#  A new scalar electrodynamics for fracton gauge theory

Sk. Moinuddin $\,{}^{\rm a,b}$, Pradip Mukherjee $\,{}^{\rm a,c}$

###### Abstract

A new representation of the scalar electrodynamics is discovered which gives a
more redundant description of electromagnetic theory and suitable to construct
an appropriate matter action which contains two global symmetries . The
symmetries of the model when localized by gauge principle , provides a co-
variant fracton gauge theory.

aDepartment of Physics, Barasat Government College,Barasat, India

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##  1 Introduction

“When I first heard about fractons, I said there is no way this could be true,
because it completely defies my prejudice of how systems behave,” - such was
the reaction of Nathan Seiberg, one of the most important physicists of our
time, about fractons [1]. No doubt he did not stop at this, but made useful
contributions to the development of fracton phase research [2]. Similarly,
another promising physicist and a leader of the research on the fracton gauge
theory , Michael Pretko, in a recent review [3], observed – “It remains an
open problem whether this theory can be rewritten in a manifestly rotationally
invariant way” . These reactions are not surprising as fractons are the most
exotic among the quasi particles — particle-like entities that emerge out of
complicated interactions between many strongly interaccting electrons inside a
material [4]. But fractons are bizarre even compared to other exotic quasi-
particles, because they are totally immobile or able to move only in a limited
way.

The fracton phase of matter is thus, one of the marvels of strongly correlated
electronic motion. Discovered only in the first decade of this century, it has
already made its place in the challenge-list of theoretical physics. Many
important research has been conducted on both gapped and gappless fracton
phases of matter [3][5]. It can be said with confidence that the theory of
fractons has reached a mature stage. Though an isolated fracton has not been
observed experimentally till date , it is believed from the behavior of the
glassy materials that the imprint of fracton has been registered as far back
as 2005 [5] [6]. But there are many gaps still remaining in the understanding
the physics of the fracton phase of matter. A prominent example is the
connection between the gapped and the gapless phases. The belief of many
theoreticians notwithstanding, the theory connecting the gapless phase to the
gapped phase of fracton [5] is still in its infancy . So we will be confined
to the gappless phase where the powerful apparatus of field theory is
applicable [7].

We see that the fractons are distinguished by their very low (often vanishing)
mobility [6] [8] in contrast with the other (quasi) particles. Due to its
immobility fractons are very slow to respond to an applied electric field .
But composites of fractons can move freely. On the theoretical side this
immobility of the fracton in isolated state and mobility in presence of other
fractons is really a challenging phenomenon. Note that the immobility of
fractons can be explained by some special symmetries . The theorists
introduced a phenomenology based on a tensor gauge theory [3]. The tensor
gauge theory is assumed to have dipole moment conservation, in addition to the
well known electric charge conservation. The conservation of dipole moment was
introduced phenomenologically . As we have mentioned earlier , the
breakthrough [7] points to another symmetry. The novelty of Pretko’s theory
consists in the correct understanding that one requires a gauge theory to
generate a conservation law. The algorithm of [7] involves a multiplet of real
scalar fields $(\phi_{1},\hskip 2.84544pt\phi_{2},...)$ which form a
representation of the gauge symmetry of the theory. A combination of the
components were found in such a way that it carries a symmetry $\phi\to
e^{i\alpha(x)}\phi$ . Thus a Lagrangian was constructed with the combinations
as the constituents in a manner so that space dependant rotation symmetry is
ensured.Expanding in an a perturbation series, upto a certain order of the
space depended rotation symmetry , the desired theory with two conserved
charges was obtained. Clearly the algorithm is based on an improvisation
rather than following from some basic premises. . The phenomenology of fracton
matter was thus explained. Additionally, many interesting interactions of
fractons were predicted by the theory in as diverse fields , from elasticity
to gravity [3].

The theory mooted in [7] was thus accepted as the basis of fracton gauge
theory and is being followed till date.

However , in the grandeur of success some disturbing points were overlooked.

  1. 1.

The interaction between the fractons is electromagnetic in nature. This is an
Abelian Lie group with a single gauge . A Hamiltonian analysis of the theory
will generate only one primary first class constraints. The gauge generator is
a one parameter operator [9] . How come two independent gauge symmetries may
be generated here?

  2. 2.

The construction of [7] does not give any idea how the modified matter
Lagrangian originates.

  3. 3.

Perhaps the biggest shortcoming of the theory of [7] is the violation of the
boost symmetry. This would make the gauge symmetry meaningless. The gauge
invariance valid in one inertial frame may be broken in another reference
frame [10].

The gauge theory which we are going to report here is a lineage of the
existing fracton gauge theory. However, the points of departures which are
required is suggested by the above. Our theory is therefore constructed right
from the beginning in a covariant way. Also we chose the fracton matter in a
way where the objections are nonexistent. How this construction is realized
will be unfolded below.

The focus of our paper is to construct an appropriate covariant charged matter
field action which has two different global symmetries. It would be convenient
if we start with a list of the new ideas that have been invoked here. .

  1. 1.

An U(1) gauge theory with usual Lagrangian structure , it is impossible to
have more than one independent gauge generator [11]. Therefore we must
consider more general Lagrangian theory that is the Lagrangian must be of
higher order [12]. In the higher order theories it has been proved that the
number of independent conservation laws may be greater than the number of
independent gauge parameters [13][14].

This is the first new idea which we invoke in our paper.

  2. 2.

The higher order Lagrangian theories are being studied for a long time [15],
but the main problem is the appearances of instability (ostrogradsky
instability ) [15]. So one may ask how such theories may be useful in the
context of real physical calculation.  Now comes the second new idea in our
work .  This may seem astounding that this new idea is borrowed from the
theory of modified gravity in the context of cosmology. To bridge the
violation of general relativity (GR) at very large distances and
simultaneously holding of GR in the solar system distances Dvali-Gabadadze-
Porrati introduce a model (DGP model)[16]. We are not concern about cosmology
here , but just note in a certain limit of DGP model the concept of a new
neutral scalar matter theory was originated [17]. We can treat it as a new
scalar matter , stripping it of all its gravitational origin. This scalar
matter has a higher order Lagrangian with the equations of motion having no
higher order term (such term cancel out as we will see below). This scalar is
known as the Galileons . The point is that the action is invariant under
linear shift

$\pi\to\pi+(a+b_{\mu}x^{\mu})$ (1)

The symmetry (1) is a global symmetry since the transformation parameters are
constants. But the elements of the symmetry group are dependent on position.
This point should be borne in mind for subsequent application.

The second new idea which we invoke here is the Galileons are the basic units
of constructing the charged fracton matter.

How these ideas can be implemented in the context of the fracton gauge theory
and how it complements the existing set of the arts to produce a covariant
gauge theory , will be the content of our report.

After the introductory section we report the construction of the scalar theory
which has both phase rotation symmetry and a linear complex shift symmetry 111
Note that the realization of the complex shift symmetry is a noval idea in the
literature.. The action was established comprehensively from the various
symmetry requirements almost uniquely. Thus we can provide not only a
‘manifestly rotational symmetry ’ but a co-variant one also. This is a
testimony that the theory presented by us reveals a deep connection with the
physics of fracton matter . It can be also appreciated that our action can
easily be generalized for any type of interaction readily . The reason is that
our theory is established on two cardinal concepts of theoretical physics,
namely, covariance and gauge symmetry. This is remarkable because a careful
survey of the literature reveals that not many papers can be found where a
derivation from the first principles is available.

In the next section a comprehensive analysis of the different symmetries have
been discussed. The conserved charges are computed using Noether’s theorem .
The connection between the two symmetries came out automatically and the
corresponding conserved charges are readily identified as the electric charge
and dipole moment. Clearly the existing phenomenology of the fracton matter
followed spontaneously.

From section IV , the applications of the gauge principle to the phase
rotation symmetry, complex shift symmetry and space time symmetries are
computed explicitly . In this process we have resolved various puzzling
aspects. The ‘gauging the symmetry approach’ will be our only tool. We start
with enunciating the precise algorithm [18] in a short review by applying it
to the complex Klein Gordon (K-G) field. The rest of the section will be
devoted to the localization of our model i.e. the un-gauged fracton model. As
a result of localization in addition to the well known Maxwell’s theory with
vector gauge field , we get another theory of a tensor gauge field
$B_{\mu\nu}$. Look how methodically the tensor gauge field appears in our
analysis as a consequence of the symmetries. The properties of this tensor
gauge field has so far been obtained from the fracton phenomenology. In the
next section we give our analysis of the behavior of fracton in external
electromagnetic field and show that the so called generalised Maxwell theory
is another way of representing the electromagnetic field. These results follow
smoothly from our algorithm. Once again we see the efficacy of it .

In section VI, we discuss the localization of the space-time symmetry
(Poincare symmetry). A surprising fallout is the emergence of another second
rank tensor gauge field. Note carefully that in the existing theory the same
tensor gauge field is assumed in both electromagnetic and gravitational
sectors. It is understandable that our theory correctly represent the two
interaction with two different tensor gauge theories . It will indeed be very
difficult to explain how a field can have both gravitational and
electromagnetic interaction. As an extra dividend, we derived the self
interaction of the fracton , a puzzling issue in the present state of art.
Note that though we are working in the backdrop of the classical field theory
, the results obtained in this paper are far more general, because our work
relies on symmetries like Lie groups of symmetries and Poincare symmetries.

##  2 A new scalar electrodynamics

This section is one of the most important parts of our work because it
introduces the construction of a new field theoretic representation of the
$U(1)$ group which would go henceforth by the name“ new scalar electrodynamics
” . To understand different intricacies of the model we divided our discussion
in a number of subsections. In the first subsection would illustrate the
application of the gauge principle to a well known field theory.

