Carrollian Conformal Fields and Flat Holography

Abstract

The null conformal boundary $\mathcal{I}$ of Minkowski spacetime $\mathbb{M}$ plays a special role in scattering theory, as it is the locus where massless particle states are most naturally defined. We construct quantum fields on $\mathcal{I}$, which create these massless states from the vacuum and transform covariantly under Poincaré symmetries. Because the latter symmetries act as Carrollian conformal isometries of $\mathcal{I}$, these quantum fields are Carrollian conformal fields. This group theoretic construction is intrinsic to $\mathcal{I}$ by contrast to existing treatments in the literature. However, we also show that the standard relativistic massless quantum fields in $\mathbb{M}$, when pulled back to $\mathcal{I}$, provide a realisation of these Carrollian conformal fields. This correspondence between bulk and boundary fields should constitute a basic entry in the dictionary of flat holography. Finally, we show that $\mathcal{I}$ provides a natural parametrisation of the massless particles as described by irreducible representations of the Poincaré group and that in an appropriate conjugate basis, they indeed transform like Carrollian conformal fields.

1. Introduction

Central to the edifice of quantum field theory is the construction of states furnishing unitary representations of symmetry groups, together with the construction of local and covariant quantum fields creating these states out of the vacuum. For the important case of the Poincaré group $ISO(1,3)$, the unitary irreducible representations (UIR) were constructed by Wigner through the method of induced representations [1]. Systematic construction of covariant quantum fields was provided only later by Weinberg [2,3,4], while Lagrangian field equations were given in [5,6,7,8]. However, it is important to emphasise that a choice is typically being made when constructing these fields, namely that they should be functions over Minkowski spacetime $\mathbb{M}$, while this is not the only available option. In this paper, we wish to consider an alternative to this standard procedure, namely the construction of local and covariant quantum fields as functions defined over null infinity $\mathcal{I}=\mathbb{R}\times {\mathbb{R}}^2$ which is the null conformal boundary of $\mathbb{M}$. Our discussion will, in fact, apply to arbitrary spacetime dimension and corresponding symmetry group $ISO(1,d)$.

The main motivation for the present work comes from the program of flat/celestial holography, which aims at a better understanding of asymptotically flat quantum gravity by exploiting the vast amount of asymptotic symmetries available, and by use of conformal methods. For an account of this subject, we refer the reader to the reviews [9,10,11] and references therein. One basic but important aspect of flat holography is to provide a dictionary between standard relativistic quantum fields in $\mathbb{M}$ and conformal quantum fields at $\mathcal{I}$. In that respect, two approaches have been pursued, either by further reducing $\mathcal{I}$ along its null direction down to the two-dimensional Riemann sphere or by keeping this null direction manifest. Most of the literature has followed the first approach, with the result that one associates a continuum of two-dimensional $SL(2,\mathbb{C})$ primary fields to a given relativistic bulk field [12,13,14,15,16,17,18,19,20,21,22,23,24]. The second approach has the definite advantage that full $ISO(1,3)$ covariance is kept manifest, with the action of the Poincaré group realised as conformal isometries of $\mathcal{I}$. In that case, a single Carrollian1 conformal field at $\mathcal{I}$ is found to correspond to a given relativistic quantum field in $\mathbb{M}$[29,31,32,33,34,35]. However, we believe that a systematic discussion of the correspondence between relativistic bulk fields, Carrollian conformal fields and particle states is to some extent incomplete, and we intend to close this gap here2. Nonetheless, our analysis is complementary and has been inspired by previous works [25,31,32,33,34]. We claim that the holographic dictionary between relativistic fields in $\mathbb{M}$ and Carrollian conformal fields on $\mathcal{I}$ can be deduced solely from group theoretic considerations. All that is required is to understand how to turn Wigner’s UIRs into local covariant quantum fields defined over either $\mathbb{M}$ or $\mathcal{I}$. The holographic correspondence then relates the two types of fields (relativistic and Carrollian) that are associated with a given UIR.

The paper is structured as follows. In Section 2, we recall the form of the covariant quantum fields in ${\mathbb{M}}^{d+1}$ which create the massless representations of $ISO(1,d)$, together with the free field equations that they satisfy. In Section 3, we independently construct the Carrollian conformal fields on $\mathcal{I}=\mathbb{R}\times {\mathbb{R}}^{d-1}$ that can be associated with the very same massless representations, exploiting the fact that $ISO(1,d)$ is the group of conformal isometries of $\mathcal{I}$. Just as in the standard case of relativistic conformal fields [36], this is done through the method of induced representations, starting from a finite-component representation of the stability subgroup of the origin of $\mathcal{I}$. We further show that Carrollian fields at $\mathcal{I}$ can only possibly create massless states, which is how group theory tells us that massive fields are supported at future timelike infinity $i^{+}$ rather than at null infinity. In Section 4, we explicitly connect the two independent constructions by showing that the Carrollian conformal fields arise as asymptotic limits of the standard relativistic massless fields. To be more specific, we adopt retarded coordinates $(r,u,x^i)$ such that null infinity $\mathcal{I}$ corresponds to the limit $r\to \infty$. In that limit, we show that the independent gauge-invariant components of the relativistic massless fields precisely behave as Carrollian conformal fields under $ISO(1,d)$ transformations. This provides an explicit dictionary between relativistic massless fields in ${\mathbb{M}}^{d+1}$ and Carrollian conformal fields at $\mathcal{I}$. In Section 5, we provide a more direct relation between the massless particle states and the Carrollian conformal fields. First, the irreducible massless representations of the Poincaré group are constructed in momentum space by boosting the rest frame states using the generators from outside the little group as originally set out by Wigner. Taking their Fourier transform to Minkowski spacetime and then the $r\to \infty$ limit, the corresponding states on the boundary $\mathcal{I}$ are found. It is shown that the parametrisation of the states is very closely related to the coordinates $(u,x^i)$ on $\mathcal{I}$. Indeed the coordinates $x^i$ are precisely the same as those in the parametrisation of the boost while the remaining coordinate u is a conjugate coordinate in the sense of Fourier transform. Indeed $\mathcal{I}$ encodes in an unconstrained way the kinematics and the states of a massless particle, essentially as they appear in the irreducible representation of the Poincaré group which characterises it. The physical states that emerge on $\mathcal{I}$ transform in a way reminiscent of the Carrollian conformal fields of Section 3. Embedding the physical states in a larger representation of the type used to display the Poincaré transformations in a manifest manner, we find states on $\mathcal{I}$ that now transform exactly like the Carrollian conformal fields constructed in Section 3.

2. Relativistic Massless Fields

Fundamental to the standard scattering theory in $(d+1)$-dimensional Minkowski spacetime ${\mathbb{M}}^{d+1}$ is the assumption that asymptotic states belong to the tensor product of unitary irreducible representations (UIR) of the isometry group $ISO(1,d)$, and more specifically those induced by finite-dimensional representations of the corresponding (short) little groups [1,37]. In the case of massless states, the helicity representations are those which carry finite-dimensional UIRs of the short little group $SO(d-1)$. Single-particle states $|p,\sigma \rangle$ belonging to a given helicity representation are labelled by a null momentum $p^{\mu }$ and a discrete label $\sigma$ running over the internal spin degrees of freedom. Furthermore, they are built out of the vacuum by the corresponding creation operators,

$$|p,\sigma \rangle \equiv a_{p,\sigma }^{†}|0\rangle .$$

(1)

In the present work, we will restrict our attention to totally symmetric tensor helicity representations. In four spacetime dimensions ($d=3$), totally symmetric representations are the only available ones, while in higher dimensions, there is also the possibility of mixed symmetry [38,39,40,41,42,43].

To a given helicity representation, one can associate a local quantum field which creates single-particle states out of the vacuum and transforms covariantly albeit non-unitarily. This local quantum field takes the generic form,

$${\varphi }_{{\mu }_1\dots {\mu }_s}\left(x\right)=\sum _{\sigma }\int d^{d}p{\varphi }_{{\mu }_1\dots {\mu }_s}^{p,\sigma }\left(x\right)a_{p,\sigma }^{†},$$

(2)

with $d^{d}p$ a Lorentz-invariant measure on the lightcone $p^2=0$. The intertwiners ${\varphi }_{{\mu }_1\dots {\mu }_s}^{p,\sigma }\left(x\right)$ are determined precisely such that the unitary transformation of the creation operators $a_{p,\sigma }^{†}$ is converted into covariant transformation of the local quantum field ${\varphi }_{{\mu }_1\dots {\mu }_s}\left(x\right)$[37]. It turns out that these intertwiners are just the (gauge-invariant) positive-frequency solutions to a set of covariant field equations. Here we recall the form of these equations and refer the reader to [43,44,45] for a complete review. The quantum field associated with a spin-s helicity representation is a totally symmetric tensor ${\varphi }_{{\mu }_1\dots {\mu }_s}$ satisfying the covariant field equations

$$\begin{array}{c}\hfill \square {\varphi }_{{\mu }_1\dots {\mu }_s}=0,{\nabla }^{\nu }{\varphi }_{\nu {\mu }_2\dots {\mu }_s}=0,{\varphi }^{\nu }{}_{\nu {\mu }_3\dots {\mu }_s}=0,\end{array}$$

(3)

subject to the gauge redundancy

$$\begin{array}{c}\hfill \delta {\varphi }_{{\mu }_1\dots {\mu }_s}={\nabla }_{({\mu }_1}{\epsilon }_{{\mu }_2\dots {\mu }_s)},\end{array}$$

(4)

where the totally symmetric gauge parameter ${\epsilon }_{{\mu }_1\dots {\mu }_{s-1}}$ itself satisfies

$$\begin{array}{c}\hfill \square {\epsilon }_{{\mu }_1\dots {\mu }_{s-1}}=0,{\nabla }^{\nu }{\epsilon }_{\nu {\mu }_2\dots {\mu }_{s-1}}=0,{\epsilon }^{\nu }{}_{\nu {\mu }_3\dots {\mu }_{s-1}}=0.\end{array}$$

(5)

The field Equation (3) constitute a partially gauge-fixed version of Fronsdal equations [7]. While the wave equation is the Casimir equation ${\mathcal{C}}_2\equiv P^{\mu }{P}_{\mu }=0$ characterising massless states, the remaining equations can be used to show that the gauge-invariant tensor components precisely reduce to that of a traceless symmetric tensor ${\varphi }_{i_1\dots i_s}$ furnishing a spin-s representation of the short little group $SO(d-1)$ [46].

