Null Infinity and Unitary Representation of The Poincare Group

Following Pasterski-Shao-Strominger we construct a new basis of states in the single-particle Hilbert space of massless particles as a linear combination of standard Wigner states. Under Lorentz transformation the new basis states transform in the Unitary Principal Continuous Series representation. These states are obtained if we consider the little group of a null momentum \textit{direction} rather than a null momentum. The definition of the states in terms of the Wigner states makes it easier to study the action of space-time translation in this basis. We show by taking into account the effect of space-time translation that \textit{the dynamics of massless particles described by these states takes place completely on the null-infinity of the Minkowski space}. We then second quantize the theory in this basis and obtain a unitary manifestly Poincare invariant (field) theory of free massless particles living on null-infinity. The null-infinity arises in this case purely group-theoretically without any reference to bulk space-time. Action of BMS on massless particles is natural in this picture. As a by-product we generalize the conformal primary wave-functions for massless particles in a way which makes the action of space-time translation simple. Using these wave-functions we write down a modified Mellin(-Fourier) transformation of the S-matrix elements. The resulting amplitude is Poincare covariant. Under Poincare transformation it transforms like products of primaries of inhomogeneous $SL(2,\mathbb{C})$ ($ISL(2,\mathbb{C})$) inserted at various points of null-infinity. $ISL(2,\mathbb{C})$ primaries are defined in the paper.

transformation. In particular, the detailed analysis of [2] shows that the "Unitary principal continuous series representation" of the Lorentz group plays a central role in this whole construction. One of the interesting points about this basis of "conformal primary wavefunctions" is that the 4-D scattering amplitudes, expressed in this basis, transform like the correlation function of products of SL(2, C) primaries inserted at various points of the celestial sphere at null-infinity. This suggests a relation between 4-D scattering amplitudes and a 2-D Euclidean CFT defined on the celestial sphere. This 4D-2D correspondence is a hint of flat-space holography [1-3, 5, 6, 39, 40, 44-46].
In this paper we further explore this construction by looking at the single-particle Hilbert space of massless particles. Following [1][2][3] we define a new basis in the single where λ ∈ R and σ is the helicity of the massless particle. It is shown in the paper that or the Poincare group on the Heisenberg-picture states are shown to be, U (Λ) h,h, u, z,z = 1 (cz + d) 2h 1 (cz +d) 2h h ,h , u (1 + zz) |az + b| 2 + |cz + d| 2 , az + b cz + d ,āz +b cz +d e −il.P h,h, u, z,z = h,h, u + f (z,z, l), z,z where l µ is the space-time translation vector and So a massless particle described by the quantum state h,h, z,z can be thought of as living in a space(-time) with three coordinates given by (u, z,z). The action of the Poincare group on the (u, z,z) space(-time) is given by, Here the coordinate u is time-like because we get it from Hamiltonian evolution of the state. Now, as far as the action of the Poincare group is concerned, we can identify the three dimensional space-time as the null-infinity in Minkowski space with (u, z,z) identified as the Bondi coordinates at infinity. So from now on we will refer to this space as null-infinity but there is of course no "bulk space-time" metric here. The null-infinity arises purely group-theoretically and the dynamics of a free massless particle takes place in this space.
We then second-quantize the theory using the basis states h,h, z,z and write down Heisenberg-Picture creation and annihilation fields which live on the null-infinity. So a manifestly Poincare-invariant and unitary many-body theory of free massless particles can be formulated completely on ((u, z,z) space-time) null-infinity without any reference to bulk space-time.
The transition from the momentum space to null-infinity is direct. (z,z) start as coordinates in the momentum-space but once we take into account the dynamics, (z,z) together with the time coordinate u transmute into space-time coordinates, albeit of the boundary. This is also consistent with the current understanding of the relation between infrared structure of gauge and gravity theories and asymptotic symmetries of flat spacetime [4, 15-26, 31-36, 38].
This free theory has some interesting properties. For example, the transition amplitude given by, is SL(2, C) (Lorentz) covariant and is manifestly invariant under space-time translation because of the Dirac delta function. In fact the transition amplitude retains its form under more general "BMS supertranslation" given by (u, z,z) → (u + g(z,z), z,z) where g(z,z) is an arbitrary smooth function on the sphere. So we get a hint of supertranslation in a purely non-gravitational context. We discuss these things in detail in the paper.
This paper consists of two parts. In the first part we do the (2 + 1) case as a warm-up and also to check the correctness the procedure. In the first part we do not discuss the action of space-time translation. The second part consists of the (3 + 1) dimensional case which is our main interest. The two parts are more or less independent.
Throughout this paper we will use the mostly positive metric signature and use the conventions of Weinberg [48].

