterative prescription [30] we derive the gauge
couplings. The dual description, expressed in terms of a vector field or a
tensor field, is reproduced. Let us consider the $u$ term in (4.2). The
conserved currents are defined analogous to (2.63), recalling that the shift
symmetry is only linear (2.6) [27]

$\displaystyle J^{0}$
$\displaystyle=j^{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\Phi)}\delta\Phi+c.c=-\Phi^{*}\Phi$
$\displaystyle J^{i}$
$\displaystyle=wj^{i}-(\partial_{j}w)j^{ji}=\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\Phi)}\delta\Phi+\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\Phi)}\partial_{j}(\delta\Phi)-\Big{(}\partial_{j}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\Phi)}\Big{)}\delta\Phi+c.c$
(4.14)

Equating coefficients of $c,c_{i}$, we find,

$j^{i}=\partial_{j}j_{ij}=i\partial_{j}\Big{(}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{)}$
(4.15)

Modulo the multiplication factor $|\Phi|^{4}$, this current was earlier
introduced in (3.8,3.9) using the map between $\phi$ and $\Phi$ fields. To
obtain the gauge couplings we adopt the iterative Noether’s scheme. According
to this scheme the first correction to the lagrangian is given by,

$\mathcal{L}_{1}=-j^{i}A_{i}=-i\Big{(}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{)}\partial_{j}A_{i}$
(4.16)

Since derivatives (on the matter field) persist, there will be another
correction. This is given by,

$j^{i}=\partial_{j}\Big{(}|\Phi|^{4}(\partial_{j}A_{i}+\partial_{i}A_{j})\Big{)}$
(4.17)

where we have used $\mathcal{L}_{1}$ (4.16) in (4.14) to get this current.
Then the contribution to the lagrangian is given by,

$\mathcal{L}_{2}=-j^{i}A_{i}=\frac{1}{4}|\Phi|^{4}(\partial_{i}A_{j}+\partial_{j}A_{i})^{2}$
(4.18)

so that the current is reproduced by,

$j^{i}=-\frac{\delta}{\delta A_{i}}\int\mathcal{L}_{2}$ (4.19)

Since there are no further derivatives on $\Phi$, the iteration terminates and
the final form of the gauged theory is given by,

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{0}+\mathcal{L}_{1}+\mathcal{L}_{2}$
$\displaystyle=|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}+\frac{i}{2}\Big{(}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{)}(\partial_{i}A_{j}+\partial_{j}A_{i})$
$\displaystyle+\frac{1}{4}|\Phi|^{4}(\partial_{i}A_{j}+\partial_{j}A_{i})^{2}$
(4.20)

which exactly matches with the form (4.13).
It is also possible to recast the above analysis in terms of the tensor field
defined in (4.9). In that case the first correction (4.16) is expressed as,

$\mathcal{L}_{1}=-i\Big{(}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{)}A_{ij}=-j^{ij}A_{ij}$
(4.21)

where the current $j^{ij}$ coupling to $A_{ij}$ is defined in (4.15). Pursuing
this line of analysis the new (tensor) current obtained from the first
iteration is given by,

$j^{ij}=-2|\Phi|^{4}A_{ij}$ (4.22)

which is consistent with (4.17). The new contribution to the lagrangian now
follows,

$\mathcal{L}_{2}=|\Phi|^{4}A_{ij}^{2}=-j^{ij}A_{ij}$ (4.23)

so that the current (4.22) is reproduced by,

$j^{ij}=-\frac{\delta}{\delta A_{ij}}\int\mathcal{L}_{2}$ (4.24)

Note that the structure (4.23) is consistent with (4.18) on using the relation
(4.9). Thus the final lagrangian takes the form,

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{0}+\mathcal{L}_{1}+\mathcal{L}_{2}$
$\displaystyle=|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-i\Big{[}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{]}A_{ij}+|\Phi|^{4}A_{ij}^{2}$
(4.25)

which is the dual version of (4.20) and is same as (4.7).

###  4.2 Full interacting theory in dual description

As we have shown in section-4 that, for complex scalar theories with linear
space dependent shift symmetry, the gauging can be done, either by making use
of tensor gauge fields or by using usual vector gauge fields. We have shown
that this is a consequence of the functional relationship between the vector
and higher rank tensor currents. However, this was only discussed
algebraically as no gauge field dynamics was involved in the earlier sections.
In this section we shall discover that the mapping is even valid at the
dynamical level. By taking the full interacting lagrangian where the gauge
fields, either vector or tensor, are also dynamical, descriptions of these
theories are possible not only at the lagranian level but also at the
hamiltonian level.

####  4.2.1 Lagrangian and hamiltonian analysis in (2+1) dimensional space-
time

The interacting theory can be presented in two distinct ways. As a symmetric
tensor theory or a mapped vector theory using (4.9) in the tensor theory. We
use the mapping in two different ways: (1) we write down a lagrangian in terms
of vector fields from the tensor theory lagrangian using the map and then find
the equation of motions of the vector fields. (2) On the other hand we can
just use the map at the level of equation of motion of the tensor theory
itself to find a set of vector field equations of motion and see that in both
cases the vector equations of motion match.
We first consider the (2+1) dimensional case. In (2+1) dimension the Maxwell
theory for U(1) vector fields modifies from that of (3+1) dimensional theory.
The electric field is a 2 dimensional vector
$E_{i}=-F_{oi}=\partial_{i}A_{0}-\partial_{0}A_{i}$ and the magnetic field is
a scalar $B=\epsilon_{ij}\partial_{i}A_{j}$.

Theory with second rank tensor gauge fields
Now let us write a (2+1) dimensional theory in terms of tensorial gauge
fields. For a tensorial theory in 2+1 dimension, the electric field is taken
as a rank-2 symmetric tensor, while the magnetic field becomes a vector. Let
us first define the electric and magnetic fields for this tensor theory in
terms of a symmetric rank-2 tensor gauge field $A_{ij}$ as,

$E_{ij}=\partial_{0}A_{ij}+\partial_{i}\partial_{j}A_{0},\quad
B_{k}=\epsilon_{ij}\partial_{i}A_{jk}$ (4.26)

One can check that both these fields are gauge invariant. The lagrangian for
the pure gauge field is written,

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{1}{2}(E_{ij}^{2}-B_{i}^{2})$ (4.27)
$\displaystyle=\frac{1}{2}\Big{[}(\partial_{i}\partial_{j}A_{0})^{2}+(\partial_{0}A_{ij})^{2}+2(\partial_{0}A_{ij})(\partial_{i}\partial_{j}A_{0})\Big{]}-\frac{1}{2}(\partial_{m}A_{nk}-\partial_{n}A_{mk})\partial_{m}A_{nk}$
(4.28)

The complete theory is now given by,

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}$
$\displaystyle-i\Big{[}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{]}A_{ij}+|\Phi|^{4}A_{ij}^{2}$
$\displaystyle=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-j_{ij}A_{ij}$
(4.29)

where,

$j_{ij}=i\Big{[}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi\partial_{j}\Phi)-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\Big{]}-|\Phi|^{4}A_{ij}$
(4.30)

The equation of motion for $A_{0}$ is given by

$\partial_{i}\partial_{j}E_{ij}=-\Phi^{*}\Phi$ (4.31)

while that of the gauge field $A_{ij}$ is given by

$\partial_{0}E_{ij}+\frac{1}{2}\partial_{k}(\epsilon_{ik}B_{j}+\epsilon_{jk}B_{i})=-j_{ij}$
(4.32)

Lastly, the equation of motion of the scalar field $\Phi$ is given by

$\displaystyle-(i\partial_{0}+A_{0})\Phi^{*}+4(\partial_{i}\partial_{j}\Phi)\mathcal{C}_{ij}+4(\partial_{i}\Phi)\partial_{j}\mathcal{C}_{ij}+\Phi\partial_{i}\partial_{j}\mathcal{C}_{ij}-i\Phi\mathcal{C}_{ij}A_{ij}+4i{\Phi^{*}}^{2}(\partial_{i}\partial_{j}\Phi)A_{ij}$
$\displaystyle+2{\Phi^{*}}^{2}\Phi
A_{ij}^{2}-\partial_{i}\Big{(}(\Phi^{*})^{2}A_{ij}\Big{)}(\partial_{j}\Phi)-\Phi\partial_{i}\partial_{j}\Big{(}(\Phi^{*})^{2}A_{ij}\Big{)}=0$
(4.33)

where

$\mathcal{C}_{ij}=\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*}$
(4.34)

Now let us compute the Hamilton’s equations of motion for the electric and
magnetic fields and deduce the respective Maxwell equations for the tensor
theory. From (4.28), one can compute the canonical momenta corresponding to
$A_{0}$ and $A_{ij}$ as

$\pi_{0}=\frac{\partial\mathcal{L}}{\partial\dot{A}^{0}}=0,\qquad\pi_{ij}=\frac{\partial\mathcal{L}}{\partial\dot{A}^{ij}}=\dot{A}_{ij}+\partial_{i}\partial_{j}A_{0}=E_{ij}$
(4.35)

and the corresponding canonical Hamiltonian as

$H_{c}=\frac{1}{2}(E_{ij}^{2}+B_{k}^{2})+...$ (4.36)

where the ellipses denote terms containing $\Phi$ and the interaction terms.
The basic Poission relation between the gauge field $A_{ij}$ and its canonical
momenta $E_{ij}$ is given by