The notion of locality and covariance discussed above is that of local tensor fields in ${\mathbb{M}}^{d+1}$. We will be interested in a distinct notion of covariance with respect to a null plane $\mathcal{I}=\mathbb{R}\times {\mathbb{R}}^{d-1}$ of codimension one, where the map between those two descriptions is exactly what should constitute the basics of the flat holographic dictionary.

3. Carrollian Conformal Fields

The Poincaré group $ISO(1,d)$ naturally acts as the group conformal isometries3 of a generic null plane $\mathcal{I}=\mathbb{R}\times {\mathbb{R}}^{d-1}$ equipped with the canonical metric

$$d{s}_{\mathcal{I}}^2=q_{\alpha \beta }d{x}^{\alpha }d{x}^{\beta }=0 d{u}^2+{\delta }_{ij}d{x}^{i}d{x}^j.$$

(6)

This metric is degenerate, i.e., there is a vector $n^{\alpha }$ such that

$$n^{\alpha }{q}_{\alpha \beta }=0.$$

(7)

In the coordinate system $x^{\alpha }=(u,x^i)$, this vector is simply given by $n^{\alpha }=(1,0^i)$.

Perhaps the easiest way to see that $ISO(1,d)$ is the conformal group associated with $\mathcal{I}$ is to obtain it as an Inönü-Wigner contraction of the conformal group $SO(2,d)$ of ${\mathbb{M}}^d$. Historically this is exactly how Lévy-Leblond introduced the notion of Carrollian isometries [53]. The relevant contraction corresponds to the ultrarelativistic/Carrollian limit $c\to 0$, upon which

$${\mathbb{M}}^d\to \mathcal{I},SO(2,d)\to ISO(1,d).$$

(8)

This contraction is explicitly performed in the Appendix A and allows to derive the subsequent formulae, some of which can also be found in [25,34].

Infinitesimally the Carrollian conformal isometries under consideration take the form $x^{\prime \alpha }=x^{\alpha }+{\zeta }^{\alpha }$ in terms of the vector field

$$\begin{array}{cc}\hfill {\zeta }^u& =a+b_i{x}^i+k{x}^2+(\lambda -2k_i{x}^i)u,\hfill \\ \hfill {\zeta }^i& =a^i+{\omega }^i{}_j{x}^j+\lambda x^i+k^i{x}^2-2k_j{x}^j{x}^i.\hfill \end{array}$$

(9)

Alternatively this can be written

$$\begin{array}{cc}\hfill x^{\prime \alpha }& =(1+iaH+i{a}^i{P}_i+\frac{i}{2}{\omega }^{ij}{J}_{ij}+i{b}^i{B}_i+i\lambda D+ikK+i{k}^i{K}_i)x^{\alpha },\hfill \end{array}$$

(10)

in terms of the differential operators

$$\begin{array}{cccc}\hfill P_i& =-i{\partial }_i,\hfill & \hfill J_{ij}& =i(x_i{\partial }_j-x_j{\partial }_i),\hfill \\ \hfill D& =-i(u{\partial }_u+x^i{\partial }_i),\hfill & \hfill K_i& =-i(x^2{\partial }_i-2x_i(u{\partial }_u+x^j{\partial }_j)),\hfill & \hfill K& =-i{x}^2{\partial }_u,\hfill \\ \hfill H& =-i{\partial }_u,\hfill & \hfill B_i& =-i{x}_i{\partial }_u,\hfill \end{array}$$

(11)

that satisfy the algebra

$$\begin{array}{cccc}\hfill \left[J_{ij},J_{mn}\right]& =-i\left({\delta }_{im}{J}_{jn}+{\delta }_{jn}{J}_{im}-{\delta }_{in}{J}_{jm}-{\delta }_{jm}{J}_{in}\right),\hfill & \hfill \left[D,P_i\right]& =i{P}_i,\hfill \\ \hfill \left[J_{ij},P_k\right]& =-i\left({\delta }_{ik}{P}_j-{\delta }_{jk}{P}_i\right),\hfill & \hfill \left[D,H\right]& =iH,\hfill \\ \hfill \left[J_{ij},K_k\right]& =-i\left({\delta }_{ik}{K}_j-{\delta }_{jk}{K}_i\right),\hfill & \hfill \left[D,K_i\right]& =-i{K}_i,\hfill \\ \hfill \left[J_{ij},B_k\right]& =-i\left({\delta }_{ik}{B}_j-{\delta }_{jk}{B}_i\right),\hfill & \hfill \left[D,K\right]& =-iK,\hfill \\ \hfill \left[B_i,P_j\right]& =i{\delta }_{ij}H,\hfill & \hfill \left[H,K_i\right]& =2i{B}_i,\hfill \\ \hfill \left[B_i,K_j\right]& =i{\delta }_{ij}K,\hfill & \hfill \left[K,P_i\right]& =2i{B}_i,\hfill \\ \hfill \left[K_i,P_j\right]& =-2i\left({\delta }_{ij}D-J_{ij}\right).\hfill \end{array}$$

(12)

This algebra is isomorphic to $\mathfrak{iso}(1,d)$, which can be seen explicitly through the identification

$${\tilde{J}}_{ij}=J_{ij},{\tilde{J}}_{i0}=-\frac{1}{2}\left(P_i+K_i\right),{\tilde{J}}_{id}=\frac{1}{2}\left(P_i-K_i\right),{\tilde{J}}_{0d}=-D,$$

(13)

and

$${\tilde{P}}_0=\frac{1}{\sqrt{2}}(H+K),{\tilde{P}}_i=-\sqrt{2}{B}_i,{\tilde{P}}_d=\frac{1}{\sqrt{2}}(K-H),$$

(14)

such that $\{{\tilde{J}}_{\mu \nu },{\tilde{P}}_{\mu }\}$ indeed satisfy the $\mathfrak{iso}(1,d)$ algebra in its standard form. Crucially this representation of the Poincaré algebra is such that the quadratic Casimir operator vanishes identically,

$${\mathcal{C}}_2={\tilde{P}}^{\mu }{\tilde{P}}_{\mu }=-{\tilde{P}}_0^2+{\tilde{P}}^i{\tilde{P}}_i+{\tilde{P}}_d^2=-(HK+KH)+2B^i{B}_i=2x^2{\partial }_u^2-2x^2{\partial }_u^2=0.$$

(15)

This is in stark contrast with the standard representation ${\mathcal{C}}_2={\partial }^{\mu }{\partial }_{\mu }$ associated with Minkowski space ${\mathbb{M}}^{d+1}$. This shows that fields at $\mathcal{I}$ can only possibly carry massless representations of $ISO(1,d)$.

We now turn to the construction of fields defined locally on $\mathcal{I}$ and transforming covariantly under $ISO(1,d)$. Just as in the standard case of relativistic conformal fields [36], we first look for finite-dimensional irreducible representations of the stability subgroup H of the origin $x^{\alpha }=0$. Looking at (11), we see that the latter is generated by the algebra

$$\mathfrak{h}=\{J_{ij},B_i,K_i,K,D\}.$$

(16)

First we note that $J_{ij}$ are $SO(d-1)$ generators, which naturally leads us to consider a symmetric and traceless spin-s tensor field $\varphi \left(0\right)\equiv {\varphi }_{i_1\dots i_s}\left(0\right)$ transforming as

$$\left[J_{ij},\varphi \left(0\right)\right]={\Sigma }_{ij}\varphi \left(0\right),$$

(17)

where ${\Sigma }_{ij}$ is the standard $SO(d-1)$ irreducible spin-s hermitian representation. This exactly coincides with the UIR of the short little group $SO(d-1)$ from which the spin-s helicity representation is induced. Because $B_i$ and $K_i$ transform like $SO(d-1)$ vectors, finite dimensionality of $\varphi \left(0\right)$ requires them to act trivially,

$$\left[B_i,\varphi \left(0\right)\right]=\left[K_i,\varphi \left(0\right)\right]=0.$$

(18)

Consistency with the algebra (12) then requires the generator K to act trivially too,

$$\left[B_i,K_j\right]=i{\delta }_{ij}K\Rightarrow \left[K,\varphi \left(0\right)\right]=0.$$

(19)

On the other hand, since it commutes with the spin generators, the action of the dilation operator can be diagonalised,

$$\left[D,\varphi \left(0\right)\right]=i\Delta \varphi \left(0\right),\Delta \in \mathbb{R}.$$

(20)

Note that $K_{\alpha }=(K,K_i)$ and $P_{\alpha }=(H,P_i)$ act as lowering and raising operators for the conformal dimension, respectively,

$$\begin{array}{cc}\hfill \left[D,\left[K_{\alpha },\varphi \left(0\right)\right]\right]& =i(\Delta -1)\left[K_{\alpha },\varphi \left(0\right)\right],\hfill \\ \hfill \left[D,\left[P_{\alpha },\varphi \left(0\right)\right]\right]& =i(\Delta +1)\left[P_{\alpha },\varphi \left(0\right)\right],\hfill \end{array}$$

(21)

and the triviality of $K_{\alpha }$ imposed in (18) and (19) amounts to the primary field condition

$$\left[K_{\alpha },\varphi \left(0\right)\right]=0.$$

(22)

As we argued this is a logical consequence of the finite dimensionality of $\varphi \left(0\right)$.

