Canonical Realization of BMS Symmetry. Quadratic Casimir

We study the canonical realization of BMS symmetry for a massive scalar field introduced in reference \cite{LM}. We will construct an invariant scalar product for the generalized momenta. As a consequence we will introduce a quadratic Casimir with the supertranslations.

Motivation and results. -Recently there has been a renewed interest in the BMS group [1]. It has been shown the BMS invariance of the gravitational scattering [2], the relation among soft graviton theorems [3] and BMS supertranslations [4]. The relation among supertranslations, gravitational memory and soft gravitons theorems has been also studied [5], There is also the proposal that BMS group could be useful to understand holography in asymptotically flat space times [6] [7] [8] [9]. Related to these developments has been the study of asymptotic symmetries in quantum field theories, in the case of QED, see [10] for the massless case and [11] [12] for the massive case. A recent overview on the whole subject is given by Strominger at Strings 2015 [13].
In this work we study the canonical realization of the BMS symmetry for a free massive real scalar field in four dimensions introduced in [14]. The Poincare generators P µ , M µν are written in terms of the Fourier modes a( k), a * ( k) of the plane wave expansion of the Klein-Gordon field. The momentum mass-shell condition q 2 − m 2 = 0 defines a 3d dimensional space like-hyperboloid H 1 3 [15]. It is useful to introduce a differential operator in this space D = −m 2 ∆ + 3, where ∆ is the Laplace-Beltrami operator on H 1 3 . It happens that the four-dimensional momenta k µ are zero modes of D, this suggest to look for the zero modes of this operator in general. These are given by an infinite set of function w l,m , defined on H 1 3 , unique up to rescalings, l = 0, 1, 2, ... | m |≤ l . The explicit expression of this functions was given in [14], see also next section. The functions w l,m can be considered as a generalization of four momenta.
This allows to define the supertranslations for massive scalar field P l,m in terms of w l,m and the Fourier modes a( k), a * ( k), see next section.
The transformation of the scalar field under spacetime translations is obtained from the scalar nature of the field under Poincare transformations. This not the case for supertranslations. The supertranslations acts on the field and their conjugated momenta as a non-local linear canonical transformation. All together P l,m , M µν give the infinite vector representation of the BMS group. Note that the appearance of the BMS symmetry introduced for the scalar field [14] is not an asymptotic gauge symmetry as in [1].
We will construct a BMS invariant scalar product for the generalized momenta w l,m , or for a rescaling of them. We write this product in terms of an infinite dimensional matrix η l,m;l ′ ,m ′ that generalizes the Minkowski metric η µν for the scalar product of 4d momenta k µ . In a suitable basis of these zero modes η l ′ ,m ′ ;l.m = δ l ′ ,l δ m ′ ,m . The convergence of this scalar product has also been studied 1 . The scalar product is the key ingredient for the definition of a configuration space, see, for example [16], and therefore to give meaning to coordinates x l.m conjugated to w l,m .
Using this infinite dimensional metric we will construct a quadratic Casimir with the supertranslations "P 2 " = η l,m;l ′ ,m ′ P * l,m P l ′ ,m ′ . Physically this Casimir allows us to define the BMS masshell constraint.
Our analysis of the scalar product and the Casimir "P 2 " will be useful to study BMS symmetries and it 1 Details and prove of the results will presented elsewhere be useful to the study of scattering S-matrix, since the in and out states are free fields, also to study particles, strings,.. with BMS symmetries. The canonical realization of BMS considered in [14] and here could be extended to other fields with non-vanishing spin and to a massless fields.
