Asymptotic Symmetries of Maxwell Theory in Arbitrary Dimensions at Spatial Infinity

TL;DR

This paper analyzes the asymptotic symmetries of Maxwell theory at spatial infinity in arbitrary dimensions, establishing well-defined action principles, conserved charges, and boundary conditions, with implications for null infinity and inertial observers.

Contribution

It provides a comprehensive analysis of Maxwell asymptotic symmetries in arbitrary dimensions, introducing a well-defined action principle and generalizing parity conditions.

Findings

Antipodal condition imposed by regularity at light cone for d≥4

Reproduction and generalization of parity conditions in higher dimensions

Derivation of boundary conditions at null infinity for 3D theory

Abstract

The asymptotic symmetry analysis of Maxwell theory at spatial infinity of Minkowski space with $d\geq 3$ is performed. We revisit the action principle in de Sitter slicing and make it well-defined by an asymptotic gauge fixing. In consequence, the conserved charges are inferred directly by manipulating surface terms of the action. Remarkably, the antipodal condition on de Sitter space is imposed by demanding regularity of field strength at light cone for $d\geq 4$. We also show how this condition reproduces and generalizes the parity conditions for inertial observers treated in 3+1 formulations. The expression of the charge for two limiting cases is discussed: Null infinity and inertial Minkowski observers. For the separately-treated 3d theory, a set of non-logarithmic boundary conditions at null infinity are derived by a large boost limit.

Equations

$$d{s}^2=\text{d}{\rho }^2+\frac{{\rho }^2}{{\sin }^{2}T}\left(-\text{d}{T}^2+q_{AB}\text{d}{x}^A\text{d}{x}^B\right),0\le T\le \pi .$$

$$\left\{\begin{array}[]{c}\rho^{2}=r^{2}-t^{2}\\ \cos{\scriptstyle T}=t/r\end{array}\right.\qquad\qquad\left\{\begin{array}[]{c}t=\rho\cot{\scriptstyle T}\\ r=\rho/\sin{\scriptstyle T}\end{array}\right.$$

$$\mathcal{A}(\rho ,x^a)=\sum _n{A}^{(n)}(x^a){\rho }^{-n}$$

$$\frac{\delta S}{\delta\Phi^{i}}\Big{|}_{\Phi^{i}_{cl.}}=0\,$$

$$S={\int }_{M_d}\sqrt{g}\left(-\frac{1}{4}{\mathcal{F}}_{\mu \nu }{\mathcal{F}}^{\mu \nu }+{\mathcal{A}}_{\mu }{J}^{\mu }\right),$$

$${\int }_B{\rho }^{d-1}\sqrt{h}\delta {\mathcal{A}}_a{\mathcal{F}}^{a\rho }$$

$$\delta S[\Phi ]\approx {\int }_{I_2}\mathcal{I}(\Phi ,\delta \Phi )-{\int }_{I_1}\mathcal{I}(\Phi ,\delta \Phi )+{\int }_B\mathcal{B}(\Phi ,\delta \Phi )$$

$${\delta }_{\lambda }S\approx {\int }_{\partial I_2}\mathcal{C}(\Phi ,{\delta }_{\lambda }\Phi )-{\int }_{\partial I_1}\mathcal{C}(\Phi ,{\delta }_{\lambda }\Phi ).$$

$$\Omega ={\int }_I\mathcal{I}(\delta \Phi ,{\delta }^{\prime }\Phi )$$

$${\int }_B\mathcal{B}(\delta \Phi ,{\delta }^{\prime }\Phi )$$

$$\int_{B}\mathcal{B}(\delta\Phi,\delta^{\prime}\Phi)\equiv-\Omega^{\text{\tiny b'dry}}\big{|}^{\partial I_{2}}_{\partial I_{1}}$$

$$D_a{\mathcal{F}}^{a\rho }$$

$$\frac{1}{{\rho }^{d-1}}{\partial }_{\rho }({\rho }^{d-1}{\mathcal{F}}^{\rho a})+D_b{\mathcal{F}}^{ba}$$

$${\partial }_{\rho }{\mathcal{F}}_{ab}+2{\partial }_{[a}{\mathcal{F}}_{b]\rho }$$

$${\partial }_{[a}{\mathcal{F}}_{bc]}$$

$${\mathcal{F}}_{a\rho }={\partial }_a\Psi ={\rho }^{3-d}{\partial }_a\psi (x^b),D^a{D}_a\Psi =J^{\rho }$$

$$D^a{D}_{[a}{A}_{b]}^{(0)}=0$$

$$(d-3)F_{ab}^{(d-3)}=(\text{d}\psi {)}_{ab},D^b(\text{d}\psi {)}_{ba}-(d-3){\psi }_a=0$$

$$Q={\int }_{S^{d-2}}\sqrt{q}{r}^{d-2}{\mathcal{F}}_{tr}\to {\mathcal{F}}_{tr}=\frac{Q}{a_{d-2}}{r}^{2-d}$$

$${\mathcal{F}}_{T\rho }=-\frac{\rho }{\sin T}{\mathcal{F}}_{tr}=-\frac{Q}{a_{d-2}}{\rho }^{3-d}{\sin }^{d-3}T$$

$${\mathcal{F}}_{\rho a}\sim \mathcal{O}({\rho }^{3-d}),{\mathcal{F}}_{ab}\sim \mathcal{O}({\rho }^{3-d}).$$