The dependence on the coordinates $x^{\alpha }=(u,x^i)$ is then obtained by use of the translation operators $P_{\alpha }=(H,P_i)$,

$$\varphi \left(x\right)\equiv U\left(x\right)\varphi \left(0\right)U{\left(x\right)}^{-1},U\left(x\right)\equiv e^{-i{x}^{\alpha }{P}_{\alpha }}=e^{-i(uH+x^i{P}_i)}.$$

(23)

To work out the action of an arbitrary generator $X\in \mathfrak{iso}(1,d)$ on the field $\varphi \left(x\right)$, we make use of

$$\left[X,\varphi \left(x\right)\right]=U\left(x\right)\left[X^{\prime },\varphi \left(0\right)\right]U{\left(x\right)}^{-1},$$

(24)

where

$$X^{\prime }=U{\left(x\right)}^{-1}XU\left(x\right)=\sum _{n=0}^{\infty }\frac{i^n}{n!}{x}^{{\alpha }_1}\dots x^{{\alpha }_n}\left[P_{{\alpha }_1},\left[\dots \left[P_{{\alpha }_n},X\right]\right]\right].$$

(25)

Explicitly using the form (12) of the $\mathfrak{iso}(1,d)$ algebra, this series truncates at order $n=2$ at most, and yields the infinitesimal action

$$\begin{array}{cc}\hfill \left[H,\varphi \left(x\right)\right]& =-i{\partial }_u\varphi \left(x\right),\hfill \\ \hfill \left[P_i,\varphi \left(x\right)\right]& =-i{\partial }_i\varphi \left(x\right),\hfill \\ \hfill \left[J_{ij},\varphi \left(x\right)\right]& =-i\left(i{\Sigma }_{ij}-x_i{\partial }_j+x_j{\partial }_i\right)\varphi \left(x\right),\hfill \\ \hfill \left[D,\varphi \left(x\right)\right]& =-i\left(-\Delta +u{\partial }_u+x^i{\partial }_i\right)\varphi \left(x\right),\hfill \\ \hfill \left[K,\varphi \left(x\right)\right]& =-i{x}^2{\partial }_u\varphi \left(x\right),\hfill \\ \hfill \left[K_i,\varphi \left(x\right)\right]& =-i\left(2x_i\Delta +2i{x}^j{\Sigma }_{ij}-2u{x}_i{\partial }_u-2x_i{x}^j{\partial }_j+x^2{\partial }_i\right)\varphi \left(x\right),\hfill \\ \hfill \left[B_i,\varphi \left(x\right)\right]& =-i{x}_i{\partial }_u\varphi \left(x\right).\hfill \end{array}$$

(26)

This can be compactly written in terms of the Lie derivative ${\mathcal{L}}_{\zeta }$,

$$\begin{array}{cc}\hfill \delta \varphi \left(x\right)& \equiv i[(aH+a^i{P}_i+\frac{1}{2}{\omega }^{ij}{J}_{ij}+b^i{B}_i+\lambda D+kK+k^i{K}_i),\varphi ]\hfill \\ \hfill & =\left({\zeta }^{\alpha }{\partial }_{\alpha }-\frac{i}{2}{\partial }_{[i}{\zeta }_{j]}{\Sigma }^{ij}-\Delta \Omega \right)\varphi =\left({\mathcal{L}}_{\zeta }-\Delta \Omega \right)\varphi \left(x\right),\hfill \end{array}$$

(27)

where the scaling factor is given by

$$\Omega =\frac{1}{d}{\partial }_{\alpha }{\zeta }^{\alpha }=\lambda -2k_i{x}^i.$$

(28)

The Carrollian conformal field $\varphi \left(x\right)$ transforms covariantly in an irreducible representation of $ISO(1,d)$. Up to this point, their relation to the spin-s helicity states is, however, unclear. To assess whether such covariant fields can carry the helicity states, we can fix a momentum frame and determine whether the independent field components transform in the UIR of the little group from which the full helicity representation is induced [43,44,45,54]. The point $x^{\alpha }=0$ corresponds to the momentum frame

$${\tilde{P}}_{\mu }=\frac{1}{\sqrt{2}}(H,0,\dots ,0,-H),$$

(29)

as follows from (17) and (22), and the massless character of the representation is again manifest. In that frame, the little group which leaves (29) invariant is therefore generated by

$$\{{\tilde{J}}_{ij},{\tilde{J}}_{id}+{\tilde{J}}_{i0}\}=\{J_{ij},K_i\}=\mathfrak{iso}(d-1).$$

(30)

Equations (17) and (22) tell us that $\varphi \left(0\right)$ carries the finite-dimensional spin-s UIR of the little group, the very same UIR from which the spin-s helicity representation is induced. Thus $\varphi \left(x\right)$ is a good candidate quantum field to create the spin-s helicity states. The simple-minded argument given above does not immediately fix the conformal dimension $\Delta$ of the Carrollian field as the dilation operator D does not belong to the little algebra $\mathfrak{iso}(d-1)$. Rather its action on massless particles is induced and therefore determined by the little group UIR. In Section 5, we directly build the Carrollian representations starting from the massless UIRs, which allows us to determine the conformal dimension from the spin of the representation,

$$\Delta \left(s\right)=s-\frac{d-1}{2}.$$

(31)

This will also be explicitly realised in Section 4 when pulling back relativistic bulk fields to future null infinity $\mathcal{I}$.

Finally, we may wish to achieve general covariance, i.e., to embed ${\varphi }_{i_1\dots i_s}$ into a Carrollian tensor ${\varphi }_{{\alpha }_1\dots {\alpha }_s}$. This is achieved almost trivially by requiring the latter to be fully symmetric and by further imposing

$$n^{\alpha }{\varphi }_{\alpha \dots {\alpha }_s}=0,q^{\alpha \beta }{\varphi }_{\alpha \beta \dots {\alpha }_s}=0.$$

(32)

Here, $q^{\alpha \beta }$ is any symmetric tensor satisfying $q^{ij}={\delta }^{ij}$, which is therefore defined only up to $q^{\alpha \beta }\mapsto q^{\alpha \beta }+l_1^{\alpha }{n}^{\beta }+n^{\alpha }{l}_2^{\beta }$ for any two vectors $l_{1,2}^{\alpha }$. However, this ambiguity is inconsequential on account of the first condition in (32). Thus the only nonzero tensor components in ${\varphi }_{{\alpha }_1\dots {\alpha }_s}$ are indeed the spatial components ${\varphi }_{i_1\dots i_s}$. he transformation (27) then takes the general covariant form

$$\begin{array}{cc}\hfill \delta {\varphi }_{{\alpha }_1\dots {\alpha }_s}& =\left({\zeta }^{\alpha }{\partial }_{\alpha }-\frac{i}{2}{\partial }_{[\alpha }{\zeta }_{\beta ]}{\Sigma }^{\alpha \beta }-\Delta \Omega \right){\varphi }_{{\alpha }_1\dots {\alpha }_s}=\left({\mathcal{L}}_{\zeta }-\Delta \Omega \right){\varphi }_{{\alpha }_1\dots {\alpha }_s},\hfill \end{array}$$

(33)

where ${\Sigma }^{\alpha \beta }$ is any completion of ${\Sigma }^{ij}$4. The last expression in terms of the Lie derivative ${\mathcal{L}}_{\zeta }$ is obviously valid provided the projection condition (32) holds.

This concludes the construction of finite-component conformal primary fields defined over a generic null plane $\mathcal{I}$ and transforming in irreducible representations of the Carrollian conformal group $ISO(1,d)$. The above considerations were intrinsic to $\mathcal{I}$ and made no reference whatsoever to a higher-dimensional Minkowskian geometry ${\mathbb{M}}^{d+1}$. Of course $\mathcal{I}$ also arises as the conformal boundary of ${\mathbb{M}}^{d+1}$. In that context, the goal of the next section will be to show that the pullback to $\mathcal{I}$ of the standard bulk quantum fields provides realisations of the Carrollian conformal fields introduced above. It will also automatically provides the intertwining relation between these Carrollian quantum fields and the creation operators $a_{p,\sigma }^{†}$ of the helicity representations.

4. The Bulk-Boundary Dictionary

We now have a closer look at the standard massless quantum fields in ${\mathbb{M}}^{d+1}$ which were briefly described in Section 2, and show that the Carrollian conformal fields constructed in Section 3 naturally arise as asymptotic values of these bulk fields. This is similar in spirit to [32,33] although the demonstration is different. To this end it is best to adopt retarded coordinates $(r,u,x^i)$, related to cartesian coordinates $X^{\mu }$ by

$$X^{\mu }=u{n}^{\mu }+r{\widehat{q}}^{\mu }\left(x^i\right),$$

(34)

where $n^{\mu }$ and ${\widehat{q}}^{\mu }\left(x^i\right)$ are null vectors with cartesian components given by

$$\begin{array}{cc}\hfill {\widehat{q}}^{\mu }\left(x^i\right)& =\frac{1}{\sqrt{2}}\left(1+x^2,2x^i,1-x^2\right),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill n^{\mu }& =\frac{1}{\sqrt{2}}\left(1,0^i,-1\right),\hfill \end{array}$$

(36)

and satisfying $n·\widehat{q}=-1$ as well as $\widehat{q}\left(x\right)·\widehat{q}\left(y\right)=-{|x-y|}^2$. In retarded coordinates, the flat metric takes the simple form

$$d{s}^2={\eta }_{\mu \nu }d{X}^{\mu }d{X}^{\nu }=-2dudr+2r^2{\delta }_{ij}d{x}^{i}d{x}^j,$$

(37)

such that future null infinity $\mathcal{I}$ lies at $r\to \infty$.