$\\{A_{ij}(\vec{x}),A_{kl}(\vec{y})\\}=0=\\{E_{ij}(\vec{x}),E_{kl}(\vec{y})\\};\quad\\{A_{ij}(\vec{x}),E_{kl}(\vec{y})\\}=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\delta^{2}(\vec{x}-\vec{y})$
(4.37)

while that between the physical variables (electric and magnetic fields) are
given by

$\\{B(\vec{x}),B(\vec{y})\\}=0,\qquad\\{E_{ij}(x),B_{k}(y)\\}=\frac{1}{2}\partial_{m}^{y}(\epsilon_{mi}\delta_{jk}+\epsilon_{mj}\delta_{ik})\delta^{2}(\vec{x}-\vec{y})$
(4.38)

Now using (4.37) we compute the time derivative of the electric field as

$\partial_{0}E_{ij}=\\{E_{ij},H_{c}\\}=\frac{1}{2}(\epsilon_{ki}\partial_{k}B_{j}+\epsilon_{kj}\partial_{k}B_{i})-j_{ij}$
(4.39)

This corresponds to Faraday’s law for tensor theory in (2+1) dimension.
Similarly the equation of motion of $B_{i}$ is given by

$\partial_{0}B_{i}=\\{B_{i},H_{c}\\}=-\epsilon_{jk}\partial_{j}E_{ki}$ (4.40)

which corresponds to Ampere’s law for tensor theory in (2+1) dimensions.

Theory with vector gauge field

We now use the map (4.9) between the rank-2 symmetric tensor gauge field and
usual vector field to write down a lagrangian in terms of vector fields. Then
we shall find out the equations of motion for the vector fields.
Using (4.9) in the lagrangian (4.28), the vector gauge field lagrangian for
(2+1) dimensional theory is given by

$\mathcal{L}_{0}=\frac{1}{4}\Big{[}2(\partial_{i}\partial_{j}A_{0})^{2}+(\partial_{0}\partial_{i}A_{j})^{2}-4(\partial_{i}\partial_{j}A_{0})(\partial_{i}\partial_{0}A_{j})+(\partial_{i}\partial_{0}A_{j})(\partial_{j}\partial_{0}A_{i})\Big{]}-\frac{1}{8}(\epsilon_{jk}\partial_{i}\partial_{j}A_{k})^{2}$
(4.41)

The full lagrangian containing scalar field part and the interaction part will
be given by

$\displaystyle\mathcal{L}=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-j_{i}A_{i}$

where (4.8) has been used to recast the interaction term in favour of vector
quantities. Now we can compute the equation of motion of the fields from this
lagrangian. First, the equation of motion of $A_{0}$ field is given by

$\partial^{2}\partial_{i}E_{i}=-\Phi^{*}\Phi$ (4.42)

which corresponds to the Gauss constraint, where $E_{i}$ is defined as before
$E_{i}=-F_{oi}$, However note that it is not the canonical momenta
corresponding to $A_{i}$. This will be shown in details in the Hamiltonian
analysis. The equation of motion for $A_{i}$ is given by

$\frac{1}{2}\partial^{2}(\partial_{0}E_{i}-\frac{1}{2}\epsilon_{ij}\partial_{j}B)+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}=-j_{i}$
(4.43)

where

$j_{i}=i\partial_{j}[\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi\partial_{j}\Phi)-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)]+\partial_{j}[|\Phi|^{4}(\partial_{i}A_{j}+\partial_{j}A_{i})]$
(4.44)

is the vector current for the theory [11] and is related to a tensor current
$j_{ij}$ as

$j_{i}=\partial_{j}j_{ij};\qquad
j_{ij}=[i\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi\partial_{j}\Phi)-i{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)+|\Phi|^{4}(\partial_{i}A_{j}+\partial_{j}A_{i})]$
(4.45)

The equation of motion of $\Phi$ is essentially the same as (4.33), because
the interaction lagrangian is not changed, apart from the fact that $A_{ij}$
will now be replaced by $-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$ in
this picture.

We shall now discuss about the hamiltonian analysis of the vector theory. From
(4.41), one can find out the canonical momenta for the vector fields as

$\pi_{0}=\frac{\partial\mathcal{L}}{\partial\dot{A^{0}}}=0\,;\qquad\pi_{i}=\frac{\partial\mathcal{L}}{\partial\dot{A^{i}}}=\frac{1}{2}(\delta_{ij}\partial^{2}+\partial_{i}\partial_{j})E_{j}$
(4.46)

The canonical Hamiltonian for the theory can be written using Legendre
transformation as

$H_{c}=\frac{1}{2\partial^{4}}(\partial_{i}\pi_{j})^{2}+\frac{1}{8}(\partial_{i}B)^{2}+...$
(4.47)

where the ellipses mean terms containing scalar fields and interactions.
The Poission algebra between the canonically conjugate fields $A_{i}$ and
$\pi_{i}$ are given by

$\\{A_{i}(\vec{x}),A_{j}(\vec{y})\\}=\\{\pi_{i}(\vec{x}),\pi_{j}(\vec{y})\\}=0;\qquad\\{A_{i}(\vec{x}),\pi_{j}(\vec{y})\\}=\delta_{ij}\delta^{2}(\vec{x}-\vec{y})$
(4.48)

while that between the electric and magnetic fields are,

$\\{\pi_{i},\pi_{j}\\}=0=\\{B,B\\},\quad\\{B(\vec{x}),\pi_{j}(\vec{y})\\}=\epsilon_{kj}\partial_{k}^{x}\delta^{2}(\vec{x}-\vec{y})$
(4.49)

Using this algebra, we can derive the equation of motion of $E_{i}$ and $B$
from the above Hamiltonian as

$\displaystyle\partial_{0}\pi_{i}=\\{\pi_{i},H_{c}\\}=\frac{\partial^{2}}{4}\epsilon_{ji}\partial_{j}$
$\displaystyle B-j_{i}$
$\displaystyle\Rightarrow\frac{\partial^{2}}{2}(\partial_{0}E_{i}-\frac{1}{2}\epsilon_{ij}\partial_{j}B)+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}$
$\displaystyle=-j_{i}$ (4.50)

and

$\partial_{0}B=\\{B,H_{c}\\}=\epsilon_{ji}\partial_{j}E_{i}$ (4.51)

Comparative study
We cannot compare the equations of motion derived in two scenarios at this
stage, as the first case provides equation of motion for tensor fields whereas
the second case considers vector fields. The first step is to find a map
connecting the physical gauge invariant variables (electric and magnetic
fields) in the two pictures. Exploiting the basic map (4.9) in (4.26), the
following identifications are obtained,

$E_{ij}=\frac{1}{2}(\partial_{i}E_{j}+\partial_{j}E_{i}),\qquad
B_{k}=\epsilon_{ij}\partial_{i}A_{jk}=-\frac{1}{2}\partial_{k}B$ (4.52)

where $E_{i}$ and $B$ were defined earlier.
The Gauss constraint may now be compared. From (4.31) we can write using
(4.52) ,

$-\Phi^{*}\Phi=\partial_{i}\partial_{j}E_{ij}=\frac{1}{2}\partial_{i}\partial_{j}(\partial_{i}E_{j}+\partial_{j}E_{i})=\partial^{2}\partial_{i}E_{i}$
(4.53)

which is same as (4.42) but now derived directly from the tensor field
equations using the mapping.
Using the map (4.52) in (4.32), we obtain,

$\frac{1}{2}\partial_{j}[\partial_{0}E_{i}-\frac{1}{2}\epsilon_{im}\partial_{j}\partial_{m}B]+\frac{1}{2}\partial_{0}\partial_{i}E_{j}=-j_{ij}$

Now using $\partial_{j}$ on the above equation from left we get

$\Rightarrow\frac{1}{2}(\partial^{2}\partial_{0}E_{i}+\partial_{0}\partial_{i}\partial_{j}E_{j})-\frac{1}{4}\partial^{2}\epsilon_{ij}\partial_{j}B=-j_{i}$
(4.54)

and (4.50) is reproduced. Moreover, it is feasible to do a consistency check
by acting a $\partial_{i}$ on (4.54) and see using (4.42) that it reproduces
the usual current conservation law for vector fields,

$\partial_{0}(\Phi^{*}\Phi)-\partial_{i}j_{i}=\partial_{0}j_{0}-\partial_{i}j_{i}=0$
(4.55)

Let us now see the comparison at the hamiltonian level. It is seen that one
can also arrive at (4.47) from the tensor theory Hamiltonian (4.36), using the
map (4.52). Now the tensorial Ampere’s law (4.40) can be mapped to the
following equation using (4.52).

