#### Multipole charge conservation and implications on electromagnetic radiation

HJE
Multipole charge conservation and implications on electromagnetic radiation
Ali Seraj 0 1
0 P. O.Box 19395-5531, Tehran , Iran
1 School of Physics, Institute for Research in Fundamental Sciences , IPM
2 ieA and its current is given by j = i e
It is shown that conserved charges associated with a speci c subclass of gauge symmetries of Maxwell electrodynamics are proportional to the well known electric multipole moments. The symmetries are residual gauge transformations surviving the Lorenz gauge, with nontrivial conserved charge at spatial in nity. These Multipole charges" receive contributions both from the charged matter and electromagnetic elds. The former is nothing but the electric multipole moment of the source. In a stationary con guration, there is a novel equipartition relation between the two contributions. The multipole charge, while conserved, can freely interpolate between the source and the electromagnetic and therefore can be propagated with the radiation. Using the multipole charge conservation, we obtain in nite number of constraints over the radiation produced by the dynamics of charged matter.
Gauge Symmetry; Global Symmetries; Space-Time Symmetries
1 Introduction 3 4
3.1
The nontrivial sector
Stationary con gurations
4.1
Electrostatics
2
Gauge symmetries and conservation laws
2.1
2.2
2.3
Charges in the covariant phase space
Noether current
Residual gauge symmetries and asymptotic symmetries
Residual gauge symmetries of Maxwell theory
4.1.1
4.1.2
Screening e ect and equi-partition relation
Symplectic symmetries
4.2
Magnetostatics
5
Electrodynamics
5.1
5.2
Conservation
A preliminary example
5.3 In nite constraints over radiation
6
Discussion A Algebra of charges in covariant phase space 1 3
constants of motion in the action formulation of particle or eld theories [1]. This can also
be rephrased in the Hamiltonian description where a symmetry can be associated with a
function over the phase space which commutes with the Hamiltonian of the system and
also generates the symmetry transformation through the Poisson bracket. These results
continue to hold in the quantum theory if the symmetry is anomaly free.
Dividing symmetries of a theory into global or local, the above theorem is usually
supposed to be restricted to the former. Local (gauge) symmetries, on the other hand, are
considered in the Noether's second theorem (also discussed in [1]) which states that the
However, simultaneous implementation of both of the above Noether theorems leads
to the association of a conserved charge to a local (gauge) symmetry as well [2{4]. The
key result in this case, is that the charges can be formulated as surface (codimension
2) integrals, instead of volume (codimension 1) integrals. Accordingly, if the
elds drop
fast enough near the boundary, the charges associated with local symmetries would vanish.
While this is usually presumed in quantum
eld theory, it is not the case in many examples
of physical interest. A more relaxed boundary condition on gauge elds, can make the
surface integral charges nonvanishing. However, one should make sure that such relaxation
does not lead to divergent charges. The \large gauge transformations" allowed by the
relaxed boundary conditions having nonvanishing charges, form a closed algebra called the
asymptotic symmetry algebra.
On the other hand, one may put the boundary conditions on gauge invariant quantities
like the eld strength, instead of gauge eld itself. This can be more physical, since the
observable quantities are gauge invariant. Such boundary condition impose no restriction
on the allowed gauge transformations. However, still a subclass of gauge transformations
can be singled out by choosing a gauge condition instead of a boundary condition. While
the gauge xing condition kills most of the gauge redundancies, it allows for residual gauge
transformations respecting a given gauge condition. Such viewpoint was stressed in [5{8].
In this paper, we will follow this approach and show that a subclass of residual gauge
transformations can be associated with nontrivial conserved charges. In the context of
Maxwell electrodynamics, we will show that such conserved quantities have a very nice
interpretation in terms of electric multipole moments.
Multipole moments in electrodynamics are obviously not conserved. For example a
point charge located at origin of space has only monopole moment, while if it starts to
leave the origin, it will obtain dipole and higher moments. However, we will show that if a
\soft multipole charge" is attributed to the electromagnetic eld, the total multipole charge
composed of hard and soft pieces will be a conserved quantity. Interestingly, it turns out
that this charge is nothing but the conserved charge associated with residual symmetries of
electrodynamics. The multipole charge can freely interpolate between the charged matter
and the electromagnetic eld.
The organization of the paper is as follows. In section 2 we derive in a systematic way,
the conserved charges associated with nontrivial gauge symmetries, using the covariant
phase space method. Those who are not interested in the details of the derivation can
easily jump to (2.18) and (2.20).1 In section 3 we determine the residual symmetries as
the physical subset of U(
1
) gauge transformations of Maxwell theory. Then in section 4 we
compute -in an electrostatic con guration- the charges associated with these symmetries
and show our main result relating the charges and electric multipole moments. In section 5
we discuss electrodynamics and show how the above conservation laws put constraints on
the radiation. We conclude in section 6. An appendix is devoted to the Poisson bracket of
charges over the covariant phase space.
1Although in Maxwell theory, these results can also be obtained by the usual Noether's approach, but
of the correct charge [3, 15].
We consider the theory of Maxwell electrodynamics sourced by an arbitrary charged matter
eld. We choose the natural units in which "0 = 0 = c = 1 and the Largrangian takes
L =
1
4
F F
j A + Lmatter;
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
Variation with respect to A leads to the Maxwell eld equations
In the next section, we give a systematic approach to compute the charges associated with
gauge symmetries.