As a first step, let us recover the Carrollian representation (11) of the Poincaré generators from their standard Minkowskian representation. Indeed in Minkowski space ${\mathbb{M}}^{d+1}$ the Poincaré algebra $\mathfrak{iso}(1,d)$ is naturally represented by the differential operators

$$J_{\mu \nu }=i\left(X_{\mu }\frac{\partial }{\partial X^{\nu }}-X_{\nu }\frac{\partial }{\partial X^{\mu }}\right),P_{\mu }=-i\frac{\partial }{\partial X^{\mu }},$$

(38)

satisfying

$$\begin{array}{cc}\hfill \left[J_{\mu \nu },J_{\rho \sigma }\right]& =-i\left({\eta }_{\mu \rho }{J}_{\nu \sigma }+{\eta }_{\nu \sigma }{J}_{\mu \rho }-{\eta }_{\mu \sigma }{J}_{\nu \rho }-{\eta }_{\nu \rho }{J}_{\mu \sigma }\right),\hfill \\ \hfill \left[J_{\mu \nu },P_{\rho }\right]& =-i\left({\eta }_{\mu \rho }{P}_{\nu }-{\eta }_{\nu \rho }{P}_{\mu }\right).\hfill \end{array}$$

(39)

In terms of retarded coordinates, they read

$$\begin{array}{cc}\hfill P_0& =-\frac{i}{\sqrt{2}}\left({\partial }_r+(1+x^2){\partial }_u-r^{-1}{x}^i{\partial }_i\right),\hfill \\ \hfill P_i& =-\frac{i}{\sqrt{2}}\left(-2x_i{\partial }_u+r^{-1}{\partial }_i\right),\hfill \\ \hfill P_d& =-\frac{i}{\sqrt{2}}\left({\partial }_r-(1-x^2){\partial }_u-r^{-1}{x}^i{\partial }_i\right),\hfill \end{array}$$

(40)

and

$$\begin{array}{cc}\hfill J_{0d}& =-i(r{\partial }_r-u{\partial }_u-x^i{\partial }_i),\hfill \\ \hfill J_{i0}& =\frac{i}{2}\left((r^{-1}u+1+x^2){\partial }_i+2x_i(r{\partial }_r-u{\partial }_u-x^j{\partial }_j)\right),\hfill \\ \hfill J_{id}& =\frac{i}{2}\left((r^{-1}u-1+x^2){\partial }_i+2x_i(r{\partial }_r-u{\partial }_u-x^j{\partial }_j)\right),\hfill \\ \hfill J_{ij}& =i(x_i{\partial }_j-x_j{\partial }_i).\hfill \end{array}$$

(41)

It is straightforward to check that their limit $r\to \infty$ exactly coincides with the expression given in (13) and (14) for the orbital part of ${\tilde{P}}_{\mu },{\tilde{J}}_{\mu \nu }$, up to the $r{\partial }_r$ terms which survive in that limit as well. This operator $r{\partial }_r$ is a geometrical bulk realisation of the conformal weight $\Delta$ as can be observed by comparison with (26). Said differently, we can already anticipate that a massless bulk field $\Phi$ with asymptotic behaviour

$$\Phi (r,x^{\alpha })=r^{\Delta }{\Phi }_{\Delta }\left(x^{\alpha }\right)+\dots$$

(42)

will induce a conformal field ${\Phi }_{\Delta }\left(x^{\alpha }\right)$ of conformal dimension $\Delta$ at $\mathcal{I}$. We expect this conformal dimension to be determined by the spin of the representation as in (31). We will come back to this point momentarily.

Let us make this more precise and study the behavior near $\mathcal{I}$ of the bulk quantum field ${\varphi }_{{\mu }_1\dots {\mu }_s}$ associated with states in the spin-s helicity representation, and satisfying the covariant field Equation (3). In retarded coordinates, the wave equation takes the form

$$\square {\varphi }_{{\mu }_1\dots {\mu }_s}=-2{\partial }_u{\nabla }_r{\varphi }_{{\mu }_1\dots {\mu }_s}+{\nabla }^i{\nabla }_i{\varphi }_{{\mu }_1\dots {\mu }_s}=0,$$

(43)

while the transversality and traceless constraints imply

$$\begin{array}{c}\hfill g^{ij}{\varphi }_{ij\dots }=2{\varphi }_{ur\dots },g^{ij}{\nabla }_i{\varphi }_{j\dots }={\partial }_u{\varphi }_{r\dots }+{\nabla }_r{\varphi }_{u\dots }.\end{array}$$

(44)

From this we show that the wave operator takes the form

$$\begin{array}{cc}\hfill \square {\varphi }_{r\left(m\right)u(s-m-k)i\left(k\right)}& =\left(-2{\partial }_u{\partial }_r+(2k+1-d-2m)r^{-1}{\partial }_u+\frac{1}{2r^2}{\partial }^2\right){\varphi }_{r\left(m\right)u(s-m-k)i\left(k\right)}\hfill \\ \hfill & -m\left(2r^{-1}{\partial }_r+(2m+d-3-2k)r^{-2}\right){\varphi }_{r(m-1)u(s-m-k+1)i\left(k\right)}\hfill \\ \hfill & -2k{r}^{-1}{\partial }_{\left(i\right(1)}{\varphi }_{i(k-1)\left)r\right(m\left)u\right(s-m-k+1)}\hfill \\ \hfill & +2k(k-1){\delta }_{\left(i\right(2)}{\varphi }_{i(k-2)\left)r\right(m\left)u\right(s-m-k+2)},\hfill \end{array}$$

(45)

where we have introduced the following shorthand notation for the various field components,

$${\varphi }_{r\left(m\right)u(s-m-k)i\left(k\right)}\equiv {\varphi }_{r_1\dots r_m{u}_1\dots u_{s-m-k}{i}_1\dots i_k}.$$

(46)

Equation (45) implies that the field component behave asymptotically as

$${\varphi }_{r\left(m\right)u(s-m-k)i\left(k\right)}(r,x^{\alpha })=r^{k-m-\frac{d-1}{2}}{\overline{\varphi }}_{r\left(m\right)u(s-m-k)i\left(k\right)}\left(x^{\alpha }\right)+\dots .$$

(47)

The components ${\overline{\varphi }}_{u(s-k)i\left(k\right)}$ are completely unconstrained, while the radial components are determined through the trace and transversality constraints (44). Up to this point, we have not used the gauge redundancy (4), which should precisely further reduce the independent physical asymptotic components ${\overline{\varphi }}_{{\mu }_1\dots {\mu }_s}$ to the tensor components ${\overline{\varphi }}_{i_1\dots i_s}$ carrying the spin-s representation of the (short) little group. Indeed the gauge parameter ${\epsilon }_{{\mu }_1\dots {\mu }_{s-1}}$ satisfies (5) and thus similarly behaves asymptotically as

$${\epsilon }_{r\left(m\right)u(s-k-m-1)i\left(k\right)}(r,x^{\alpha })=r^{k-m-\frac{d-1}{2}}{\overline{\epsilon }}_{r\left(m\right)u(s-m-k-1)i\left(k\right)}\left(x^{\alpha }\right)+\dots .$$

(48)

Because the components of ${\overline{\epsilon }}_{u(s-k-1)i\left(k\right)}$ are completely unconstrained, they allow us to gauge away all retarded time components ${\overline{\varphi }}_{u(s-k)i\left(k\right)}$ ($k\ne s$) since the latter transform as

$${\delta }_{\epsilon }{\overline{\varphi }}_{u(s-k)i\left(k\right)}={\partial }_u{\overline{\epsilon }}_{u(s-k-1)i\left(k\right)}.$$

(49)

Thus, the independent gauge-invariant components are just the spatial components ${\overline{\varphi }}_{i_1\dots i_s}$.

We now have a look at the transformation of the gauge-invariant field components ${\overline{\varphi }}_{i_1\dots i_s}$ under Poincaré transformations. We know that the bulk field ${\varphi }_{{\mu }_1\dots {\mu }_s}$ transforms covariantly, such that it can be expressed in arbitrary coordinates in terms of the Lie derivative

$${\delta }_{\xi }{\varphi }_{{\mu }_1\dots {\mu }_s}={\mathcal{L}}_{\xi }{\varphi }_{{\mu }_1\dots {\mu }_s}={\xi }^{\rho }{\partial }_{\rho }{\varphi }_{{\mu }_1\dots {\mu }_s}+{\partial }_{{\mu }_1}{\xi }^{\rho }{\varphi }_{\rho \dots {\mu }_s}+\dots +{\partial }_{{\mu }_s}{\xi }^{\rho }{\varphi }_{{\mu }_1\dots \rho },$$

(50)

where ${\xi }^{\mu }$ is a Killing vector field. Adopting the standard parametrisation

$${\xi }^{\mu }=A^{\mu }+{\Omega }^{\mu }{}_{\nu }{X}^{\nu },$$

(51)

with $A^{\mu }$ and ${\Omega }_{\mu \nu }=-{\Omega }_{\nu \mu }$ corresponding to translations and Lorentz rotations, respectively, its components in retarded coordinates are explicitly given by

$$\begin{array}{cc}\hfill {\xi }^r& =-n^{\mu }{\xi }_{\mu }=-r{n}^{\mu }{\Omega }_{\mu \nu }{\widehat{q}}^{\nu }-n^{\mu }{A}_{\mu },\hfill \\ \hfill {\xi }^u& =-{\widehat{q}}^{\mu }{\xi }_{\mu }=-{\widehat{q}}^{\mu }{A}_{\mu }+u{n}^{\mu }{\Omega }_{\mu \nu }{\widehat{q}}^{\nu },\hfill \\ \hfill {\xi }^i& =\frac{1}{2r}{\partial }_i{\widehat{q}}^{\mu }{\xi }_{\mu }=-\frac{1}{2}\left({\widehat{q}}^{\mu }{\Omega }_{\mu \nu }{\partial }_i{\widehat{q}}^{\nu }+r^{-1}{\partial }_i{\xi }^u\right).\hfill \end{array}$$