$\partial_{0}\partial_{i}B=\epsilon_{jk}\partial_{i}\partial_{j}E_{k}$ (4.56)

The solution of the above equation gives us the corresponding Ampere’s law for
vector theory (4.51).
The above comparison can be captured using the following chart.
Basic gauge field Electric field Magnetic field Fields $A_{i},\,\,A_{ij}$
$E_{i}=\partial_{i}A_{0}-\partial_{0}A_{i},\,\,E_{ij}$
$B=\epsilon_{jk}\partial_{j}A_{k},\,\,B_{i}$ Derived Map
$A_{ij}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$
$E_{ij}=\frac{1}{2}(\partial_{i}E_{j}+\partial_{j}E_{i})$
$B_{i}=-\frac{1}{2}\partial_{i}B$

Table I : Comparison of various fields in (2+1) dimension for linear shift
symmetric theory

This table shows how the basic fields in the rank two theory
$(A_{ij},E_{ij},B_{i})$ are related to the corresponding variables
$(A_{i},E_{i},B)$ in the vector theory.To complete the picture we provide
another table.

Gauss law Faraday’s law Ampere’s law Tensor theory
$\partial_{i}\partial_{j}E_{ij}+\Phi^{*}\Phi=0$
$\partial_{0}E_{ij}+\frac{1}{2}(\epsilon_{ik}\partial_{k}B_{j}+\epsilon_{jk}\partial_{k}B_{i})=-j_{ij}$
$\partial_{0}B_{i}=-\epsilon_{jk}\partial_{j}E_{ki}$ Using (4.52) in tensor
theory $\partial^{2}\partial_{i}E_{i}+\Phi^{*}\Phi=0$
$\frac{1}{2}\partial_{j}[\partial_{0}E_{i}-\frac{1}{2}\epsilon_{im}\partial_{j}\partial_{m}B]+\frac{1}{2}\partial_{0}\partial_{i}E_{j}=-j_{ij}$
$\partial_{i}\partial_{0}B=\epsilon_{jk}\partial_{i}\partial_{j}E_{k}$ Using
$\partial_{j}$ on above equation Acting with $\frac{1}{\partial_{i}}$ on above
eqn Vector theory $\partial^{2}\partial_{i}E_{i}+\Phi^{*}\Phi=0$
$\frac{1}{2}\partial^{2}[\partial_{0}E_{i}-\frac{1}{2}\epsilon_{ij}\partial_{j}B]+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}=-j_{i}$
$\partial_{0}B=\epsilon_{jk}\partial_{j}E_{k}$

Table II : Comparison of various equations of motion in (2+1) dimension for
linear shift symmetric theory

To use this table one has to adopt the following approach. The first and last
rows give the results obtained from the tensor coupling and vector coupling,
respectively. The results for the vector case are reproduced from the tensor
theory using the maps given in table I. It is either obtained directly (Gauss
law) or by applying suitable derivative operators (Faraday’s and Ampere’s
law). It is expected as Faraday’s and Ampere’s law for tensor and vector
theories differ by one rank whereas Gauss law being a scalar equation in both
cases is reproduced automatically.

The different descriptions of gauge theories presented here need some
elaboration. Starting from a rank-2 tensor theory (4.28), the vector theory
(4.41) was constructed using the map (4.9). These theories were connected in
the sense that the physical equations- Gauss law, Faraday’s law, and Ampere’s
law - were identical, irrespective of the fact that these were directly
obtained from the vector theory or derived from the tensor theory exploiting
suitable maps.

It should be pointed out that the map (4.9) allows a passage from the tensor
theory to the vector theory but not the other way around. All correspondences
have to be interpreted in this way, either in the action formulation or in the
equation of motion. While the tensor Faraday’s equations pass to the vector
ones, the converse does not hold. In passing from the vector to the tensor
equations, the transverse components remain undetermined. This is bacause,
there is a mismatch in the number of degrees of freedom of the gauge fields.
To see this more elaborately, we recall that the two degrees of freedom of the
vector field are the transverse and longitudinal components, while the three
degrees of freedom of the tensor field can be decomposed as a fully transverse
component, a partially transverse component, and a longitudinal component.
While the longitudinal components of the two gauge fields may be mapped, this
is not true for the transverse components.

The map (4.9) was used since the primary motivation was to discuss a
connection among various gauge theories, apart from the fact that it occurs
naturally in the present analysis and has also been mentioned earlier in the
literature [9, 11]. Similar discussion is also valid for the subsequent
sections when we discuss about 3+1 dimensional case and quadratic shift
symmetric theory.

####  4.2.2 Lagrangian and hamiltonian analysis in (3+1) dimensional space-
time

Theory with second rank tensor gauge field

Here we shall write down the gauge field lagrangian for tensorial gauge fields
in (3+1) dimension, which also has to be gauge invariant. So, first let us
write the gauge invariant electric and magnetic field tensor [31, 11]
corresponding to $A_{ij}$ as,

$\displaystyle
E_{ij}=F_{0ij}=\partial_{i}\partial_{j}A_{0}+\partial_{0}A_{ij}$ (4.57)
$\displaystyle
B_{ij}=\partial_{k}F_{kij}=\partial_{k}(\partial_{j}A_{ik}-\partial_{i}A_{jk})$
(4.58)

Note that the magnetic field is now changed from the (2+1) dimensional case,
which is now a two rank anti-symmetric tensor.
The pure gauge field lagrangian, in analogy with the Maxwell example, is given
by

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{1}{2}(E_{ij}^{2}-B_{ij}^{2})$
$\displaystyle=\frac{1}{2}\Big{[}(\partial_{i}\partial_{j}A_{0})^{2}+(\partial_{0}A_{ij})^{2}\Big{]}+(\partial_{0}A_{ij})(\partial_{i}\partial_{j}A_{0})-(\partial_{i}\partial_{k}A_{kj})^{2}+(\partial_{i}\partial_{k}A_{kj})(\partial_{j}\partial_{l}A_{li})$
(4.59)

We can construct the full lagrangian in terms of $A_{ij}$ fields as

$\mathcal{L}=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-j_{ij}A_{ij}$
(4.60)

Note that terms involving $A_{0}$ and pure matter and matter-gauge interaction
will remain unchanged from the previous case of (2+1) dimension. The tensor
current $j_{ij}$ is already defined before in (4.30).
The equation of motion for $A_{0}$ is given by

$\partial_{i}\partial_{j}E_{ij}+\Phi^{*}\Phi=0$ (4.61)

which corresponds to the Gauss law, and the equation of motion of $A_{ij}$ as
given by

$\partial_{0}E_{ij}-\frac{1}{2}\partial_{k}(\partial_{i}B_{kj}+\partial_{j}B_{ki})=-j_{ij}$
(4.62)

The equation of motion of $\Phi$ is same as that of (4.33).

One can compute the canonical momenta from lagrangian (4.59) corresponding to
$A_{0}$ and $A_{ij}$ as,

$\pi_{0}=\frac{\partial\mathcal{L}}{\partial\dot{A}^{0}}=0,\qquad\pi_{ij}=\frac{\partial\mathcal{L}}{\partial\dot{A}^{ij}}=\dot{A}_{ij}+\partial_{i}\partial_{j}A_{0}=E_{ij}$
(4.63)

So the canonical Hamiltonian can be written as

$H_{C}=\frac{1}{2}(E_{ij}^{2}+B_{ij}^{2})+...$ (4.64)

The ellipses denote terms containing the pure matter sector and interactions.
Now let us compute the equation of motion of the tensorial electric field
$E_{ij}$ and magnetic field $B_{ij}$ directly using the canonical Hamiltonian
$H_{c}$ (4.64). So the time derivative of the electric field tensor is given
by

$\displaystyle\partial_{0}E_{ij}(x)$ $\displaystyle=\\{E_{ij},H_{C}\\}=\int
d^{3}y[\\{E_{ij}(x),B_{mn}(y)\\}B_{mn}(y)+\\{E_{ij}(x),A_{kl}(y)j_{kl}(y)\\}]$
$\displaystyle=\frac{1}{2}(\partial_{k}\partial_{i}B_{kj}+\partial_{k}\partial_{j}B_{ki})(x)-j_{ij}(x)$
(4.65)

where we have made use of (4.37). One can identify the right hand side of the
above relation as curl of the tensor field $B_{ij}$. So the above equation
(4.65) represents the counterpart of Faraday’s law for tensorial gauge theory.
Similarly we can derive the time derivative of the magnetic field, given by

$\displaystyle\partial_{0}B_{ij}(x)$ $\displaystyle=\\{B_{ij},H_{C}\\}=\int
d^{3}y\\{B_{ij}(x),E_{mn}(y)\\}E_{mn}(y)$
$\displaystyle=-(\partial_{k}\partial_{i}E_{kj}-\partial_{k}\partial_{j}E_{ki})$
(4.66)

which corresponds to the Ampere’s law studied in ususal Maxwell theory.