2.1
Charges in the covariant phase space
In order to be able to study the conservation laws associated with gauge symmetries, one
can use the Hamiltonian formulation of gauge theories [9{12] which is well established.
However, this has the drawback that it breaks the manifest covariance of the theory, and
potentially leads to cumbersome expressions. Instead, one can use a pretty mathematical
construction called the \covariant phase space" to study gauge symmetries and associated
conserved charges [2, 3, 13{16] (see also [17] for a review). This is the setup we use in
this paper.
To start, one needs to de ne a symplectic form on the space of eld con gurations.
The symplectic form can then be used to de ne a Poisson bracket between functionals
(of dynamical elds). Moreover, one can associate a Hamiltonian to each gauge symmetry,
which generates that gauge transformation through the Poisson bracket. The on-shell value
of the Hamiltonian will de ne the charge of that gauge symmetry.
According to the action principle, the on-shell variation of the Lagrangian is by
construction a total derivative
L
d ( ) ;
where L is the Lagrangian as a top form, and
stands collectively for all dynamical
elds in the theory (in our case the gauge eld A and the matter eld ). Taking another
antisymmetric variation of
de nes the symplectic current ! (as a d 1 form of spacetime
and a two form over the phase space)
!( ; 1 ; 2 ) = 1 ( 2 )
2 (
1
) :
{ 3 {
The pre-symplectic form
( ; 1 ; 2 ) is de ned through the symplectic current !
Z
or global) is then de ned through
HJEP06(217)8
It is proved [2, 3] that for a gauge transformation in a gauge invariant theory one has
that is the symplectic current contracted with a gauge transformation is necessarily an
exact form. Accordingly
H
=
( ;
;
) :
!( ;
;
) = d k ( ;
) ;
I
Therefore one nds the important result that the conserved charge associated with a gauge
symmetry is given by a co-dimension 2 integral. Meanwhile the conserved charge of a global
symmetry is given by a volume integral. This explains why the electric charge is given by
the Gauss' surface integral. On the other hand it shows why energy and angular momenta
are given by volume integrals in Special Relativity (where Lorentz symmetries are global)
while in General Relativity, where di eomorphisms are local symmetries, similar quantities
are given by surface integrals.
Now let us compute the charges corresponding to gauge symmetries of Maxwell
electrodynamics. For explicit computation, let us take the Lagrangian of scalar QED
L =
1
4
F
F
+ D
(D
) ;
where D
(D
) + c:c. As we will see,
the result is independent of the speci c form of the matter eld, and hence the rest of the
paper is general for any type of electrically charged matter eld.
The theory of scalar QED is invariant under the transformations
(x);
(x) = ie (x) (x) ;
To compute the charges, let us de ne the dual quantities
It can be checked that
= ?( dx );
! = ?(! dx ) :
(
and work in the general case (2.1).
2.2
Noether current
the Noether current
which is a total derivative on-shell
and accordingly conserved
and integrating over variations, one nds the nite charges
We stress again that the charges are written only in terms of the gauge eld and does not
have explicit dependence on matter elds and from now on we forget the Lagrangian (2.10)
One can simply integrate over the variation in the symplectic current (2.15) to arrive at
and hence
(D
)
+ (D
)
+ c:c:
Using the Maxwell equations for the linearized perturbations, we obtain the following
HJEP06(217)8
simple form for the symplectic current
! ( ;
;
F
);
on-shell ;
which con rms the general theorem (2.8). Accordingly the charges are de ned
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
I
I
F
;
Q
=
d
F
(x) :
J
(x)j + F
J
= 0 :
This continuity equation ensures the conservation of charges de ned in (2.18) provided
that there is no ux of current at the boundary of spacetime. We will elaborate more on
this in section 5.
The current (2.19) and the corresponding charge can be decomposed into hard and soft
pieces, which include the contribution of electrically charged elds and the electromagnetic
eld respectively. Explicitly Q
= Q(h) + Q(s), where
(2.22)
{ 5 {
In this paper, we consider the four dimensional at spacetime with the metric
though we expect that the arguments can be generalized to asymptotically at spacetimes
without much e ort. If we take the hypersurface
to be the t = const surface, we can use
the identi cations j = ( ; j), F 0i =
Ei and Fij =
ijkBk, where E; B are the spatial
electric and magnetic elds respectively. Accordingly, the expressions for the charges can
be simpli ed to
Q =
where S can be chosen as a sphere of constant raduis R ! 1. Note that throughout this
paper r without explicit latin index refers to the three dimensional spatial gradient.
2.3
Residual gauge symmetries and asymptotic symmetries
Existence of gauge symmetries in a gauge theory provides a covariant description of the
theory, at the cost of bringing in an in nite redundancy in the system. This redundancy
is then removed through \gauge
xing". However, a speci c class of gauge symmetries
may survive this gauge xing which we call residual gauge symmetries. It turns out that
a subset of residual symmetries, can be \large" near the boundary. It is argued in many
di erent ways that \large gauge symmetries" can play important physical role in di erent
theories. Most famously, in the context of gravity, large gauge transformations provide
basic understanding of holography [10, 18], microscopic counting of black hole entropy [5,
19, 20], and even an identi cation of black hole microstates [21]. In QED and gravity,
they are recently used to prove Weinberg's soft theorems [7, 22{25] and more generally
Low's subleading soft theorems [26{28] and its gravitational counterpart [29, 30]. Also
large gauge transformations are used to describe the so called \edge states" in quantum
Hall e ect [31, 32]. The role of such symmetries in theories with Weyl scaling is still
unclear [33].
