BMS in higher space-time dimensions and Non-relativistic BMS
D. G. Delmastro






	BMS in higher space-time dimensions and Non-relativistic BMS.


	
    < g r a p h i c s >


    
Author: Delmastro, D.G. 

Supervisors:  Batlle, C. and Gomis, J.
 

    June 2017
=============================================================================================






.9
BMS in higher space-time dimensions and Non-relativistic BMS.

		by

		Delmastro, D.G.

		under the supervision of

		Batlle, C. and Gomis, J.

		Submitted to the department of physics in partial
 fulfilment of the requirements for the degree of

		Master of science

		in Astrophysics, Particle Physics and Cosmology

		at the Universitat de Barcelona.

		

[1]email: ddelmade7@alumnes.ub.edu
[2]email: carles.batlle@upc.edu
[3]email: joaquim.gomis@icc.ub.edu





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CHAPTER: ACKNOWLEDGMENTS.

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Needless to say, nothing in this document – or in my career as a whole – would have been possible without the continuous support of my family. I will forever be grateful to them.

I am also thankful to friends and colleagues, with whom I have shared many priceless conversations and unforgettable experiences, and who not only have helped shape my scientific view of the world but also made the process enjoyable.

In addition, I am also in debt to the many professors I have had the pleasure to interact with and from whom I have learnt invaluable things, both from the Universidad Autónoma de Madrid, where I studied my Bachelor's Degree, and from the Universitat de Barcelona, where I studied my Master's Degree. I would also like to thank the very attentive staff from both universities.

The professors that supervised my final projects in both cycles deserve a special mention. I owe a lot to Antonio González-Arroyo España, from Madrid, and to Carles Batlle Arnau and to Joaquím Gomis Torné, from Barcelona. Last but not least, I am also in debt to the coordinator of the Master's Degree, Bartomeu Fiol Núñez.


I have refrained from making other names explicit for fear of forgetting someone. I am sure the relevant people will know I am talking about them.




CHAPTER: OUTLINE.
tocchapterOutline.



The Bondi-Metzner-Sachs (BMS) Group is the group of asymptotic isometries of asymptotically flat space-times. This group is given by the semi-direct product of the Lorentz Group with the infinite-dimensional group of the so-called super-translations, which constitutes an abelian sub-group.

The isometries of a strictly flat manifold are, as is well-known, Poincaré. If one admits a non-trivial geometry, but such that it becomes Minkowski in some of its asymptotic directions, then one may try to find the transformations that leave the asymptotic form of the metric invariant. If the metric is asymptotically flat, the most natural result one would expect is that the asymptotic isometries are Poincaré.






    < g r a p h i c s >







Indeed, this was the programme that BMS undertook, as a part of their study of gravitational waves, in the early sixties <cit.>. In an attempt to rederive the Poincaré Group as the group of isometries of a manifold that is asymptotically flat, these authors discovered, much to their surprise, that the actual group of asymptotic isometries does indeed contain Poincaré, but it contains much more: it is in fact infinite-dimensional. The new set of symmetries, today called super-translations, can be regarded as a generalisation of the standard translations. We will review the details of the BMS Group in Chapter <ref>.

The new physics discovered by these authors soon transcended by far their original intents, to the point that we nowadays consider it to be a just a part of a much larger and richer structure; for a recent review, see the lecture notes <cit.>. For example, there is compelling evidence that the BMS Group is an essential ingredient of quantum gravitational scattering <cit.>, black holes and the information paradox <cit.>, flat holography <cit.>, soft theorems <cit.>, the memory effect <cit.>, etc.