Theory with vector gauge fields

We now use the map (4.9) between the rank-2 symmetric tensor gauge field and
usual vector field to write down a a lagrangian in terms of vector fields just
like we did in (2+1) dimensional theory. Then we shall find out the equation
of motions for the vector fields.
Using (4.9) in the lagrangian (4.59) the gauge field lagrangian for (3+1)
dimensional theory is given by

$\mathcal{L}_{0}=\frac{1}{4}\Big{[}2(\partial_{i}\partial_{j}A_{0})^{2}+(\partial_{0}\partial_{i}A_{j})^{2}-4(\partial_{i}\partial_{j}A_{0})(\partial_{i}\partial_{0}A_{j})+(\partial_{i}\partial_{0}A_{j})(\partial_{j}\partial_{0}A_{i})\Big{]}-\frac{1}{8}(\partial^{2}(\partial_{i}A_{j}-\partial_{j}A_{i}))^{2}$
(4.67)

The full lagrangian containing scalar field part and the interaction part will
be given by

$\displaystyle\mathcal{L}=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}-j_{i}A_{i}$

Now we can compute the equation of motion of the fields from this lagrangian.
First, the equation of motion of $A_{0}$ field is given by

$\partial^{2}\partial_{i}E_{i}=-\Phi^{*}\Phi$ (4.68)

which corresponds to the Gauss constraint. Now the equation of motion
corresponding to $A_{i}$ is given by

$\frac{\partial^{2}}{2}\Big{[}\partial_{0}E_{i}-\frac{\partial^{2}}{2}\epsilon_{ijk}\partial_{j}B_{k}\Big{]}+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}=-j_{i}$
(4.69)

One can check that the canonical momenta corresponding to the field $A_{\mu}$
obtained from the lagrangian (4.67) are given by the same expression as (4.46)
as there is no change in the electric field.
The canonical hamiltonian corresponding to the theory is written as

$H_{c}=\frac{1}{2\partial^{4}}(\partial_{i}\pi_{j})^{2}+\frac{1}{8}(\epsilon_{ijk}\partial^{2}B_{k})^{2}+....$
(4.70)

The ellipses denote the terms containing scalar fields and interaction terms.
Using the basic Poission relation between gauge field $A$ and its canonical
momentum $\pi$ given in (4.48), we can write down the equation of motion of
$E_{i}$ from the above Hamiltonian as

$\displaystyle\partial_{0}\pi_{i}(x)=\\{\pi_{i},H_{C}\\}$
$\displaystyle\Rightarrow$
$\displaystyle\frac{\partial^{2}}{2}[\partial_{0}E_{i}-\frac{\partial^{2}}{2}\epsilon_{ikj}\partial_{k}B_{j}]+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}=-j_{i}$
(4.71)

Similarly the equation of motion for the field $B_{i}$ is given by

$\partial_{0}B_{i}(x)=\\{B_{i},H_{C}\\}=-\epsilon_{ijk}(\partial_{j}E_{k}-\partial_{k}E_{j})$
(4.72)

Comparative study

Now to compare the equations of motion in both pictures we must bring them at
the same level. We cannot compare two pictures as the former equations of
motion are of tensor fields whereas the later are equations of motion of
vector fields. So first we find a map between the physical variables (electric
and magnetic field) of both pictures. The physical variables of tensor theory
can be expressed in favour of the Maxwell field $E_{i}$ and $B_{i}$ as

$\displaystyle E_{ij}=$
$\displaystyle\frac{1}{2}(\partial_{i}F_{j0}+\partial_{j}F_{i0})=\frac{1}{2}(\partial_{i}E_{j}+\partial_{j}E_{i});\qquad
E_{i}=-F_{0i}$ (4.73) $\displaystyle B_{ij}=$
$\displaystyle\frac{1}{2}\partial^{2}F_{ij}=\frac{1}{2}\,\epsilon_{ijk}\,\partial^{2}B_{k}$
(4.74)

Using (4.73), we can write from the tensor equation corresponding to $A_{0}$
(4.61) as

$-\Phi^{*}\Phi=\partial_{i}\partial_{j}E_{ij}=\partial^{2}\partial_{i}E_{i}$
(4.75)

which is same as we derived in (4.68). We can also map the equation of motion
of tensor field $A_{ij}$ (4.62) to a vector field equation of motion. From
(4.62) we can write

$\frac{\partial_{j}}{2}[\partial_{0}E_{i}-\frac{\partial^{2}}{2}\epsilon_{ikl}\partial_{k}B_{l}]+\frac{1}{2}\partial_{0}\partial_{i}E_{j}=-j_{ij}$

By acting $\partial_{j}$ operator from left on the above and using
$j_{i}=\partial_{j}j_{ij}$, we reproduce the same equation (4.69).
Now let us take the Ampere’s law for tensor theory (4.66) and use the map
(4.73,4.74) to get

$\partial^{2}\Big{[}\frac{1}{2}\epsilon_{ijk}\partial_{0}B_{k}+(\partial_{i}E_{j}-\partial_{j}E_{i})\Big{]}=0$
(4.76)

By acting $\epsilon_{ijm}$ on the above equation we get

$\partial^{2}\Big{[}\partial_{0}B_{m}+\epsilon_{ijm}(\partial_{i}E_{j}-\partial_{j}E_{i})\Big{]}=0$
(4.77)

the solution of which is the vector equation of motion of $B_{m}$ given by
(4.72) from the vector theory. Now the comparison can be illustrated from the
following chart:

|  Gauge field |  Electric field |  Magnetic field
---|---|---|---
Fields |  $A_{i},\,\,A_{ij}$ |  $E_{i}=\partial_{i}A_{0}-\partial_{0}A_{i},\,\,E_{ij}$ |  $B_{i}=\epsilon_{ijk}\partial_{j}A_{k},\,\,B_{ij}$
Mapping |  $A_{ij}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$ |  $E_{ij}=\frac{1}{2}(\partial_{i}E_{j}+\partial_{j}E_{i})$ |  $B_{ij}=\frac{1}{2}\epsilon_{ijk}\partial^{2}B_{k}$

Table III : Comparison of various fields in (3+1) dimension for linear shift
symmetric theory

|  Gauss law |  Faraday’s law |  Ampere’s law
---|---|---|---
Tensor theory |  $\partial_{i}\partial_{j}E_{ij}+\Phi^{*}\Phi=0$ |  $\partial_{0}E_{ij}-\frac{1}{2}\partial_{k}(\partial_{i}B_{kj}+\partial_{j}B_{ki})=-j_{ij}$ |  $\partial_{0}B_{ij}+\partial_{k}(\partial_{i}E_{kj}-\partial_{j}E_{ki})=0$
Using map (4.73,4.74) in tensor theory |  $\partial^{2}\partial_{i}E_{i}+\Phi^{*}\Phi=0$ |  $\frac{\partial_{j}}{2}[\partial_{0}E_{i}-\frac{\partial^{2}}{2}\epsilon_{ikl}\partial_{k}B_{l}]+\frac{1}{2}\partial_{0}\partial_{i}E_{j}=-j_{ij}$ |  $\partial^{2}[\frac{1}{2}\epsilon_{ijk}\partial_{0}B_{k}+(\partial_{i}E_{j}-\partial_{j}E_{i})]=0$
|  |  Acting $\partial_{j}$ on the above equation |  Acting $\frac{\epsilon_{ijm}}{\partial^{2}}$ on the above equation
Vector theory |  $\partial^{2}\partial_{i}E_{i}+\Phi^{*}\Phi=0$ |  $\frac{\partial^{2}}{2}[\partial_{0}E_{i}-\frac{\partial^{2}}{2}\epsilon_{ikj}\partial_{k}B_{j}]+\frac{1}{2}\partial_{0}\partial_{i}\partial_{j}E_{j}=-j_{i}$ |  $\partial_{0}B_{m}+\epsilon_{ijm}(\partial_{i}E_{j}-\partial_{j}E_{i})=0$

Table IV : Comparison of various equations of motion in (3+1) dimension for
linear shift symmetric theory

Above table should be interpreted in the same way as was described after table
II.

##  5 Dual description of theory with space-dependent quadratic shift symmetry

The interplay between a tensor formulation and the usual vector formulation is
generic, not confined to the linear shift symmetry. Infact it is more general.
In this sub-section we show that the non-standard gauging prescription for a
lagrangian having quadratic space dependent shift symmetry involves a rank-3
tensor gauge field which is either described by the usual minimal prescription
based on a vector gauge field or by a rank-2 tensor field.

Quadratic space dependent shift transformation on the complex scalar field can
be written as,

$\Phi(x)\to e^{iw(x)}\Phi(x)\quad;\quad
w(x)=c+c_{i}x^{i}+\frac{1}{2}c_{ij}x^{i}x^{j},\quad
c,c_{i},c_{ij}=\textrm{constant parameters}$ (5.1)

A specific form for the matter lagrangian having the above symmetry (5.1) is
given by [4]

$\displaystyle\mathcal{L}$
$\displaystyle=u\Big{|}{\Phi^{*}}^{3}\Big{(}\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)\Big{)}\Big{|}^{2}$
$\displaystyle+v\Big{|}\Big{(}\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)\Big{)}\Big{|}^{2}$
$\displaystyle+t\Big{|}\Phi^{2}\partial_{i}\partial_{j}\partial_{j}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{j}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{j}\Phi+\partial_{j}\Phi\partial_{j}\partial_{i}\Phi\partial_{j}\Phi\partial_{i}\partial_{j}\Phi)\Big{|}^{2}-m^{2}\Phi^{*}\Phi$
(5.2)

Now let us concentrate only on the term that yields the $v$ part of the
lagrangian ,

$\mathcal{L}^{v}_{ijk}(\Phi)=\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)$
(5.3)

which was also derived here in (3.11).
Under a general local transformation,

$\Phi\to\Phi^{\prime}=e^{iw(x)}\Phi$ (5.4)

(5.3) transforms as,

$\mathcal{L}^{v}_{ijk}(\Phi)\to\tilde{\mathcal{L}}^{v}_{ijk}(\Phi)=e^{3iw(x)}(\mathcal{L}^{v}_{ijk}(\Phi)+i\Phi^{3}\partial_{i}\partial_{j}\partial_{k}w)$
(5.5)

So (5.3) does not transform covariantly under the local gauge transformation
(5.4). However, it is evident if $w$ depends quadratically on position
$x_{i}$, as given in (5.1), $|\mathcal{L}^{v}_{ijk}|^{2}$ will be invariant.