$$F_{a\rho }^{(d-3)}={\partial }_a\psi ,D_a{D}^a\psi =j_{\rho }^{(1-d)}$$

$${\mathcal{A}}_a\sim \mathcal{O}({\rho }^{3-d}){\mathcal{A}}_{\rho }\sim \mathcal{O}({\rho }^{3-d})$$

$$Q={\int }_{S^{d-2}}\ast \mathcal{F}$$

$${\int }_B\sqrt{h}\delta A_b^{(0)}{F}_{(d-3)}^{b\rho }={\int }_B\sqrt{h}\delta A_b^{(0)}{\partial }^b\psi$$

$$D^a{A}_a^{(0)}+\alpha (d-2)A_{\rho }^{(1)}=0,\alpha =1$$

$$S_b=-\alpha {\int }_{B_3}\sqrt{h}{\psi }^2\text{for}d=4$$

$$S\approx {\int }_{I_2}\sqrt{\gamma }{n}_T{\mathcal{A}}_{\mu }{\mathcal{F}}^{\mu T}-{\int }_{I_1}\sqrt{\gamma }{n}_T{\mathcal{A}}_{\mu }{\mathcal{F}}^{\mu T}+{\int }_B\sqrt{h}{\partial }_a\varphi {\partial }^a\psi$$

$${\delta }_{\Lambda }S-{\int }_{I_2}\Lambda J^T+{\int }_{I_1}\Lambda J^T\approx -{\int }_{\partial I_2}\sqrt{g}\lambda {\mathcal{F}}^{\rho T}+{\int }_{\partial I_1}\sqrt{g}\lambda {\mathcal{F}}^{\rho T}+{\int }_B\sqrt{h}{\partial }_a\lambda {\partial }^a\psi .$$

$${\delta }_{\Lambda }S-{\int }_{I_2}\Lambda J^T+{\int }_{I_1}\Lambda J^T={\int }_B\sqrt{h}{\rho }^{d-1}\lambda J^{\rho }$$

$$\delta A_a^{(0)}={\partial }_a\lambda ,D^a{D}_a\lambda =0,$$

$$Q_{\lambda }={\int }_{\partial I}\sqrt{g}\left(\lambda {\mathcal{F}}^{T\rho }-{\partial }^T\lambda \psi \right)$$

Full text

1 Introduction

In Lagrangian theories with a Lie group of global symmetries, Noether’s first theorem establishes a conserved current for every generator of the corresponding Lie algebra. Noether’s method, however, fails to assign conserved currents to gauge symmetries Compere:2007az . Instead, several methods have been proposed to associate 2-form currents $k^{\mu\nu}$ to gauge symmetriesRegge:1974zd ; Lee:1990nz ; Anderson:1996sc ; Barnich:2001jy , which yield conserved surface charges. In gauge theories (and gravity), the asymptotic symmetry group (ASG) is the group of gauge transformations with finite surface charge111Quotiented by transformations with vanishing charge., and the elements are called large (or improper) gauge transformations (in gravity, large (or improper) diffeomorphisms). To obtain the asymptotic symmetry group, one fixes an appropriate gauge, and imposes certain fall-off behavior on the fields. Large gauge transformations are then the residual gauge transformations i.e. those which preserve both the boundary conditions and the gauge.

Recent interest in ASG of Maxwell theory in flat space emanated from the discovery that soft photon theorem in QED is the Ward identity of the asymptotic symmetry group He:2014cra ; Campiglia:2015qka . Concerned with that motivation, research on asymptotic symmetries is mostly performed in null slicing (Bondi coordinates) of flat space, where the surface of integration is (almost) light-like-separated from the scattering event He:2014cra ; Campoleoni:2018uib ; Campoleoni:2017qot ; Kapec:2014zla ; Satishchandran:2019pyc ; Hosseinzadeh:2018dkh ; Hirai:2018ijc . The charges “at null infinity” have also been generalized to subleading orders Seraj:2016jxi ; Campiglia:2016hvg ; Campiglia:2018dyi .

The asymptotic symmetry group of Maxwell theory at spatial infinity is, as far as we know, restricted to three and four dimensions Barnich:2015jua ; Seraj:2016jxi ; Campiglia:2017mua ; Henneaux:2018gfi ; Prabhu:2018gzs . Spatial infinity examination allows applying the canonical methods and define the ASG in a standard way. In Seraj:2016jxi , the multipole moments of a static configuration were exhibited as asymptotic symmetry charges. In Campiglia:2017mua , the charges were defined in de Sitter slicing of flat space (explained later), and the null infinity charges would be recovered if the integration surface approached null infinity. We will follow much similar path in this paper, recovering Strominger:2014pwa ; Satishchandran:2019pyc at null infinity.

In this work, we study the asymptotic structure of Maxwell field in arbitrary dimensions at spatial infinity, and identify a set of boundary conditions with non-trivial ASG, generalizing previous works in four dimensions. The ASG with our prescribed boundary conditions is local-$U(1)$ on celestial sphere $S^{d-2}$, parametrized by arbitrary functions on $S^{d-2}$. The surface charges are obtained by manipulating surface terms arising from variation of the action, circumventing standard methods. To do this, we make the action principle well-defined, by making the timelike boundary term vanish, as done in Marolf:2000cb ; Mann:2006bd ; Goto:2018iay ; Castro:2008ms . As it was shown in Afshar:2018apx , demanding the action principle to be well-defined determines the asymptotic gauge almost completely. This condition automatically ensures conservation of the charges for residual gauge transformations.