###  2.1 Application of the gauge principle illustrated

As declared above we start with the concept of gauge principle and how to use
it by reviewing the deduction of the scalar electromagnetic theory i.e. to
derive the well known model of the usual scalar electrodynamics, which is
given by

$\displaystyle{\mathcal{L}_{kg}}=\partial_{\mu}\phi\partial^{\mu}\phi^{*}-J^{\mu}A_{\mu}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$
(2)

where $F_{\mu\nu}=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})$ , $A_{\mu}$
is the 4 potential and $J^{\mu}$ is the conserved current;
$J^{\mu}=-i(\phi\partial^{\mu}\phi^{*}-\phi^{*}\partial_{\mu}\phi)$ . A
beautiful illustration of the interplay of different symmetries including the
gauge symmetry is revealed in our way towards establishing the Lagrangian (2).
The free Lagrangian ($\partial_{\mu}\phi\partial^{\mu}\phi^{*}$) representing
the matter field has a global symmetry

$\displaystyle\phi$ $\displaystyle\to$ $\displaystyle e^{i\Lambda}\phi$
$\displaystyle\phi^{*}$ $\displaystyle\to$ $\displaystyle e^{-i\Lambda}\phi*$
(3)

Note that here $\partial_{\mu}\phi$ transform in the identical way as the
field $\phi$. Technically this is called covariance . This covariance of the
gradient however is lost if we localize the symmetry that is making $\Lambda$
as function of space and time . Now this lost symmetry can be regained by
defining a covariant derivative
($D_{\mu}\phi=\partial_{\mu}\phi+iA_{\mu}\phi$) and replacing the gradient
everywhere by the covariant derivative. Here $A_{\mu}$ is a one form which
transform appropriately so that $D_{\mu}\phi$ is covariant. $A_{\mu}$ is the
gauge field which is identified with the electromagnetic potential. So using
the gauge principle we get the modified Lagrangian as $D_{\mu}\phi
D^{\mu}\phi^{*}$. Simplification of this gives the Lagrangian (2) without the
last term (dynamical term) that is the form of $\phi$ interacting with
$A_{\mu}$. This procedure is well known and term as gauge principle. To apply
the gauge principle the following points should be noted

  1. 1.

The free theory Lagrangian must have a global symmetry which becomes the
symmetry of the interacting field on localization.

  2. 2.

Matter interacts with the electromagnetic field through the charge. Also there
is one independent gauge field as the symmetry group is a one parameter Lie
group. Remember that the number of gauge symmetry is equal to number of first
class constraint in phase space [19] .

###  2.2 A new field theoretic representation for U(1) symmetry

Now we turn back to our problem of interest, In the introductory remarks it
has been mentioned that one of the most bizarre aspects is the existence of
two conservation laws . This means that there are two primary first class
constraints in the theory. But we know that the fundamental interaction here
is Max-well’s electrodynamics which is U(1) gauge theory that is one parameter
abelian group . From the Hamiltonian point of view this implies that there
should be not more that one primary first class constraint. This again states
that the number of independent conservation laws cannot exceed one. This
apparent paradox actually lies in the representation chosen for the group. In
scalar electrodynamics the usual action contains a Lagrangian which may at-
most depend on $\partial_{\mu}\phi$ . Thus the usual Lagrangian is of first
order, so that equation of motion are maximum second order. Naturally , there
is no scope of such matter theory to have more than one independent
conservation law. This paradox disappears if we allow higher order theories to
enter in our scenario . We know that the relation between the number of
independent conservation laws may be greater than the number of primary first
class constraints. So the new scalar electrodynamics must have a matter sector
which is a theory containing higher order derivatives in th Lagrangean.But
then one may legitimately argue that the higher order theories are known to
have in general higher order equations of motion. These are plagued with
essential instability (ostrogradsky instability) [12], [15]. So it appears
that we are in a fix.

The way-out from this fix is obtained surprisingly from a distant quarter. In
the research on modified gravity in a certain limit of Dvali-Gabadadze-
Porrati (DGP model there emerged a scalar ( Galileon scalar) theory which has
ordinary second order equations of motion in-spite of having higher order
action. In their seminal paper [17] Rattazzi at-all demonstrated that a very
special combination of first and second derivatives of the scalar field may
serve as the corresponding Galileon Lagrangian. In (3+1) dimension there are
five Galileon actions $L_{1}$to $L_{5}$. Of them $L_{1}=\pi$ is trivial and
$L_{2}=\partial_{\mu}\pi\partial^{\mu}\pi$ is the usual Lagrangian . The first
nontrivial Gallileon Lagrangian is

$\displaystyle L_{3}=\Box\pi\partial_{\mu}\pi\partial^{\mu}\pi$ (4)

$L_{4}$ and $L_{5}$ are the other nontrivial Galileon term. Among them $L_{4}$
is given by 222This action is not identical to the usual expression in [17] ,
but one can easily check that they are equivalent.

$\displaystyle
L_{4}=\bigg{(}\partial_{\mu}\pi\partial^{\mu}\partial_{\nu}\pi\partial^{\nu}\partial_{\rho}\pi\partial^{\rho}\pi-\Box\pi\partial_{\nu}\pi\partial^{\rho}\pi\partial^{\nu}\partial_{\rho}\pi\bigg{)}$
(5)

All the nontrivial Galileon Lagrangian satisfy the following properties :

  1. 1.

Despite being higher derivative theories, they are free of Ostrogradsky type
ghosts, since the equation of motion following from Galileons action are
second order.

  2. 2.

The Galileon field is a scalar under Poincare transformations and has a unique
symmetry under the shift,

$\pi\to\pi+a+b_{\mu}x^{\mu}$ (6)

where a and $b_{\mu}$ are real numbers. A common mistake should be avoided
here. The symmetry transformation given above is a global symmetry, in spite
of its $x$ dependence. Note also that we some times indicate the first term
$a$ as a translation and second part $b_{\mu}x^{\mu}$ as shift while some time
we will refer the whole transformation as shift. It will be understood from
the context, which meaning is implied.

Let us note that though the concept of Galileonic theories are very important
for our purpose, there is no question of following it top to bottom . We have
to build an action in terms of a complex scalars. But they have a
correspondence. A shift of the Galileon space corresponds to a complex shift
in the group space of the complex scalars. So armed with the novel scalar
field we may pass to the model building.

###  2.3 Building the model

We can define a complex scalar field using the doublet of Galileon scalar as a
basic elements.

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\pi_{1}+i\pi_{2}$
$\displaystyle\phi^{*}$ $\displaystyle=$ $\displaystyle\pi_{1}-i\pi_{2}$ (7)

where $\pi_{1}$ and $\pi_{2}$ are Galileon scalars . $\phi$ and $\phi^{*}$ are
linear in $\pi_{1}$ and $\pi_{2}$ .

We will show now that just as we may construct the usual scalar
electrodynamics by taking a scalar doublet, both this scalars ($\pi_{1}$ and
$\pi_{2}$) satisfy identical dynamics but no charge. A charged scalar doublet
now allows one to map this doublet in the complex space.

In our usual representation by complex scalar doublet we chose the dynamics of
the system by adding the individual Lagrangian as

$\mathcal{L}=\mathcal{L_{1}}+{\mathcal{L_{2}}}$ (8)

Note however it will be wrong that the above prescription is the only choice .

The new scalar electrodynamics will be built on combination of two elements
from the space of the Galileons. How should the elements combine to meet the
very stringent symmetry requirements for the fracton is definitely an
intriguing question.

We start with enlisting the symmetries that the new action must posses,

  1. a.

The Lagrangian must be real.

  2. b.

It must be Poincare invariant.

  3. c.

It must be symmetric under global phase rotations i.e. we must have scalar
Lagrangian with respect to the transformations $\phi\to e^{i\Lambda}\phi$

  4. d.

It will have shift symmetry with the parameters extended to complex values.

  5. e.

It must be a higher order theory with the equation of motion must be second
order in time derivative of $\phi$ and $\phi^{*}$.

  6. f.

It must be symmetric under the interchange of $\phi$ and $\phi^{*}$ (charge
conjugation symmetry).

We start from a general form

$\displaystyle{\mathcal{L}}={\mathcal{L}}(\phi,\phi^{*},\partial_{\mu}\phi,\partial_{\mu}\phi^{*},\partial_{\mu}\partial^{\nu}\phi,\partial_{\mu}\partial^{\nu}\phi^{*})$
(9)

  1. 1.

Due to condition (b) above $\phi$ and $\phi^{*}$ cannot occur explicitly in
the Lagrangian otherwise the translation in-variance (which is the
inhomogeneous part of the Poincare in-variance) will be lost.

  2. 2.

Again ${\mathcal{L}}$ must be invariant under Lorentz transformation (which is
the homogeneous part of the Poincare in-variance). So all the indices must be
properly contracted .

  3. 3.

To implement phase rotation symmetry as par (c) $\phi$ and $\phi^{*}$ term
must occur in pair.

So we are left with condition (d) and (e) . To tackle this point we look at
the list of Galileon Lagrangian given above. The structure of these Galileons
indicate the process of implementation of the shift symmetry . Read with the
condition (c) above implies that the fourth Galileon Lagrangian ($L_{4}$) is
only relevant in our construction because $L_{3}$ and $L_{5}$ contain odd
number of $\pi$ term. So these two structure fail to satisfy our recruitment
that $\phi$ and $\phi^{*}$ term must occur in pair .

All these condition keep in mind and from the idea of fourth Galileon
Lagrangian we proposed the Lagrangian as ,

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\bigg{[}2\bigg{(}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi\partial^{\nu}\partial_{\rho}\phi^{*}\partial^{\rho}\phi^{*}-\Box\phi\partial_{\nu}\phi\partial^{\rho}\phi^{*}\partial^{\nu}\partial_{\rho}\phi^{*}\bigg{)}+2\bigg{(}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\partial^{\nu}\partial_{\rho}\phi\partial^{\rho}\phi^{*}-$
(10)
$\displaystyle\qquad\qquad\Box\phi^{*}\partial_{\nu}\phi^{*}\partial^{\rho}\phi\partial^{\nu}\partial_{\rho}\phi\bigg{)}+\bigg{(}\partial_{\mu}\phi^{*}\partial^{\mu}\partial_{\nu}\phi\partial^{\nu}\partial_{\rho}\phi\partial^{\rho}\phi^{*}-\Box\phi\partial_{\nu}\phi^{*}\partial^{\rho}\phi^{*}\partial^{\nu}\partial_{\rho}\phi\bigg{)}$
$\displaystyle\qquad\qquad+\bigg{(}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\partial^{\nu}\partial_{\rho}\phi^{*}\partial^{\rho}\phi-\Box\phi^{*}\partial_{\nu}\phi\partial^{\rho}\phi\partial^{\nu}\partial_{\rho}\phi^{*}\bigg{)}\bigg{]}$

See that only mimicking the form of $L_{4}$ (5) does not work because our
Lagrangian must have the symmetry under the interchange of $\phi$ and
$\phi^{*}$. A little reflection shows that this symmetry may be gain by
replicating the form of $L_{4}$ in the indicated manner. Thus the conditions
(a) to (f) uniquely define the fracton Lagrangian (10). We can thus say that
this construction satisfy both co-variance and gauge in-variances. This is
definitely hitherto novel and is marked with the universal generalization that
make physics so interesting.