II. LORENTZ GROUP IN (2+1) DIMENSIONS
In (2+1) dimensions Lorentz group is SO(2, 1). In this paper we shall consider SL(2, R) which is the double cover of SO(2, 1). SL(2, R) is the group of real two by two matrices with determinant 1. The group elements will be denoted by, Let us now consider a momentum vector P = (P 0 , P 1 , P 2 ) and associate a real symmetric two by two matrix, denoted by the same letter P , defined as So the determinant gives the norm of the three vector P .
The SL(2, R) matrix Λ acts on P according to the following rule, Under this transformation P is again a real symmetric matrix and, det P = det P .
Therefore {±Λ} induces a Lorentz transformation of the three vector P .

A. Null Momenta
Let us now specialize to the case of null momenta. When P is null we can define a real variable z as, The variable z has the property that when P transforms under Lorentz transformation as P → P = ΛP Λ T , z transforms as, The geometrical interpretation of the variable z is that it parametrizes the space of Geometrically the space of null momentum directions in (2+1) dimensions is a circle parametrized by the angle θ and z is the stereographic coordinate on the circle. The Lorentz group SL(2, R) acts on the z coordinate according to Eq-2.5.

B. Standard Rotation
A standard rotation, R(z), is a rotation in the 1 − 2 plane which takes the refer- In matrix form, In the standard Wigner construction little group of a null vector plays a central role.
We want to consider instead the little group of a null direction. This is the subgroup of the Lorentz group which does not change the direction of a null vector. As a result, the set of all null vectors which point in the same direction, is closed under the action of this subgroup. Let us now explicitly write down the subgroup in (2+1) dimensions.
The generators of SL(2, R) can be written as (J 0 , K 1 , K 2 ), where J 0 is the generator of rotation in the 1 − 2 plane, K 1 is the generator of boost in the 1-direction and K 2 is the generator of boost in the 2-direction. Algebra satisfied by these generators is, Let us now consider the little group of the reference null direction given by the set
Let Λ be an arbitrary Lorentz transformation and U (Λ) its unitary representation in the Hilbert space which acts on |p as, If we take Λ to be the boost in the 2-direction with velocity v then where η is the rapidity defined as, tanh η = v.
Now the SL(2, R) matrix which generates boost in the 2-direction is given by , The transformation of the z variable under boost in the 2-direction is, z → e η z. So it is a scale transformation for which z = 0 is a fixed point. This is another way of seeing that

Change of Basis
The works of [1-3, 5, 6] suggest the introduction of the following states parametrized by a complex number ∆, Now if we apply a boost in the 2-direction with rapidity η we get, Let us now define states |∆, z which are related to |∆, z = 0 by standard rotation We can write down the states explicitly as, The inner product between two such states can be written as, where * denotes complex conjugation. This integral is convergent if we choose ∆ to be of the form ( 1 2 + iλ, λ ∈ R) [2]. With this choice of ∆ the states that we have defined are delta function normalizable, i.e,