The last two of these topics, the soft theorems and the memory effect, have a major role in the study of the BMS Group. The soft theorems describe the universal behaviour of scattering amplitudes that include massless particles with vanishing momentum. These soft particles can be understood as the Goldstone bosons of spontaneously broken large gauge symmetries; in particular, a broken BMS symmetry gives rise to soft gravitons. On the other hand, the memory effect is the permanent displacement of two detectors as the consequence of passing-by gravitational waves; in particular, the relative positions and clock times of the detectors before and after the radiation transit differ by a super-translation. As stressed by Strominger and collaborators <cit.>, the memory effect, the soft theorems, and the concept of asymptotic symmetries are the three corners of an infra-red triangle – a triality that is present in any theory with non-trivial infra-red dynamics.

1.0 



[thick,blue,<->,>=stealth] (1,0) – (3,0);
[thick,blue,<->,>=stealth] (3.5, 0.866) – (2.5, 2.598);
[thick,blue,<->,>=stealth] (0.5, 0.866) –  (1.5, 2.598);

at (0,.2) [align=center] Soft
Theorem;
at (4,.2) [align=center] Memory
Effect;
at (2,3.3) [align=center] Asymptotic
Symmetry;








This triangular equivalence interweaves these three seemingly unrelated topics and makes them interdependent. This is relevant to our work for the following reason: on the one hand, it has recently been argued that the memory effect vanishes for d>3, where d is the number of space dimensions <cit.>. On the other hand, the soft theorems exist for any number of space-time dimensions <cit.>. This puts the third corner, the asymptotic symmetries, in an uncertain position: to what extent should we expect a non-trivial BMS Group in higher dimensions?


This apparent contradiction has recently been addressed by Strominger et al <cit.>. According to these authors, the classical analysis of asymptotic symmetries (e.g. <cit.>, etc.) imposed unnecessarily strong conditions for asymptotic flatness, thus eliminating the degrees of freedom corresponding to super-translations. If one slightly relaxes these conditions, the whole family of super-translations emerges, and the paradox concerning the triangle is resolved.

This provides the main motivation for the first part of our thesis (Chapter <ref>): we will argue that one can construct the algebra of the BMS Group in any number of space-time dimensions, by a different method than the standard approach: we will not consider the gravitational problem, but we will consider instead the canonical construction of the 𝔟𝔪𝔰 algebra <cit.>. This construction is based on the study of the symmetries of the equations of motion of a free field in flat space-time. As is well-known, the set of point symmetries of such equations contains Poincaré transformations; but, as we will show, it is much larger than that, and it contains, in particular, super-translations. The algebra of these symmetries coincides with the one of the gravitational problem, which is one more piece of evidence that suggests that the relevance of the BMS Group goes beyond the gravitational problem.

The existence of a canonical BMS Group for any number of space-time dimensions means that either we should expect a non-trivial gravitational BMS for any d, or that the agreement between the canonical and gravitational BMS in d=2,3 is just a fortuitous coincidence. Of course, the first of these two options is not only more attractive, but also much more natural.




The second part of this thesis (Chapter <ref>) concerns the problem of a non-relativistic BMS Group. One could define this group as the set of isometries of asymptotically flat Newtonian space-times, in the sense of Cartan <cit.>. Even though some interesting results in this area have been obtained (e.g., <cit.>), the overall picture is much less clear than in the relativistic case.


In order to construct the non-relativistic analogue of 𝔟𝔪𝔰 there are several approaches one could take. The simplest one is to consider a suitable İnönü-Wigner contraction of the corresponding relativistic 𝔟𝔪𝔰 algebra, in the same sense that the Galilei algebra can be obtained by contracting the Poincaré one. An alternative to obtain a possible non-relativistic 𝔟𝔪𝔰 algebra is to mimic the aforementioned canonical construction, but using a scalar field with Galilean space-time symmetries instead of a relativistic field. Finally, a third alternative is, naturally, to follow the same steps that led to the original 𝔟𝔪𝔰 algebra, but in a non-relativistic setting, i.e., one could try to study the set of isometries of asymptotically flat Newtonian space-times. In this work, we will only consider the first two possibilities.