##  3 Justification of the proposed Lagrangian

So we have actually constructed a space-time invariant and gauge invariant
action for fracton matter. Our demand of the space-time and gauge invariant
Lagrangian is so brilliant and gratifying that it calls for a reverse
verification that the Lagrangian (10) really manifest the symmetry condition
(a) to (f). Thus, for instance the shift symmetry of the model is claimed to
be satisfied by a complex extension of the well known shift symmetry of
theories investigated by many stalwarts [2][17][20][21][22]. It is natural for
the sceptic to think that the claims are vacuous. So we have verified these
issues from different points of view.

So the first task will be to show by explicit calculations from our model the
consequences of all the symmetries. On the top of that we have to show that
the equations of motion do not contain any higher derivative term. For the
clarity of presentation we will again divide our results in subsections.

###  3.1 Equation of motion

The specific Lagrangian (10) which is a function $\phi$ and $\phi^{*}$, of the
fields and there derivatives upto the second order . Note that one will expect
that the equations of motion would contain time derivatives of the fields of
third order. But the Galileon model is so constructed that such terms will
cancel out. The equations of motion are, computed as in the following. The
action is,

$\displaystyle
S=\int{\mathcal{L}}(\phi,\phi^{*},\partial_{\mu}\phi,\partial_{\mu}\phi^{*},\partial_{\mu}\partial^{\nu}\phi,\partial_{\mu}\partial^{\nu}\phi^{*})d^{4}x$

As usual, let there be a variation $\delta\phi$ which vanishes at the initial
and final instants as well as the spatial boundaries (which is usually at
infinite). Hence the equation of motion is,

$\displaystyle\frac{\delta
S}{\delta\phi}=\bigg{[}\frac{\partial{\mathcal{L}}}{\partial\phi}-\partial_{\mu}\bigg{(}\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\phi)}\bigg{)}+\partial_{\mu}\partial^{\nu}\bigg{(}\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\partial^{\nu}\phi)}\bigg{)}\bigg{]}=0$
(11)

Using the Lagrangian (10) , we get the equation of motion of our model as

$\displaystyle\Box{\phi}\bigg{(}\partial^{\nu}\partial_{\rho}\phi^{*}\partial_{\nu}\partial^{\rho}\phi^{*}-\Box{\phi^{*}}\Box{\phi^{*}}\bigg{)}+2\partial_{\mu}\partial^{\nu}\phi\bigg{(}\partial^{\mu}\partial_{\nu}\phi^{*}\Box{\phi^{*}}-\partial^{\mu}\partial_{\rho}\phi^{*}\partial_{\nu}\partial^{\rho}\phi^{*}\bigg{)}=0$
(12)

Similarly the equation of motion for $\phi^{*}$ may be obtained. If we
substitute $\pi_{2}=0$ and $\pi_{1}=\pi$ in the equation (12) , then we get

$\displaystyle\Box{\pi}\bigg{(}\partial^{\nu}\partial_{\rho}\pi\partial_{\nu}\partial^{\rho}\pi-\Box{\pi}\Box{\pi}\bigg{)}+2\partial_{\mu}\partial^{\nu}\pi\bigg{(}\partial^{\mu}\partial_{\nu}\pi\Box{\pi}-\partial^{\mu}\partial_{\rho}\pi\partial_{\nu}\partial^{\rho}\pi\bigg{)}=0$
(13)

This is a remarkable thing that in this limit our equation of motion reduces
in a form which is well known in the Galileon physics [17]. This can be made
more explicit by rearranging terms , we get

$\displaystyle(\Box{\pi})^{3}-3\Box{\pi}(\partial_{\mu}\partial^{\nu}\pi)^{2}+2(\partial_{\mu}\partial^{\nu}\pi)^{3}=0$
(14)

This is the same equation which we see in [17]333 See page 9 , equation (42)
of [17].. Such a neat cycle of logic is indeed commendable. We have contracted
the fracton field by a combination of the Galileons. But then the Lagrangian
for the fracton was derived uniquely from the symmetries of the fracton phase
by standard algorithm . The equation of motion (12) thus obtained is shown to
pass the Galileon equation of motion in the appropriate limit. Thus the theory
proposed here can be considered as the complex continuation of Galileon theory
[17]. As far as we know this is the first occurrence of such extension of
Galileon model in the literature.

###  3.2  Global phase rotation symmetry

As stated about our model (10) is endowed with various symmetries. The
${\mathcal{U}}(1)$ symmetry is obvious. If we substitute $\phi\to
e^{i\Lambda}\phi$ and $\phi^{*}\to e^{-i\Lambda}\phi^{*}$ in (10) then we see
that the Lagrangian is unchanged. Carefully note that this will only be true
when the parameter $\Lambda$ is constant (i.e only for global phase rotation).
Now under infinitesimal phase rotation $\phi$ and $\phi^{*}$ transform as

$\displaystyle\phi$ $\displaystyle\to$
$\displaystyle\phi\bigg{(}1+i\Lambda\bigg{)}$ $\displaystyle\phi^{*}$
$\displaystyle\to$ $\displaystyle\phi^{*}\bigg{(}(1-i\Lambda\bigg{)}$ (15)

putting these value in (10) and after some simple but lengthy calculation we
easily verified that Lagrangian (10) is invariant under infinitesimal phase
rotation (15).

###  3.3  Space time symmetry

Slightly more elaborate is the space time symmetry. Under space time
transformation the coordinates, fields and their derivatives transforms as

$\displaystyle\delta x_{\mu}$ $\displaystyle=$ $\displaystyle\xi_{\mu}$
$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\phi$
$\displaystyle\delta\phi^{*}$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\phi^{*}$
$\displaystyle\delta(\partial_{\mu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\lambda}\phi+\theta_{\mu}{}^{\lambda}\partial_{\lambda}\phi$
$\displaystyle\delta(\partial_{\mu}\phi^{*})$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\lambda}\phi^{*}+\theta_{\mu}{}^{\lambda}\partial_{\lambda}\phi^{*}$
$\displaystyle\delta(\partial_{\mu}\partial_{\nu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\nu}\partial_{\lambda}\phi+\theta_{\mu}{}^{\lambda}\partial_{\nu}\partial_{\lambda}\phi+\theta_{\nu}{}^{\lambda}\partial_{\mu}\partial_{\lambda}\phi$
$\displaystyle\delta(\partial_{\mu}\partial_{\nu}\phi^{*})$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\nu}\partial_{\lambda}\phi^{*}+\theta_{\mu}{}^{\lambda}\partial_{\nu}\partial_{\lambda}\phi^{*}+\theta_{\nu}{}^{\lambda}\partial_{\mu}\partial_{\lambda}\phi^{*}$
(16)

Here , $\xi^{\lambda}=\epsilon^{\lambda}+\theta^{\lambda}{}_{\rho}x^{\rho}$
are the infinitesimal Poincare transformation parameters. Under the
transformation (16) the total change of Lagrangian (10) is

$\displaystyle\Delta{\mathcal{L}}=\delta{\mathcal{L}}+\xi^{\lambda}\partial_{\lambda}{\mathcal{L}}+\partial_{\lambda}\xi^{\lambda}{\mathcal{L}}$
(17)

where $\delta{\mathcal{L}}$ is the form variation of ${\mathcal{L}}$ . Using
the anti-symmetric property of $\theta^{\lambda}{}_{\rho}$ we can easily show
that ,

$\displaystyle\Delta{\mathcal{L}}=0$ (18)

Thus our Lagrangian (10) is symmetric under full Poincare group.

###  3.4  Shift symmetry

Now the most nontrivial question is whether the theory (10) has shift symmetry
or not. Here the Galileon scalars $\pi_{1}$ and $\pi_{2}$ transform as,

$\displaystyle\pi_{1}\to\pi_{1}+a+b_{\mu}x^{\mu}$
$\displaystyle\pi_{2}\to\pi_{2}+a+b_{\mu}x^{\mu}$ (19)

Using the defining relations (7) we obtain

$\displaystyle\phi$ $\displaystyle\to$
$\displaystyle\bigg{(}\phi+(1+i)(a+b_{\mu}x^{\mu})\bigg{)}$
$\displaystyle\phi^{*}$ $\displaystyle\to$
$\displaystyle\bigg{(}\phi^{*}+(1-i)(a+b_{\mu}x^{\mu})\bigg{)}$ (20)

Thus under shift transformation (20) the fields and their derivatives
transforms as

$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle(1+i)(a+b_{\mu}x^{\mu})\hskip
28.45274pt\delta\phi^{*}=(1-i)(a+b_{\mu}x^{\mu})$
$\displaystyle\delta(\partial_{\mu}\phi)$ $\displaystyle=$
$\displaystyle(1+i)b_{\mu}\hskip
56.9055pt\delta(\partial_{\mu}\phi^{*})=(1-i)b_{\mu}$
$\displaystyle\delta(\Box{\phi})$ $\displaystyle=$ $\displaystyle 0\hskip
85.35826pt\delta(\Box{\phi^{*}})=0$ (21)

The variation of (10) due to shift transformation is given by

$\displaystyle{\delta\mathcal{L}}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\bigg{[}\\{b_{\nu}\partial_{\mu}\partial^{\nu}\phi\partial^{\rho}\phi^{*}-b_{\mu}\Box{\phi}\partial^{\rho}\phi^{*}\\}\\{(1+i)\partial_{\rho}\phi^{*}+2(1-i)\partial_{\rho}\phi\\}$
(22)
$\displaystyle\qquad\qquad+\\{b_{\nu}\partial_{\mu}\partial^{\nu}\phi^{*}\partial^{\rho}\phi-
b_{\mu}\Box{\phi^{*}}\partial^{\rho}\phi\\}\\{2(1+i)\partial_{\rho}\phi^{*}+(1-i)\partial_{\rho}\phi\\}\bigg{]}$

We observed that the shift invariance of Lagrangian (10) is at quasi level
because ${\delta\mathcal{L}}$ changes by a partial derivative.