A key result of this paper is that we provide a rationale for imposing the antipodal matching condition in arbitrary dimensions. Previous works on gauge theories in flat space advocate a matching condition Strominger:2014pwa for the fields at spatial infinity $i^{0}$, when it is approached from future and past null boundaries $\mathcal{I^{+}},\,\mathcal{I}^{-}$. On the asymptotic de Sitter space, this condition relates the states at past and future boundaries $\mathcal{I^{\pm}}$. In dS/CFT studies, various antipodal conditions are proposed to make the Hilbert space well-defined Bousso:2001mw . We show that an antipodal condition is necessary to ensure regularity of field strength at light cone for $d\geq 4$.

We will work in de Sitter slicing Ashtekar:1991vb ; Prabhu:2018gzs of Minkowski space which makes the boundary conditions manifestly Lorentz invariant. In the 3+1 Hamiltonian approach of Henneaux:2018gfi , the formalism loses manifest Lorentz symmetry and the ASG is presented as the product of two opposite-parity subgroups. We will show how their results regarding conserved charges and parities are recovered and generalized, by focusing on specific slices of de Sitter space.

Finally, 3-dimensional theory is covered in section 4. Asymptotic symmetries of 3d Einstein-Maxwell theory was studied in Barnich:2015jua at null infinity and in Donnay:2018ckb in near-horizon geometries. We will show by taking null infinity limit that the same set of charges (in Maxwell sector) can be obtained in a non-logarithmic expansion. In addition, our hyperbolic setup fits completely with Compere:2017knf on BMS3 symmetry at spatial infinity. Thus, we expect that the combined hyperbolic analysis will reproduce the results of Barnich:2015jua in its non-radiative sector.

2 Rindler patch, action principle and conserved charges

Given an arbitrary point $\mathcal{O}$ in Minkowski space, one can define null coordinates $u=t-r$ and $v=t+r$. The future light cone $\mathcal{L}^{+}$ of $\mathcal{O}$ is the $u=0$ hypersurface, while the past light cone $\mathcal{L}^{-}$ is at $v=0$. $\mathcal{L}^{+}$ and $\mathcal{L}^{-}$ intersect at the origin $\mathcal{O}$. We call the set of points with space-like distance to $\mathcal{O}$, the Rindler patch and denote it by Rindd-1 (see figure 1). The Rindler patch is conveniently covered by coordinates $(\rho,{\scriptstyle T},x^{\scriptscriptstyle A}),\,A=1,\cdots,d-2$, in which the metric is

[TABLE]

where

[TABLE]

The origin is at $\rho=0$ and undefined ${\scriptstyle T}$. Future light cone $\mathcal{L}^{+}$ is at $(\rho=0,{\scriptstyle T}=0)$ and past light cone $\mathcal{L}^{-}$ is at$(\rho=0,{\scriptstyle T}=\pi)$.222A point at radius $r$ on $\mathcal{L}^{+}$ is at $(\rho=0,{\scriptstyle T}=0)$, by taking the limit ${\scriptstyle T}\to 0$ such that $\rho=r{\scriptstyle T}$. Spatial infinity $i^{0}$ defined as the destination of spacelike geodesics is at $(\rho\to\infty,0<{\scriptstyle T}<\pi)$, shown as the intersection of future and past null infinities on the Penrose diagram. The limit $(\rho\to\infty,{\scriptstyle T}\to 0,\pi)$ covers the portion of null infinity outside the light cone.333The point at retarded time $u$ on future null infinity is reached by taking the limit $\rho\to\infty$ such that $u=-\rho{\scriptstyle T}/2$. Similarly, taking the limit with fixed $v=\rho(\pi-{\scriptstyle T})/2$, one arrives at the advanced time $v$ on past null infinity.

The constant $\rho$ hypersurfaces are $(d-1)$-dimensional de Sitter spaces with radius $\rho$, invariant under Lorentz transformations about $\mathcal{O}$. We will show de Sitter coordinates by $x^{a}$, $a=2,\cdots,d$, and the unit $dS_{d-1}$ metric by $h_{ab}$.

The study is restricted to solutions with asymptotic power expansion in $\rho$

[TABLE]

In some cases, we drop the superscript $(n)$ for the leading term (the least $n$) in each component to reduce clutter.

2.1 The action principle

In the Lagrangian formulation of physical theories, the classical trajectories of the dynamical variables $\Phi^{i}$ are stationary points of an action functional

[TABLE]

for fixed initial and final values. In field theories, the functional derivative of the action is well-defined, if variation of dynamical fields leaves no boundary terms. In our setup, there are two spacelike boundaries $I_{1,2}$ and one timelike boundary $B$ lying on asymptotic de Sitter space (see figure 2). Data on spacelike boundaries is fixed, so we must ensure that the boundary term on $B$ either vanishes or itself is a total derivative.

For Maxwell theory with action444Notation: $g=|\text{det}g_{\mu\nu}|$ for all metrics involved.

[TABLE]

the timelike boundary term is

[TABLE]

We will show that for specific boundary conditions and an asymptotic gauge fixing, the boundary term does vanish.

2.2 Conserved charges

For the specific example of Maxwell theory, we show that with a well-defined action principle at hand, one can define conserved charges for gauge transformations of the theory, and identify the asymptotic symmetry group as the group of gauge transformations having finite charge.

Consider variation of the action around a solution to equations of motion555Notation: $\approx$ is equality when equations of motion hold.