We should make a comparison here between the notions of residual gauge symmetry
and the more familiar asymptotic symmetry. An asymptotic symmetry is de ned through a
consistent boundary condition on the gauge elds. Boundary conditions rule out too large
gauge transformations which break the BCs. The remaining ones, are called nontrivial
(trivial) if the associated charge is nonvanishing (vanishing). Asymptotic symmetries are
de ned as the quotient of nontrivial modulo trivial gauge transformations. Through the
Lie bracket, they form the asymptotic symmetry algebra. While many intriguing results
have been obtained from this approach, it has the drawback that the boundary conditions
must be imposed on gauge elds (like A ) and not on gauge invariant quantities (like F
).
However, the physical signi cance of such boundary conditions is not clear, since local
physics involves gauge invariant quantities. The situation is di erent in gravity, since in
that case such conditions can be interpreted as the choice of \observers" at in nity.
{ 6 {
On the other hand, if the boundary conditions are imposed on gauge invariant
quantities, no gauge transformation is ruled out by the boundary conditions. Then the \physical"
gauge symmetries are the nontrivial residual symmetry transformations that respect the
gauge condition. This has the advantage that the form of residual symmetries are
determined all-over the bulk, and not only at the boundary. Unlike the asymptotic symmetries,
the radial dependence of these symmetries can be completely di erent from one to
another. This has interesting implications that we will discuss in our problem. However,
this approach can have its own drawbacks. The form of residual symmetries depend on
the choice of gauge condition which cannot be singled out by physical considerations. Still
some gauge conditions are favored e.g. by requiring causality in the propagation of gauge
eld. We expect that the results must be eventually independent of the gauge condition,
however a general proof of such claim remains an open problem (more comments can be
found in [6]). An interesting relevant result appeared recently in [34], proving that the
independence of classical physics from the choice of gauge condition is equivalent to the
Anti-BRST symmetry of the corresponding quantum theory.
3
Residual gauge symmetries of Maxwell theory
In this section, we determine the residual gauge symmetries of Maxwell theory in Lorenz
gauge. Then we single out the nontrivial sector of these symmetries.
To remove the in nite redundancy in Maxwell theory due to U(
1
) gauge degrees of
freedom, we impose the Lorenz gauge condition
In this gauge, Maxwell equations become wave equations
A = j . However, there are
still residual gauge transformations that respect the Lorentz gauge and therefore are not
killed by the gauge xing. They satisfy the equation
Expanding the time dependence of in Fourier modes
the equation becomes the Helmholtz equation
r A = 0 :
(x) = 0 :
(t; x) = e i!t !(x) ;
(r2 + !2) ! = 0 :
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
There are two qualitatively di erent sets of solutions: those with ! 6= 0 and those with
! = 0. They have drastically di erent behavior at large radial coordinates.2 The general
solution to the wave equation with nonvanishing frequency is
!(x) = X
1
`
X
A`;mh(`+)(kr) + B`;m h(` )(kr) Y`;m( ; ') ;
grow linearly in x.
{ 7 {
in which h(+); h( ) are respectively the spherical Hankel functions of rst and second kind,
which express outgoing and ingoing waves, respectively. The asymptotic expansion of these
functions are given by
h
(` )(x) =
( i)`+1 e ix
x
+ O
1
x2
:
As we will see in the next section, any solution of the form of (3.5) is trivial in the sense that
the corresponding charge is vanishing. Meanwhile, the story is di erent for the vanishing
frequency modes, which satisfy the Laplace equation r
2 (x) = 0 whose solutions are
Note that a minus sign is conventionally absorbed in `;m in order to cancel the minus signs
appearing in (2.24). We will see shortly that while the negative power modes are again
trivial, the positive modes have non-vanishing charge with interesting physical interpretation.
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
3.1
The nontrivial sector
Using (2.24), we can associate a charge to each of the above residual symmetries
I
where the integral is taken over a sphere of constant raduis R !
1. To compute this
integral, we need to specify the asymptotic behavior of r^ E = F r0. To this end, we note
that the scalar r E solves the same equation as (3.2) whose solutions are given by (3.5)
which fallo like O(1=r). Therefore assuming that the sources are localized, the reasonable
boundary condition is
This can be considered as a Neumann boundary condition which allows most of
physically interesting situations including radiating systems.3 It is important to note that this
boundary condition does not impose any restriction over gauge transformations, since F
is gauge invariant. Therefore all of
`;m transformations are allowed. This makes our
approach di erent with the usual asymptotic symmetry group analysis.
3Note that in electromagnetic radiation, only the transverse components of the electric and magnetic
eld fall o as 1=r.
r^ E
1
O( r2 ) :
{ 8 {
the gauge
section 4.1.2.
corresponding to `;m
+
This implies that total electric charge is the charge of constant gauge transformation of
electromagnetic theory. Note that this gauge transformation is special, since it leaves
eld intact, i.e. A ! A + d
= A. We will come back to this point later in
Therefore the conservation of electric charge is a direct consequence of gauge invariance
of the theory. However, this is not the whole information that can be inferred from the
gauge invariance. This is what we show in the next sections by computing the charge
As was promised, we can now show that only the \soft part", i.e. the zero frequency
subset of the residual symmetries lead to nontrivial charges. It is enough to use (3.10)
and (3.7) in (3.9) to arrive at
Q !