(52)

The various Lorentz contractions can be evaluated in terms of the constant cartesian components of $A^{\mu }$ and ${\Omega }_{\mu \nu }$,

$$\begin{array}{cc}\hfill {\widehat{q}}^{\mu }{A}_{\mu }& =-\frac{1}{\sqrt{2}}\left(A^0-A^d+(A^d+A^0)x^2-2A_i{x}^i\right),\hfill \\ \hfill n^{\mu }{\Omega }_{\mu \nu }{\widehat{q}}^{\nu }& ={\Omega }_{0d}+({\Omega }_{0i}-{\Omega }_{di})x^i,\hfill \\ \hfill {\widehat{q}}^{\mu }{\Omega }_{\mu \nu }{\partial }_i{\widehat{q}}^{\nu }& =-2{\Omega }_{0d}{x}_i+({\Omega }_{0i}+{\Omega }_{di})+({\Omega }_{0i}-{\Omega }_{di})x^2-2({\Omega }_{0j}-{\Omega }_{dj})x^j{x}_i-2{\Omega }_{ij}{x}^j.\hfill \end{array}$$

(53)

In the limit $r\to \infty$ we recover the Carrollian vector field (9)

$$\begin{array}{cc}\hfill {\zeta }^u& \equiv \underset{r\to \infty }{lim}{\xi }^u=a+k{x}^2+b_i{x}^i+u(\lambda -2k_i{x}^i),\hfill \\ \hfill {\zeta }^i& \equiv \underset{r\to \infty }{lim}{\zeta }^i=a^i+{\omega }^i{}_j{x}^j+\lambda x^i+k^i{x}^2-2k_j{x}^j{x}^i,\hfill \end{array}$$

(54)

upon identifying the Carrollian symmetry parameters

$$\begin{array}{cccccccc}\hfill a& =-\frac{1}{\sqrt{2}}(A_0+A_d),\hfill & \hfill a_i& =-\frac{1}{2}({\Omega }_{0i}+{\Omega }_{di}),\hfill & \hfill \lambda & ={\Omega }_{0d},\hfill & \hfill b_i& =-\sqrt{2}{A}_i,\hfill \\ \hfill k& =-\frac{1}{2}(A_0-A_d),\hfill & \hfill k_i& =-\frac{1}{2}({\Omega }_{0i}-{\Omega }_{di}),\hfill & \hfill {\omega }_{ij}& ={\Omega }_{ij}.\hfill \end{array}$$

(55)

In addition, we find that the asymptotic limit of the r-component is related to the scaling factor (28),

$${\xi }^r=-r(\lambda -2k_i{x}^i)+O\left(r^0\right)=-r\Omega +O\left(r^0\right).$$

(56)

The extra radial direction, or holographic direction, gives a geometrical encoding of the dilation operator D. Putting these equations together, we obtain the transformation of the gauge-invariant tensor components, which can be simply written

$$\delta {\overline{\varphi }}_{i_1\dots i_s}=\left[{\mathcal{L}}_{\zeta }-\left(s-\frac{d-1}{2}\right)\Omega \right]{\overline{\varphi }}_{i_1\dots i_s}.$$

(57)

This is nothing but the transformation (27) of a Carrollian conformal primary field of conformal dimension 5

$$\Delta \left(s\right)=s-\frac{d-1}{2}.$$

(58)

Hence, relativistic massless fields on ${\mathbb{M}}^{d+1}$ are dual to Carrollian conformal primary fields at $\mathcal{I}$. Note also that the combination $-\Delta \Omega$ in (57) comes from the radial derivative ${\xi }^r{\partial }_r=-\Omega r{\partial }_r$ appearing in the Lie derivative (50), in agreement with the geometrisation $\Delta \mapsto r{\partial }_r$ described around (42).

Using the above correspondence, it is easy to express the Carrollian conformal fields in terms of the creation operators $a_{p,\sigma }^{†}$. We simply need to consider the expression (2) in the limit $r\to \infty$. For that, we note that the intertwiner wavefunctions are of the form [37]

$${\varphi }_{{\mu }_1\dots {\mu }_s}^{p,\sigma }\left(X\right)={ϵ}_{{\mu }_1\dots {\mu }_s}^{p,\sigma }{e}^{-ip·X},$$

(59)

where $e^{-ip·X}$ carries the spacetime dependence, and ${ϵ}_{{\mu }_1\dots {\mu }_s}^{p,\sigma }$ is a polarisation tensor with constant cartesian components. If we adopt the convenient parametrisation of a generic null momentum

$$p^{\mu }(\omega ,y^i)=\omega {\widehat{q}}^{\mu }\left(y^i\right),$$

(60)

with $\widehat{q}$ given in (35), such that the Lorentz-invariant measure becomes

$$d^{d}p={\omega }^{d-2}d\omega d^{d-1}y,$$

(61)

then the phase of the plane wave takes the simple form

$$p(\omega ,y)·X(r,u,x)=-\omega (u+r|x-y{|}^2).$$

(62)

In the limit $r\to \infty$, the $d^{d-1}y$ integral in (2) can be performed by stationary phase approximation, which localises at $y=x$ and yields

$${\varphi }_{{\mu }_1\dots {\mu }_s}={\left(\frac{-i\pi }{r}\right)}^{\frac{d-1}{2}}\sum _{\sigma }{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{ϵ}_{{\mu }_1\dots {\mu }_s}^{p,\sigma }{e}^{i\omega u}{a}_{p,\sigma }^{†}{|}_{p=\omega \widehat{q}\left(x^i\right)}+\dots$$

(63)

The components of interest are then given by

$${\varphi }_{i_1\dots i_s}=r^s{\partial }_{i_1}{\widehat{q}}^{{\mu }_1}\dots {\partial }_{i_s}{\widehat{q}}^{{\mu }_s}{\varphi }_{{\mu }_1\dots {\mu }_s},$$

(64)

which, in the limit $r\to \infty$, provide the nonzero components of the Carrollian conformal field,

$${\overline{\varphi }}_{i_1\dots i_s}=\underset{r\to \infty }{lim}{r}^{\frac{d-1}{2}-s}{\varphi }_{i_1\dots i_s}={(-i\pi )}^{\frac{d-1}{2}}\sum _{\sigma }{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{\overline{ϵ}}_{i_1\dots i_s}^{p,\sigma }{e}^{i\omega u}{a}_{p,\sigma }^{†}{|}_{p=\omega \widehat{q}\left(x^i\right)},$$

(65)

with the polarisation tensors

$${\overline{ϵ}}_{i_1\dots i_s}^{p,\sigma }\equiv {\partial }_{i_1}{\widehat{q}}^{{\mu }_1}\dots {\partial }_{i_s}{\widehat{q}}^{{\mu }_s}{ϵ}_{{\mu }_1\dots {\mu }_s}^{p,\sigma }.$$

(66)

By contrast to massless bulk fields, the dual Carrollian fields are not constrained by a wave equation or analogue. The group theoretical reason for this is that the massless Casimir equation ${\mathcal{C}}_2=0$ is automatically satisfied in the Carrollian representation of the $ISO(1,d)$ algebra, as shown in (15).

5. The Particles States at Null Infinity

In this section, we will show how the particle states appear on the boundary ${\mathcal{I}}^{+}$ of Minkowski spacetime. More precisely, we will consider the particles as the irreducible representation of the Poincaré group, as formulated by Wigner in 1939 [1], and push them to the boundary by taking the limit $r\to \infty$. We will find that the physical states in momentum space are naturally encoded in ${\mathcal{I}}^{+}$ even though this is part of spacetime.

To construct an irreducible representation of the Poincaré group, we choose a reference momentum for a massless particle and in particular $p_{\mu }^{\left(0\right)}=(-1/\sqrt{2},0,\dots ,0,1/\sqrt{2})$, or in light-cone notation $p^{\left(0\right)}{}^{+}\equiv p_{-}^{\left(0\right)}=1$ with all other components being zero. We use the notation $V^{\pm }\equiv \frac{1}{\sqrt{2}}(V^d\pm V^0)=V_{\mp }$. The little algebra $\tilde{\mathcal{H}}$ that preserves this choice has the generators $\tilde{\mathcal{H}}=\{{\tilde{J}}_{ij},{\tilde{J}}_{i+},{\tilde{P}}_{\mu }\}$. We begin by taking an irreducible unitary representation of the little group $\tilde{\mathcal{H}}$ which acts on the states ${\psi }_{\sigma }\left(p^{\left(0\right)}\right)\equiv |p^{\left(0\right)},\sigma \rangle$ with the chosen momentum. As such, we take

$${\tilde{P}}_{-}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)={\psi }_{\sigma }\left(p^{\left(0\right)}\right),{\tilde{P}}_{+}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)=0={\tilde{P}}_i{\psi }_{\sigma }\left(p^{\left(0\right)}\right),$$

(67)

and

$$i{\tilde{J}}_{ij}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)=-\sum _{{\sigma }^{\prime }}{\left(D_{ij}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{\psi }_{{\sigma }^{\prime }}\left(p^{\left(0\right)}\right),{\tilde{J}}_{i+}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)=0.$$

(68)

The last equation is required by unitarity of the representation as the ${\tilde{J}}_{i+}$ form an Abelian algebra. Note that the translation generators ${\tilde{P}}_{\mu }$ should rightly be thought of as part of the little group as the states ${\psi }_{\sigma }\left(p^{\left(0\right)}\right)$ also carry a representation of these. In what follows, we will take this to be understood and use the passive action for the generators. The discussion of the irreducible representation of the Poincaré group in this section has considerably benefitted from unpblished notes by P. Van Nieuwenhuizen and P. West.