[TABLE]

If the field variation is a gauge transformation (or a diffeormorphism in gravity theories), then, the $\mathcal{I}$ integrands in (2.7) become total derivatives, so the first two terms becomes codimension-2 integrals on $\partial I_{1}$ and $\partial I_{2}$. This can be checked in specific examples, and a proof for gravity case is given in Iyer:1994ys

If the action principle is well-defined, the $B$-integral on the timelike boundary is either vanishing, or a total derivative on the hyperboloid (so that it becomes a surface integral on boundaries of $B$). As a result, the gauge transformation of the action becomes the difference of two codimension-2 integrals on shell

[TABLE]

The left-hand-side depends on the the explicit form of the action. If the action is gauge invariant ($\delta_{\lambda}S=0$), (2.8) shows that the integral $\int_{\partial I}\mathcal{C}$ is independent of the surface of integration; thus we can identify the codimension-2 integrals as the conserved charges corresponding to the gauge transformation $\delta_{\lambda}$.

Covariant phase space method

Let us compare the procedure above with covariant phase space method. The symplectic form of the theory is nothing but variation of the action surface terms

[TABLE]

for two field variations $\delta,\delta^{\prime}$, and $\mathcal{I}$ is defined in (2.7). Taking a second variation of (2.7) shows that in general $\Omega$ is not conserved since its flux at timelike boundary $B$ is non-vanishing and given by

[TABLE]

where the integrand $\mathcal{B}$ is again defined in (2.7). Therefore, eliminating the symplectic flux is equivalent to making the action principle well-defined. For the conservation of the symplectic form, the flux (2.10) need not be strictly vanishing. It is enough, if possible, to make it a total divergence reducing the expression to codimension-2 integrals on $\partial B$:

[TABLE]

Finally, $\Omega^{\text{\tiny b'dry}}$ can be added to $\Omega$ as a surface term, leading to conserved charges. This subtraction is a Y ambiguity in covariant phase space terminology Lee:1990nz . This procedure was done in Campiglia:2017mua for 4d Maxwell theory. It can be readily generalized to arbitrary dimensions by appropriate choice of boundary conditions. However, we decide to bypass the symplectic form by working directly with the action.

2.3 Field equations in de Sitter slicing

Written in coordinates $(\rho,x^{a})$, the field equations and Bianchi identities are

[TABLE]

where $D$ is the covariant derivative on dSd-1. Analyzing the solutions suggests appropriate boundary conditions for the theory. Note that $\mathcal{F}_{a\rho}$ and $\mathcal{F}_{ab}$ are distinct Lorentz invariant components. First we ask if there are solutions to equations of motion once either of them is set to zero.

If we set $\mathcal{F}_{ab}=0$, by (2.12b) we have 666Notation: $\sim\mathcal{O}(\rho^{n})$ means all powers not exceeding $n$, while $\propto\rho^{n}$ means the $n$-th power of $\rho$ exclusuvely. $\mathcal{F}^{a\rho}\propto\rho^{1-d}$ . Furthermore, by (2.12c) and (2.12a),

[TABLE]

In this case, the solution consists of a scalar degree of freedom $\psi$. 2. 2.

In general, $\mathcal{F}_{ab}$ is closed on de Sitter space by (2.12d), thus it is locally exact $\mathcal{F}_{ab}=(\text{d}\mathcal{A})_{ab}$. Switching $\mathcal{F}_{a\rho}$ off, fixes the $\rho$-dependence by (2.12c) to $\mathcal{F}_{ab}\propto\rho^{0}$. Finally, the field equation (2.12b) reduces to

[TABLE]

(Notation is explained in (2.3)).

Any other solution involves both $\mathcal{F}_{ab}$ and $\mathcal{F}_{a\rho}$. The solutions with power-law fall-off in $\rho$ correspond to multipoles of electric and magnetic branes. Electric monopoles generate the independent solution (2.13) for $\mathcal{F}_{a\rho}$, while magnetic mono-poles(-branes) generate the independent solution (2.14) for$\mathcal{F}_{ab}$. Their multi-poles generate fields of lower fall-off which mix $\mathcal{F}_{a\rho}$ and $\mathcal{F}_{ab}$. On the contrary, arranging monopoles to build lines of charge will generate stronger fields at infinity, but in any case mix $\mathcal{F}_{a\rho}$ and $\mathcal{F}_{ab}$. 777 For example, for an electric dipole, $\mathcal{F}_{\rho a}\propto\rho^{2-d}$, and by Bianchi identity (2.12c), $\mathcal{F}_{ab}\propto\rho^{3-d}$. Define $\psi_{a}\equiv F^{(d-2)}_{a\rho}$, Then,

$\displaystyle(d-3)F^{(d-3)}_{ab}=(\text{d}\psi)_{ab},\qquad D^{b}(\text{d}\psi)_{ba}-(d-3)\psi_{a}=0$

(2.15a) away from sources.