It bears mentioning that some partial results concerning the non-relativistic 𝔟𝔪𝔰_4 algebra have already been published, cf. <cit.>. In this thesis we extend the results therein and contextualise them by comparing them to the relativistic algebra.





CHAPTER: THE BMS GROUP.


In this chapter we will provide a brief but hopefully self-contained introduction to the BMS Group. A more detailed discussion can be found in <cit.>.

The problem that BMS addressed is the following: given a certain metric g_μν(x) that describes the geometry of an asymptotically flat (lorentzian) manifold,

    g_μν(x)η_μν,

we want to find the vector fields ξ such that they leave the asymptotic form of the metric invariant,

    ℒ_ξ g_μν(x)0,

where ℒ denotes a Lie derivative (see e.g. <cit.>). Of course, to make this problem well-defined we must properly specify what we mean by the asymptotic limit x∼∞.


To make the discussion above more precise, let us consider a flat manifold with cartesian coordinates x^μ=(x^0,…,x^3), and let us define the radial coordinate r and the retarded time u_+ as

    r   √(x_1^2+x_2^2+x_3^2)
    
    u_+    x^0-r



The coordinate u_+ is the parameter that appears in the solution to Maxwell's equations in the Lorenz gauge <cit.>, ∂^2 A^μ= j^μ, viz.

    A^μ(x)   =∫Δ_ret(x-y) j^μ(y) dy
       =∫1/2πδ((x_0-y_0)^2-( x- y)^2)Θ(y^0-x^0) j^μ(y) dy
       =∫j^μ(x^0-| x- y|, y)/4π| x- y| d y

and as such, it has the following interpretation: if there is a perturbation in the source at a time u_+, then a detector placed at a distance r will receive the signal at a time x^0. In other words, the world-line of a massless particle is of the form u_+=const.


If we emit massless particles radially away from the origin, at a time u_+, then the sphere they reach at null infinity, r→∞, is called the celestial sphere 𝒞𝒮^2. We let z^A, with A=1,2, parametrise this sphere. Some common choices for these coordinates are: spherical coordinates, z^A=(θ,φ), defined by

    x^1=rsinθcosφ,    
    x^2=rsinθsinφ,    
    x^3=rcosθ

and the stereographic (complex) coordinates z^A=(z,z̅), defined by

    z= x^1+i x^2/r+x^3,    z̅= x^1-i x^2/r+x^3


We will ultimately be interested in a possible extension of the upcoming discussion to higher dimensional space-times. Therefore, the parametrisation in terms of angles seems to be more convenient than the stereographic one. We will come back to this point later on; for now, we keep the coordinates z^A unspecified.


The coordinates u_+,r,z^A are called retarded Bondi coordinates. The set of all celestial spheres,

    ℐ^+={(u_+,z^A): u_+∈ℝ, z^A∈𝒞𝒮^2}

is called future null infinity, and it is topologically a cone ℝ× S^2.

There exists an analogous construction using advanced Bondi coordinates, where one introduces an advanced time u_- x^0+r. In the context of electrodynamics, this coordinate appears in the advanced propagator Δ_adv, and is relevant to, for example, the Wheeler-Feynman absorber theory <cit.>. 





As in the retarded case, one defines past celestial spheres, which foliate past null infinity ℐ^-. All massless particles originate in ℐ^- and end up in ℐ^+. On the other hand, massive particles originate in past timelike infinity i^-, and end up in future timelike infinity i^+. Finally, spatial infinity is denoted by i^0 (see Fig. <ref>).


1.0 




In Bondi coordinates, the Minkowski metric reads

    ds^2=-du^2_±∓2dr du_±+r^2γ_ABdz^Adz^B

where γ_AB is the metric on the unit sphere. In spherical coordinates, γ=(1,sin^2θ), and in stereographic coordinates, γ_zz̅=γ_z̅z=2/(1+zz̅)^2.