Canonical Realization of BMS Symmetry. -We consider a free real scalar field Φ(t, x) of mass m in four dimensions. The Fourier expansion is given by , The realization of the Poincare group in terms of Fourier modes of the scalar field Φ(t, x) is given by as one check the algebra by using the Poisson brack- The mass-shell condition for Φ(t, x) is given by q 2 − m 2 = 0; it defines the 3d space like hyperboloid, H 1 3 , in the space of momenta, with coordinates k. The relation with the embedding momenta q µ is q 0 = k 2 + m 2 , q i = k i . The induced metric on H 1 3 is given by In polar coordinates ds 2 = −m 2 ( 1 r 2 +1 dr 2 + r 2 ds 2 S 2 ) where r = | k| m , and ds 2 S 2 is the metric of the sphere. This expression, apart dimensions, is used in [11] to give a foliation of the Minkowski space-time.
The coordinates are a global parametrization of the Euclidean AdS 3 space. The Laplace Beltrami operator on H 1 3 is given by It is an elliptic operator and has the property ∆k µ = 3 m 2 k µ . We introduce the operator D D = −m 2 ∆ + 3, the four momenta are zero modes of D, Dk µ = 0. Since we want to find a generalization of four momenta to construct the supertranslations this property suggest to study all zero modes of Df (k) = 0. Introducing spherical coordinates, these are given by the functions [14] w l,m ( k) = u l (r)Y l,m (k), u l (r) = r l F ((l − 1)/2, (l + 3)/2; l + 3/2; −r 2 ), (9) where F is the Hypergeometric function 2 The functions w l,m span an infinite dimensional non unitary representation of the Lorentz group. This representation has the following properties i) it is the only representation [14] with an invariant subspace of dimension 4, ii) for l = 0, 1 w l,m is the four-vector k µ in the spherical basis iii) the functions w lm (r, θ, φ) have an asymptotic behavior like k µ , for all values of l = 0, 1, 2, ....
The presence of the zero modes w l,m ( k) enables us to define the supertranslations in terms of Fourier modes In reference [14] it is proved that these integrals are well defined and that M µν , P l,m verifiy the BMS algebra. BMS Invariant Scalar product and Quadratic Casimir. -Since we have an infinite set of function w l,m that generalizes the four momenta k µ , it is natural to ask if there exists a scalar product, invariant under BMS, which generalizes the usual Minkowski metric.
The strategy to construct this scalar product will be to require the hermicity of Lorenz generators acting on the set w l,m or a rescaling of them. We first consider the boost K 3 = M 0j . We look for a new basis of zero modes {k l,m } where the hermiticity is studied using the diagonal scalar product, for l > 1, with For l = 0, 1 we will use the ordinary Minkowski metric.
The .
(15) The factor M (l) is introduced in order to have simple behaviour for l → ∞ since we have |w l,m | ≤ 2l+1 4π √ 1 + r 2 M (l). The factor N (l) is unknow it is determined by imposing the hermiticity of K 3 . We have where a l,m = −i(l − 1)C l+1,m , b l,m = i(l + 2)C l,m and The generator K 3 will be self adjoint if which implies If we define E(l) = [N (l)] 2 , we get the recurrence relation This recurrence equation is meaningful only for l ≥ 2. The solution is given by where E(2) has an arbitrary value. The action of K 3 on {k l,m }, apart from a phase factor, is where A l,m = −i (l − 1)(l + 3)C l+1,m , Note that The action of the other generartosrs can be obtained from K 1 = i[K 3 , L 2 ], K 2 = −i[K 3 , L 1 ], and the standard action of L 3 and L ± We obtain where A and B are defined in equation (21) while D and E are and The hermicity of the Lorentz generartors implies the invariance of the scalar product under Lorentz transformations. It is also invariant under supertranslations, since these form an abelian algebra.
If we define the supertranslations using the set of zero modes {k l,m } P l,m = R 3d k k l,m ( k)a * ( k)a( k).
The BMS algebra becomes [M i,j , P l,m ] = ǫ ijk P l ′ ,m ′ (L k ) l ′ ,m ′ ;l,m [M 0,j , P l,m ] = −P l ′ ,m ′ (K j ) l ′ ,m ′ ;l,m , where the matrices L k and K j are defined in equations (21), (23), (24) and (25). This representation gives the vector representation of the BMS group. Let us observe that the supertranslations act as a canonical, but non local, transformation of the scalar field and its momentum, see [14]. The invariant scalar product allows to define a quadratic Casimir for the supertranslations "P 2 " = η l,m;l ′ ,m ′ P * l,m P l ′ ,m ′ One can check ["P 2 ", BM S] = 0 using (29). Therefore the BMS masshell condition is given by η l,m;l ′ ,m ′ P * l,m P l ′ ,m ′ = m 2

BMS
(31) This condition will be be useful to construct particles, strings,.. with BMS symmetries.