Denote the set of solutions for electric monopoles given in (2.13) by $\mathcal{E}$. This space covers moving electric charges in space, which are passing the origin simultaneously at $t=0$, hence their worldlines cross $\mathcal{O}$. The field strength is $\mathcal{F}_{\rho a}\propto\rho^{3-d}$ with no subleading terms. For an arbitrary configuration of freely moving charges, the leading component of asymptotic field is an element of $\mathcal{E}$, but subleading terms are generally present. In other words, the definition of $\mathcal{E}$ is Lorentz invariant, but not Poincaré invariant. $\mathcal{E}$ encodes the information of charge values $q_{n}$ and their velocities $\vec{\beta}_{n}$. The space $\mathcal{E}$ is isomorphic to the space of boost vectors $\vec{\beta}$, that is $\mathbb{R}^{d-1}$. The space of conserved electric charges we will construct is also isomorphic to $\mathbb{R}^{d-1}$; each point of this space with coordinate vector $\vec{\beta}$ is a conserved charge and gives the total electric charge in space, moving with that specific boost.

The set of solutions (2.14) covers magnetic monopoles moving freely in space and crossing the origin at $t=0$. In dimensions larger than 4, the magnetic monopoles are replaced by extended magnetic branes since the dual field strength $\ast F$ is a $(d-2)$-form in that case. We are considering boundary conditions which exclude magnetic charges in this work.

3 Four and higher dimensions

In this section, we exploit the asymptotic symmetries of Maxwell theory in dimensions higher than three. First, we present a set of well-motivated boundary conditions on field strength tensor. Nonetheless, existence of large gauge transformations demand that the gauge field be finite at infinity. That will necessitate an asymptotic gauge choice to make the action principle well-defined. Finally, we find the conserved charges of the theory at spatial infinity by computing the on-shell action.

3.1 Boundary conditions and the action principle

The electromagnetic field of a static electric charge is 888 For a spherically symmetric field we have

$Q=\int_{S^{d-2}}\sqrt{q}r^{d-2}\mathcal{F}_{tr}\quad\rightarrow\quad\mathcal{F}_{tr}=\frac{Q}{a_{d-2}}r^{2-d}$

where $a_{d-2}$ is the area of a $(d-2)$-sphere. In hyperbolic coordinates we have

$\mathcal{F}_{{\scriptscriptstyle T}\rho}=-\frac{\rho}{\sin{\scriptstyle T}}\mathcal{F}_{tr}=-\frac{Q}{a_{d-2}}\rho^{3-d}\sin^{d-3}{\scriptstyle T}$

$\mathcal{F}_{{\scriptscriptstyle T}\rho}\propto\rho^{3-d}$. Applying a boost (which belongs to the isometry group of the hyperboloid) will turn on other de Sitter components of $\mathcal{F}_{a\rho}$ with the same fall-off; so one generally has $\mathcal{F}_{a\rho}\propto\rho^{3-d}$. Therefore, we propose the following boundary conditions for $d$-dimensional theory

[TABLE]

The $\mathcal{F}_{ab}$ components arise because of electric multipoles (c.f. §2.3). The leading component of $\mathcal{F}_{a\rho}$ is in $\mathcal{E}$ of $\S$ 2.3 and satisfies

[TABLE]

Components of gauge field that saturate (3.16) behave like

[TABLE]

. Plugging into (2.6), the boundary term falls like $\mathcal{O}(\rho^{3-d})$. For $d>3$, the action principle is well-defined. However, this choice will make the charges for all gauge transformations vanish. For instance, the Gauss law

[TABLE]

is regarded as the charge for gauge transformation with $\lambda=1$, which is excluded if $\mathcal{A}_{a}\sim\mathcal{O}(\rho^{3-d})$ in dimensions higher than three. The theory enjoys non-trivial ASG, only if $\delta\mathcal{A}_{a}\sim\mathcal{O}(1)$. Thus, our prescribed boundary condition is as follows: $\mathcal{A}_{a}\sim\mathcal{O}(1)$ but the first few terms in the asymptotic expansion of $\mathcal{A}_{a}$ are pure gauge999By “pure gauge” we mean a flat connection; a configuration gauge equivalent to $\mathcal{A}_{\mu}=0$, although it may involve an improper gauge transformation (that with non-zero charge)., such that $\mathcal{F}_{ab}\sim\mathcal{O}(\rho^{3-d})$. Previous works in four dimensional Maxwell theory allow magnetic monopoles. That would make $\mathcal{F}_{ab}\sim\mathcal{O}(1)$ so the leading term of the gauge field would not be pure gauge. Here we are not taking account of magnetic charges though.

With the aforementioned boundary condition, the boundary term of the action will be finite

[TABLE]

According to (3.16), $F_{ab}^{(0)}=0$ so the leading term is (locally) pure gauge $A^{(0)}_{b}=\partial_{b}\phi$. Consequently, after integration by parts, the boundary term of the action vanishes on shell, by equation of motion $D^{2}\psi=0$ (up to a total divergence on $B$). However, we request off-shell vanishing of the boundary term, since the variational principle must entail the equations of motion, and they can not be used a priori.

One way out is to fix the asymptotic gauge $\delta D^{a}A_{a}^{(0)}=0$, for which the boundary term becomes a total divergence on $B$ after an integration by parts. There are also other possibilities. The Lorenz gauge at leading order is

[TABLE]

and by our boundary conditions on field strength, $A_{\rho}^{(1)}=0$ for $d>4$. Thus, the Lorenz gauge, or its extension to general $\alpha$ will make the action principle well-defined in dimensions strictly higher than 4. In four spacetime dimensions, $A_{\rho}^{(1)}=\psi$ (up to a constant number which drops from derivatives) so it is necessary to add a boundary term

[TABLE]

to make the action well-definedAfshar:2018apx .