Under a general local transformation, covariance can be achieved by adding an
extra term in $\mathcal{L}^{v}_{ijk}(\Phi)$ itself, which also transforms
appropriately along with (5.4) under local transformation,

$\mathcal{L}^{v}_{ijk}(\Phi,A_{ijk})=(\mathcal{L}^{v}_{ijk}(\Phi)-i\Phi^{3}A_{ijk})\quad\longrightarrow\quad
e^{3iw(x)}(\mathcal{L}^{v}_{ijk}(\Phi)-i\Phi^{3}A_{ijk})$ (5.6)

where $A_{ijk}$ is a rank-3 symmetric tensor gauge field and should transform
under local transformation as

$A_{ijk}\to A^{\prime}_{ijk}=A_{ijk}+\partial_{i}\partial_{j}\partial_{k}w$
(5.7)

to preserve the covariance of the lagrangian. So the total interacting
lagrangian is now given by

$\mathcal{L}^{v}=|\mathcal{L}^{v}_{ijk}|^{2}-i\Big{(}\Phi^{3}\mathcal{L}_{ijk}^{v}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}A_{ijk}+|\Phi|^{6}A_{ijk}^{2}$
(5.8)

Now at this point we again refer to section-2 where we have shown that for
lagrangian having quadratic shift symmetry, the vector current and the fully
symmetric rank-3 tensor current can be related as (2.68),

$j_{i}=\partial_{j}\partial_{k}j_{ijk}$

Now coupling the vector current with a vector gauge field will give

$A_{i}j_{i}=A_{i}(\partial_{j}\partial_{k}j_{ijk})=\frac{1}{3}\Big{(}\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{k}\partial_{i}A_{j}\Big{)}j_{ijk}=:-A_{ijk}j_{ijk}$
(5.9)

which enables us to identify

$A_{ijk}=-\frac{1}{3}\Big{(}\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{k}\partial_{i}A_{j}\Big{)}$
(5.10)

One important point to note here is that, $A_{ijk}$ can also be recast in
terms of rank-2 tensor gauge fields. The coupling between the rank-2 tensor
current and rank-2 tensor gauge field can alternatively be written as

$A_{ij}j_{ij}=A_{ij}(\partial_{k}j_{ijk})=-\frac{1}{3}(\partial_{k}A_{ij}+\partial_{j}A_{ik}+\partial_{i}A_{kj})j_{ijk}=-A_{ijk}j_{ijk}$
(5.11)

which enables us to identify,

$A_{ijk}=\frac{1}{3}\Big{(}\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij}\Big{)}=-\frac{1}{3}\Big{(}\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{k}\partial_{i}A_{j}\Big{)}$
(5.12)

One can check easily using the gauge transformation rule of vector field
$A_{i}$’s (2.30) or $A_{ij}$ (4.5), that $A_{ijk}$ (5.10,5.12) indeed
transforms as (5.7), which is consistent with our construction.

Now the minimal gauging prescription with $A_{i}$ is a generalisation of that
discussed in the earlier sub-section where a rank-2 tensor field was required.
As happened earlier, we show that the non-standard gauging (5.6) using a
rank-3 tensor field is rephrased in favour of the standard minimal
prescription using a vector field. The first step is to symmetrize the triple
and double derivatives in (5.2) as shown below,

$\displaystyle\mathcal{L}_{ijk}^{v}(\Phi)=$
$\displaystyle\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)$
$\displaystyle=$
$\displaystyle\frac{1}{3}\Phi^{2}(\partial_{i}\partial_{j}\partial_{k}\Phi+\partial_{j}\partial_{k}\partial_{i}\Phi+\partial_{k}\partial_{i}\partial_{j}\Phi)+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi$
$\displaystyle-\frac{1}{2}\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{i}\Phi\partial_{k}\partial_{j}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{j}\Phi\partial_{i}\partial_{k}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi+\partial_{k}\Phi\partial_{j}\partial_{i}\Phi)$
(5.13)

Next, minimal prescription is imposed by replacing ordinary derivatives
$\partial_{i}$ by their covariant expressions $(\partial_{i}+iA_{i})$, where
$A_{i}$ is the usual vector gauge field. Then (5.13) will take the form

$\displaystyle\mathcal{L}_{ijk}^{v}(\Phi,A)=\frac{1}{3}\Phi^{2}\Big{(}(\partial_{i}+iA_{i})(\partial_{j}+iA_{j})(\partial_{k}+iA_{k})\Phi+....\Big{)}+2(\partial_{i}+iA_{i})\Phi(\partial_{j}+iA_{j})\Phi(\partial_{k}+iA_{k})\Phi$
$\displaystyle-\frac{1}{2}\Phi\Big{(}(\partial_{i}+iA_{i})\Phi(\partial_{j}+iA_{j})(\partial_{k}+iA_{k})\Phi+...+(\partial_{k}+iA_{k})\Phi(\partial_{j}+iA_{j})(\partial_{i}+iA_{i})\Phi\Big{)}$
(5.14)

A series of remarkable simplifications reduces this to,

$\mathcal{L}_{ijk}^{v}(\Phi,A)=\mathcal{L}_{ijk}^{v}(\Phi)+\frac{i}{3}\Phi^{3}\Big{(}\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{k}\partial_{i}A_{j}\Big{)}$
(5.15)

So we have reproduced the same result as shown in (5.6) using standard minimal
coupling, if one identifies $A_{ijk}$ as (5.10).

The important point is that the gauging prescription in terms of a tensor
field, given in the literature is exactly reproducible by the standard minimal
prescription $\partial_{i}\,\to\,\partial_{i}+iA_{i}$ with an appropriate
symmetrization of the higher derivatives. On top of that there is an
additional freedom of developing the theory in terms of a rank-2 gauge field.
It is therefore instructive to carry out a detailed analysis of the full
interacting theory in all these formulations. As we show, some nice features
are uncovered.

###  5.1 Gauge coupling from iterative Noether prescription

We now discuss the explicit realization of the currents for quadratic shift
symmetric lagrangians, directly using Noether’s prescription. So let us choose
the $v$ part of the lagrangian (5.2).The conserved currents are defined
analogous to (2.63), which are given by

$\displaystyle J^{i}$
$\displaystyle=wj^{i}-(\partial_{j}w)j^{ij}+(\partial_{j}\partial_{k}w)j^{jki}=\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\Phi)}\delta\Phi+\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\Phi)}\partial_{j}(\delta\Phi)-\Big{(}\partial_{j}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\Phi)}\Big{)}\delta\Phi$
$\displaystyle+\partial_{j}\partial_{k}\Big{(}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\Big{)}\delta\Phi-\partial_{j}\Big{(}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\Big{)}\partial_{k}(\delta\Phi)+\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{j}\partial_{k}(\delta\Phi)+c.c$
(5.16)

Equating the coefficients of $c,c_{i},c_{ij}$ on opening up $w$, we find,

$j^{i}=\partial_{j}j_{ij}=\partial_{j}\partial_{k}j_{ijk}=i\partial_{j}\partial_{k}\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}$
(5.17)

where $\mathcal{L}^{v}_{ijk}(\Phi)$ is defined in (5.3) and
$\mathcal{L}^{v}_{ijk}(\Phi^{*})$ is its complex conjugate.