###  3.5 Other symmetries

Right from the beginning of our construction we have worked with variables
that have a well defined transformations under the Poincare group and our
Lagrangian is fully contracted. Hence the space time symmetries are all
satisfied.

Of the discrete symmetries, one may be interested to know whether the charge
conjugation symmetry which exist in electrodynamics is satisfied by our model
(10). The answer to it will be obtained simply by looking at the Lagrangian.

We are now satisfied about the symmetries of the model which are seen to be
identical to the phenomenology of fracton . So the achievement of our action
is at par with the existing fracton gauge theory [3][5]. But the possibilities
of our model exceeds any existing theory because it is both co-variant and
gauge invariant. In this paper we will cite two examples - namely the
electromagnetic interaction and gravitational interaction of the fractons.
This analysis will show the power and efficacy of the Lagrangian (10) in
fracton phases.

Before we discuss the applications it will be appropriate to find the
conservation laws quantitatively. So in the next section we will present these
calculations from our model (10).

##  4 Conservation Laws from different symmetries

The fact that we have a covariant as well as gauge invariant theory of fracton
phase is itself a great news. We will give an elaborate analysis of the
symmetries of our model so that one can understand that why it is obvious that
our model may have two independent gauge symmetries. Also this will explain
these two symmetries as natural outcome of our model. Needless to say this
achievement is provided the strongest reason for considering our works .

We have seen that Galileon charged matter theory has two gauge symmetries, the
first of these is the well known phase rotation symmetry and the other one is
analytic continuation of the shift symmetry. We must see what conservation
laws are indicated by it. A useful procedure is to apply Noether’s theorems.
Due to higher derivatives in the Lagrangian the Noether’s formula will be
appropriately modified. So the conservation law is better to be derived from
the functional derivative with respect to the parameters of transformation.
First let us derive a general formula. This can then be specialized to
different symmetry transformations of (10).

If $\delta\phi$ be the change in $\phi$ then the action changes by,

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\bigg{[}\frac{\partial{\mathcal{L}}}{\partial\phi}\delta\phi+\frac{\partial{\mathcal{L}}}{\partial\phi^{*}}\delta\phi^{*}+\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\phi)}\delta(\partial_{\mu}\phi)+\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\phi^{*})}\delta(\partial_{\mu}\phi^{*})$
$\displaystyle\qquad\qquad+\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\partial^{\nu}\phi)}\delta(\partial_{\mu}\partial^{\nu}\phi)+\frac{\partial{\mathcal{L}}}{\partial(\partial_{\mu}\partial^{\nu}\phi^{*})}\delta(\partial_{\mu}\partial^{\nu}\phi^{*})\bigg{]}d^{4}x$

Note that $\delta\phi$ not necessarily vanish on the boundary. Using equations
of motion we obtain

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\partial_{\sigma}\bigg{[}\frac{\partial{\mathcal{L}}}{\partial(\partial_{\sigma}\phi)}\delta\phi+\frac{\partial{\mathcal{L}}}{\partial(\partial_{\sigma}\phi^{*})}\delta\phi^{*}-\partial^{\lambda}\bigg{(}\frac{\partial{\mathcal{L}}}{\partial(\partial_{\sigma}\partial^{\lambda}\phi)}\bigg{)}\delta\phi$
(23)
$\displaystyle\qquad\qquad-\partial^{\lambda}\bigg{(}\frac{\partial{\mathcal{L}}}{\partial(\partial_{\sigma}\partial^{\lambda}\phi^{*})}\bigg{)}\delta\phi^{*}\bigg{]}d^{4}x$

Equation (23) is the master formula for calculating the conserved currents and
charges. Now, we will specialize for different symmetries. Start with the
shift symmetry . For shift transformation (20) we see ,
$\delta\phi=(1+i)(a+b_{\mu}x^{\mu})$ ,
$\delta\phi^{*}=(1-i)(a+b_{\mu}x^{\mu})$, substituting this variations of
$\phi$ and $\phi^{*}$ in (23) and after some algebra we obtain expressions

$\displaystyle{\theta}^{\sigma}$ $\displaystyle=$ $\displaystyle
3(1+i)(a+b_{\rho}x^{\rho})\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(24)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.$

where $c.c.$ denotes complex conjugation. Thus the current corresponding to
$b_{\mu}$ is given by

$\displaystyle{\theta^{\sigma\rho}}_{b}$ $\displaystyle=$ $\displaystyle
x^{\rho}\bigg{[}3(1+i)\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(25)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

Note that the current is a two indexed quantity . So the corresponding charge
is

$\displaystyle{J^{k}}_{b}$ $\displaystyle=$
$\displaystyle\int{\theta^{0k}}_{b}d^{3}x=\int
x^{k}\bigg{[}3(1+i)\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(26)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

Again the conserved current corresponding to $a$ is given by

$\displaystyle{\theta^{\sigma}}_{a}$ $\displaystyle=$
$\displaystyle\bigg{[}3(1+i)\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(27)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

So the corresponding charge is

$\displaystyle\int{\theta^{0}}_{a}d^{3}x$ $\displaystyle=$
$\displaystyle\int\bigg{[}3(1+i)\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(28)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

Now for infinitesimal phase rotation (15) , $\delta\phi=i\Lambda\phi$ and
$\delta\phi^{*}=-i\Lambda\phi^{*}$, substituting this variations of $\phi$ and
$\phi^{*}$ in (23) and after some algebra we obtain expressions for the
conserved current corresponding to phase rotation symmetry which is given by ,

$\displaystyle{\theta^{\sigma}}_{\Lambda}$ $\displaystyle=$
$\displaystyle\bigg{[}3i\phi\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(29)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

So the corresponding charge is

$\displaystyle\int{\theta^{0}}_{\Lambda}d^{3}x$ $\displaystyle=$
$\displaystyle\int\bigg{[}3i\phi\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(30)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

An important point to be noted that when the symmetries are localized , there
is no difference in the effect of phase rotation and the ‘translation’ part of
shift symmetry 444Similar correspondence was reported earlier [20].. We see
that things may be arranged so that corresponding to a phase rotation there is
a shift translation . Not only that the quantitative relation between the two
parameters can easily be taken out as

$\displaystyle(1+i)a({\bf{x}},t)$ $\displaystyle=$ $\displaystyle
i\Lambda({\bf{x}},t)\phi({\bf{x}},t)$ $\displaystyle(1-i)a({\bf{x}},t)$
$\displaystyle=$ $\displaystyle-i\Lambda({\bf{x}},t)\phi^{*}({\bf{x}},t)$ (31)

If we consider $A=(1+i)a$ ,$A^{*}=(1-i)a$, $\lambda=i\Lambda$ ,
$\lambda^{*}=-i\Lambda$ , $B_{\mu}=(1+i)b_{\mu}$ and
$B_{\mu}^{*}=(1-i)b_{\mu}$ then from (31) we can write ,

$\displaystyle A({\bf{x}},t)$ $\displaystyle=$
$\displaystyle\lambda({\bf{x}},t)\phi({\bf{x}},t)$ $\displaystyle
A^{*}({\bf{x}},t)$ $\displaystyle=$
$\displaystyle\lambda^{*}({\bf{x}},t)\phi^{*}({\bf{x}},t)$ (32)

Now the corresponding conserved current for the phase rotation parameter
($\Lambda$) (29) can be written as ,

$\displaystyle\theta^{\sigma}$ $\displaystyle=$ $\displaystyle\frac{\delta
S}{\delta\lambda}+\frac{\delta
S}{\delta\lambda^{*}}=\bigg{[}3\phi\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(33)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

The charge density is computed by putting $\sigma=0$. So electric charge is

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\bigg{[}3\phi\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(34)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

Naturally this is the charge of the fracton. This is quite like the usual
theory and does not depend on the choice of the Galileons as the basic units.

Similarly for the translation part of shift ($a$) the conserved current (27)
can be written as ,

$\displaystyle{\theta^{\prime\sigma}}$ $\displaystyle=$
$\displaystyle\frac{\delta S}{\delta A}+\frac{\delta S}{\delta
A^{*}}=\bigg{[}3\bigg{(}\partial^{\sigma}\partial_{\mu}\phi\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial^{\sigma}\partial_{\mu}\phi^{*}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(35)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\partial^{\sigma}\phi^{*}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

So the corresponding charge is

$\displaystyle Q^{\prime}$ $\displaystyle=$
$\displaystyle\int\bigg{[}3\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(36)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

obtained by putting $\sigma$ = zero .

We have argued that the symmetry corresponding to the phase rotation is
equivalent to the symmetry corresponding to the shift translation. This
equivalence becomes an equality if we note that, on account of (32),

$\displaystyle\frac{\delta S}{\delta A}$ $\displaystyle=$
$\displaystyle\frac{\delta S}{\delta\lambda}\frac{\delta\lambda}{\delta
A}=\frac{1}{\phi}\frac{\delta S}{\delta\lambda}$ $\displaystyle\frac{\delta
S}{\delta A^{*}}$ $\displaystyle=$ $\displaystyle\frac{\delta
S}{\delta\lambda^{*}}\frac{\delta\lambda^{*}}{\delta
A^{*}}=\frac{1}{\phi^{*}}\frac{\delta S}{\delta\lambda^{*}}$ (37)

thus $Q$ and $Q^{\prime}$ are equal. Also from (26) the conserved charge
belonging to shift parameter ($b_{\mu}$) is given by ,

$\displaystyle J^{k}$ $\displaystyle=$ $\displaystyle\int
x^{k}\bigg{[}3\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(38)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}d^{3}x$

The implication of (38) is indeed gratifying. From (36) we can write

$\displaystyle Q=Q^{\prime}=\int\rho({\bf{x}},t)d^{3}{\bf{x}}$ (39)

where,

$\displaystyle\rho({\bf{x}},t)$ $\displaystyle=$
$\displaystyle\bigg{[}3\bigg{(}\partial_{\mu}\dot{\phi}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi^{*}-\partial^{\mu}\phi^{*}\Box{\phi^{*}}\\}+\partial_{\mu}\dot{\phi^{*}}\\{\partial^{\nu}\phi^{*}\partial_{\nu}\partial^{\mu}\phi-$
(40)
$\displaystyle\qquad\qquad\partial^{\mu}\phi^{*}\Box{\phi}\\}+\dot{\phi^{*}}\\{\Box{\phi^{*}}\Box{\phi}-\partial^{\nu}\partial_{\mu}\phi\partial^{\mu}\partial_{\nu}\phi^{*}\\}\bigg{)}+c.c.\bigg{]}$

its the charge density. From (38) we can now write the second conserved charge
as

$\displaystyle J^{k}=\int x^{k}\rho({\bf{x}},t)d^{3}{\bf{x}}$ (41)

Evidently, the second conserved charge is the dipole moment of the system. As
promised in the beginning, our choice of fracton matter action along with the
application of the gauge principle derives systematically the conserved
quantities.