Action of U (Λ)
Let us now calculate U (Λ) |∆, z where Λ is an arbitrary Lorentz transformation. If we use the definition of the state given in Eq-(2.15) we get, where W (Λ, z) is the Lorentz transformation given by, Now using the definition of the state |∆, z = 0 we get, So we have to find out the effect of the transformation R −1 (Λz)ΛR(z) on the set of null vectors of the form (E, 0, E), pointing in the positive 2-direction which is our reference direction. Since Lorentz transformation is linear it is sufficient to consider its effect on the vector k µ = (1, 0, 1). The transformations act on k µ in the following way : 1) R(z) sends the vector k µ = (1, 0, 1) to the vector k µ = (1, sin θ, cos θ). The corresponding z transforms from z = 0 to z = tan θ 2 . 2) Now we make an arbitrary Lorentz transformation Λ given by the SL(2, R) matrix, This transformation sends k to Λk and z = tan θ 2 to where Λθ is the angle made by Λk with the positive 2-direction.
The components of Λk are given by, We also have the relation, 3) Now the final rotation R −1 (Λz) rotates the vector Λk in 1-2 plane through an angle −Λθ and the resulting vector points again in the positive 2-direction. Similarly the rotation brings Λz = tan Λθ 2 back to z = 0 which is the reference null direction Therefore the resulting vector can be written as, Here we have made use of the facts that the 0-th component of a vector does not change under rotation in the 1 − 2 plane and the transformed vector is a null vector.
For a general vector k µ of the form (E, 0, E) we can write, where we have defined the boost factor e −η as, With this information we can now derive the transformation law as, So a general Lorentz transformation Λ acts on the states |∆, z like, Now using this it is easy to see that the states |∆, Λ , for a fixed ∆, form a representation of the Lorentz group. In order to see that let us consider two Lorentz transformations Where, and so U (Λ) form a representation of the Lorentz group when acting on the states |∆, z according to Eq-2.33. So we have representations of the Lorentz group parametrized by ∆ = 1 2 + iλ, λ ∈ R . This is also a unitary representation as one can easily check that, The representation labelled by a fixed ∆ is known as the "Unitary principal continuous series" representation of the Lorentz group [10] which has also appeared in several recent investigations of CFT [11][12][13].

III. REPRESENTATION ON WAVE FUNCTIONS (PACKETS)
The inner product between the states is given by, So we can write the completeness relation as, where |Ψ is an arbitrary massless one-particle state in the QFT. Using this the inner product between two states can be written as, where * denotes complex conjugation. We want to find out the effect of an arbitrary Lorentz transformation Λ on the wave function Ψ(∆, z). In order to do that we can write, Therefore the action of the Lorentz group on the wave-functions is, A. Differential Operators and Casimir The Lorentz group SL(2, R) is the group of conformal transformations of the line (or circle) parametrized by z. So let us consider dilatation, special conformal transformation and translation separately.

Translation
Let us consider translation given by, Now after a straightforward calculation, which we describe in the Appendix, we get, So, Now using Eq-3.5, we get,

Special Conformal Transformation (SCT)
Let us consider the SCT given by, After a starightforward calculation we get, So, In differential operator form we get,

Scale Transformation (ST)
Consider the scale transformation given by, It is easy to check that, So in differential operator form,

Value of the Casimir for ∆-Representation
The Casimir of the Lorentz group is given by, Let us now consider a representation with a specific value of ∆. According to our previous calculation, in such a representation, Now one can easily check that, This is the value of the Casimir obtained in [10].