We now try to adopt the iterative Noether’s prescription to obtain the gauge
coupling for the above theory. According to this scheme the first correction
to the lagrangian is given by,

$\mathcal{L}_{1}=j^{i}A_{i}=i\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}\partial_{k}\partial_{j}A_{i}$
(5.18)

As can be seen from the above expression, the derivative on the matter fields
still persists in $\mathcal{L}^{v}_{ijk}(\Phi)$ and its complex conjugate. So
we expect yet another correction to the lagrangian, which is obtained by first
finding the correction to the current,

$j^{i}=\frac{2}{3}\partial_{k}\partial_{j}\Big{(}|\Phi|^{6}(\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{i}\partial_{k}A_{j})\Big{)}$
(5.19)

where $\mathcal{L}_{1}$ (5.18) is used in (5.16) to get this current. So the
contribution to the lagrangian is now given by,

$\mathcal{L}_{2}=\frac{1}{9}|\Phi|^{6}(\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{i}\partial_{k}A_{j})^{2}$
(5.20)

so that the current is reproduced by (4.19). The iteration method terminates
here as there is no further derivatives on $\Phi$. So the total gauged
lagrangian is given by,

$\displaystyle\mathcal{L}$
$\displaystyle=\Big{|}\mathcal{L}_{ijk}^{v}|^{2}+\frac{i}{3}\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}(\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{i}A_{j}+\partial_{k}\partial_{j}A_{i})$
$\displaystyle+\frac{1}{9}|\Phi|^{6}(\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{i}\partial_{k}A_{j})^{2}$
(5.21)

We can also recast the above analysis in terms of the tensor field defined in
(4.9). Then (5.18) can be re-expressed as,

$\mathcal{L}_{1}=-\frac{i}{3}\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}(\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij})=j_{ij}A_{ij}$
(5.22)

As $\mathcal{L}_{1}$ consists derivatives of $\phi$, we can derive the current
again using $\mathcal{L}_{1}$ in (5.16) and the new (rank-2 tensor) current is
given by

$j_{ij}=-\frac{2}{3}\partial_{k}\Big{(}|\Phi|^{6}(\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij})\Big{)}$
(5.23)

and the new contribution to the lagrangian is now given by

$\mathcal{L}_{2}=\frac{1}{9}|\Phi|^{6}(\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij})^{2}$
(5.24)

so that the current (5.23) is given by the definition (4.24). Hence the final
lagrangian takes the form

$\mathcal{L}=|\mathcal{L}^{v}_{ijk}|^{2}-\frac{i}{3}\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}(\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij})+\frac{1}{9}|\Phi|^{6}(\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij})^{2}$
(5.25)

Now let us proceed to realise the above theory in terms of rank-3 tensor
fields. The lagrangian (5.22) can be re-expressed as

$\mathcal{L}_{1}=-i\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}A_{ijk}=-j_{ijk}A_{ijk}$
(5.26)

where this coupling of $j^{ijk}$ to $A_{ijk}$ is defined in (5.11). From this
lagrangian we get a new rank-3 tensor current as

$j_{ijk}=-2|\Phi|^{6}A_{ijk}$ (5.27)

The contribution to the lagrangian is as follows,

$\mathcal{L}_{2}=|\Phi|^{6}A_{ijk}^{2}$ (5.28)

so that the current (5.27) can be reproduced by

$j^{ijk}=-\frac{\delta}{\delta A_{ijk}}\int\mathcal{L}_{2}$ (5.29)

Thus the total lagrangian can be written as

$\mathcal{L}=|\mathcal{L}^{v}_{ijk}|^{2}-i\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}A_{ijk}+|\Phi|^{6}A_{ijk}^{2}$
(5.30)

Here we have shown that how the theory described by (5.8) in the section-5 can
be derived from a scalar theory coupled with vector gauge fields using the
relationship between various bootstrapped currents or equivalently the various
gauge fields.

###  5.2 Full interacting theory and equations of motion in (3+1) dimension

In this section we shall derive the equations of motion of quadratic shift
symmetric lagrangian in 3+1 dimension and try to establish a connection
between such equations of various gauge fields such as $A_{i},A_{ij},A_{ijk}$
which are interrelated with each other as shown in section-5, although the
relations were established at only algebraic level. With this dynamical
treatment we can establish that the relations (5.10,5.12) shown in previous
sections are robust in nature.

Theory with rank-3 tensor gauge fields
We have already shown in section-5, that for a quadratic shift symmetric
lagrangian the gauging prescription can be carried out with a rank-3 symmetric
tensor gauge field $A_{ijk}$. We shall now make the gauge fields dynamic and
calculate the equations of motion for all the fields. For that let us first
define the electric and magnetic fields as666Note that as both the magnetic
fields of rank-3 and rank-2 tensor theories are tensors of rank-2, we have
used $\tilde{B}$ and $B$ to distinguish. $\tilde{B}$ is magnetic field for
rank-3 theory and $B$ is the magnetic field for rank-2 theory.,

$\displaystyle E_{ijk}$
$\displaystyle=\partial_{0}A_{ijk}+\partial_{i}\partial_{j}\partial_{k}A_{0}$
$\displaystyle\tilde{B}_{ij}$
$\displaystyle=\partial_{k}\partial_{m}(\partial_{i}A_{jkm}-\partial_{j}A_{ikm})$
(5.31)

The pure gauge field lagrangian can be written as

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{1}{2}(E_{ijk}^{2}-\tilde{B}_{ij}^{2})$
$\displaystyle=\frac{1}{2}\Big{[}(\partial_{i}\partial_{j}\partial_{k}A_{0})^{2}+(\partial_{0}A_{ijk})^{2}+2(\partial_{0}A_{ijk})(\partial_{i}\partial_{j}\partial_{k}A_{0})\Big{]}-\frac{1}{2}\Big{[}\partial_{m}\partial_{k}(\partial_{i}A_{jkm}-\partial_{j}A_{ikm})\Big{]}^{2}$
(5.32)

So the total lagrangian is written as

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+\Big{|}\mathcal{L}^{v}_{ijk}(\Phi)-i\Phi^{3}A_{ijk}\Big{|}^{2}$ (5.33)
$\displaystyle=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+\Big{|}\mathcal{L}^{v}_{ijk}(\Phi)\Big{|}^{2}-j_{ijk}A_{ijk}$ (5.34)

where $\mathcal{L}^{v}_{ijk}$ is defined in (5.3) and tensor current $j_{ijk}$
is given by

$j_{ijk}=i\Big{(}\Phi^{3}\mathcal{L}^{v}_{ijk}(\Phi^{*})-{\Phi^{*}}^{3}\mathcal{L}^{v}_{ijk}(\Phi)\Big{)}+|\Phi|^{6}A_{ijk}$
(5.35)

Now the equation of motion of $A_{0},A_{ijk}$ are, respectively, given by

$\displaystyle\partial_{i}\partial_{j}\partial_{k}E_{ijk}$
$\displaystyle=-\Phi^{*}\Phi$ (5.36)
$\displaystyle\partial_{0}E_{ijk}-\frac{1}{3}\partial_{l}\Big{(}\partial_{i}\partial_{j}\tilde{B}_{lk}+\partial_{j}\partial_{k}\tilde{B}_{li}+\partial_{k}\partial_{i}\tilde{B}_{lj}\Big{)}$
$\displaystyle=-j_{ijk}$ (5.37)

The canonical momentum corresponding to $A_{ijk}$ is given by

$\pi_{ijk}=\frac{\partial\mathcal{L}}{\partial(\partial^{i}\partial^{j}\partial^{k}\Phi)}=E_{ijk}$
(5.38)

and the algebra between the basic fields is defined as,

$\displaystyle\\{A_{ijk}(\vec{x}),E_{mnp}(\vec{y})\\}=\frac{1}{6}[\delta_{im}(\delta_{jn}\delta_{kp}+\delta_{jp}\delta_{kn})+\delta_{jn}(\delta_{im}\delta_{kp}+\delta_{ip}\delta_{km})+\delta_{kp}(\delta_{im}\delta_{jn}+\delta_{in}\delta_{jm})]\delta(\vec{x}-\vec{y})$
$\displaystyle\\{A_{ijk}(\vec{x}),A_{mnl}(\vec{y})\\}=0,\qquad\\{\pi_{ijk}(\vec{x}),\pi_{mnl}(\vec{y})\\}=0$
(5.39)

The corresponding canonical Hamiltonian can be written as,

$H_{c}=\frac{1}{2}(E_{ijk}^{2}+\tilde{B}_{ij}^{2})+...$ (5.40)

Using the Poission algebra (5.39) we can find out the Hamiltonian equation of
motion as

$\partial_{0}\tilde{B}_{ij}+\frac{2}{3}\partial_{n}\partial_{k}(\partial_{i}E_{jnk}-\partial_{j}E_{ink})=0$
(5.41)

and the other Hamiltonian equation involving time derivative of $E_{ijk}$ is
same as that of (5.37).