##  5  Localization of phase rotation symmetry and complex shift symmetry

We will employ the celebrated gauge principle to localize the different
symmetry of our model. It will be convenient to introduce the concept of
“gauge principle” in the familiar context of Maxwell’s electromagnetic theory
. We start with a free theory namely the zero mass Klein-Gordon Lagrangian

$\displaystyle{\mathcal{L}_{kg}}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi^{*}$
(42)

This is symmetric under the global gauge transformation $\phi\to\phi
e^{i\Lambda}$, where $\Lambda$ is a constant. The construction of the un-
gauged theory is most important for localization process . Note that here the
particular transformations of $\phi$ and $\partial_{\mu}\phi$ are such that
the Lagrangian (42) is invariant. Indeed, under the infinitesimal phase
rotation,

$\displaystyle\delta\phi=i\Lambda\phi\hskip
5.69046pt\delta{(\partial_{\mu}\phi)}=-i\Lambda\partial_{\mu}\phi$
$\displaystyle\delta\phi^{*}=i\Lambda\phi^{*}\hskip
5.69046pt\delta{(\partial_{\mu}\phi^{*})}=-i\Lambda\partial_{\mu}\phi^{*}$
(43)

and this is the way the fields and their derivatives should transform so as to
ensure invariance.

When the phase rotation symmetries are localized the parameters of the
transformation are no longer constant but arbitrary functions of space time.
So we consider $\Lambda$ to be a function of $x$ and $t$. Then the
transformation of the field $\delta\phi$ retains it form but derivatives of it
deviates from how they would do earlier. To compensate this change we have to
replace the partial derivatives by

$\displaystyle D_{\mu}{\phi}=(\partial_{\mu}+iA_{\mu})\phi$ (44)

where $A_{\mu}$ is called a gauge field . Its transformation under the local
gauge transformation is assumed so that $\nabla_{\mu}$ transform just as
$\partial_{\mu}$ transform in un - gauged theory (43). Invariance is thus
retained . This is the gauge principle which we see is instrumental to obtain
a Lagrangian invariant under local transformation. It is observed that the
commutator of $\nabla_{\mu}$ and $\nabla_{\nu}$ satisfy

$\displaystyle[D_{\mu},D_{\nu}]=F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
(45)

and the dynamics of the field $A_{\mu}$ is completely given by $F_{\mu\nu}$.
Also it is anti-symmetric over the interchange of $\mu$ and $\nu$. It is then
straight forward to identify the interaction generated by $A_{\mu}$ as the
electromagnetic force . This is nothing new but a well known fact. The gauge
principle is thus a very useful tool in model building. In the following it
will be seen that the gauge principle has similar ubiquity in constructing the
fracton gauge theory 555From what has been stated in the above, one should not
get the impression that the whole electromagnetic theory can be thus
constructed. It is rather the opposite. It took more than 200 years of
experimenting and theorizing to know $U(1)$ is the appropriate symmetry..

In order to understand the dynamics of the fracton phase of matter in the
electromagnetic field, we have to put the system in space and watch its
development in time. Now, if the symmetry elements are uniform i.e the
symmetries are global then there is no interaction. However, if the elements
change from one point to another new gauge fields will appear as we have seen
in the above example. Of the symmetries derived in section 2, the phase
rotation symmetry and shift symmetry are manifested in the flat background
where the theory is formulated . So confining ourselves in flat background the
two internal symmetries should be simultaneously localized because both
transformation takes place in the $\phi$ space. Also at the first glance it
appears that there should be three gauge fields corresponding to the three
parameter ($\Lambda$ , $a$, $b_{\mu}$). However both $\Lambda$ and $a$ produce
the same effects . Hence we have to introduce only one gauge field
corresponding to these two parameter. So the new gauge fields will be two in
number which we are denoted by $A_{\mu}$ and $B_{\mu\nu}$. Thus the co-variant
derivatives is given by

$\displaystyle
D_{\mu}{\phi}=\partial_{\mu}\phi+iA_{\mu}\phi+(1+i)B_{\mu\nu}x^{\nu}\phi$ (46)

Here $A_{\mu}$ is a vector under Lorentz transformations and $B_{\mu\nu}$ is
second rank tensor . A little exercise proves that $B_{\mu\nu}$ may be assumed
symmetric, as is shown in the following .

Let it be a general second rank tensor. Consider a virtual displacement
${\delta x^{\mu}}$ of the system then from (46) the contraction

$\displaystyle\delta x^{\mu}D_{\mu}{\phi}=\delta
x^{\mu}\left(\partial_{\mu}\phi\right)+i\delta x^{\mu}A_{\mu}\phi+(1+i)\delta
x^{\mu}B_{\mu\nu}x^{\nu}\phi$ (47)

Now any general second rank tensor can be split up in anti-symmetric and
symmetric parts

$\displaystyle B_{\mu\nu}=\left[B^{(A)}_{\mu\nu}+B^{(S)}_{\mu\nu}\right]$ (48)

where

$\displaystyle
B^{(A)}_{\mu\nu}=\frac{1}{2}\bigg{(}B_{\mu\nu}-B_{\nu\mu}\bigg{)}$
$\displaystyle
B^{(S)}_{\mu\nu}=\frac{1}{2}\bigg{(}B_{\mu\nu}+B_{\nu\mu}\bigg{)}$ (49)

The left hand side of (47) is interpreted as the change of the field in the
local translation $\delta x^{\mu}$. One or two steps of algebra show that the
anti symmetric part contributes nothing to the factor $\delta
x^{\mu}D_{\mu}{\phi}$. So any anti symmetric part of $B_{\mu\nu}$ is
irrelevant. It implies that $B_{\mu\nu}$ may be assumed symmetric in (46) .

Coming back to the discussion ongoing from the co-variant derivatives (46), we
calculate variations as

$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle\bigg{(}i\Lambda\phi+(1+i)b_{\mu}x^{\mu}\bigg{)}\hskip
28.45274pt\delta\phi^{*}=\bigg{(}-i\Lambda\phi^{*}+(1-i)b_{\mu}x^{\mu}\bigg{)}$
$\displaystyle\delta(\partial_{\mu}\phi)$ $\displaystyle=$
$\displaystyle\bigg{(}i\Lambda\partial_{\mu}\phi+(1+i)b_{\mu}\bigg{)}\hskip
28.45274pt\delta(\partial_{\mu}\phi^{*})=\bigg{(}-i\Lambda\partial_{\mu}\phi^{*}+(1-i)b_{\mu}\bigg{)}$
$\displaystyle\delta(\partial^{\mu}\partial_{\mu}\phi)$ $\displaystyle=$
$\displaystyle\bigg{(}i\Lambda\partial^{\mu}\partial_{\mu}\phi\bigg{)}\hskip
85.35826pt\delta(\partial^{\mu}\partial_{\mu}\phi^{*})=\bigg{(}-i\Lambda\partial^{\mu}\partial_{\mu}\phi^{*}\bigg{)}$
(50)

The above analysis gives us the general methods of gauging the symmetry and we
have listed the variations of the fields and their derivatives under the
global phase rotation symmetry and the complex shift symmetry .

After localization the transformations of the new gauge fields $A_{\mu}$ and
$B_{\mu\nu}$ must ensure that the“new derivative ($D_{\mu}\phi$)” will
transform in the same way as the usual derivatives (50) do in the global
theory. Thus the variation of $D_{\mu}\phi$ must be

$\displaystyle\delta(D_{\mu}\phi)$ $\displaystyle=$
$\displaystyle\bigg{(}i\Lambda D_{\mu}\phi+(1+i)b_{\mu}\bigg{)}$ (51)

But from definition of $D_{\mu}\phi$ (46) we can compute it in terms of the
fields independently which should be given by (51), according to gauge
principle. In that process by equating the real and imaginary part separately
we get two equation involving $\delta A_{\mu}$ and $\delta B_{\mu\nu}$, from
which we get,

$\displaystyle\phi\delta A_{\mu}$ $\displaystyle=$
$\displaystyle-\bigg{(}\phi\partial_{\mu}\Lambda+2b_{\nu}x^{\nu}A_{\mu}+2B_{\mu\nu}b_{\alpha}x^{\alpha}x^{\nu}\bigg{)}$
$\displaystyle\phi\delta B_{\mu\nu}$ $\displaystyle=$
$\displaystyle\bigg{(}A_{\mu}b_{\nu}-\partial_{\mu}b_{\nu}\bigg{)}$ (52)

This conditions (52) also ensure the co-variance of $D_{\mu}\phi^{*}$.

Our complex Galileon model (10) also contains second derivative of $\phi$ and
$\phi^{*}$. The transformations of the second derivatives (50), however, do
not have any shift symmetry contribution . Consequently, the second derivative
$\partial_{\mu}\partial^{\mu}\phi$ should be replaced by
${\bar{D}}_{\mu}D^{\mu}\phi$ .

Where ${\bar{D}}\phi$ is defined as

$\displaystyle{\bar{D_{\mu}}}{\phi}=\partial_{\mu}\phi+iA_{\mu}\phi$ (53)

Explicit calculation shows that under complex shift and phase rotation
transformations ${\bar{D}}_{\mu}D^{\mu}\phi$ transform in the same way as
$\partial_{\mu}\partial^{\mu}\phi$ i.e.