In cartesian coordinates, the Killing vectors of this manifold are given by

    ξ=a^μ∂_μ+Ω^μνx_ν∂_μ,

with a^μ,Ω^μν∈ℝ a set of constants, with Ω^μν=-Ω^νμ. In Bondi coordinates, these Killing vectors are given by a more complicated and not particularly illuminating expression (see e.g. <cit.>). In the particular case of r→∞, that is, at ℐ^±, translations correspond to

    u→ u+α(z^A)

where, in spherical coordinates,

    α(θ,φ)=a^0+a^1sinθcosφ+a^2sinθsinφ+a^3cosθ


For future reference, one should note that these are just the spherical harmonics Y_0^0,Y_1^m, with m=0,±1. In stereographic coordinates,

    α(z,z̅)∼linear combinations of 1,z,z^2,z̅,z̅^2




On the other hand, (homogeneous) Lorentz transformations correspond to conformal transformations on the celestial sphere, given by

    z   →ω_11z+ω_12/ω_21z+ω_22    with    ω=ω(Ω)∈SL(2,ℂ)
    
    u   → (1+δ(z,z̅))u

for a certain function δ (whose form shall not concern us here).

The infinitesimal generators M_μν of these transformations are given by <cit.>

    M_12   =z∂_z-z̅∂_z
    
    M_13   =-i/2((z^2+1)∂_z+(z̅^2+1)∂_z̅)
    
    M_23   =-1/2((z^2-1)∂_z-(z̅^2-1)∂_z̅)
    
    M_01   =+i/2((z^2-1)∂_z+(z̅^2-1)∂_z̅)-iuz+z̅/1+zz̅ ∂_u
    
    M_02   =+1/2((z^2+1)∂_z-(z̅^2+1)∂_z̅)-iui(z̅-z)/1+zz̅ ∂_u
    
    M_03   =-i(z∂_z+z̅∂_z)-iu1-zz̅/1+zz̅ ∂_u


These vector fields satisfy the Lorentz algebra. As a matter of fact, the angular part can be written as linear combination of the generators of the Witt algebra <cit.>, l_n=-z^n+1∂_z,l̅_n=-z̅^n+1∂_z̅, with n=0,±1.


On the other hand, in spherical coordinates the Lorentz generators are given by

    M_12   =-i∂_φ
    
    M_13   =icosφ ∂_θ-iθsinφ ∂_φ
    
    M_23   =isinφ ∂_θ+iθcosφ ∂_φ
    
    M_01   =icosφcosθ ∂_θ-isinφθ ∂_φ-iucosφsinθ ∂_u
    
    M_02   =isinφcosθ ∂_θ+icosφθ ∂_φ-iusinφsinθ ∂_u
    
    M_03   =isinθ ∂_θ+iucosθ ∂_u


Upon Lie bracketing, one verifies that these Killing vectors satisfy the Poincaré algebra,

    [P^μ, P^ν]   =0
    
    [P_α, M^μν]   =2iδ^[μ_α P^ν]
    
    [M_αβ,  M^μν]   =4iδ^[μ_[α   M^ν]_β]





Let us now move on to the non-flat case. With our previous definitions at hand, we we can now properly define what it means for a manifold to be asymptotically flat at ℐ^±: it means that it admits local coordinates u_±,r,z^A, such that, as r→∞, the metric takes the form (<ref>) up to terms subleading in r. Apart from the metric becoming flat at null infinity, g_μν(ℐ^±)=η_μν, we also assume it to be smooth there. One can prove <cit.> that this is equivalent to assuming that the components of the metric admit a well-defined 1/r expansion around r→∞ (however, see <cit.>).

A much more technical and elegant definition of asymptotic flatness can be found in <cit.> and references therein. This definition is based on the existence of a particular Penrose compactification of the manifold and, as such, it is much more geometrical and coordinate-independent than ours. As discussed in <cit.>, the Penrose definition is not easily extended to higher (odd) dimensions, due to a potential non-analiticity of the metric at null infinity. On the other hand, the extension to higher dimensions of the Bondi definition in terms of an explicit 1/r expansion is in principle much more direct. This latter definition will be enough for our purposes. 