Theory with rank-2 tensor gauge fields

We shall now use the map (5.12) between rank-3 gauge field and rank-2 gauge
field to write down the gauge field lagrangian in terms of rank-2 tensor gauge
fields. Using (5.12) the lagrangian can be recast as

$\mathcal{L}_{0}=(\partial_{i}\partial_{j}\partial_{k}A_{0})^{2}+\frac{1}{3}(\partial_{i}\partial_{0}A_{jk})^{2}+2(\partial_{i}\partial_{0}A_{jk})(\partial_{i}\partial_{j}\partial_{k}A_{0})+\frac{2}{3}(\partial_{i}\partial_{0}A_{jk})(\partial_{j}\partial_{0}A_{ik})-\frac{4}{9}[\partial^{2}\partial_{m}(\partial_{i}A_{jm}-\partial_{j}A_{im})]^{2}$
(5.42)

The electric ($E_{ij}$) and magnetic ($B_{ij}$) fields are given by
(4.57,4.58). The total lagrangian is written as

$\mathcal{L}=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+\Big{|}\mathcal{L}^{v}_{ijk}(\Phi)\Big{|}^{2}-j_{ij}A_{ij}$ (5.43)

where $j_{ij}=\partial_{k}j_{ijk}$. The equations of motion of $A_{0},A_{ij}$
are, respectively, given by

$\displaystyle\partial^{2}\partial_{j}\partial_{k}E_{jk}=-\Phi^{*}\Phi$ (5.44)
$\displaystyle\frac{1}{3}\partial_{0}\Big{(}\partial^{2}E_{ij}+\partial_{i}\partial_{k}E_{jk}+\partial_{j}\partial_{k}E_{ik}\Big{)}+\frac{2}{9}\partial^{4}\partial_{k}(\partial_{i}B_{kj}+\partial_{j}B_{ki})=-j_{ij}$
(5.45)

The canonical momentum conjugate to $A_{ij}$ is,

$\pi_{ij}=\frac{\partial\mathcal{L}}{\partial(\partial^{i}\partial^{j}\Phi)}=\frac{1}{3}(\partial^{2}E_{ij}+\partial_{k}\partial_{i}E_{jk}+\partial_{k}\partial_{j}E_{ik})=\partial_{k}E_{ijk}$
(5.46)

satisfying the following Poission algebra

$\\{A_{ij}(x),\pi_{kl}(y)\\}=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\delta(x-y),\quad\\{A_{ij}(x),A_{kl}(y)\\}=0=\\{\pi_{ij}(x),\pi_{kl}(y)\\}$
(5.47)

The corresponding canonical Hamiltonian is given by

$H_{c}=\frac{1}{2}\Big{(}\frac{1}{\partial^{2}}\partial_{k}\pi_{ij}\Big{)}^{2}+\frac{2}{9}(\partial^{2}B_{ij})^{2}+...$
(5.48)

Using (5.47) one can find out the Hamiltonian equation of motion as

$\partial_{0}B_{ij}-\frac{2}{3}\partial_{k}(\partial_{i}E_{jk}-\partial_{j}E_{ik})=0$
(5.49)

Equation of motion for usual vector gauge fields

Now we shall use the map (5.10) to write down the lagrangian (5.32) in terms
of usual vector fields $A_{i}$. The corresponding electric $E_{i}$ and
magnetic $B_{i}$ fields follow their usual definitions. The gauge lagrangian
is given by

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{1}{2}\Big{[}(\partial_{i}\partial_{j}\partial_{k}A_{0})^{2}+\frac{1}{9}(\partial_{0}\partial_{i}\partial_{j}A_{k}+\partial_{0}\partial_{i}\partial_{k}A_{j}+\partial_{0}\partial_{k}\partial_{j}A_{i})^{2}$
$\displaystyle-\frac{2}{3}\partial_{0}(\partial_{i}\partial_{j}A_{k}+\partial_{i}\partial_{k}A_{j}+\partial_{k}\partial_{j}A_{i})\Big{]}-\frac{1}{18}(\partial^{4}\epsilon_{ijk}\epsilon_{klm}\partial_{l}A_{m})^{2}$
(5.50)

so that the total lagrangian can be written as

$\mathcal{L}=\mathcal{L}_{0}+\frac{i}{2}\Phi^{*}\overleftrightarrow{\partial}_{0}\Phi-\Phi^{*}\Phi
A_{0}+\Big{|}\mathcal{L}^{v}_{ijk}(\Phi)\Big{|}^{2}-j_{i}A_{i}$ (5.51)

The equations of motion of $A_{0},A_{i}$ are given by

$\displaystyle\partial^{4}\partial_{i}E_{i}$ $\displaystyle=-\Phi^{*}\Phi$
(5.52)
$\displaystyle\frac{\partial^{4}}{3}\Big{[}\partial_{0}E_{i}-\frac{\partial^{4}}{3}\epsilon_{ijk}\partial_{j}B_{k}\Big{]}+\frac{2}{3}\partial^{2}\partial_{0}\partial_{i}(\partial.E)$
$\displaystyle=-j_{i}$ (5.53)

The canonical momentum conjugate to $A_{i}$ is given by

$\pi_{i}=\frac{\partial\mathcal{L}}{\partial\dot{A}^{i}}=\frac{1}{3}\partial^{2}(\partial^{2}\delta_{ij}+2\partial_{i}\partial_{j})E_{j}$
(5.54)

which satisfies the algebra

$\\{A_{i}(\vec{x}),\pi_{j}(\vec{y})\\}=\delta_{ij}\delta(\vec{x}-\vec{y}),\quad\\{A_{i}(\vec{x}),A_{j}(\vec{y})\\}=0=\\{\pi_{i}(\vec{x}),\pi(\vec{y})\\}$
(5.55)

while the canonical hamiltonian is obtained by a Legendre transformation from
(5.50),

$H_{c}=\frac{1}{2}\Big{(}\frac{\partial_{j}\partial_{k}\pi_{i}}{\partial^{4}}\Big{)}^{2}+\frac{1}{2}\Big{(}\frac{1}{3}\epsilon_{ijk}\partial^{4}B_{k}\Big{)}^{2}$
(5.56)

which yields the equation of motion as

$\partial_{0}B_{i}+\frac{1}{3}\epsilon_{mni}(\partial_{m}E_{n}-\partial_{n}E_{m})=0$
(5.57)

Comparative study

We now use the relations between various gauge fields and electric fields to
find out the mapping between the equations in various theories discussed
above. First we use the mapping between rank-3 tensor fields to rank-2 tensor
fields given by (5.12). The corresponding mapping between the electric and
magnetic fields are given by

$E_{ijk}=\frac{1}{3}(\partial_{i}E_{jk}+\partial_{j}E_{ik}+\partial_{k}E_{ij})$
(5.58)
$\tilde{B}_{ij}=\partial_{k}\partial_{m}(\partial_{i}A_{jkm}-\partial_{j}A_{ikm})=\frac{2}{3}\partial^{2}\partial_{m}(\partial_{i}A_{jm}-\partial_{j}A_{im})=-\frac{2}{3}\partial^{2}B_{ij}$
(5.59)

Now using (5.58,5.59) in (5.36) and (5.37), we get

$\displaystyle\partial^{2}\partial_{j}\partial_{k}E_{jk}$
$\displaystyle=-\Phi^{*}\Phi$ (5.60)
$\displaystyle\frac{1}{3}\partial_{0}(\partial_{i}E_{jk}+\partial_{j}E_{ik}+\partial_{k}E_{ij})+\frac{2\partial^{2}}{9}\partial_{l}(\partial_{i}\partial_{j}B_{lk}+\partial_{j}\partial_{k}B_{li}+\partial_{k}\partial_{i}B_{lj})$
$\displaystyle=-j_{ijk}$ (5.61)

Acting $\partial_{k}$ on (5.61), we get (5.45) which is the equation of motion
of $E_{ij}$ for rank-2 tensorial theory.
The canonical momenta of the respective theories are given by

$\pi_{ijk}=E_{ijk}=\frac{1}{\partial^{2}}\partial_{k}\pi_{ij}$ (5.62)

Exploiting this result the hamiltonian (5.40) exactly maps to (5.48). Further,
using the maps (5.58,5.59), equation (5.41) can be recast in the following
form

$\partial^{2}\Big{[}\partial_{0}B_{ij}-\frac{2}{3}\partial_{k}(\partial_{i}E_{jk}-\partial_{j}E_{ik})\Big{]}=0$
(5.63)

which is compatible with the Hamiltonian equation for rank-2 tensor fields
given by (5.49).

Now we use the map (5.10) from rank-3 tensor gauge fields to rank-1 vector
gauge fields to write down the following relations between the corresponding
electric and magnetic fields as

$\displaystyle E_{ijk}=$
$\displaystyle\frac{1}{3}(\partial_{i}\partial_{j}E_{k}+\partial_{j}\partial_{k}E_{i}+\partial_{k}\partial_{i}E_{j})$
(5.64) $\displaystyle\tilde{B}_{ij}=$
$\displaystyle-\frac{1}{3}\partial^{4}\epsilon_{ijk}B_{k}$ (5.65)

Using these mapping the lagrangian equations of motion for rank-3 tensor
fields (5.36,5.37) can be mapped to usual vector field equations as

$\displaystyle\partial^{4}(\partial.E)$ $\displaystyle=-\Phi^{*}\Phi$ (5.66)
$\displaystyle\frac{\partial_{0}}{3}[\partial_{i}\partial_{j}E_{k}+\partial_{j}\partial_{k}E_{i}+\partial_{k}\partial_{i}E_{j}]+\frac{\partial^{4}\partial_{l}}{9}[\partial_{i}\partial_{j}\epsilon_{lkm}+\partial_{j}\partial_{k}\epsilon_{lim}+\partial_{k}\partial_{i}\epsilon_{ljm}]B_{m}$
$\displaystyle=-j_{ijk}$ (5.67)

By acting $\partial_{j}\partial_{k}$ on (5.67), we get (5.53), the equation of
motion of $E_{i}$ of vector theory, while the Gauss law is exactly reproduced
(5.53,5.66).
The canonical momenta corresponding to the respective theories can be mapped
as

$\pi_{ijk}=\frac{\partial_{j}\partial_{k}\pi_{i}}{\partial^{4}}$ (5.68)