$\delta({\bar{D}}_{\mu}D^{\mu}\phi)=\bigg{(}i\Lambda{\bar{D}}_{\mu}D^{\mu}\phi\bigg{)}$
(54)

In the above we have found out that both the phase rotations and the complex
shift symmetries have appeared in global gauge theory. These symmetries
together define fracton interaction with the applied electromagnetic fields.

##  6  Maxwell’s theory in the fracton representation

In the last section we have seen that the localization process applied to the
phase rotation symmetry and the shift symmetry produces two independent gauge
fields of which $A_{\mu}$ represent the Maxwell’s electrodynamics and another
is $B_{\mu\nu}$ whose physical content is yet to be relieved . This may be the
counter part of the symmetric tensor gauge field in the existing theory. Here
we derived it in full tonsorial form without any ad-hoc approximation. For
instance we have proved that this tensor must be symmetric (see the text below
(46)) whereas in the existing theory it was assumed only.

Proceeding with the localization process as above we get the covariant
derivative (46). Now the commutator of two covariant derivative give the field
tensor from which the dynamics can be obtained. Now the commutator of
$D_{\mu}$ and $D_{\nu}$ are given by

$\displaystyle[D_{\mu},D_{\nu}]=i\bigg{(}\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\bigg{)}+(1+i)(\partial_{\mu}B_{\nu\rho}-\partial_{\nu}B_{\mu\rho})x^{\rho}+(B_{\mu\nu}-B_{\nu\mu})$
(55)

As $B_{\mu\nu}$ is symmetric tensor thus

$\displaystyle[D_{\mu},D_{\nu}]=i\bigg{(}\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\bigg{)}+(1+i)(\partial_{\mu}B_{\nu\rho}-\partial_{\nu}B_{\mu\rho})x^{\rho}$
(56)

To construct the dynamics of the gauge field in our theory we will introduce
$\mathcal{F_{\mu\nu}}$ where $\mathcal{F_{\mu\nu}}$ is given by

$\displaystyle\mathcal{F_{\mu\nu}}=[D_{\mu},D_{\nu}]$ (57)

Using the equation (56), we get

$\displaystyle\mathcal{F_{\mu\nu}}=F_{\mu\nu}+G_{\mu\nu\rho}x^{\rho}$ (58)

Where

$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
i(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})$ $\displaystyle G_{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle(1+i)(\partial_{\mu}B_{\nu\rho}-\partial_{\nu}B_{\mu\rho})$ (59)

From (59) we see that $F_{\mu\nu}$ is anti-symmetric and after some little
calculation we can prove that $G_{\mu\nu\rho}$ is also totally anti-symmetric.
To show this we proceed as below.

If we interchange $\mu$ and $\nu$ then we get

$\displaystyle G_{\nu\mu\rho}$ $\displaystyle=$
$\displaystyle(1+i)(\partial_{\nu}B_{\mu\rho}-\partial_{\mu}B_{\nu\rho})=-G_{\mu\nu\rho}$
(60)

again from the definition of $G_{\nu\mu\rho}$ we get

$\displaystyle G_{\rho\mu\nu}$ $\displaystyle=$
$\displaystyle(1+i)(\partial_{\rho}B_{\mu\nu}-\partial_{\mu}B_{\rho\nu})$
$\displaystyle G_{\nu\rho\mu}$ $\displaystyle=$
$\displaystyle(1+i)(\partial_{\nu}B_{\rho\mu}-\partial_{\rho}B_{\nu\mu})$ (61)

adding these two term we get

$\displaystyle G_{\rho\mu\nu}+G_{\nu\rho\mu}$ $\displaystyle=$
$\displaystyle(1+i)\bigg{[}(\partial_{\rho}B_{\mu\nu}-\partial_{\rho}B_{\nu\mu})-(\partial_{\mu}B_{\rho\nu}-(\partial_{\nu}B_{\rho\mu})\bigg{]}$

as $B_{\mu\nu}$ is a symmetric tensor thus we get

$\displaystyle G_{\rho\mu\nu}+G_{\nu\rho\mu}=G_{\nu\mu\rho}$ (62)

using (60)

$\displaystyle G_{\rho\mu\nu}+G_{\nu\rho\mu}+G_{\mu\nu\rho}=0$ (63)

which can be compactly written as

$\epsilon^{\lambda\mu\nu\rho}G_{\mu\nu\rho}=0$ (64)

Thus $G_{\mu\nu\rho}$ must be a anti-symmetric tensor. Now from (58) we can
say that $\mathcal{F_{\mu\nu}}$ is also a anti-symmetric tensor. We now
defined the dual of $\mathcal{F_{\mu\nu}}$, which is given by

$\displaystyle\tilde{\mathcal{F^{\mu\nu}}}=\epsilon^{\mu\nu\alpha\beta}\mathcal{F_{\alpha\beta}}$
(65)

using (58) the partial derivative of $\tilde{\mathcal{F^{\mu\nu}}}$ can be
written as

$\displaystyle\partial_{\mu}\tilde{\mathcal{F^{\mu\nu}}}=\epsilon^{\mu\nu\alpha\beta}\bigg{[}\partial_{\mu}F_{\alpha\beta}+\partial_{\mu}(G_{\alpha\beta\rho}x^{\rho})\bigg{]}=\epsilon^{\mu\nu\alpha\beta}\bigg{[}\partial_{\mu}F_{\alpha\beta}+\partial_{\mu}(G_{\alpha\beta\rho})x^{\rho}+G_{\alpha\beta\mu}\bigg{]}$
(66)

using the definition of $F_{\alpha\beta}$ and $G_{\alpha\beta\rho}$ and using
(64) we can easily show that

$\displaystyle\partial_{\mu}\tilde{\mathcal{F^{\mu\nu}}}=0$ (67)

Thus in analogy with Maxwell’s theory we can define electric field and
magnetic field as

$\displaystyle\mathcal{E}^{l}$ $\displaystyle=$
$\displaystyle\mathcal{F}^{{l0}}=F^{l0}+G^{l0k}x_{k}=i(\partial^{l}A^{0}-\partial^{0}A^{l})+(1+i)(\partial^{l}B^{0k}-\partial^{0}B^{lk})x_{k}$
$\displaystyle\mathcal{B}^{l}$ $\displaystyle=$
$\displaystyle\tilde{\mathcal{F}^{{l0}}}=\epsilon^{l0mn}\bigg{[}F_{mn}+G_{mnk}x^{k}\bigg{]}=\epsilon^{l0mn}\bigg{[}i(\partial_{m}A_{n}-\partial_{n}A_{m})+$
(68)
$\displaystyle\qquad\qquad(1+i)(\partial_{m}B_{nk}-\partial_{n}B_{mk})x^{k}\bigg{]}$

In terms of field tensor $\mathcal{F}^{{\mu\nu}}$ , two in-homogeneous
Maxwell’s equation can be written as

$\displaystyle\partial_{\mu}\mathcal{F}^{{\mu\nu}}=J^{\nu}$ (69)

While the two homogeneous equations can be written as

$\displaystyle\partial_{\mu}\tilde{\mathcal{F^{\mu\nu}}}=0$ (70)

again as $\mathcal{F}^{{\mu\nu}}$ is anti-symmetric, thus

$\displaystyle\partial_{\nu}\partial_{\mu}\mathcal{F}^{{\mu\nu}}=0$ (71)

thus using (69) we can write

$\displaystyle\partial_{\nu}J^{\nu}=0$ (72)

This is the equation of continuity. We can then write all the relevant
equations of electromagnetism in terms of the matrix ${\mathcal{F}}^{\mu\nu}$
as they are expressed in equation (67) ,(69) . Hence the theory generated by
the tensor gauge field $B_{\mu\nu}$ is nothing but the electromagnetic theory
but expressed in a representation different from the usual one. In this
representation the complex scalar field introduced by us determine the matter
action. So the concept of generalized Maxwell’s theory is a misnomer .
Actually it is the electromagnetic interaction in the different, Galileon
based representation. This difference with the existing theory is not to
contradict but compliment the existing fracton theory.

##  7  Gravitational effect of fractons

There is a conjecture in the existing fracton gauge theory that as fracton
theory is formulated in terms of a second rank symmetric tensor , it may have
a gravitational effect [3] [23]. Interestingly they used the same tensor field
introduced in connection with electromagnetic interaction of fractons ,as
responsible for the emergence of gravity . This is a singular assumption.
Moreover , in combination with the other conclusions it should be looked upon
as a very difficult paradox that whether it suggests a connection between
electromagnetic theory and general theory of relativity or not? It is a
paradox of the existing fracton gauge theory which can never be resolved in
the confines of the same theory.

All these questions and paradoxes associated with the fracton - gravity
interaction in the existing theory are absent in our formulation, based as it
is on an action which has two general principles co-variance and gauge in-
variances. To see how gravitational interaction of the fractons originate in
our theory the following discussion may be helpful. A very quick revision of
certain concepts will be necessary for a proper understanding of our work,
specially for those who are not well versed in the gauge theories of gravity .

Einstein, in the formulation of General relativity (GR) considered space-time
, the arena where massive bodies move in a self consistent way. In other
words, space-time geometry is considered as a dynamical object. Massive bodies
otherwise free from interaction follow a ‘straight line’ (geodesic) in the
space-time. If space time is curved, the geometry is non euclidean. The
curvature of the geometry is obtained from a fourth rank tensor, called the
Riemann tensor ${R^{\gamma}}_{\mu\nu\rho}$. In the Gauge theoretic formulation
of gravity Riemann tensor is defined by the commutator of two covariant
derivative [9]. From the Riemann tensor, we get the Richie tensor

$\displaystyle R_{\mu\nu}={R^{\lambda}}_{\mu\lambda\nu}$ (73)

A nonzero Riemann tensor is a guarantee that the space-time is curved space
time which is a signature of gravity. How the matter interacts with the
gravitational field is given by the Einstein’s equation(in natural units
$c=\hbar=1$) ,

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}$ (74)

where the symmetric energy-moment tensor is defined by $T_{\mu\nu}$

$\displaystyle T_{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta
g^{\mu\nu}}$ (75)

here $S$ is the action of the matter theory. Often the energy momentum tensor
obtained from Noether’s theorem agrees with that obtain from (75). But in many
cases they may differ . This difference can again be exploited to obtained
significant result [24]. We will come to it in some later communication.