1
O( r ) ! 0 :
Similar reasoning implies that the charge of `;m in (3.7) is vanishing. That is, `;m and
! are pure gauge transformations, since their charge cannot be used to label di erent
con gurations of the phase space. In [7], the same conclusion was arrived at, using another
argument based on the notion of adiabatic modes [35].4
On the other hand, `+;m transformations grow badly in large radius and one may expect
that the corresponding charges diverge. However, as we will see, the existence of spherical
harmonics kills all divergent terms and leads to well de ned, physically meaningful charges.
+
The zero mode 0;0 = 41 corresponds to
Note that hereafter, we drop the plus index of Q`+;m, as the minus sector is trivial.
We close this section with two comments:
Algebra of charges. We show in the appendix that the Poisson bracket between the
Hamiltonian generators H of symmetry transformation parametrized by
is given by (for
a more detailed discussion, see [17])
Z
Using the symplectic form if Maxwell theory (2.14), and the transformation rules (2.11),
we nd that the Poisson bracket of charges vanish.
4Also it is interesting to note that, in vacuum, the above pure gauge transformations can be removed
further by the additional gauge xing conditions like A0 = 0 without a ecting the physical residual symmetries.
But this is not what we follow here.
fH 1 ; H 2 g = 0 :
{ 9 {
The charges, being the on-shell value of the Hamiltonian generators, obey the same algebra.
This agrees with the general theorem that the Poisson bracket of charges is a central
extension of the Lie algebra of symmetries up to possibly a central extension [2, 16, 36]
fH 1 ; H 2 g = H[ 1; 2] + C( 1; 2) :
(3.16)
Here the Lie algebra of U(
1
) gauge symmetries is trivial and no central extension arise at
the level of charges.
Physical symmetries in static gauge. One may wonder if the physical symmetries
obtained above crucially depend on the choice of Lorenz gauge or they appear in other
gauge conditions as well. In [37], we will perform a general analysis of the space of vacua of
Yang-Mills theories in the static gauge. Here, we will brie y report some results showing
that the same symmetries appear in the static gauge, and refer the interested reader to the
original reference [37] for details.
obtains the form
Let us
x the static gauge A0 = 0, in which the Lagrangian L = 12 R d3x(Ei2
B2)
i
L =
g(A_ ; A_ )
where g( 1A; 2A) = R d3x 1Ai 2Ai. That is, the Lagrangian induces a positive de nite
metric g on the space of eld con gurations.
The residual gauge symmetries in the static gauge are parametrized by time
independent functions (x). However, analysis of adiabatic motion on the space of vacuum
con gurations reaveals that those residual symmetries whose parameter
vanishes at the
boundary are unphysical (pure) gauge transformations. Let us denote these by
0 and
take the (IR regulated) boundary to be given by a sphere of radius R. It can then be
argued that physical gauge symmetries are those which are orthogonal to all local gauge
transformations with respect to the metric introduced above. Explicitly
(3.17)
(3.18)
(3.19)
whose non-singular solutions coincide with the physical symmetries of Lorenz gauge.
4
Stationary con gurations
In this section, we compute the charges corresponding to nontrivial residual symmetries in
a stationary solution of Maxwell theory. This will provide us a physical interpretation of
these symmetries.
1
2
Z
I
R
r
2
= 0 ;
g( A; 0 A) =
Z
=
da r
0
d3x 0 r
2
= 0 :
The rst term in the last line vanishes as 0 R = 0. Therefore, the orthogonality condition
implies
In this case, B = 0 and we can write E =
whose solution can be expanded as
r . The potential obeys the Laplace equation
The coe cients q`;m are called the \electric multipole moments" which are determined by
the distribution of charged matter as
Hence an electrostatic con guration is completely determined, given the multipole
moments. Note also that higher order moments fall o
more and more rapidly in large
distances. Now let's compute the charges (3.13) an electrostatic con guration of the above
form,
Using (4.1) and the fact that the integral is taken over a sphere at large R we have
Q(`s;m) = Q`;m
Q(`h;m) =
`
2` + 1 q`;m :
5In SI units, the above would read Q`;m = 2``++11 q`";0m where "0 is the vacuum permittivity.
(4.1)
(4.2)
(4.3)
(4.5)
(4.6)
(4.7)
I
R`+2 I
d Y`0;m0 ( ; ') Y`;m( ; ') :
(4.4)
Given the orthogonality of spherical harmonics H d Y`0;m0 ( ; ') Y`;m( ; ') = `;`0 m;m0 , we
conclude our main result
This result implies that the charges associated with physical residual gauge symmetries are
proportional to the electric multipole moments.5 This gives the classical interpretation of
residual symmetries of electrodynamics. Accordingly, we call Q`;m the multipole charge.
The factor 2``++11 is also important. For a better understanding of this factor, let's
compute the hard and soft contributions to the charge Q`;m as de ned in (2.22),
Q(`h;m) =
Z
d3x
`;m =
Z
d3x r`Y`;m( ; ') = q`;m :
Therefore the hard piece of charge exactly reproduces the multipole moment of order (`; m).
However, there is also a contribution from the elds, i.e. the soft piece
As we mentioned, the hard piece is the contribution from matter elds to the charges
Q`;m, while the soft piece is the contribution from the surrounding electromagnetic eld.