Comments on The Representation of The Conformal Algebra
The algebra of Lorentz generators in (2 + 1) dimensions can be written as, This is also the algebra of global conformal group SL(2, R) in one dimension. But as we have seen the reality properties of the generators are different if we think of the SL(2, R) as acting unitarily on the Hilbert space of massless single particle states of a (2 + 1) dimensional QFT. For example in the standard highest weight representation of the conformal group, whereas in our case unitarity requires that, In four dimensions the same argument goes through unchanged except that the states now acquire an extra helicity index. For the sake of convenience we will collect the necessary formulas.
In four dimensions we can associate a hermitian matrix with a four momentum P µ = (P 0 , P 1 , P 2 , P 3 ) as, In (3 + 1) dimensions SL(2, C) is the double cover of the Lorentz group SO(3, 1) and an SL(2, C) matrix Λ acts on P as, In four dimensions the space of null directions is a two-sphere (a space-like cross section of the future light-cone in the momentum space). The stereographic coordinate of the two-sphere can be defined as, Here the projection is from the south-pole of the sphere so that the north-pole has coordinate z = 0, which corresponds to the family of null vectors, (E, 0, 0, E) E > 0 , pointing in the positive 3-direction. One can also check that under Lorentz transformation Λ, z transforms as, Now if we introduce spherical polar coordinates in momentum space then we can write a null vector P µ as, In this prametrization (θ, φ) become coordinates on the two-sphere. Its relation to the stereographic coordinates z is given by, Let us define the following state, where σ is the helicity. The momentum states have the standard normalisation given by, The action of the little group on the state |λ, σ, z = 0,z = 0 is given by, A |λ, σ, z = 0,z = 0 = B |λ, σ, z = 0,z = 0 = 0 (4.10) U R 3 (φ) |λ, σ, z = 0,z = 0 = e iσφ |λ, σ, z = 0,z = 0 (4.11) U B 3 (η) |λ, σ, z = 0,z = 0 = e η∆ |λ, σ, z = 0,z = 0 , ∆ = 1 + iλ (4.12) where φ is the angle of rotation around the 3-axis and η is the rapidity of the boost in the 3-direction. Now let us define the states |λ, σ, z,z as, where U R(z,z) is a unitary rotation operator defined as, 14) The rotation operator we have defined takes the standard null direction (E, 0, 0, E) z = 0 to the direction given by (E, E sin θ cos φ, E sin θ sin φ, E cos θ) z = tan θ 2 e iφ . Now with our choice of normalization the inner product between the states is given by,
We will evaluate this by multiplying the corresponding SL(2, C) matrices. Please see the appendix for our convention.
The matrices are given by, where, If we multiply these matrices the little group element W (Λ, z,z) can be written as, where e α = 1 + |z| 2 1 + |Λz| 2 1 (cz + d) 2 (4.22) We do not need the expression of β. So we get, Now it is easy to check that , Similarly, U (S(β)) = e βL 1 +βL 1 , S(β)z = z 1 − βz (4.26) where, For later use let us also write down the generator of translation in z. It is given by, Now let us take note of the fact that the generator of the special conformal transformations on z can be written as, where A and B are the elements of the little group of the null vector (E, 0, 0, E).
These are the generators which are set to zero on the states |E, 0, 0, E; σ to satisfy the requirement of a finite number of polarization states of a massless particle. Using these facts we get, where we have used the definition of the states |λ, σ, z,z and have defined, Hereh is not the complex conjugate of h.
Therefore, if we rename the state |λ, σ, z,z as h,h, z,z , we can write, This will also give us the physical/geometrical interpretation of these states.
There are four space-time translation operators given by, P µ = P 0 = H, P 1 , P 2 , P 3 .
Here Poincare action is realized completely geometrically modulo the little group factors. In fact we can think of a space parametrized by three coordinates (u, z,z) on which the Poincare group action is given by, where f is given above in Eq-5.6. This action of the Poincare group on the (u, z,z) space is the same as the action of the Poincare group at null-infinity in Minkowski space if we identify (u, z,z) with the Bondi coordinates. We would like to emphasize that the coordinates (u, z,z) arise purely group-theoretically and the coordinate u has a time-like character because we get it by unitary Hamiltonian evolution. Also there is no Poincare invariant metric in the (u, z,z) space.
Therefore the Heisenberg picture states h,h, u, z,z transform like the primary of the (Asymptotic) Poincare or inhomogeneous SL(2, C) denoted by ISL(2, C).
From a physical point of view, in the basis h,h, z,z for single-particle quantum states, massless particles can be thought of as living at null-infinity. More importantly, the Poincare invariant dynamics takes place at infinity. This has the flavour of "holography". We will come back to this later.

A. Derivation
We will not give the details of the algebra -leading to Eq-5.4 and Eq-5.5 -which is identical to whatever we have done before in the case Lorentz transformations. Let us just mention that the state h,h, z = 0,z = 0 has the following crucial property which is required for this thing to work. It satisfies, e ia(H−P 3 ) h,h, z = 0,z = 0 = e iaP 1 h,h, z = 0,z = 0 = e iaP 2 h,h, z = 0,z = 0 = h,h, z = 0,z = 0 where a is a real number and , This is the same standard state given in Eq-4.8 defined at the beginning except that we have replaced (λ, σ) with (h,h).