For concreteness, we will now focus on the asymptotic future ℐ^+, so we will drop the `+' label on the retarded time. The corresponding analysis at ℐ^- is analogous.



Following Bondi, we take u to be a null coordinate, so that g^uu=0, and we take the angular coordinates z^A to be constant along null rays, so that g^uA=0. Imposing that the two-spheres spanned by z^A have area 4π r^2, we get g_AB=r^4γ_AB. These conditions fix the diffeomorphism invariance of the metric, and are usually known as the Bondi gauge conditions. One should note that some of these conditions are sometimes modified in the literature; for example, Penrose <cit.> replaces the metric on the sphere, γ_AB by one conformally equivalent to it, with an arbitrary conformal factor (see <cit.> for a comparison of the Bondi-Sachs and the Penrose approaches). In any case, we shall stick to the Bondi conditions in what follows.


After fixing the gauge, the metric takes the so-called Bondi-Sachs form <cit.>:

    ds^2=-V/re^2βdu^2-2 e^2βdr du+r^2h_AB(dz^A-1/2 U^Adu)(dz^B-1/2 U^Bdu)

where we have defined h_AB r^-2g_AB, so that the Bondi area gauge condition reads h_AB=γ_AB. The indices on the sphere A,B,… are lowered and raised with the metric h_AB and its inverse h^ABh_BC=δ^A_C. We also denote the covariant derivatives with respect to h_AB by the letter D_A.

Asymptotic flatness, in Bondi coordinates, means

    lim_r→∞β   =lim_r→∞U^A=0
    lim_r→∞V/r   =1
    lim_r→∞h_AB   =γ_AB





If, as before, we denote the covariant derivative with respect to h_AB by D_A, and we let ƪ_A be the covariant derivative with respect to γ_AB, then the asymptotic condition h_AB→γ_AB implies that D_A→ƪ_A. The precise correspondence is

    D_Av^B=ƪ_A v^B+1/2rγ^BF(ƪ_A c_FE+ƪ_Ec_FA-ƪ_Fc_AE)v^E+𝒪(r^-2)

for an arbitrary vector field v^B. Here, c_AB=c_AB(z) is defined as the subleading part of the metric on the sphere:

    h_ABγ_AB+c_AB/r+𝒪(r^-2)


The rest of the components of the metric must have a well-defined falloff too. There is a certain freedom in the choice of these falloffs, and certain consistency conditions that determine some coefficients in terms of the rest. The (physically motivated) choice for the subleading behaviour of the metric made by Bondi, van der Burg, Metzner and Sachs was

    V   =r-2M+𝒪(r^-1)
    β   =-1/32c^ABc_AB/r^2+𝒪(r^-3)
    
    U^A   =-ƪ_B c^AB/2r^2+2L^A/r^3+𝒪(r^-4)

where M,L are arbitrary functions defined on ℐ^+; M=M(u,z) is called the mass aspect, and L^A=L^A(u,z) is called the angular momentum aspect.


We are now in position to tackle the problem of characterising the isometries of asymptotically flat space-times; the set of these diffeomorphisms is called the BMS Group. These are, by definition, the diffeomorphisms that preserve the Bondi gauge conditions as well as the metric falloffs (<ref>). Preservation of the gauge conditions

    g_rr=g_rA=0,     h_AB=γ_AB

requires

    ℒ_ξ g_rr=ℒ_ξ g_rA=ℒ_ξ h_ABh^AB=0

while preservation of the metric falloffs requires

    ℒ_ξ g_ur=𝒪(r^-2),  ℒ_ξ g_uA=𝒪(1),  ℒ_ξ g_AB=𝒪(r),  ℒ_ξ g_uu=𝒪(r^-1)



The general solution to the first set of equations (<ref>) is

    ξ(r,u,z)=α(u,z)∂_u+f^A(u,z)∂_A+ξ^r∂_r+𝒪(r^-1)

where α,f^A are a pair of arbitrary functions, and where the actual form of ξ^r and of the 𝒪(r^-1) term shall not concern us here.