The states in the full representation are found by boosting the above states by the action of the generators of the Lorentz algebra, which are not in the little algebra, that is, the generators ${\tilde{J}}_{-i}$ and ${\tilde{J}}_{+-}$. We define

$${\psi }_{\sigma }\left(p\right)\equiv e^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}{\psi }_{\sigma }\left(p^{\left(0\right)}\right).$$

(69)

To determine the momentum $p_{\mu }$ of this state we just act with ${\tilde{P}}_{\mu }$, namely

$${\tilde{P}}_{\mu }{\psi }_{\sigma }\left(p\right)=\left(e^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}{e}^{-i\varphi {\tilde{J}}_{+-}}{e}^{-i{\phi }^i{\tilde{J}}_{-i}}{\tilde{P}}_{\mu }{e}^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}\right){\psi }_{\sigma }\left(p^{\left(0\right)}\right)=p_{\mu }{\psi }_{\sigma }\left(p\right).$$

(70)

Using the commutation relations of the Poincaré group and Equation (67) we find

$$p^{+}=e^{\varphi },p^{-}=-\frac{1}{2}{\phi }^i{\phi }_i{e}^{\varphi },p^i=-{\phi }^i{e}^{\varphi }.$$

(71)

Thus we find how the $d+1-1=d$ components of the massless momenta are parametrised by the d group parameters $\varphi$ and ${\phi }^i$. Identifying $e^{\varphi }=\omega$ and ${\phi }^i=-\sqrt{2}{y}^i$ we recognise precisely the parametrisation of Equation (60). It is interesting to see how the parametrisation of the momenta, that is, $(\omega ,y^i)$ associated with ${\mathcal{I}}^{+}$ arises naturally in the construction of the irreducible representations of the Poincaré group.

The states in the irreducible representation transform under the Lorentz group, $g\in SO(1,d)$, as

$$U\left(g\right){\psi }_{\sigma }\left(p\right)=D{\left(h^{-1}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{\psi }_{{\sigma }^{\prime }}\left(p^{\prime }\right)$$

where $h\in \tilde{\mathcal{H}}$ is defined by the coset relation

$$g{e}^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}=e^{i{\phi }^i{}^{\prime }{\tilde{J}}_{-i}}{e}^{i{\varphi }^{\prime }{\tilde{J}}_{+-}}h,$$

(72)

and $p^{\mu }{}^{\prime }$ is the momentum corresponding to ${\phi }^i{}^{\prime }$ and ${\varphi }^{\prime }$. Indeed if we take our group element g to be of the form

$$g=e^{-\frac{i}{2}{\Omega }^{\mu \nu }{\tilde{J}}_{\mu \nu }}=e^{-i\lambda {\tilde{J}}_{+-}-i\sqrt{2}{a}^i{\tilde{J}}_{-i}-\frac{i}{2}{\omega }^{ij}{\tilde{J}}_{ij}+i\sqrt{2}{k}^i{\tilde{J}}_{+i}},$$

(73)

we find that

$$\begin{array}{cc}\hfill {\phi }^i{}^{\prime }& ={\phi }^i-\sqrt{2}{a}^i+{\omega }^i{}_j{\phi }^j-\lambda {\phi }^i+\sqrt{2}\left({\phi }^j{k}_j\right){\phi }^i-\frac{1}{\sqrt{2}}\left({\phi }^j{\phi }_j\right)k^i,\hfill \\ \hfill {\varphi }^{\prime }& =\varphi +\lambda -\sqrt{2}\left(k^i{\phi }_i\right),\hfill \end{array}$$

(74)

with

$$h=e^{-\frac{i}{2}{\omega }^{ij}{\tilde{J}}_{ij}-i\sqrt{2}{k}^i{\phi }^j{\tilde{J}}_{ij}+i{e}^{-\varphi }\sqrt{2}{k}^i{\tilde{J}}_{+i}}.$$

(75)

We can also compute the action of the translation $g=e^{-ia{\tilde{P}}_{-}-ik{\tilde{P}}_{+}-\frac{i}{\sqrt{2}}{b}^i{\tilde{P}}_i}$ on ${\psi }_{\sigma }\left(p\right)$ by passing it though the factor $e^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}$ and using Equation (67), one finds that

$$e^{-ia{\tilde{P}}_{-}-ik{\tilde{P}}_{+}-\frac{i}{\sqrt{2}}{b}^i{\tilde{P}}_i}\psi \left(p\right)=e^{-i\omega (a+\frac{k}{2}{\phi }^j{\phi }_j-\frac{1}{\sqrt{2}}{\phi }^j{b}_j)}\psi \left(p\right)$$

(76)

To find this result, we used the identity

$$e^{A+B}=e^B{e}^{A+\frac{1}{2}[A,B]-\frac{1}{12}[A,[A,B]]+\dots }=e^{A-\frac{1}{2}[A,B]-\frac{1}{12}[A,[A,B]]+\dots }{e}^B,$$

(77)

valid for any for two generators A and B but for only first order in B. In this particularly simple case, one can also use Equation (25) rather than the above more complicated identity.

In order to push these particle states to the boundary, we require them in the Minkowski spacetime, and so we take the Fourier transform

$${\psi }_{\sigma }\left(X\right)\equiv \int \frac{d^{d+1}p}{{\left(2\pi \right)}^{d+1}}2\pi \delta \left(p^2\right)\theta \left(p^0\right)e^{ip·X}{\psi }_{\sigma }\left(p\right)=\int \frac{d^{d}p}{2p^0{\left(2\pi \right)}^d}{e}^{ip·X}{\psi }_{\sigma }\left(p\right).$$

(78)

Using relation between the coordinates $X^{\mu }$ of Minkowski space ${\mathbb{M}}^{d+1}$ and the coordinates $(r,u,x^i)$ of Equation (34), which are suited to the emergence of ${\mathcal{I}}^{+}$, we find that

$$e^{ip·X}=e^{-i\omega u}{e}^{-ir\omega {(x-y)}^2},$$

(79)

while the change of variable from $p^i$ to $\omega ,y^i$ has Jacobian $\frac{d^{d}p}{p^0}={\left(\sqrt{2}\right)}^{d-1}{\omega }^{d-2}d\omega d{y}^i$. Using these results we find that

$${\psi }_{\sigma }\left(X\right)=\int \frac{d{y}^i}{{\left(2\pi \right)}^d}\frac{d\omega }{\omega }{\left(\sqrt{2}\right)}^{d-3}{\omega }^{d-1}{e}^{-i\omega u}{e}^{-ir\omega {(x-y)}^2}{e}^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}{\psi }_{\sigma }\left(p^{\left(0\right)}\right).$$

(80)

We will now take the limit of $r\to \infty$ of ${\psi }_{\sigma }\left(X\right)$ to obtain the states at the boundary ${\mathcal{I}}^{+}$ by using the formula

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\int }_a^{b}dxf\left(x\right)e^{irg\left(x\right)}=\underset{r\to \infty }{lim}f\left(z\right)e^{irg\left(z\right)}\sqrt{\frac{2\pi i}{r{g}^{\prime }{}^{\prime }\left(z\right)}},\end{array}$$

(81)

where z is a point in the interval $[a,b]$ where $g^{\prime }\left(z\right)=0$ and $g^{″}\left(x\right)=\frac{d^{2}g\left(x\right)}{d{x}^2}$. We find that

$${\psi }_{\sigma }(u,x^i)\equiv \underset{r\to \infty }{lim}{r}^{\frac{d-1}{2}}{\psi }_{\sigma }\left(X\right)=c_1{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{e}^{-i\omega u}{e}^{i{x}^i{P}_i}{e}^{-iln\omega D}{\psi }_{\sigma }\left(p^{\left(0\right)}\right),$$

(82)

where $c_1=\frac{1}{2}{e}^{-i\frac{(d-1)}{2}\pi }{\left(2\pi \right)}^{-\frac{d+1}{2}}$ and we have relabelled the generators by $P_i=\sqrt{2}{\tilde{J}}_{i-}$ and ${\tilde{J}}_{+-}=-D$. In the above equation, we can write $e^{i{x}^i{P}_i}{e}^{-iln\omega D}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)={\psi }_{\sigma }\left(p\right)={\psi }_{\sigma }({\phi }^i,\varphi )={\psi }_{\sigma }(x^i,\omega )$ whereby abuse of notation we use the same symbol $\psi$ for the function even though we have changed the variables.

To better understand what these states are, we consider them at the origin of ${\mathcal{I}}^{+}$, namely at $u=0=x^i$. One readily finds that

$$K_i{\psi }_{\sigma }(0,0)=0=K{\psi }_{\sigma }(0,0)=B_i{\psi }_{\sigma }(0,0),$$

(83)

where we have relabelled the Poincaré generators as follows

$$K_i=-\sqrt{2}{\tilde{J}}_{i+},K={\tilde{P}}_{+},\sqrt{2}{B}_i=-{\tilde{P}}_i.$$

(84)

Acting with D we find that

$$iD{\psi }_{\sigma }(0,0)=-c_1\int d\varphi e^{\frac{(d-1)}{2}\varphi }\frac{d}{d\varphi }{e}^{-i\varphi D}{\psi }_{\sigma }\left(p^{\left(0\right)}\right)=\frac{(d-1)}{2}{\psi }_{\sigma }(0,0).$$

(85)

This is essentially the same as (20) together with (31), however the two expressions differ by the spin s factor. The reason for this difference comes from the r-dependence of the relativistic bulk fields and their tensorial transformations, to which we will come back at the end of this section. We also note that

$$e^{\frac{u}{\sqrt{2}}iH+x^{i}i{P}_i}{\psi }_{\sigma }(0,0)={\psi }_{\sigma }(u,x^i),i{J}_{ij}{\psi }_{\sigma }(0,0)=-\sum _{{\sigma }^{\prime }}{\left(D_{ij}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{\psi }_{{\sigma }^{\prime }}(0,0),$$

(86)

where $H=-{\tilde{P}}_{-}$. To derive the last equation we used the identity $e^{-i\varphi {\tilde{J}}_{+-}}{\tilde{P}}_{-}{e}^{i\varphi {\tilde{J}}_{+-}}=e^{\varphi }{\tilde{P}}_{-}$.