For the case ` = 0, which corresponds to the total electric charge, we see from (4.7) that
electromagnectic eld does not carry any electric charge. This is what we expect from a
U(
1
) gauge theory. However, it does carry higher multipole charges.
The minus sign in (4.7) means that in the equilibrium, there is an screening e ect
from the elds reducing the e ective multipole charge Q`;m compared to the bare (hard)
multipole charge Q(`h;m) . The above result can also be written in the suggestive form
which resembles a special \equipartition" relation between the soft and hard pieces of the
multipole charge. Note that for `
1 this approaches
12 . It is tempting to
nd a deeper
understanding of this equipartition relation of multipole charges in equilibrium between
electromagnetic eld and matter source.
4.1.2
Symplectic symmetries
In [38{40] the notion of symplectic symmetry was de ned by the condition that the
symplectic current (2.16) vanishes locally outside sources. Accordingly, using (2.8), (2.9) it can
be easily shown that the charges can be computed at any surface sorrounding the sources,
not only at the boundary [41].
Among the multipole charges, only Q0;0 corresponding to the total electric charge
is precisely symplectic, since according to (2.19) the current vanishes identically
outside sources.
For other multipole charges, the situation is di erent. In the electrostatic case, it can
be checked that although they are not symplectic in the strict sense, but they can still
be computed at any closed sphere containing the source. The reason is that, while the
symplectic current (2.16) or its
nite version (2.20), is not vanishing locally, its volume
integral over regions free of charged matter vanishes, i.e.
Q
S2
Q
S1
=
Z
12
d
J
= 0 ;
(4.8)
(4.9)
where S1;2 are two spheres with di erent radii, both containing the source, and
12 is the
volume enclosed between the two spheres. Therefore the soft piece of multipole charge
only gets nontrivial contribution within the charge horizon [42], i.e. the smallest sphere
containing sources.
However, this does not continue to hold when radiation enters in the game, which
carries nontrivial multipole charges except Q0;0. Therefore, symplectic symmetries appear
in the nondynamical sector of the phase space in accordance with results of [39, 40].
4.2
We showed in previous section that the nontrivial gauge symmetries are associated to
electric multipole charges. These charges are blind to the magnetic eld, and hence cannot
uniquely
x the eld. In order to overcome this de ciency, we need to de ne a new set
of charges that correspond to magnetic multipoles.6 Such charges were introduced in [43]
(see also [30]) through the electic-magnetic duality
~
Q
I
S
d
I
S
"
F
(x) =
da B (x) ;
(4.10)
HJEP06(217)8
where "
is the Levi-Civita symbol, and the gauge parameter (x) is any combination of
solutions of Laplace equation of the form r`Y`;m. In di erential forms language where F is a
two form, the charges (2.18) can be related to the three form Noether current J
= d( ? F )
which is conserved on-shell, while the above charge corresponds to the o -shell conserved
current
In the last equation, we have used the Bianchi identity dF = 0. The charges can accordingly
be written as volume integrals
J = d ( F ) = d ^ F :
Z
~
Q
=
d3xB
r :
We observe that this is similar to the electric result (2.24), but without the hard
contribution. Speci cally, the monopole charge corresponding to
= 1 vanishes. In the following,
we compute the magnetic charge (4.10) for a magnetostatic con guration. In this case,
outside the source, the magnetic eld can be written as B =
r M where [46]
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
The coe cients M`;m are the \magnetic multipole moments" given by
Using (4.13) in (4.10), we nd
M (x) =
M`;m
4
Let us nish this section by noting that since Maxwell theory is linear, one can
superpose arbitrary magnetostatic and electrostatic solutions to obtain a general stationary
solution. The multipole charges (2.24) and (4.10) detect the electric and magnetic
distributions respectively, and are blind to the other. We will use these results in section 5.
6The author is grateful to Jarah Evslin, Temple He, and Shahin Sheikh-Jabbari for useful discussions
on this section.
for `;m symmetries in more detail.
Before studying the charges in electrodynamics, let us discuss the conservation law (2.21)
Let us expand the continuity equation (2.21), using j = ( ; j)
+ E
r ) + r ( j + B
r ) = 0 :
where
is a combination of nontrivial residual symmetries found in section 3. Integrating
this over a t = const hypersurface and using the de nition of hard and soft pieces of
charge (2.22), we obtain
d
dt
(
d
dt
d
dt
Q
=
Q(h) + Q(s)
=
I
da ( j + B
r )
F :
de ne J (h)
gives Q(h). Then multiply (2.2) by
This implies that the time rate of change of multipole charge stored in both the source
and EM
eld equals minus the ux of multipole current at the boundary. This is the
statement of the conservation of multipole charge. Note that the usual expression for
multipole moment i.e. Q(h) is not conserved individually, since the multipole charge can
freely interpolate between electromagnetic eld and charged matter. To see this explicitly,
j , representing only the hard piece of multipole current whose integral
and use the time independence of
to arrive at
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
This gives the transfer rate of multipole charge from the electromagnetic eld to the source,
and makes clear why the hard charge is not conserved.
5.2
A preliminary example
A simple intuition that may stop one to think of multipole moments as conserved charges
is that a point charge with constant velocity has an increasing dipole moment growing with
time. Before studying the real dynamical situations, let us discuss this example. While this
is trivial given the fact that it can be reverted to the electrostatic case through a Lorentz
transformation, it will be illuminating in some aspects.