The method described so far is the metric approach which is not very suitable
for formulating the gauge theory of gravity [9]. So, the approach to gravity
useful for us are however, vielbein based Poincare gauge theory (PGT) . The
PGT formulates gravity in terms of vielbeins and spin connections. But there
are well defined procedures in the literature which connects the metric with
the vielbein and the affine connection [9] in terms of both vielbeins and spin
connections. The essence of PGT is gauging of symmetries of a Poincare
invariant matter theory [18] , [25], [26]. From the localization of the shift
symmetry of a Poincare invariant theory we get a geometrical manifold
(Einstein-Cartan manifold ) which goes to Riemannian geometry under the
assumption of symmetric connections [27]. This approach is sometimes referred
to as the ‘seventh route to geometrodynamics’ [28].

Now in PGT corresponding to the basic fields $\Sigma^{i}_{\ \mu}$ and
$M^{ij}_{\ \ \mu}$ the Lorentz field strength $R^{ij}_{\ \ \mu\nu}$ and the
translation field strength $T_{i\mu\nu}$ are obtained following the usual
procedure in gauge theory. The commutator of two covariant derivatives gives
[9]

$\displaystyle\left[\nabla_{k},\nabla_{l}\right]\phi=\frac{1}{2}~{}\Sigma_{k}^{\
\mu}\Sigma_{l}^{\ \nu}~{}R^{ij}_{\ \ \mu\nu}\sigma_{ij}\phi-\Sigma_{k}^{\
\mu}\Sigma_{l}^{\ \nu}~{}T^{i}_{\ \mu\nu}\nabla_{i}\phi$ (76)

These defining equations give the following expressions for the field-
strengths

$\displaystyle T^{i}_{\ \mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma^{i}_{\ \nu}+M^{i}_{\ \,k\mu}\Sigma^{k}_{\
\nu}-\partial_{\nu}\Sigma^{i}_{\ \mu}-M^{i}_{\ \,k\nu}\Sigma^{k}_{\ \mu}$
$\displaystyle R^{ij}_{\ \ \mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}M^{ij}_{\ \ \nu}-\partial_{\nu}M^{ij}_{\ \
\mu}+M^{i}_{\ k\mu}M^{kj}_{\ \ \nu}-M^{i}_{\ k\nu}M^{kj}_{\ \ \mu}.$ (77)

and also

$\displaystyle
R^{\rho\sigma}{}_{\mu\nu}=\Sigma_{i}{}^{\rho}\Sigma_{j}{}^{\sigma}R^{ij}{}_{\mu\nu}$
(78)

So far the theory is in the Minkowski space and has been developed as a gauge
theory. From the point of view of geometric interpretation, the Lorentz field
strength $R^{ij}_{\ \ \mu\nu}$ and the translation field strength
$T_{i\mu\nu}$, correspond to the Riemann tensor and the torsion (see [9]).
Using these basic structures, gravity can be formulated in the framework of
PGT. Now for zero torsion (76) becomes

$\displaystyle\left[\nabla_{k},\nabla_{l}\right]\phi=\frac{1}{2}~{}\Sigma_{k}^{\
\mu}\Sigma_{l}^{\ \nu}~{}R^{ij}_{\ \ \mu\nu}\sigma_{ij}\phi$ (79)

But before localizing the space time symmetry we first show that our
Lagrangian (10) is invariant under combined Poincare and shift
transformations. Now under combined Poincare and shift transformation the
fields and their derivative transforms as [27]

$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\phi+(1+i)b_{\mu}x_{\mu}$
$\displaystyle\delta\phi^{*}$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\phi^{*}+(1-i)b_{\mu}x_{\mu}$
$\displaystyle\delta(\partial_{\mu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\lambda}\phi+\theta_{\mu}{}^{\lambda}\partial_{\lambda}\phi+(1+i)b_{\mu}$
$\displaystyle\delta(\partial_{\mu}\phi^{*})$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\lambda}\phi^{*}+\theta_{\mu}{}^{\lambda}\partial_{\lambda}\phi^{*}+(1-i)b_{\mu}$
$\displaystyle\delta(\partial_{\mu}\partial_{\nu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\nu}\partial_{\lambda}\phi+\theta_{\mu}{}^{\lambda}\partial_{\nu}\partial_{\lambda}\phi+\theta_{\nu}{}^{\lambda}\partial_{\mu}\partial_{\lambda}\phi$
$\displaystyle\delta(\partial_{\mu}\partial_{\nu}\phi^{*})$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\mu}\partial_{\nu}\partial_{\lambda}\phi^{*}+\theta_{\mu}{}^{\lambda}\partial_{\nu}\partial_{\lambda}\phi^{*}+\theta_{\nu}{}^{\lambda}\partial_{\mu}\partial_{\lambda}\phi^{*}$
(80)

Here , $\xi^{\lambda}=\epsilon^{\lambda}+\theta^{\lambda}{}_{\rho}x^{\rho}$
are the infinitesimal Poincare transformation parameters. Under the
transformation (80) the total change of Lagrangian (10) is

$\displaystyle\Delta{\mathcal{L}}=\delta{\mathcal{L}}+\xi^{\lambda}\partial_{\lambda}{\mathcal{L}}+\partial_{\lambda}\xi^{\lambda}{\mathcal{L}}$
(81)

where $\delta{\mathcal{L}}$ is the form variation of ${\mathcal{L}}$ . Now
using the variation (80) we easily shows that

$\displaystyle\Delta{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\bigg{[}\\{b_{\nu}\partial_{\mu}\partial^{\nu}\phi\partial^{\rho}\phi^{*}-b_{\mu}\Box{\phi}\partial^{\rho}\phi^{*}\\}\\{(1+i)\partial_{\rho}\phi^{*}+2(1-i)\partial_{\rho}\phi\\}$
(82)
$\displaystyle\qquad\qquad+\\{b_{\nu}\partial_{\mu}\partial^{\nu}\phi^{*}\partial^{\rho}\phi-
b_{\mu}\Box{\phi^{*}}\partial^{\rho}\phi\\}\\{2(1+i)\partial_{\rho}\phi^{*}+(1-i)\partial_{\rho}\phi\\}\bigg{]}$

Thus our Lagrangian (10) is quasi-invariant under combined Poincare and shift
transformation.

Now we are ready to localized the space-time symmetry . In order to localized
Poincare symmetry we have to replace $\partial_{\mu}$ by

$\displaystyle\nabla_{a}\phi=\Sigma_{a}{}^{\mu}D_{\mu}\phi=\Sigma_{a}{}^{\mu}\bigg{(}\partial_{\mu}\phi+\frac{1}{2}M_{\mu}{}^{bc}\sigma_{bc}\phi\bigg{)}$
(83)

where $\sigma_{ab}$ is the Lorentz spin matrix and $\Sigma_{a}{}^{\mu}$ and
$M_{\mu}{}^{bc}$ are the new gauge fields corresponding to translation and
Lorentz transformation 666As $\phi$ is a Lorentz scalar , the gauge fields
corresponding to the Lorentz rotation do not appear . . As The Galileon field
has an additional shift symmetry , so we require to introduce further gauge
field $P_{\mu\nu}$. The new covariant derivatives are now defined as,

$\displaystyle{\bar{\nabla}}_{a}\phi=\Sigma_{a}{}^{\mu}{\bar{D}}_{\mu}\phi$
(84)

where

$\displaystyle{\bar{D}}_{\mu}\phi=\left(D_{\mu}\phi+F_{\mu}\phi\right);\left(F_{\mu}=(1+i)P_{\mu\nu}x^{\nu}\right)$
(85)

The transformations of the new fields are obtained by demanding that the
covariant derivatives (84) transform as the ordinary ones do in the ungauged
theory (80),

$\delta({\bar{\nabla}}_{a}\phi)=-\xi^{c}\partial_{c}({\bar{\nabla}}_{a}\phi)+{\theta_{a}}^{b}{\bar{\nabla}}_{b}\phi+(1+i)b_{a}$
(86)

This yields,

$\displaystyle\delta\Sigma_{a}{}^{\mu}$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\Sigma_{a}{}^{\mu}+\partial_{\lambda}\xi^{\mu}\Sigma_{a}{}^{\lambda}+\theta_{a}{}^{b}\Sigma_{b}{}^{\mu}$
$\displaystyle\delta M_{\mu}{}^{ab}$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}M_{\mu}{}^{ab}-\partial_{\mu}\theta^{ab}-\partial_{\mu}\xi^{\lambda}M_{\mu}{}^{ab}+\theta^{a}{}_{c}M_{\mu}{}^{cb}+\theta^{b}{}_{c}M_{\mu}{}^{ac}$
$\displaystyle\delta\left(x^{\nu}P_{\mu\nu}\phi\right)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}\left(x^{\nu}P_{\mu\nu}\phi\right)-\partial_{\mu}\xi^{\lambda}\left(x^{\nu}P_{\lambda\nu}\phi\right)-x^{\nu}\partial_{\mu}b_{\nu}$
(87)

The Galileon symmetries manifested through these relations are compatible with
analogous results in [27][29] where the algebra of Galileon generators has
been defined. Also note that

$\displaystyle\delta({\bar{D}}_{\mu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}({\bar{D}}_{\mu}\phi)-\partial_{\mu}\xi^{\lambda}{\bar{D}}_{\lambda}\phi+(1+i)b_{\mu}$
$\displaystyle\delta(F_{\mu}\phi)$ $\displaystyle=$
$\displaystyle-\xi^{\lambda}\partial_{\lambda}(F_{\mu}\phi)-\partial_{\mu}\xi^{\lambda}(F_{\lambda}\phi)-(1+i)x^{\nu}\partial_{\mu}b_{\nu}$
(88)

Thus both $F_{\mu}\phi$ and ${\bar{D}}_{\mu}\phi$ transform as four vectors
under local Poincare transformations.

The transformations of the second derivatives, however, do not have any
Galileon contribution (see the last two eq. of (80). Consequently, the second
derivative $\partial_{\mu}\partial_{\nu}\phi$ should be replaced by
$\nabla_{a}{\bar{\nabla}}_{b}\phi$ 777A general discussion of localising the
Poincare sector of a higher derivative theory is given in [30].. Explicit
calculation shows that under local Poincare plus Galileon transformations
$\nabla_{a}{\bar{\nabla}}_{b}\phi$ transform in the same way as
$\partial_{\mu}\partial_{\nu}\phi$ i.e.