This relation is used to show that the Hamiltonian (5.40) can exactly be
mapped to the hamiltonian of vector theory (5.56). The Hamiltonian equation of
motion (5.41) is then mapped to the following equation using (5.64,5.65),

$\partial^{2}(\partial_{0}B_{i}+\frac{1}{3}\epsilon_{mni}(\partial_{m}E_{n}-\partial_{n}E_{m}))=0$
(5.69)

the solution of which is exactly (5.57). The comparison can be clearly
understood using the following chart:
Gauge field Electric field Magnetic field Fields $A_{i},\,A_{ij},\,A_{ijk}$
$E_{i}=\partial_{i}A_{0}-\partial_{0}A_{i},\,E_{ij},\,E_{ijk}$
$B_{i}=\epsilon_{ijk}\partial_{j}A_{k},B_{ij},\tilde{B}_{ij}$ Derived maps
$A_{ijk}=\frac{1}{3}\Big{(}\partial_{i}A_{jk}+\partial_{j}A_{ik}+\partial_{k}A_{ij}\Big{)}$
$E_{ijk}=\frac{1}{3}\Big{(}\partial_{i}E_{jk}+\partial_{j}E_{ik}+\partial_{k}E_{ij}\Big{)}$
$\tilde{B}_{ij}=-\frac{2}{3}\partial^{2}B_{ij}$
$A_{ijk}=-\frac{1}{3}\Big{(}\partial_{i}\partial_{j}A_{k}+\partial_{j}\partial_{k}A_{i}+\partial_{k}\partial_{i}A_{j}\Big{)}$
$E_{ijk}=\frac{1}{3}\Big{(}\partial_{i}\partial_{j}E_{k}+\partial_{j}\partial_{k}E_{i}+\partial_{k}\partial_{i}E_{j}\Big{)}$
$\tilde{B}_{ij}=-\frac{1}{3}\partial^{4}\epsilon_{ijk}B_{k}$

Table V : Comparison of various fields in (3+1) dimension for quadratic shift
symmetric theory

The above table shows the relationship of rank-3 tensor fields
($A_{ijk},E_{ijk},\tilde{B}_{ij}$) with rank-2 tensor fields
($A_{ij},E_{ij},B_{ij}$) and with vector fields $(A_{i},E_{i},B_{i})$.
Gauss Law Faraday’s Law Ampere’s Law Rank-3 tensor theory
$\partial_{i}\partial_{j}\partial_{k}E_{ijk}=-\Phi^{*}\Phi$
$\partial_{0}E_{ijk}-\frac{1}{3}\partial_{l}\Big{(}\partial_{i}\partial_{j}\tilde{B}_{lk}+\partial_{j}\partial_{k}\tilde{B}_{li}+\partial_{k}\partial_{i}\tilde{B}_{lj}\Big{)}=-j_{ijk}$
$\partial_{0}\tilde{B}_{ij}+\frac{2}{3}\partial_{n}\partial_{k}(\partial_{i}E_{jnk}-\partial_{j}E_{ink})=0$
Using map (5.58,5.59) in rank-3 theory
$\partial^{2}\partial_{j}\partial_{k}E_{jk}=-\Phi^{*}\Phi$
$\frac{1}{3}\partial_{0}(\partial_{i}E_{jk}+\partial_{j}E_{ik}+\partial_{k}E_{ij})+\frac{2\partial^{2}}{9}\partial_{l}(\partial_{i}\partial_{j}B_{lk}+\partial_{j}\partial_{k}B_{li}+\partial_{k}\partial_{i}B_{lj})=-j_{ijk}$
$\partial^{2}\Big{[}\partial_{0}B_{ij}-\frac{2}{3}\partial_{k}(\partial_{i}E_{jk}-\partial_{j}E_{ik})\Big{]}=0$
Acting $\partial_{k}$ on the above equation Acting $\frac{1}{\partial^{2}}$ on
the above equation Rank-2 tensor theory
$\partial^{2}\partial_{j}\partial_{k}E_{jk}=-\Phi^{*}\Phi$
$\frac{1}{3}\partial_{0}\Big{(}\partial^{2}E_{ij}+\partial_{i}\partial_{k}E_{jk}+\partial_{j}\partial_{k}E_{ik}\Big{)}+\frac{2}{9}\partial^{4}\partial_{k}(\partial_{i}B_{kj}+\partial_{j}B_{ki})=-j_{ij}$
$\partial_{0}B_{ij}-\frac{2}{3}\partial_{k}(\partial_{i}E_{jk}-\partial_{j}E_{ik})=0$
Using map (5.64,5.65) in rank-3 theory
$\partial^{4}\partial_{i}E_{i}=-\Phi^{*}\Phi$
$\frac{\partial_{0}}{3}[\partial_{i}\partial_{j}E_{k}+\partial_{j}\partial_{k}E_{i}+\partial_{k}\partial_{i}E_{j}]+\frac{\partial^{4}\partial_{l}}{9}[\partial_{i}\partial_{j}\epsilon_{lkm}+\partial_{j}\partial_{k}\epsilon_{lim}+\partial_{k}\partial_{i}\epsilon_{ljm}]B_{m}=-j_{ijk}$
$\partial^{2}(\partial_{0}B_{i}+\frac{1}{3}\epsilon_{mni}(\partial_{m}E_{n}-\partial_{n}E_{m}))=0$
Acting $\partial_{j}\partial_{k}$ on above equation Acting
$\frac{1}{\partial^{2}}$ on above equation Vector theory
$\partial^{4}\partial_{i}E_{i}=-\Phi^{*}\Phi$
$\frac{\partial^{4}}{3}\Big{[}\partial_{0}E_{i}-\frac{\partial^{4}}{3}\epsilon_{ijk}\partial_{j}B_{k}\Big{]}+\frac{2}{3}\partial^{2}\partial_{0}\partial_{i}(\partial.E)=-j_{i}$
$\partial_{0}B_{i}+\frac{1}{3}\epsilon_{mni}(\partial_{m}E_{n}-\partial_{n}E_{m})=0$

Table VI : Comparison of various equations of motion in (3+1) dimension for
quadratic shift symmetric theory

The first, fourth and last row represent the equations of motion for theories
with rank-3,rank-2 and vector couplings respectively. It is also shown using
the relations of table V, that the rank-3 tensor equations are mapped to
rank-2 and rank-1 tensor equations. It is either mapped directly (Gauss law)
or by using some appropriate derivative operators (Faraday’s and Ampere’s
law).

##  6 Conclusion

Study of polynomial space dependent shift symmetric theories introduced in
[32, 33], is appealing and is vastly studied recently, especially due to their
n-pole conservation laws apart from usual charge conservation stemming from
standard invariance under global (constant) transformations.
In this paper using a generalised Noether type prescription, we have given a
definition yielding all the n-pole currents. Also, various conservation laws
associated with these currents were derived at one stroke using a master
equation. It was shown that various tensor currents are intertwined with each
other and were related by the action of a tower of spatial derivatives.

Using this result it was possible to establish a duality between tensor gauge
fields with different ranks. Introduction of interactions was the precise
mechanism by which these dual relations were derived (see for example,
discussion around (4.8)). Thus it was possible to write down the interaction
between a gauge field and a current in different ways (using tensors of
different ranks) exploiting the duality relations between the currents and the
gauge fields.
On the other hand, it was shown in [11], that using the iterative Noether
prescription of Deser [30] one can introduce matter-gauge interactions in two
different ways, the reason of which is in the dual picture stated above. In
one case they are introduced by a vector gauge field and the additional terms,
besides the free piece, can be written so that the complete theory is the
standard minimally coupled theory. Alternatively, one can also express the
interactions in terms of a symmetric tensor gauge field recently discovered in
fractonic theories. This method of introducing interactions naturally led to a
nonstandard ‘minimal prescription’ where the complete theory was expressed by
replacing the ordinary double derivatives by additional terms involving tensor
fields which was explicitly derived previously in [11] for linear shift
symmetric theory. Symmetrisation of derivatives plays an important role in
this prescription. We have expanded this possibility and shown that this non-
standard minimal scheme is valid for a general shift symmetric theory. The
calculations also indicate that the non-standard minimal coupling prescription
is obtained from the standard one by using the map that relates the vector
(standard) and tensor (non-standard) gauge fields discussed above.
We have considered in details theory of complex scalars interacting either
with vector or symmetric tensor gauge fields. It is important to mention that
the pure gauge part has two possible options: (a) using the vector gauge
fields (b) using the tensor gauge fields. To start with we considered the
theory in terms of the tensor fields. Using our maps this was also expressed
in favour of the vector fields. Both these theories admit equations of motion
(Gauss law, Ampere’s law and Faraday’s law) which were suitably compared. All
computations were done for linear shift symmetric theories in 2+1 and 3+1
dimensions and quadratic shift symmetric theories in 3+1 dimension.

## Acknowledgement

RB acknowledges support from a (DAE) Raja Ramanna Fellowship (grant no:
1003/(6)/2021/RRF/R$\&$D -II/4031, dated: 20/03/2021). AC acknowledges DST,
India for financial support in the form of fellowship.

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