3.2 Conserved charges from action

The action with a solution to equations of motion plugged in, is a functional of initial and final field values (or boundary values in Euclidean versions); That is how classical trajectories are defined. For Maxwell theory,

[TABLE]

$\gamma$ is the induced metric on $I$ and $n^{\mu}$ is its future-directed normal vector. Varying (3.23) by gauge transformations $\delta\mathcal{A}_{\mu}=\partial_{\mu}\Lambda$, and using field equations following an integration by parts gives101010In equation (3.24), the induced metric on $\partial I$ yields a determinant factor $\rho^{d-2}\sin^{2-d}{\scriptstyle T}$. On the other hand, $n_{\scriptscriptstyle T}=-\frac{\rho}{\sin{\scriptstyle T}}(0,1,\vec{0})$. The combination is equal to $-\sqrt{g}$.

[TABLE]

where $\lambda=\Lambda^{(0)}$. The explicit form of Maxwell action (2.5) shows that the left-hand-side above is the flux through spatial boundary:

[TABLE]

We can make this “charge flux” vanish asymptotically by the additional assumption $J^{\rho}\sim\mathcal{O}(\rho^{-d})$. This condition ensures that the system is localized and the charges are conserved. So far we made the left-hand-side in (3.24) vanish; let us look at the other side.

Recall that the action principle necessitated fixing the asymptotic Lorenz gauge (3.21), leaving residual gauge transformations

[TABLE]

with arbitrary subleading terms. The condition on $\lambda$ allows us to turn the very last term in the right-hand-side of (3.24) into a total divergence on $B$. As a result we manage to prove that the quantity

[TABLE]

is independent of $I$; i.e. conserved.

3.3 Light cone regularity and antipodal identification

$\lambda$ and $\psi$ both satisfy

[TABLE]

and the solution is obtained by spectral decomposition of Laplace operator on $S^{d-2}$, being $\mathcal{D}^{2}Y_{\ell}(\hat{x})=-\ell(\ell+d-3)Y_{\ell}(\hat{x})$. Then, (3.28) will simplify to

[TABLE]

The general solution is

[TABLE]

where $P_{l}^{m}$ and $Q_{l}^{m}$ are associated Legendre functions of the first and second kind respectively. For $\ell=0$, the solutions are

[TABLE]

As far as field equations are concerned, the whole set of solutions in (3.30) with two sets of coefficients are admissible both for $\psi$ and $\lambda$. In previous works in four dimensional Maxwell theory, a boundary condition, the antipodal matching condition Strominger:2014pwa , was imposed such that one of branch of the solutions in (3.30) was allowed for $\psi$ and the other for $\lambda$. Here we will provide a rationale for the antipodal matching condition in higher dimensions.

The field strength tensor $\mathcal{F}$ being a physical field must be regular at light cone $\mathcal{L}^{\pm}$ (i.e. $u=0$ and $v=0$ surfaces in advanced/retarded Bondi coordinates). Recall that in $\mathcal{E}$ space, $\mathcal{F}_{a\rho}=\rho^{3-d}\partial_{a}\psi$ in $d$ dimensions, which diverges at $\rho=0$ in dimensions larger than three. Near $\mathcal{L}^{+}$ (located at $\rho=0,{\scriptstyle T}=0$), $\psi$ must decay at least like ${\scriptstyle T}^{d-2}$, to make $\mathcal{F}_{{\scriptscriptstyle T}\rho}$ finite.

The light cone behavior of solutions (3.30) is111111In four dimensions, the subleading term for $f_{+}$ is $\mathcal{O}({\scriptstyle T}^{2}\log{{\scriptstyle T}})$

[TABLE]

Request for light cone regularity leads us to take $f_{-}$ for $\psi$ as a boundary condition on $\mathcal{L}^{+}$, hence the notation $\bar{\psi}$. Similar argument can be made at $\mathcal{L}^{-}$ at ${\scriptstyle T}\to\pi$. Extracting $f_{-}$ from (3.30) amounts to setting $b_{\ell}=0$ in even dimensions, and setting $a_{\ell}=0$ in odd dimensions (and keeping $b_{0}$ in all dimensions). These conditions can be summarized as antipodal identification of solutions on $dS_{d-1}$

[TABLE]

This is a well-known condition in dS/CFT studies Strominger:2001pn . Gauge parameters with non-vanishing charge (3.27) must reside in $f_{+}$ set. These are even under de Sitter antipodal map

[TABLE]

Note that the conditions (3.33) and (3.34) hold on the entire de Sitter space and in particular for ${\scriptstyle T}=0$, relating the fields on future and past boundaries of the hyperboloid

[TABLE]

The fields on left-hand-side live on the past of future null infinity $\mathcal{I}^{+}_{-}$ while those on right-hand-side live on the future of past null infinity $\mathcal{I}^{-}_{+}$. Therefore $\lambda$ and $F_{a\rho}=\partial_{a}\psi$ are both even under antipodal map between future and past null infinity.