Consider a particle moving with a constant velocity v along the z direction. Its current
is given by j =
ddxt and
where r0(t) = v t z^ is the position of the point charge. The particle produces electric and
magnetic elds
= q 3 (r
r0(t)) ;
E =
B = v
E ;
E
1 + ( v n)2 3=2 ;
3 qR
1
- 3 qR
R
v
t
HJEP06(217)8
with constant velocity along the z direction. The discontinuities occur when the particle enters and
exits the integration surface.
where E = 4 j
q(rr rr00)j3 ,
dipole charge) is
= (1
v2) 1=2 and n is a unit radial vector from the charge's
present position to the observation point. Now the dipole moment (the hard piece of
Q(1h;0)(t) = q1;0 = qvt ;
jtj <
R
v
;
Z
Z
d3x j r 1;0 =
d3xJz = qv ;
which is linearly growing in time. Meanwhile, according to (5.3), the rate of transfer of
dipole charge between the source and the eld is
which is exactly the time rate of change of (5.7). Therefore the change in dipole moment
of the point charge is due to the absorption of dipole charge from the electromagnetic eld.
The total dipole charge is o by a factor 2=3 as in the previous section due to the soft part
stored in the eld. Therefore the total charge is not constant in time. This is possible only
if there is a ux of dipole charge at the boundary. To see this, assume that the integration
surface is a large sphere at r = R. There is no
ux of charged particles at the boundary
when jtj < R=v. The ux of dipole charge in this period, can be obtained using (5.6)
in (5.2), leading to F1;0 =
23 qv as expected. After a period
ux is 23 qR. At t = R=v, when the particle escapes the integration surface, there is a
sudden outgoing
ux of hard charge by the amount +qR. The di erence is nothing but
the soft charge remaining in the integration surface. As the particle gets farther, the soft
t = R=v, the total incoming
charge within the integration surface decays as
Q1;0(t) = Q(1s;0)(t) =
R3
1
q
3 z(t)2
jtj >
R
v
:
These are summarized in gure 1. { 15 { (5.7) (5.8)
(5.9)
(regions I,III) and radiating in the region II. The charges are computed at two time slices
and
+ before and after the radiation phase, and are given by surface integrals over S
of radius R
where S
. These two surfaces are connected by a timelike hypersurface
B of constant
radius (not drawn). Note that both S
reside in region I as R ! 1.
5.3
In nite constraints over radiation
In this section, we study how the multipole charge conservation constrains the radiation
generated by a dynamical charged system. Throughout this section, we assume that the
dynamics takes place in a region of spacetime with compact support. This is reasonable
since an eternal dynamical radiating system requires an in nite source of energy. Therefore
we consider the matter con guration which is stationary in the region jtj > T and the
dynamics happens in the interval jtj < t0 (t0 can be any given number). We rst prove
that the charge is conserved in time, and then show how this conservation laws impose
in nite number of constraints over the radiation produced by the source. Figure 1 is a
schematic picture of the problem.
As we mentioned before, associated with any residual symmetry
`;m = r`Y`;m, there
is a charge Q`;m(t; R) computed at a sphere of radius R at a constant time slice t. Further
as before, we assume that R ! 1. Now let us compute the charge at two di erent times
t =
T where T > t0, that is before and after the dynamics of the source. As mentioned
before, the charge at t =
T is given by an integral over its boundary S . The important
point is that since R is taken to in nity i.e. R
cT , both S
fall in the region A of the
spacetime. Using this we can show that Q+ = Q . The reason is that
Q(+)
Q
( ) =
Z
B
F ;
(5.10)
where
B is the timelike boundary between S ; S+ and the
However, the
ux is zero since no radiation can reach
ux F is given by (5.2).
B as R ! 1, as it is clear from
gure 2. Note that we have assumed that the source of radiation has compact support
in space and time. Accordingly, we obtain the conservation of multipole charges in the
presence of radiation
Moreover, the charge at t =
T is given by (4.5), i.e.
Q(`+;m) = Q`;m :
( )
Q(`;m) =
` + 1 ( )
2` + 1 q`;m ;
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
where q`(;m) denotes the multipole moments in the stationary phase before the dynamics.
Now let us compute Q(+) through a volume integral over
face
+ naturally divides into three regions
+. The constant time
hypersurI ; I+I; I+II as shown in gure 2. Therefore
+
Q(+) =
Z
J =
J +
J +
Z
I
+
Z
II
+
Z
III
+
J :
According to the discussion in section 4.1.2, the rst term on the right hand side is zero,
while the last term is
+
Z
+
III J`;m =
2``++11 q`(;+m) :
Q(`r;mad) =
` + 1
q`(;m)
q`(;+m) :
Therefore we nd that the total multipole charge Q`r;amd carried by the radiation is
Similarly, the same argument can be repeated for the magnetic multipole charges Q~`;m
discussed in section 4.2, to obtain another set of constraints over the radiation
Q~(`r;mad) =
` + 1
Given merely the initial and nal stationary con guration of the matter, determined by
q`(;m); M`(;m), we have found in nitely many constraints over the radiation produced during
the dynamical phase of the system, without solving the equations of motion. This result
resembles the recent developments relating the asymptotic symmetries of QED with
Weinberg soft photon theorem [23, 25, 43]. However, nding a precise relation is beyond the
scope of this paper.