With this, the first equation in (<ref>) implies that

    α(u,z)=α(z)+1/2∫_0^uƪ_Bf^B(u',z) du'

while the second equation implies that f^B(u,z)=f^B(z). The third equation is equivalent to requiring f^B to be  a conformal Killing vector of the sphere,

    ƪ_(Af_B)-1/2γ_ABƪ_Cf^C=0

while the fourth equation is automatically satisfied.


The infinitesimal generators of the BMS Group thus read <cit.>

    ξ(ℐ^+)=[α(z)+1/2u ƪ_Bf^B(z)]∂_u+f^A(z)∂_A

where α(z) is an arbitrary function, and f^A is a conformal Killing vector of the sphere.

The Killing vectors with α(z)≡ 0 correspond to local conformal transformations on the celestial spheres. In complex coordinates, the conformal Killing equation becomes the Cauchy-Riemann equations, which means that f^z (resp. f^z̅) is given by an arbitrary holomorphic (resp. anti-holomorphic) function on the Riemann sphere S^2=ℂ∪{∞}. One can always write such a function as

    f^z∂_z=∑_n∈ℤ f_n l_n,    with   l_n=-z^n+1∂_z

and a similar relation for f^z̅. If f_n≠ 0 for n<-1, then the Killing vector is singular at z→ 0, while if f_n≠0 for n>1, then it is singular at z^-1→ 0. If we restrict ourselves to non-singular functions (so that the global transformation is well-defined) we recover the Lorentz generators (<ref>), that is, the values n=0,±1 (for more details, see <cit.>). It has recently been argued <cit.> that there is no reason to require global well-definedness. If we abandon such condition, the general set of generators is l_n,l̅_n, for n∈ℕ. The new transformations are called super-rotations. We will not be interested in them in this thesis.

for d>3, and due to Liouville's theorem <cit.>, the solutions to the conformal Killing equation are just the Lorentz generators, with no super-rotations. A simplified argument for the augmented group of conformal diffeomorphisms on the d-1=1,2 spheres is the following <cit.>: let n=d-1 be the dimension of the celestial spheres S^n. Then the conformal Killing equation has n unknowns f^A, and 1/2(n-1)(n+2)  independent equations. For n=1 we get no constraints, so the general solution depends on one arbitrary function. For n=2, the two equations for the two unknowns are the Cauchy-Riemann equations, so the general solution depends on two arbitrary functions. For n> 2, the system is over-determined, and the general solution has a finite number of real parameters: 𝔠𝔬𝔫𝔣(S^n)<∞. Indeed, 𝔠𝔬𝔫𝔣(S^n) coincides with the orientation-preserving part of Spin(1,d), which is the double cover of the Lorentz Group Möb(n)≅SO(1,d).

On the other hand, the vector fields with f^A(z)≡ 0 are called super-translations, and they include standard translations for a specific choice of the function α. Indeed, if we let

    α(θ,ϕ)=a^0+a^1sinθcosφ+a^2sinθsinφ+a^3cosθ

we recover equation (<ref>), i.e., we find the expected isometries of Minkowski. The surprising fact is that α(z) can be an arbitrary function, which means that the group of isometries is now infinite-dimensional.

one could argue that Minkowski is itself an asymptotically flat space-time, and therefore its group of asymptotic symmetries includes super-translations as well. In this sense, there would be no enlargement of the group of symmetries when one goes from flat space-time to asymptotically flat space-times: in both cases the group of symmetries is BMS. This is not really the case: the formal definition of an asymptotic symmetry is one such that ℒ_ξ g_μν≈ 0 “as good as possible”. In flat space-time, the best approximation to the Killing equation is the Killing equation itself, ℒ_ξη_μν≡ 0, whose solutions are exactly Poincaré. In non-flat space-times, in general the Killing equation doesn't hold anywhere except possibly at ℐ^±. There, the solutions include an arbitrary function α, and there is no general way (or reason) to single out the particular case (<ref>). If we happen to find translations, they must come hand in hand with super-translations.