A Geometric Argument
Let us now give a geometric argument which makes it clear that why Here P is a representative from the equivalence class P ∼ αP α > 0 specifying the direction of the null normal. The null geodesic generators of the null hyperplane are also parallel to P . Now let us consider a Lorentz transformation Λ, in the active sense, which maps the point X lying on the null hyperplane to ΛX. Now if ΛX also belongs to the null hyperplane then, (ΛX).P = 0 = X.(Λ −1 P ). Therefore we must have, This has three solutions corresponding to translations in the direction of P itself and in the two space-like directions orthogonal to P . These are generated by H − P 3 , P 1 , P 2 if we take P to be (1, 0, 0, 1).
The remaining three generators of the Poincare group act non-trivially on the space of null-hyperplanes. Now a null-hyperplane can be thought of as the past light-cone of a point of the future null-infinity (or the future light-cone of a point of the past nullinfinity) 1 . So the space of null-hyperplanes in Minkowski space can either be thought of as the future null-infinity or the past null-infinity. The number of parameters also match.
There are three-parameter family of null hyperplanes corresponding to the three Poincare 1 See for example [47].
generators which act non-trivially on the space of null hyperplanes and the null infinity is also three dimensional. This shows that we can think of the massless particle, described by a quantum state associated with a null-hyperplane, as sitting at one point at null infinity.
Let us now analyse the state h,h, u, z,z = e iHu U (R(z,z)) h,h, z = 0,z = 0 . As we have already described, we can associate the state h,h, z = 0,z = 0 with the nullhyperplane X.P = 0 with P ∼ (1, 0, 0, 1) which passes through the origin. Now U (R(z,z)) rotates the state and the rotated state corresponds to the null-hyperplane X.P = 0 where P = R(z,z)P . In this way we generate the 2-parameter family of null-hyperplanes all of which pass through the origin. They represent the family of states h,h, z,z with varying (z,z). Now the rest of the null hyperplanes are generated by time translating this two parameter family of null hyperplanes. This is essentially the action of e iHu on the states h,h, z,z . This explains that why the states h,h, u, z,z transform under the asymptotic Poinacre group.
In passing we would like to point out that the generators J 3 , K 3 , A = J 2 − K 1 , B = −J 1 − K 2 , H − P 3 , P 1 , P 2 of the Poincare group are also the kinematic generators in the Light-front quantization. It will be interesting to see if the states in the lightfront quantization are related to the asymptotic states h,h, u, z,z by some non-local transformation.

VI. CREATION AND ANNIHILATION FIELDS AT NULL-INFINITY
In this section we change our notation a little bit and define, Let us now introduce Heisenberg-Picture creation operator A † λ,σ (u, z,z) corresponding to the states |λ, σ, u, z,z such that, and Similarly the transformation property of the corresponding annihilation operator A λ,σ (u, z,z) is given by, ,āz +b cz +d (6.5) and e −il.P A λ,σ e il.P = A λ,σ (u + f (z,z, l), z,z) (6.6) where * denotes complex conjugation.
Now the question is how are these creation/annihilation operators related to the standard creation/annihilation operators in the momentum eigenstate basis ? The simplest anwer is given by, and a † (p, σ) is the creation operator in the momentum-helicity basis |p, σ . This is essentially a rewriting of the relation between the basis states |λ, σ, u, z,z = 1 √ 8π 4 in second-quantized notation.
The (anti) commutator between the creation and annihilation operators is given by, Eq-(6.7, 6.8) and the transformation properties under Poincare group show that we should interpret A λ,σ (u, z,z) and A † λ,σ (u, z,z) as the positive and negative frequency annihilation and creation fields living on null infinity in the Minkowski space. So we have gone directly from the momentum space to the boundary of the Minkowski space-time. In other words, (z,z) start as coordinates in the momentum-space but once we take into account the dynamics, (z,z) together with the time coordinate u transmute into coordinates of the boundary of the Minkowski space-time.
It will be interesting to relate this to the notion of asymptotic quantization [14]. One difference is that here we do not arrive at the quantum theory on the null-infinity by quantizing a classical theory living on null-infinity. It will be interesting to clarify these things.
Now for the sake of convenience we define a primary operator of (Poincare) ISL(2, C) as any Heisenberg picture operator φ h,h (u, z,z) transforming as, The (h,h) are defined to be, We emphasize that the "primary" here does not refer to any highest-weight representation. The basic creation and annihilation fields at null-infinity that we have defined in the last section are examples of such primary operators.