6
Discussion
In this paper, we studied the conservation laws associated with residual symmetries of
Maxwell theory. We showed that among the residual gauge transformations surviving the
Lorenz gauge, only those solving the Laplace equation and are growing in large radius
correspond to nontrivial conserved charges. Interestingly, these charges turned out to be
proportional to the multipole moments of the charged matter distribution, hence dubbed
as \multipole charges". The multipole charge is not equal to electric multipole moment,
since the electromagnetic eld gives a soft contribution to the multipole charge, and this
is exactly what makes the multipole charge conserved, while the multipole moment is
obviously not conserved. The only exception is the electric monopole which is only stored
in the charged matter. Using the electric-magnetic duality, we also de ned the magnetic
multipole charges proportional to magnetic multipole moments.
Using the conservation of multipole charges, we found in nite number of constraints
over the radiation produced by the charged matter, without knowing about the dynamics
of the source which can be in general complicated.
This analysis can be followed in di erent directions that we mention in the
following. While the electric multipole charges are Noether charges derived from residual gauge
symmetries, a rst principle derivation of magnetic charges remains as an open issue.
Although the analysis in this paper was done for
at spacetime in four dimensions,
we expect that similar analysis can be carried out for asymptotically
at spacetimes in
arbitrary dimensions. In case of asymptotically
at black holes geometries, one should
note that a part of radiation can be absorbed by the black hole. In this case, it was
shown in [44] that the absorption rates of long wavelength radiation is determined by the
conservation of energy and large gauge symmetry charges.
The same study may be repeated in gravity where the gravitational multipole expansion
is well established [45], but to our knowledge never studied in relation with conservation
laws of residual symmetries. The constraints over the gravitational radiation can be
especially interesting due to the recent detection of gravitational waves from black hole mergers.
While there is qualitative di erence in radial dependence of our residual symmetries
with the asymptotic symmetries considered in [22, 23, 25, 27], we still expect that there is a
close link between equation (5.15) and their results. The reason is that there is a one to one
correspondence between the smooth solutions to the Laplace equation inside a sphere and
an arbitrary function on the sphere. Also the extension of asymptotic symmetries inside
the bulk may lead to a deeper understanding of the \antipodal matching" used in [23] to
relate asymptotic symmetries of future and past null in nity.
As we mentioned, multipole charges do arise in other non-covariant gauge
conditions [7]. It will be shown in [37] that in static gauge, the multipole symmetries generate
a \vacuum moduli space", whose geodesics represent physical electrostatic con gurations.
Interestingly, the conserved momenta of these geodesics coincide with the multipole charges.
An investigation of the problem in other gauge
xing conditions would be appealing. A
general argument on the equivalence of physical results in di erent gauge conditions can
be found in [34].
Acknowledgments
The author would like to thank specially M.M. Sheikh-Jabbari for many discussions in the
course of this project. Also I am grateful to H. Afshar, S. Avery, G. Compere, J. Evslin, G.
Giribet, T. He, M. Mirbabayi, M. Pate and D. Van den Bleeken for their comments on the
paper. I also appreciate the organizers of the workshop on quantum aspects of black holes
in Yerevan during August 2016. The author would like to thank Bonyad Melli Nokhbegan
(BMN) and SarAmadan club of Iran for the partial support.
A
Algebra of charges in covariant phase space
Here we brie y discuss the algebra of charges in the covariant phase space. De ning a
coordinate system over the in nite dimensional phase space, we can write
The right hand side is the symplectic two form contracted with two vectors [ i ]A tangent
to the phase space. Also equation (2.7) is translated to
Z
!( ; 1 ; 2 ) =
AB [ 1 ]A [ 2 ]B :
(A.1)
HJEP06(217)8
The inverse of the symplectic form
AB de nes a Poisson bracket between functions over
the phase space through
Fortunately the Poisson bracket of charges can be computed without knowing the explicit
form of the inverse. This is because
AB[
fF; Gg =
fH 1 ; H 2 g =
fH 1 ; H 2 g =
fH 1 ; H 2 g =
!( ; 1 ; 2 )
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
Using (A.2) and the fact that
AC
CB = AB, we arrive at
Translating back to spacetime notation using (A.1), we conclude that
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[INSPIRE].
[2] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws
and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
[3] V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical
black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
correspondence, JHEP 05 (2009) 077 [arXiv:0902.1001] [INSPIRE].
[arXiv:1603.07862] [INSPIRE].
[6] S.G. Avery and B.U.W. Schwab, Noether's second theorem and Ward identities for gauge
symmetries, JHEP 02 (2016) 031 [arXiv:1510.07038] [INSPIRE].
[7] M. Mirbabayi and M. Simonovic, Weinberg soft theorems from Weinberg adiabatic modes,
arXiv:1602.05196 [INSPIRE].
[8] G. Compere, P. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries
of AdS3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080
[arXiv:1511.06079] [INSPIRE].
Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
[10] J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic
symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986)
[11] M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press,
207 [INSPIRE].
Princeton U.S.A. (1992).
[INSPIRE].
[gr-qc/9307038] [INSPIRE].
5849 [hep-th/0209067] [INSPIRE].
[hep-th/9712251] [INSPIRE].
82 (1999) 2828 [hep-th/9812013] [INSPIRE].
[16] G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with
applications to the mechanics of black holes and to gravity in three dimensions, Ph.D. thesis,
Vrije Universiteit Brussles, Brussels, Belgium (2007), arXiv:0708.3153 [INSPIRE].