Let us summarise the situation. We began with an irreducible representation of the Poincaré algebra with the generators ${\tilde{J}}_{\mu \nu },{\tilde{P}}_{\mu }$ corresponding to a massless particle in Minkowski spacetime. We pushed these states to the boundary ${\mathcal{I}}^{+}$ of Minkowski spacetime by taking the $r\to \infty$ on their Fourier transform. We found the states ${\psi }_{\sigma }(u,x^i)$ living on ${\mathcal{I}}^{+}$ which obey Equations (83), (85) and (86) and also carry a representation of the Poincaré algebra but with the generators identified as

$$\mathcal{H}=\{J_{ij}={\tilde{J}}_{ij},K_i=-\sqrt{2}{\tilde{J}}_{i+},K={\tilde{P}}_{+},\sqrt{2}{B}_i=-{\tilde{P}}_i,D=-{\tilde{J}}_{+-}\},$$

(87)

as well as

$$H=-{\tilde{P}}_{-},P_i=-\sqrt{2}{\tilde{J}}_{-i}.$$

(88)

We recognise ${\psi }_{\sigma }(u,x^i)$ as a representation induced from ${\psi }_{\sigma }(0,0)$ which carries a representation of the subalgebra $\mathcal{H}$ which is boosted by the generators H and $P_i$. We observe that the subalgebra $\mathcal{H}$ has the extra generator D compared to the little algebra $\tilde{\mathcal{H}}$ for the states in the Wigner construction. The construction involving ${\psi }_{\sigma }(u,x^i)$ is typically associated with a representation of the conformal group, but in this case, this group is the Poincaré group. In this process, the translation generators on ${\mathcal{I}}^{+}$ are some of the Lorentz generators in Minkowski spacetime, see Equations (87) and (88).

It is instructive to find a general Lorentz transformation of ${\psi }_{\sigma }(x^i,u)$, using Equation (73) we find that

$$\begin{array}{c}\hfill U\left(g\right){\psi }_{\sigma }(x^i,u)=c_1{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{e}^{-i\omega u}{e}^{i{\phi }^{i\prime }{\tilde{J}}_{-i}}{e}^{i{\varphi }^{\prime }{\tilde{J}}_{+-}}D{\left(h^{-1}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{\psi }_{{\sigma }^{\prime }}\left(p^{\left(0\right)}\right)\\ \hfill =c_1{\int }_0^{\infty }d{\omega }^{\prime }{\left({\omega }^{\prime }\right)}^{\frac{d-3}{2}}{e}^{-i{\omega }^{\prime }{e}^{(\lambda +\sqrt{2}{\phi }^i{k}_i)}u}{e}^{\frac{(d-1)}{2}(\lambda +\sqrt{2}{\phi }^i{k}_i)}D{\left(h^{-1}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{e}^{i{\phi }^{i\prime }{\tilde{J}}_{-i}}{e}^{i{\varphi }^{\prime }{\tilde{J}}_{+-}}{\psi }_{{\sigma }^{\prime }}\left(p^{\left(0\right)}\right),\end{array}$$

(89)

where ${\omega }^{\prime }=e^{{\varphi }^{\prime }}=e^{-(\lambda +\sqrt{2}{\phi }^i{k}_i)}\omega$. Using the above identification $x^i=y^i=-\frac{{\phi }^i}{\sqrt{2}}$ and relabelling the integration over $\omega$ we find that

$$\begin{array}{c}\hfill U\left(g\right){\psi }_{\sigma }(x^i,u)={\psi }_{\sigma }(x^{i\prime },u^{\prime })+\frac{(d-1)}{2}(\lambda -2x^i{k}_i){\psi }_{\sigma }(x^i,u)\\ \hfill +(\frac{{\omega }^{ij}}{2}-2k^i{x}^j)D{\left(J_{ij}\right)}_{\sigma }{}^{{\sigma }^{\prime }}{\psi }_{{\sigma }^{\prime }}(x^i,u),\end{array}$$

(90)

where $x^{i\prime }$ and $u^{\prime }$ are given by

$$\begin{array}{cc}\hfill x^i{}^{\prime }& =x^i+a^i+{\omega }^i{}_j{x}^j+\lambda x^i-2\left(x^j{k}_j\right)x^i+\left(x^j{x}_j\right)k^i,\hfill \\ \hfill u^{\prime }& =u+\lambda u-2\left(k^i{x}_i\right)u+a+k{x}^j{x}_j+b_j{x}^j,\hfill \end{array}$$

(91)

where we have used Equation (76) to find the last transformations of u. These results agree with those of Equation (9).

The boost is parametrised by the d variables ${\phi }^i$ and $\varphi$ while ${\mathcal{I}}^{+}$ has the d coordinates $(x^i,u)$. The ${\phi }^i$ and the coordinates $x^i$ are related by ${\phi }^i=-\sqrt{2}{x}^i$ while u and $\omega =e^{\varphi }$ are conjugate variables in the Fourier transform. Hence even though the ${\phi }^i$ arise in momentum space and the $x^i$ parametrise the asymptotic region of spacetime, they are the same up to a multiplicative constant. Indeed ${\mathcal{I}}^{+}$ encodes the kinematics of the particle in a way that is not subject to any constraint, and it carries, in essence, the same irreducible representation of the Poincaré group.

In relativistic quantum field theory, we usually prefer to work with quantities that transform in a way which makes their Poincaré symmetry manifest. To do this, we consider a finite-dimensional, albeit non-unitary, representation of the Lorentz group that contains the above irreducible representation when we restrict to the little group $\tilde{\mathcal{H}}$. Let us denote these fields by ${\Psi }_n$ and the matrix of the finite-dimensional non-unitary representation by $\tilde{D}{\left(g\right)}_n{}^m$. However, we only use this representation for transformations of the little group $\tilde{\mathcal{H}}$, and we construct the induced representation in the same way as above. For the chosen momenta $p_{-}^{\left(0\right)}=1$ we demand that under the little group $\tilde{\mathcal{H}}$ it transforms as

$$i{\tilde{J}}_{ij}{\Psi }_n\left(p^{\left(0\right)}\right)=-\sum _m{\left({\tilde{D}}_{ij}\right)}_n{}^m{\Psi }_m\left(p^{\left(0\right)}\right),i{\tilde{J}}_{+i}{\Psi }_n\left(p^{\left(0\right)}\right)=-\sum _m{\left({\tilde{D}}_{+i}\right)}_n{}^m{\Psi }_m\left(p^{\left(0\right)}\right),$$

(92)

and we define the states with any momentum by the same boost as before

$${\Psi }_n\left(p\right)\equiv e^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}{\Psi }_n\left(p^{\left(0\right)}\right).$$

(93)

The momentum $p^{\mu }$ is given by Equation (71).

One then finds that under a general group element $g\in SO(1,3)$, the states transform as

$$U\left(g\right){\Psi }_n\left(p\right)=\sum _m\tilde{D}{\left(h^{-1}\right)}_n{}^m{\Psi }_m\left(p^{\prime }\right),$$

(94)

where h is given in Equation (75). Because h only involves the generators ${\tilde{J}}_{ij}$ and ${\tilde{J}}_{+i}$, which involve Lorentz transformations with parameters ${\Lambda }_{ij}$ and ${\Lambda }_{-j}$, the component ${\Psi }_{+}$ does not transform, illustrating the fact that the ${\Psi }_n$ do not form an irreducible representation.

To obtain the corresponding fields on ${\mathcal{I}}^{+}$, we have to take the Fourier transform and take the limit $r\to \infty$, as above, to find that

$${\Psi }_n(x^i,u)\equiv \underset{r\to \infty }{lim}{c}_1{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{e}^{-i\omega u}{\Psi }_{\mu }\left(p\right)$$

(95)

where ${\Psi }_{\mu }\left(p\right)$ is given in Equation (91) and can be written after the $r\to \infty$ as ${\Psi }_{\mu }\left(p\right)=e^{i{x}^i{P}_i}{e}^{-iln\omega D}{\Psi }_n\left(p^{\left(0\right)}\right)$. Carrying out a general Lorentz transformation in the same way as we did for the physical states alone, we find that

$$\begin{array}{c}\hfill U\left(g\right){\Psi }_n(x^i,u)={\psi }_n(x^{i\prime },u^{\prime })+\frac{(d-1)}{2}(\lambda -2x^i{k}_i){\Psi }_n(x^i,u)\\ \hfill +(\frac{{\omega }^{ij}}{2}-2k^i{x}^j)\tilde{D}{\left(J_{ij}\right)}_n{}^m{\Psi }_m(x^i,u)+\frac{i\sqrt{2}{k}^i}{{\partial }_u}\tilde{D}{\left(J_{+i}\right)}_n{}^m{\Psi }_m(x^i,u)\end{array}$$

(96)

where $x^{i\prime }$ and $u^{\prime }$ are as in Equation (91) and the expression $\frac{1}{i{\partial }_u}$ just leads to the factor $\frac{1}{\omega }$ in momentum space that occurs in the group element h. As we will see, this last term does not occur in the transformation of the physical states.