VIII. CONSTRAINTS FROM TRANSLATIONAL INVARIANCE
We will now work out some kinematical constraints on correlation functions of ISL(2, C) primaries in Minkowski vacuum which follow from "bulk" translational invariance. Under space-time translation by an arbitrary four vector l µ , u shifts by, whereas z does not change.
The correlation functions are invariant under (Poincare) ISL(2, C) transformation, i.e, where φ i (P i ) is some ISL(2, C) primary operator inserted at the point P i = (u i , z i ,z i ) and U (l, Λ) is a (Poincare) ISL(2, C) transformation. |Ω is the Poincare invariant vacuum.
Let us start from the 4-point function.

4-point function
We denote the 4-point function by A(P 1 , P 2 , P 3 , P 4 ). We first make a Lorentz transformation, Λ, to map the points (z 1 , z 2 , z 3 , z 4 ) → (z, 1, 0, ∞) where z is the cross ratio of the four points given by, Under this Lorentz transformation the u's also transform to some other values say u .
So we can write, and G has no u dependence. Therefore we need to consider the 4-point function at these special values of z i 's. Now translational invariance requires that, A(P 1 , P 2 , P 3 , P 4 ) = A(P 1 , P 2 , P 3 , P 4 ) (8.6) where Here we have used the fact that under translation z i does not change. For z = 0, 1, ∞ we have, Now if we take the translations to be infinitesimal then we get four differential equations given by, Therefore the most general non-trivil solution for the 4-point function can be written as, where z is the the cross ratio of the four points (z 1 , z 2 , z 3 , z 4 ). So the four point function will be zero unless the cross ratio of the four insertion points are real. This is the same constraint obtained in [3] from the study of Gluon scattering amplitude in the basis of conformal primary wave functions. Recently this has also been studied in detail in [43].

3-point function
In the case of the 3-point function one can repeat the same procedure except that now there is no cross ratio. We get 3 differential equations given by, This has the trivial solution A 1 = A 2 = A 3 = 0. Therefore the most general non-trivial solution of the 3-point function is, This corresponds to the fact that in (− + ++) signature 3-point scattering amplitude of massless particles vanish for generic null momenta.

2-point function
The general solution of the 2-point function can be written as, where N (λ 1 , λ 2 ) is a prefactor which cannot be determined solely from the Poincare invariance. There should also be an i prescription to take care of the singularity at u 1 = u 2 but this cannot be determined just from symmetry consideration.