3.4 Charge at null infinity

In the Rindler patch, one can approach the light cone hypersurface $\mathcal{L}^{+}\cup\mathcal{L}^{-}$ from outside. The charge (3.27) takes a simpler form in that limit: The second term in (3.27) vanishes, while the first term becomes

[TABLE]

The leading field strength at null infinity becomes

[TABLE]

Hence, the familiar expression for surface charges at future null infinity is recovered

[TABLE]

3.5 Inertial observers

Consider a Minkowski observer with coordinates $(t,r,x^{\scriptscriptstyle A})$, who advocates a “3+1 formulation” of $d$-dimensional theory. Boundary conditions restrict Cauchy data residing in constant-time hypersurfaces at large $r$. It is implicitly presumed that time interval $\Delta t$ between Cauchy surfaces is much smaller that the radius $r$ beyond which is conceived as “asymptotic region”. This $\Delta t/r\to 0$ condition makes all Cauchy surfaces to converge at ${\scriptstyle T}=\pi/2$ “throat” on the asymptotic de Sitter space. Infinitesimal Lorentz boosts will incline this surface, though, to ${\scriptstyle T}=\vec{\beta}\cdot\hat{x}+\pi/2$.

The solutions to (second order) equations of motion on dSd-1 are specified by initial/final data on past/future boundaries of de Sitter space $\mathcal{I}^{-}_{+}/\mathcal{I}^{+}_{-}$. When an additional antipodal condition is imposed, only one set of data on either boundary suffices (and the other one is determined by e.o.m.). When the spacetime is restricted to a cylinder around ${\scriptstyle T}=\pi/2$, the solution can be specified by a couple of independent data $\Phi$ and $\partial_{\scriptscriptstyle T}\Phi$ (and higher time derivatives determined by e.o.m.). The antipodal condition then halves the possibilities in each one by a restriction on angular dependence, as explained below.

Here, we would like to focus around ${\scriptstyle T}=\pi/2$ surface and translate previous results to a canonical language. First of all, the coordinates are related as

[TABLE]

Next, recall that $A_{a}^{(0)}=\partial_{a}\phi$, which implies that

[TABLE]

In four dimensions, $A_{t}$ receives an additional contribution $-\psi(\frac{\pi}{2},\hat{x})/r$. The radial components may be written as

[TABLE]

where

[TABLE]

The field strength is given by

[TABLE]

The “momenta” $\pi^{i}$ are symbolic in this discussion, but they are equal to momenta in a true Hamiltonian formulation. Finally, the gauge parameter divides into

[TABLE]

The antipodal conditions (3.33) and (3.34) imply

[TABLE]

and121212Parity of $\pi^{B}$ can not be inferred from leading fields. For electric dipoles, $\pi^{B}(-\hat{x})=+\pi^{B}(\hat{x})$.

[TABLE]

In even spacetime dimensions, these are parity conditions, cause the antipodal map $\hat{x}\to-\hat{x}$ reverses the orientation of $S^{d-2}$ (the volume form shifts sign). In odd dimensions, however, the map is a rotation about the origin, preserving the orientation. These conditions are preserved under boosts. The connected part of Lorentz group $SL(d-1,1)$, commutes with parity and time-reversal, thus the antipodal conditions (3.33) and (3.34) hold in any Lorentz frame. Explicitly, for an infinitesimally boosted frame and keeping the terms at zeroth order of ${\scriptstyle T}$ we have

[TABLE]

In the second equality we have used the antipodal conditions and the temporal argument is found by $\pi-(\frac{\pi}{2}-\vec{\beta}\cdot(-\hat{x}))=\frac{\pi}{2}-\vec{\beta}\cdot\hat{x}$.

The conserved charge (3.27) is rewritten as

[TABLE]

One must note that $\mu$ transforms like a vector under boosts, for it is the $T$-derivative of a scalar.

3.6 Finite action and symplectic form

Here we will show that the symplectic form is finite in dimensions higher than 4. In analogy with mechanical systems, the symplectic 2-form $\Omega$ in field theories is defined from the boundary term of the Lagrangian. For Maxwell theory in Rindler patch, it is

[TABLE]

with $\Omega^{\text{\tiny b'dary}}$ being a surface term introduced in Campiglia:2017mua for $d=4$ 131313It exists also in higher dimensions. We did not need to introduce it for the charges were derived from the action.. In four dimensions, this is logarithmically divergent, since

[TABLE]

The second term which correponds to magnetic monopoles is excluded in our boundary condition (3.16). The first term, however has the form $\int\psi\partial_{\scriptscriptstyle T}\psi$. If the integration surface is ${\scriptstyle T}=\pi/2$, this term vanishes by antipodal condition (3.33). This remains true for boosted frames too. Nevertheless, it is not clear if the divergence cancels for arbitrary spacelike surfaces $I$, and we are not aware of any resolution. Similar divergence occurs in computing the on-shell action, where the cancellation around ${\scriptstyle T}=\pi/2$ surface is again ensured by antipodal conditions.

In higher dimensions, $\delta A^{(1)}_{\rho}=0$, and no large $\rho$ divergence appears.

4 Three dimensions

This section is devoted to three dimensional Maxwell theory. The asymptotic symmetry at null infinity was discussed in Barnich:2015jua . The reason for separate consideration of three dimensional case is the simple form of solutions: dS2 is conformally flat and the solution is a whole set of left- and right-moving scalar modes. For this simplest case, we will translate the boundary conditions to Bondi coordinates $(u,r,\varphi)$.

4.1 Boundary conditions and solution space

The boundary conditions (3.16) for $d=3$ become 141414The static Coulomb solution is $\mathcal{F}_{tr}=q/r\,.$ The electric field in hyperbolic coordinates becomes $\mathcal{F}_{{\scriptscriptstyle T}\rho}=-q\,$.