Following Sachs <cit.> (in spherical coordinates) and Barnich and  Troessaert <cit.> (in stereographic coordinates), we may expand the function α(z) in spherical harmonics Y(θ,φ), or as a Laurent series

    α(z)   =∑_ℓ∈ℕ∑_m=-ℓ^+ℓα^ℓ_m Y_ℓ^m(θ,φ)
       =∑_n,n̅∈ℤα_nn̅  z^nz̅^n̅

which leads to the (countable) infinite-dimensional family of super-translations, generated by

    P_ℓ^m(ℐ^+)    Y_ℓ^m∂_u
    
    P_n,n̅(ℐ^+)    z^nz̅^n̅∂_u



Denoting by M_μν the generators of (homogeneous) Lorentz transformations, one may show that the generators of the BMS Group satisfy the algebra

    [M_23,P_ℓ^m]   =i/2√((ℓ+m) (ℓ-m+1)) P_ℓ^m-1-i/2√((ℓ-m) (ℓ+m+1)) P_ℓ^m+1
    
    [M_31,P_ℓ^m]   =1/2√((ℓ-m) (ℓ+m+1)) P^m+1_ℓ+1/2√((ℓ+m) (ℓ-m+1)) P^m-1_ℓ
    
    [M_12,P_ℓ^m]   =m  P^m_ℓ
    [M_01,P_ℓ^m]   =ℓ +2/2[ √((ℓ-m) (ℓ-m-1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m+1+√( (ℓ +m)(ℓ+m -1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-1]
       +ℓ -1/2[√((ℓ +m+1) (ℓ +m+2)/(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^m+1+√((ℓ-m+1)(ℓ-m+2) /(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^m-1]
    
    [M_02,P_ℓ^m]   =i(ℓ +2)/2[  √((ℓ-m) (ℓ-m-1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m+1-√( (ℓ +m)(ℓ+m -1)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-1]
       +i(ℓ -1)/2[√((ℓ +m+1) (ℓ +m+2)/(2 ℓ +1) (2 ℓ +3))  P_ℓ+1^m+1-√((ℓ-m+1)(ℓ-m+2) /(2 ℓ +1) (2 ℓ +3))  P_ℓ+1^m-1]
    
    [M_03,P_ℓ^m]   =i (ℓ +2) √((ℓ -m) (ℓ +m)/(2 ℓ -1) (2 ℓ +1)) P_ℓ-1^m-i (ℓ -1)√((ℓ -m+1) (ℓ +m+1)/(2 ℓ +1) (2 ℓ +3)) P_ℓ+1^m

or, in stereographic coordinates,

    [l_n,l_n']   =(n-n')l_n+n'
    
    [l̅_n̅,l̅_n̅']   =(n̅-n̅')l̅_n̅+n̅'
    
        
    [l_p,P_n,n̅]   =(12(p+1)-n)P_n+p,n̅
    
    [l̅_p,P_n,n̅]   =(12(p+1)-n̅)P_n,n̅+p



This is the 𝔟𝔪𝔰_4 algebra. In stereographic coordinates, the result is much more compact and transparent, but it has the disadvantage that its generalisation to higher dimensions is not straightforward. We shall henceforth focus on the parametrisation in terms of spherical coordinates.