$\delta(\nabla_{a}{\bar{\nabla}}_{b}\phi)=-\xi^{d}\partial_{d}(\nabla_{a}{\bar{\nabla}}_{b}\phi)+{\theta_{a}}^{d}(\nabla_{d}{\bar{\nabla}}_{b}\phi)+{\theta_{b}}^{d}(\nabla_{a}{\bar{\nabla}}_{d}\phi)$
(89)

provided,

$\nabla_{a}b_{c}=0$ (90)

This is a natural generalization of the condition $\partial_{a}b_{c}=0$ in
flat space. We can define $F_{a}$ as,

$F_{a}={\Sigma_{a}}^{\mu}F_{\mu}$ (91)

and,

$\nabla_{a}(F_{b}\phi)={\Sigma_{a}}^{\mu}[\partial_{\mu}({\Sigma_{b}}^{\nu}F_{\nu}\phi)+\frac{1}{2}M_{\mu}{}^{cd}\sigma_{cd}{\Sigma_{b}}^{\nu}F_{\nu}\phi]$
(92)

for zero torsion,

$\displaystyle\nabla_{a}{\Sigma_{c}}^{\mu}-\nabla_{c}{\Sigma_{a}}^{\mu}=0$
(93)

thus we find,

$\nabla_{a}(F_{b}\phi)-\nabla_{b}(F_{a}\phi)={\Sigma_{a}}^{\mu}{\Sigma_{b}}^{\nu}[\partial_{\mu}(F_{\nu}\phi)-\partial_{\nu}(F_{\mu}\phi)]$
(94)

The variation under Galileon transformations now yields,

$\displaystyle\delta[\nabla_{a}(F_{b}\phi)-\nabla_{b}(F_{a}\phi)]$
$\displaystyle=(1+i){\Sigma_{a}}^{\mu}{\Sigma_{b}}^{\nu}(\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu})$
$\displaystyle=(1+i)(\nabla_{a}b_{b}-\nabla_{b}b_{a})=0$ (95)

on account of (90). This implies that the choice,

$[\nabla_{a}(F_{b}\phi)-\nabla_{b}(F_{a}\phi)]=0$ (96)

may be consistently implemented.

Now the commutator of ${\bar{\nabla}}_{a}$ and ${\bar{\nabla}}_{b}$ are given
by

$\displaystyle[{\bar{\nabla}}_{a},{\bar{\nabla}}_{b}]\phi=[{\nabla}_{a},{\nabla}_{b}]\phi+[{\nabla}_{a},F_{b}]\phi+[F_{a},{\nabla}_{b}]\phi+[F_{a},F_{b})]\phi$
(97)

After some calculation , using the symmetric property of $P_{\mu\nu}$ and the
condition (96) we get

$\displaystyle[{\bar{\nabla}}_{a},{\bar{\nabla}}_{b}]\phi$ $\displaystyle=$
$\displaystyle\Sigma_{a}{}^{\mu}\Sigma_{b}{}^{\nu}\bigg{[}\frac{1}{2}\sigma_{\alpha\beta}\bigg{(}\partial_{\mu}M_{\nu}{}^{\alpha\beta}-\partial_{\nu}M_{\mu}{}^{\alpha\beta}+M^{\alpha}{}_{\rho\mu}M^{\rho\beta}{}_{\nu}-M^{\alpha}{}_{\rho\nu}M^{\rho\beta}{}_{\mu}\bigg{)}$
(98)
$\displaystyle\qquad\qquad+(1+i)x^{\rho}(\partial_{\mu}P_{\nu\rho}-\partial_{\nu}P_{\mu\rho})$
$\displaystyle\qquad\qquad+\frac{1}{2}(1+i)x^{\rho}\sigma_{cd}(P_{\mu\rho}M_{\nu}{}^{cd}-P_{\nu\rho}M_{\mu}{}^{cd})\bigg{]}\phi$

compearing this with (79) and using (78) we get

$\displaystyle
R_{\mu\nu}{}^{\lambda\sigma}=\Sigma^{\lambda}{}_{\alpha}\Sigma^{\sigma}{}_{\beta}R_{\mu\nu}{}^{\alpha\beta}$
$\displaystyle=$
$\displaystyle\Sigma^{\lambda}{}_{\alpha}\Sigma^{\sigma}{}_{\beta}\bigg{[}(\partial_{\mu}M_{\nu}{}^{\alpha\beta}-\partial_{\nu}M_{\mu}{}^{\alpha\beta}+M^{\alpha}{}_{\rho\mu}M^{\rho\beta}{}_{\nu}-M^{\alpha}{}_{\rho\nu}M^{\rho\beta}{}_{\mu})$
(99)
$\displaystyle\qquad\qquad+2(1+i)x^{\rho}\sigma^{\alpha\beta}(\partial_{\mu}P_{\nu\rho}-\partial_{\nu}P_{\mu\rho})$
$\displaystyle\qquad\qquad+(1+i)x^{\rho}(P_{\mu\rho}M_{\nu}{}^{\alpha\beta}-P_{\nu\rho}M_{\mu}{}^{\alpha\beta})\bigg{]}$

Now from (73) we get the Ricci tensor as,

$\displaystyle R_{\nu\sigma}=R_{\mu\nu}{}^{\mu}{}_{\sigma}$ $\displaystyle=$
$\displaystyle\Sigma^{\mu}{}_{\alpha}\Sigma_{\sigma\beta}\bigg{[}(\partial_{\mu}M_{\nu}{}^{\alpha\beta}-\partial_{\nu}M_{\mu}{}^{\alpha\beta}+M^{\alpha}{}_{\rho\mu}M^{\rho\beta}{}_{\nu}-M^{\alpha}{}_{\rho\nu}M^{\rho\beta}{}_{\mu})$
(100)
$\displaystyle\qquad\qquad+2(1+i)x^{\rho}\sigma^{\alpha\beta}(\partial_{\mu}P_{\nu\rho}-\partial_{\nu}P_{\mu\rho})$
$\displaystyle\qquad\qquad+(1+i)x^{\rho}(P_{\mu\rho}M_{\nu}{}^{\alpha\beta}-P_{\nu\rho}M_{\mu}{}^{\alpha\beta})\bigg{]}$

The equation (100) is to say the least, extraordinary. The implication of the
different terms are easy to recognized . In this connection it may be noted
that the fracton action (10) does not contain any mass term still it is
interacting with gravity . Hence we can conclude that the gravity of fractons
is an emergent one. The second point is obtained from the second term which
shows that there is a self interaction of the fractons as we see if the
external gravity which is contain in the first and the third term put to zero
there is still a non zero curvature . We also see to our satisfaction that
these interactions have nothing to do with the symmetric tensor gauge field
$B_{\mu\nu}$ introduced in connection with the electromagnetic interaction.
The gravitational interaction of the fracton seems to be a deep concept which
requires further investigations.

##  8 Conclusion :

In this paper we have provided a co-varinat and gauge invarient version of the
fracton gauge theory . This construction is built on the existing fracton
gauge theory proposed in [7] . The seminal concepts of the gauge principle and
the importance of matter Lagrangian are the two premises on which the fracton
gauge theory [7] is universally viewed and ours is not an exception . However,
so far the question of space-time symmetries were not much emphasized . In
fact model mooted in [7] is reported to have no boost in-variance. But the
boost invariant is related to the equivalence of inertial frame, a concept
which is so dear in physics that this point must be re-examine . As far as we
have followed the literature of fracton gauge theory we don’t see any major
analysis of this issue , but a gauge invariant theory in a particular inertial
frame of reference may break down in another boosted frame. We have emphasized
the issue of co-variance and gauge in-variance simultaneously in this paper.

Another very important issue is that the fracton matter must be represented by
a Lagrangian field theory which has two conserved current . The physical
interaction here is the electromagnetic interaction which is a one parameter
Abelian gauge theory , will allow only one gauge degree of freedom. It is
surprising that this question was never in the foreground . A related question
is how to construct the Lagrangian of a free fracton matter with two global
gauge symmetries . Answer to this question is hard to get with in the premises
of the existing fracton gauge theory . We have shown in this paper that this
comes quite naturally if we remember that the number of gauge parameters may
be less than the number of conservation laws for a higher order Lagrangian
theory [13][14]. The accompanying theory is essentially unstable as the
equations of motion are of higher order. It is well known that in general the
higher order theories are infested with ostrogradsky instabilities. The way-
out to this problem has been worked out in this paper by the use of a very
special kind of scalar field which in-spite of being a higher order theory has
usual second order equation of motion. The example of such scalar fields came
from the research on modified gravity! This is the Galileon matter , the
properties of which were determine [17] from a certain limit of DGP model
[16]. Though Galileons are a hotbed of intense research all those are in
connection with gravity theory , the use which we made here has no precedent
888That is not to say Galileonic symmetries were unknown in this field .
However the point is no body proposed that the Galileons are the source of
fracton matter..

We have postulated a complex scalar field consisting of the Galileon-like
scalar [17], so that the theory may have a hitherto unknown complex shift
symmetry . This theory is potent enough in explaining many crucial aspects of
the fracton phase of matter which is demonstrated here. Among this the origin
of two conservation laws , appearance of the tensor gauge theory , the
physical content of the“generalized Maxwell’s theory” , the gravitational
interaction of the fractons are included. All these are possible due to the
availability of the co-variant and gauge in-variant model established in this
paper . Also the complex scalar field appearing here is a unique construction
which may have significant role in other front ranking problems of theoretical
physics .999 For instance, during the writing of the paper we are informed
that in connection with an alternative suggestion to quark confinement, the
term fracton was first used [31]. Apparently, the symmetries of their problem
has remarkable similarities with ours. Conclusions based on symmetry are often
seen to be realized.

In short this paper introduces a new way of looking at the fracton gauge
theory which may very important in future research.

##  9 Acknowledgments

Sk. moinuddin would like to thank CSIR India for the fellowship provided to
him.(File no: 08/606(0005)/2019-EMR-I).

Both SM and PM thanks R. Banerjee for useful comments.

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