Clearly, these fields do not transform in a way that makes the symmetry manifest, but we can construct some that do, specifically

$$A_n\left(p\right)\equiv \tilde{D}{\left(e^{i{\phi }^i{\tilde{J}}_{-i}}{e}^{i\varphi {\tilde{J}}_{+-}}\right)}_n{}^m{\Psi }_m\left(p\right).$$

(97)

Indeed we find that they transform as $U\left(g\right)A_n\left(p\right)=\tilde{D}{\left(g^{-1}\right)}_n{}^m{A}_m\left(p^{\prime }\right)$ under a Lorentz transformation. Taking the Fourier transform of $A_n\left(p\right)$ and the $r\to \infty$ limit we find $A_n(x^i,u)$ on ${\mathcal{I}}^{+}$, namely

$$A_n(x^i,u)\equiv \underset{r\to \infty }{lim}{c}_1{\int }_0^{\infty }d\omega {\omega }^{\frac{d-3}{2}}{e}^{-i\omega u}{A}_n\left(p\right),$$

(98)

where $A_n\left(p\right)$ is given by equation (97). Carrying out a Lorentz transformation we find that $U\left(g\right)A_n(x^i,u)=\tilde{D}{\left(g^{-1}\right)}_n{}^m{A}_m(x^{i\prime },u^{\prime })$.

The procedure is best illustrated by an example, and we choose that for a spin one particle. The corresponding irreducible representation has the $d-1$ states ${\psi }_i\left(p^{\left(0\right)}\right)$ which transform under $SO(d-1)$, obey Equations (67) and (68), and are boosted as in Equation (69). The simplest embedding to find a covariant description is in a vector representation which we denote by ${\Psi }_{\mu }$. Using Equation (92) we find that in momentum space, it transforms under the Lorentz transformation as

$$\begin{array}{c}\hfill U\left(g\right){\Psi }_i\left(p\right)={\Psi }_i\left(p^{\prime }\right)-{\omega }_i{}^j{\Psi }_j\left(p\right)-\sqrt{2}(k^i{\phi }^j-k^i{\phi }^j){\Psi }_j\left(p\right)-\sqrt{2}{k}^i{e}^{-\varphi }{\Psi }_{+}\left(p\right),\\ \hfill U\left(g\right){\Psi }_{-}\left(p\right)={\Psi }_{-}\left(p^{\prime }\right)-e^{-\varphi }\sqrt{2}{k}^j{\Psi }_j\left(p\right),U\left(g\right){\Psi }_{+}\left(p\right)={\Psi }_{+}\left(p^{\prime }\right).\end{array}$$

(99)

where $p^{\mu }$ is parametrised by ${\phi }^i$ and $\varphi$ as in Equation (71) and their transformation is given in Equation (75). The transformation of ${\Psi }_{\mu }(x^i,u)$ on ${\mathcal{I}}^{+}$ is easily read off from Equation (96).

The corresponding covariant field is denoted by $A_{\mu }$ which is the familiar Maxwell field. It is related to ${\Psi }_{\mu }$ by Equation (97) which in this case is given by

$$\begin{array}{cc}\hfill A_i\left(p\right)& ={\Psi }_i\left(p\right)-{\phi }_i{e}^{\varphi }{\Psi }_{-}\left(p\right),A_{-}\left(p\right)=e^{\varphi }{\Psi }_{-}\left(p\right),\hfill \\ \hfill A_{+}\left(p\right)& =e^{-\varphi }{\Psi }_{+}\left(p\right)+{\phi }^i{\Psi }_i\left(p\right)-\frac{1}{2}{\phi }^i{\phi }_i{e}^{\varphi }{\Psi }_{-}\left(p\right).\hfill \end{array}$$

(100)

The inverse transform is given by

$$\begin{array}{cc}\hfill {\Psi }_i\left(p\right)& ={\tilde{A}}_i\left(p\right)+{\phi }_i{A}_{-}\left(p\right),{\Psi }_{-}\left(p\right)=e^{-\varphi }{A}_{-}\left(p\right),\hfill \\ \hfill {\Psi }_{+}\left(p\right)& =e^{\varphi }(A_{+}\left(p\right)-{\phi }^i{A}_i\left(p\right)-\frac{1}{2}{\phi }^i{\phi }_i{e}^{\varphi }{A}_{-}\left(p\right)).\hfill \end{array}$$

(101)

Using Equations (99) and (100) one can easily verify that $\delta A_{\mu }={\Lambda }_{\mu }{}^{\nu }{A}_{\nu }$. Using Equation (100) in the Equation (98) for $A_n(x^i,u)$ for we find that

$$\begin{array}{c}\hfill A_i(x^i,u)={\Psi }_i(x^i,u)+i\sqrt{2}{x}_i{\partial }_u{\Psi }_{-}(x^i,u),A_{-}(x^i,u)=i{\partial }_u{\Psi }_{-}(x^i,u),\\ \hfill A_{+}(x^i,u)=\frac{1}{i{\partial }_u}{\Psi }_{+}(x^i,u)-\sqrt{2}{x}^i{\Psi }_i(x^i,u)-x^i{x}_{i}i{\partial }_u{\Psi }_{-}(x^i,u).\end{array}$$

(102)

Having embedded the irreducible representation into a larger representation, we must implement conditions that ensure that it really only contains the original $d-1$ states. For the case of spin one, we should impose that $p^{\mu }{A}_{\mu }\left(p\right)=0$. Using Equations (71) and (100), we find this implies that ${\Psi }_{+}\left(p\right)=0$, the other fields being unaffected. Thus, we are left with the d fields ${\Psi }_i\left(p\right)$ and ${\Psi }_{-}\left(p\right)$, which transform into each other under the general Lorentz transformation of Equation (99). We also require the gauge symmetry $\delta A_{\mu }\left(p\right)=p_{\mu }\Lambda \left(p\right)$ and using Equations (71) and (101) we find that it implies that $\delta {\Psi }_{-}\left(p\right)=\Lambda$, $\delta {\Psi }_i\left(p\right)=0$ and $\delta {\Psi }_{+}\left(p\right)=0$. We can use this gauge transformation to set $\delta {\Psi }_{-}\left(p\right)=0$, leaving us with the $d-1$ physical states. Carrying out the Fourier transform to Minkowski spacetime, we find that ${\Psi }_{+}\left(p\right)=0$ implies that ${\Psi }_{+}(x^i,u)=0$ and taking the $r\to \infty$ limit, we find that on ${\mathcal{I}}^{+}$ we have the d unconstrained fields ${\Psi }_i(x^i,u)$ and ${\Psi }_{-}(x^i,u)$ which transform into each other under Lorentz transformations and also inherit the gauge symmetry.

We will now make the connection with the fields studied in Section 4 where the usual coordinates $X^{\mu }$ of Minkowski spacetime were exchanged for the coordinates $x^a=(r,x^i,u)$ and taking the $r\to \infty$ the boundary ${\mathcal{I}}^{+}$ emerged with the coordinates $x^i,u$. The gauge fields $A_{\mu }^{\mathcal{I}}$ on ${\mathcal{I}}^{+}$ are then found from those on Minkowski spacetime by the change of coordinates $A_{\mu }=\frac{\partial x^a}{\partial X^{\mu }}{A}_a^{\mathcal{I}}$. One finds that

$$\begin{array}{cc}\hfill A_i(x^i,u)& =\frac{1}{\sqrt{2}r}{A}_i^{\mathcal{I}}(x^i,u)-\sqrt{2}{x}^i{A}_u^{\mathcal{I}}(x^i,u),A_{-}(x^i,u)=-A_u^{\mathcal{I}}(x^i,u),\hfill \\ \hfill A_{+}(x^i,u)& =A_r^{\mathcal{I}}(x^i,u)-\frac{1}{r}{x}^i{A}_i^{\mathcal{I}}(x^i,u)+x^i{x}_i{A}_u^{\mathcal{I}}(x^i,u).\hfill \end{array}$$

(103)

The inverse transformation is given by

$$\begin{array}{cc}\hfill A_i^{\mathcal{I}}(x^i,u)& =\sqrt{2}r{A}_i(x^i,u)-2r{x}^i{A}_{-}(x^i,u),A_u^{\mathcal{I}}(x^i,u)=-A_{-}(x^i,u),\hfill \\ \hfill A_r^{\mathcal{I}}(x^i,u)& =A_{+}(x^i,u)+\sqrt{2}{x}^i{A}_i(x^i,u)-x^i{x}_i{A}_{-}(x^i,u).\hfill \end{array}$$

(104)

Using equation (102) in equation (104) we find that

$$A_i^{\mathcal{I}}(x^i,u)=\sqrt{2}r{\Psi }_i(x^i,u),A_u^{\mathcal{I}}(x^i,u)=i{\partial }_u{\Psi }_{-}(x^i,u),A_r^{\mathcal{I}}(x^i,u)=\frac{1}{i{\partial }_u}{\Psi }_{+}(x^i,u).$$

(105)

Because ${\Psi }_{+}=0$ we find $A_r^{\mathcal{I}}(x^i,u)=0$ leaving us with the fields ${\Psi }_i$ and ${\Psi }_{-}$. As the above equation makes clear, these are very closely related to the fields $A_i^{\mathcal{I}}$ and $A_u^{\mathcal{I}}$ discussed in Section 3 and Section 4. Note that the extra factor of r in $A_i^{\mathcal{I}}$ compared to ${\Psi }_i$ explains the difference in the expression for the dilation operator D acting on $A_i^{\mathcal{I}}(x^i,u)$ in Equation (31) and on ${\Psi }_i(x^i,u)$ in Equation (96). Indeed the operator D acts non-trivially on r as we discussed around (42).

The last term in the transformation of ${\Psi }_{\mu }$ in Equation (96) contains a $\frac{1}{i{\partial }_u}$ and corresponds to a transformation with a parameter of the generic form ${\Lambda }_{-i}$. It transforms ${\Psi }_i$ into ${\Psi }_{+}$, but as ${\Psi }_{+}$ vanishes, this contribution vanishes. As such, it does not affect the physical states ${\Psi }_i$. It can also transform ${\Psi }_{-}$ into ${\Psi }_i$. However, the $\frac{1}{i{\partial }_u}$ factor disappears in the corresponding transformation of $A_u^{\mathcal{I}}(x^i,u)$ as Equation (105) has a $i{\partial }_u$ in the relation between the two fields.

Although we have only explicitly carried out the analysis for the spin of one particle, the general picture is clear.