[TABLE]

This boundary condition is realized by following fall-off on the gauge field

[TABLE]

The asymptotic behavior adopted here allows for moving charges in $2+1$ dimensions. At leading order, $F^{(0)}_{a\rho}=\partial_{a}A^{(0)}_{\rho}$. Let us denote $A^{(0)}_{\rho}$ by $\psi$ for notational harmony with higher dimensions. Its equation of motion is

[TABLE]

The differential operator is the Laplacian on $dS_{2}$, which takes a nicer form in coordinates $x^{\pm}=\varphi\pm{\scriptstyle T}$. The metric on $dS_{2}$ is

[TABLE]

The field equation(4.54) becomes

[TABLE]

The general solution with periodic boundary condition $\psi({\scriptstyle T},\varphi)=\psi({\scriptstyle T},\varphi+2\pi)$ is the following.

[TABLE]

4.2 Action principle and charges

The boundary term with fall-off (4.53) is finite

[TABLE]

Integration by parts and fixing the asymptotic gauge $D^{a}A^{(0)}_{a}=0$ makes the integrand a total divergence. In contrast to higher dimensions, fixing the Lorenz gauge $\nabla_{\mu}\mathcal{A}^{\mu}$ is not possible, because it implies either $\psi=0$ or $\mathcal{A}_{a}\sim\mathcal{O}(\rho)$.

The asymptotic gauge fixing leaves residual gauge transformations satisfying $D_{a}D^{a}\lambda=0$. The conserved charges are obtained by the same method explained before.

[TABLE]

Antipodal condition

The whole set of solutions (4.57) are regular at light cone. Nevertheless, we opt to impose conditions (3.33) which include physical solutions.

[TABLE]

The antipodal map $({\scriptstyle T},\varphi)\to(\pi-{\scriptstyle T},\varphi+\pi)$ is equivalent to $x^{+}\leftrightarrow x^{-}$. As a result, (4.57) is divided into even and odd parts

[TABLE]

By this boundary condition, the field strength is obtained by taking a derivative of $\psi$. One can explicitly check that for a boosted electric charge, the gauge field lies in (4.61a).

Charge and boundary conditions at null limit

Define

[TABLE]

Close to the future null infinity at ${\scriptstyle T}=0$, the fields behave as follows

[TABLE]

At null infinity, only the second term of the charge remains non-vanishing, so the charge is

[TABLE]

To make contact with results Barnich:2015jua let us rewrite the boundary conditions near null infinity $(\rho\to\infty,T\to 0)$ in Bondi coordinates $(u,r,x^{A})$. The coordinates are related by $u=-\rho{\scriptstyle T}/2$ and $r=\rho/{\scriptstyle T}$. Expanding the asymptotic gauge $D^{a}A^{(0)}_{a}=0$ we have

[TABLE]

This condition can be solved by introducing a scalar $\alpha({\scriptstyle T},\varphi)$ .

[TABLE]

We have to assign a fall-off for $\alpha$ around ${\scriptstyle T}=0$. Analyzing the dipole solutions, the appropriate condition is $\alpha({\scriptstyle T},\varphi)=\bar{\alpha}{\scriptstyle T}+\mathcal{O}({\scriptstyle T}^{2})$. Now we can find $\mathcal{A}_{u},\,\mathcal{A}_{r}$ and $\mathcal{A}_{\varphi}$ at leading order.

[TABLE]

These results must not be interpreted as null infinity boundary conditions. To account for electromagnetic radiation, there should exist one arbitrary function both of $u$ and $\varphi$, corresponding to the single helicity state of photon in three dimensions. Nontheless, (4.67) provides a consistent boundary condition at past of future null infinity, where the radiation has not yet started.

5 Discussion

In this paper, we considered asymptotic symmetries of Maxwell theory in three and higher dimension at spatial infinity. We tried to bypass standard methods for computing surface charges, by making the action principle well-defined, applying a gauge transformation on it, and interpreting the resulting conserved quantity as the charge. This work excludes magnetic charges to avoid technical difficulties, although they are discussed in various four dimensional treatments.

We showed that regularity of field strength tensor at light cone implies a certain antipodal condition on de Sitter space in four and higher dimensions, which was familiar in dS/CFT context. In addition, the charges depend on the scalar field $\psi$ on de Sitter space in all dimensions. It is interesting if dS/CFT quantum considerations applied to $\psi$ have implications on Maxwell theory.

In three dimension, the solution space is more transparent as the asymptotic de Sitter space is conformally flat. The light cone regularity argument does not work in three dimension, although it is satisfied by the solution for moving electric charges. For this simple model, we could solve the gauge condition and translate the boundary conditions into Bondi coordinates which are better suited for null infinity discussions.

As an interesting generalization, note that in three dimensions, non-trivial vorticity for gauge field is possible. Gauge transformations considered here are regular, so preserve vorticity. Addition of singular gauge transformations which lead to vorticity might lead to an unexpected relation with electric charges considered here; as is the case in four dimensions Hosseinzadeh:2018dkh ; Freidel:2018fsk .

Finally, we compared our treatment with Hamiltonian formulations of the theory. Symplectic form and on-shell action are finite in $d>4$ and their divergences $d=3,4$ cancel in inertial frames by virtue of parity conditions. Nonetheless, cancellation in arbitrary slice of asymptotic de Sitter space remains elusive.

Acknowledgements.

The author is greatly indebted to Shahin Sheikh-Jabbari and Hamid Afshar for illuminating discussions. He also thanks Miguel Campiglia, Vahid Hoseinzadeh, Roberto Oliveri and especially Ali Seraj for helpful comments on the draft.