IX. A HINT OF SUPERTRANSLATION
Let us first consider the transition amplitude for a free massless particle in the |λ, σ, z,z basis, given by λ, σ, z,z| e −iH(u−u ) |λ , σ , z ,z = λ, σ, u, z,z|λ , σ , u , z ,z The Dirac delta function arises because a free particle does not change its direction 2 of motion. Due to the presence of δ 2 (z − z ) the amplitude is manifestly invariant under space-time translations under which u → u + f (z,z, l) and z remains unchanged. Now it is easy to see that for the same reason the amplitude is in fact invariant under more general transformation u → u + g(z,z) where g(z,z) is an arbitrary smooth function on the sphere. These can be identified with BMS supertranslations [7]. At this stage supertranslation invariance is just an accidental symmetry which is manifest in this basis.
The dependence of a general two point function on the coordinates of the null-infinity is completely fixed by the Poincare invariance. It is clear that the 2-point function given in Eq-8.18 is also invariant under BMS supertranslations. This is also an accidental symmetry of the correlation function.
The supertranslation invariance of the transition amplitude of a free particle or the 2point function is trivial from physical point of view [4]. In fact this cannot be the case for higher-point functions (in some interacting theory) because BMS is spontaneously broken in the Minkowski vacuum [4]. But, in some sense, this shows that perhaps it is "natural" to consider extension of the Poincare group to the BMS group once the field theory is formulated on null-infinity. This is also suggested by Strominger's conjecture about BMS invariance of the gravitational S-matrix [4] and recent advances in understanding the relation between the infrared structure of gravity and asymptotic symmetry  of flat space-time.
In fact the conjecture of [8,9] and the recent works [17-24, 39, 40] suggest that that the superrotation symmetry plays a central role in a holographic reformulation of flat space physics. So the full symmetry group should perhaps be the (extended) BMS including superrotation. The representation of the BMS group in three space-time dimensions has been studied in [41,42].
We would like to emphasize that this BMS should not be thought of as an asymptotic symmetry group [7][8][9]. There is no dynamical gravity in this picture and in fact we did not require the presence of any "bulk space-time". The transition from momentum space to null-infinity was direct. Here the Supertranslation is some geometric transformation of the background space-time parametrized by (u, z,z) under which 2-point function or the commutator (Eq-6.11) turns out to be symmetric. But so far the symmetry group which acts on states in the Hilbert space is still the four dimensional Poincare group or ISL(2, C).
The fact that we get a hint of supertranslation in a purely non-gravitational setting is perhaps an indication of holography in asymptotically flat space-time. This is somewhat similar to the situation in AdS 3 -CFT 2 correspondence. The infinite dimensional Virasoro symmetry of a two-dimensional conformal field theory can be understood in a purely field theory setting without dynamical gravity. But in asymptotically AdS 3 spaces Virasoro can also be understood as the asymptotic symmetry group of the bulk gravity theory. These two facts nicely fit together in AdS 3 -holography. Similarly the goal here is to understand BMS in a purely non-gravitational setting. If such an understanding can be reached in a consistent manner then that will perhaps be an indication of flat-space holography.

X. CONFORMAL PRIMARY WAVE-FUNCTIONS WITH TRANSLATIONAL INVARIANCE
For simplicity we consider the massless scalar conformal primary wave functions [2,5,6].
The above procedure has a simple geometric interpretation. As we have described in section-(V A 1) the states h,h, u, z,z are associated with null-hyperplanes in the Minkowski space. The original 2-parameter family of conformal primary wave functions, given by Φ ± ∆ (X µ |z,z) (Eq-10.1), are singular along the null hyperplanes X · Q(z,z) = 0 passing through the origin. They correspond to the states h,h, u = 0, z,z . So to get the rest of the states with nonzero u we have translate the family X · Q(z,z) = 0 along the time axis. This gives rise to the wave functions Φ ± ∆ (X µ |u, z,z) (Eq-10.5) singular along the null-hyperplanes X · Q(z,z) + u = 0. This 3-parameter family exhausts all the nullhyperplanes in the Minkowski space and correspond to the family of states h,h, u, z,z located on null-infinity. This makes it clear that why the Poincare group has simple action on the wave-functions Φ ± ∆ (X µ |u, z,z).
Now suppose instead of plane-waves we use the wave functions Φ ± ∆ i (X µ |u i , z i ,z i ) as the wave functions of the external particles then we can write down a modified Mellintransform as, where h i =h i = ∆ i 2 and we have also defined i = +1 for an outgoing particle and i = −1 for an incoming particle. This is a slightly modified form of the Mellin amplitude defined in [1,3].
A amplitude behaves like correlation function of ISL(2, C) primaries inserted at points at null infinity and transform under the Asymptotic Poincare Group as, where Λu i = (1 + z izi )u i |az i + b| 2 + |cz i + d| 2 and f (z,z, l) is given by Eq-10.8.
It will be interesting to compute theÃ amplitude for some field theories along the lines of [3,[44][45][46].
TheÃ amplitudes can be thought of as the "scattering amplitude with asymptotic states" given by h i ,h i , u i , z i ,z i . It is very likely that a relation of the type,