[17] A. Seraj, Conserved charges, surface degrees of freedom and black hole entropy, Ph.D. thesis,
IPM, Teheran, Iran (2016), arXiv:1603.02442 [INSPIRE].
[18] K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002)
[19] A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009
[20] S. Carlip, Black hole entropy from conformal eld theory in any dimension, Phys. Rev. Lett.
[21] H. Afshar, D. Grumiller and M.M. Sheikh-Jabbari, Black Hole Horizon Flu s: Near Horizon
Soft Hairs as Microstates of Three Dimensional Black Holes, arXiv:1607.00009 [INSPIRE].
[22] A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151
[arXiv:1308.0589] [INSPIRE].
JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
[24] T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg's soft
graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
[INSPIRE].
QED, Phys. Rev. Lett. 113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].
[27] E. Conde and P. Mao, Remarks on asymptotic symmetries and the subleading soft photon
theorem, Phys. Rev. D 95 (2017) 021701 [arXiv:1605.09731] [INSPIRE].
[28] M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations,
JHEP 11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
[29] M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem,
[30] M. Campiglia and A. Laddha, Sub-subleading soft gravitons and large di eomorphisms,
JHEP 01 (2017) 036 [arXiv:1608.00685] [INSPIRE].
[31] A.P. Balachandran, G. Bimonte, K.S. Gupta and A. Stern, Conformal edge currents in
Chern-Simons theories, Int. J. Mod. Phys. A 7 (1992) 4655 [hep-th/9110072] [INSPIRE].
[32] A.P. Balachandran, L. Chandar and A. Momen, Edge states in gravity and black hole
physics, Nucl. Phys. B 461 (1996) 581 [gr-qc/9412019] [INSPIRE].
[33] R. Jackiw and S.Y. Pi, New setting for spontaneous gauge symmetry breaking?, Fundam.
Theor. Phys. 183 (2016) 159 [arXiv:1511.00994].
[34] A.A. Varshovi, Gauge xing invariance and anti-BRST symmetry, arXiv:1611.08856
[INSPIRE].
[astro-ph/0302326] [INSPIRE].
[35] S. Weinberg, Adiabatic modes in cosmology, Phys. Rev. D 67 (2003) 123504
[arXiv:1503.07861] [INSPIRE].
[arXiv:1512.05584] [INSPIRE].
[36] J.D. Brown and M. Henneaux, On the Poisson brackets of di erentiable generators in
classical eld theory, J. Math. Phys. 27 (1986) 489 [INSPIRE].
[37] A. Seraj and D. Van den Bleeken, Strolling around gauge theory vacua, to appear.
[38] G. Compere, L. Donnay, P.-H. Lambert and W. Schulgin, Liouville theory beyond the
cosmological horizon, JHEP 03 (2015) 158 [arXiv:1411.7873] [INSPIRE].
[39] G. Compere, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, Wiggling throat of extremal
black holes, JHEP 10 (2015) 093 [arXiv:1506.07181] [INSPIRE].
[40] G. Compere, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, Extremal rotating black holes in
the near-horizon limit: phase space and symmetry algebra, Phys. Lett. B 749 (2015) 443
[41] K. Hajian and M.M. Sheikh-Jabbari, Solution phase space and conserved charges: a general
formulation for charges associated with exact symmetries, Phys. Rev. D 93 (2016) 044074
[42] G. Compere and J. Long, Classical static nal state of collapse with supertranslation
memory, Class. Quant. Grav. 33 (2016) 195001 [arXiv:1602.05197] [INSPIRE].
031602 [arXiv:1509.00543] [INSPIRE].
04 (2017) 053 [arXiv:1609.04397] [INSPIRE].
[1] E. Noether , Invariant variation problems, Gott. Nachr. 1918 ( 1918 ) 235 [physics/0503066] [4] G. Compere , K. Murata and T. Nishioka , Central charges in extreme black hole/CFT [5] M.M. Sheikh-Jabbari , Residual di eomorphisms and symplectic soft hairs: the need to re ne strict statement of equivalence principle , Int. J. Mod. Phys. D 25 ( 2016 ) 1644019 [9] R.L. Arnowitt , S. Deser and C.W. Misner , The dynamics of general relativity, Gen. Rel. [12] M. Blagojevic , Gravitation and gauge symmetries , CRC Press, U.S.A. ( 2002 ). [13] J. Lee and R.M. Wald , Local symmetries and constraints , J. Math. Phys. 31 ( 1990 ) 725 [14] C. Crnkovic and E. Witten , Covariant description of canonical formalism in geometrical theories , Princeton Print- 86 - 1309 ( 1986 ). [15] R.M. Wald , Black hole entropy is the Noether charge , Phys. Rev. D 48 ( 1993 ) R3427 [25] D. Kapec , M. Pate and A. Strominger , New symmetries of QED, arXiv: 1506 .02906 [26] V. Lysov , S. Pasterski and A. Strominger , Low's subleading soft theorem as a symmetry of [43] A. Strominger, Magnetic corrections to the soft photon theorem , Phys. Rev. Lett . 116 ( 2016 ) [44] S.G. Avery and B.U.W. Schwab , Soft black hole absorption rates as conservation laws , JHEP [45] K.S. Thorne , Multipole expansions of gravitational radiation , Rev. Mod. Phys . 52 ( 1980 ) 299 [46] J. Bronzan , The magnetic scalar potential , Amer. J. Phys . 39 ( 1971 ) 1357 .