One of the most prominent features of the 𝔟𝔪𝔰_4 algebra is that it contains, as one would expect, a Poincaré sub-algebra. Indeed, if we restrict ourselves to the super-translations with ℓ=0,1, then the algebra above is easily seen to contain a closed sub-algebra, because the coefficients of the terms with ℓ→ℓ+1 are proportional to (ℓ-1), which vanishes at ℓ=1. Under the identification

    P^1   =P^1_1+P^1̅_1/i√(2)
    
    P^2   =P_1^1-P_1^1̅/√(2)
    
    P^3   =P^0_1
        inverse⟺    
    P_1^1̅   =-P^2+iP^1/√(2)
    
    P_1^1   =+P^2+iP^1/√(2)
    
    P^0_1   =P^3

one may check that the ℓ=0,1 sub-set of the 𝔟𝔪𝔰_4 algebra coincides with the Poincaré algebra. This in turns means that there is a well-defined notion of energy and momentum for asymptotically flat manifolds; this is the Bondi momentum p_B <cit.>.


the Killing equation that we used to determine the super-translations is homogeneous, which means that the normalisation of P_ℓ^m is in principle arbitrary. Here we have followed the standard convention P_ℓ^m=Y_ℓ^m∂_u, which we shall call the gravitational normalisation. Had we included some ℓ-dependent coefficient in front of the spherical harmonics, the actual value of the structure constants of the algebra (<ref>) would have been different. In the next chapter we will rederive the 𝔟𝔪𝔰 algebra in a different context, and there we will find that the most natural normalisation for the super-translations is different; we will call it the canonical normalisation. These two normalisations will be related through an ℓ-dependent rescaling.


To extend the discussion above to higher-dimensional space-times, we would have to make a careful analysis of the falloff conditions, and repeat the calculation of the Killing vector fields. From a  more pragmatic and perhaps naïve point of view, we could simply anticipate that the solution takes the same form as in the four dimensional case,

    ξ(ℐ^+)?=[α(z)+u/d-1ƪ_Bf^B(z)]∂_u+f^A(z)∂_A

where α(z) is an arbitrary function, and f is a conformal Killing vector on S^d-1,

    ƪ_(Af_B)-1/d-1γ_AB ƪ_Cf^C=0


It has recently been argued <cit.> that in d>3 the most natural falloff conditions for the metric are actually too strong to admit a non-trivial BMS Group. In such a case, the only admissible solutions to the BMS problem would be Poincaré, that is, the Killing vectors ξ would correspond to Lorentz transformations and standard translations. The case is not closed yet though: <cit.> counter-argues that these conditions are unnecessarily strong, and that the BMS Group exists and is non-trivial in any number of space-time dimensions.

In any case – be it physically realistic or not – we can dismiss these arguments and impose whatever falloff conditions we may need in order for ξ(ℐ^+)=α(z) to be a valid asymptotic isometry. 
In this case, the calculation of the 𝔟𝔪𝔰_d+1 algebra becomes straightforward: we just need to expand the super-translations in (higher dimensional) spherical harmonics, Y_ℓ, where ℓ=(ℓ_1,ℓ_2,…,ℓ_d-1) is a multi-index:

    P_ℓ(ℐ^+) Y_ℓ ∂_u

and calculate the different Lie brackets among the P_ℓ,M_μν. We will come back to this problem, from a different perspective, in Chapter <ref>. There we will see that the 𝔟𝔪𝔰 algebra is a natural construction in any number of space-time dimensions, which suggests that, irrespective of its applicability to the gravitational problem, the BMS Group exists and is non-trivial for any d.

To close this chapter, it is worthwhile mentioning that the group that we have been studying is BMS^+, corresponding to the symmetries that act on future null infinity ℐ^+. The analogous group acting on past null infinity ℐ^- is denoted by BMS^-. Both groups admit extensions in the sense of super-rotations (and even an extension to include conformal transformations <cit.>). Moreover, it has recently been conjectured <cit.> that the so-called diagonal subgroup of BMS^+×BMS^- is the group of symmetries of (both classical and quantum) gravitational scattering, giving rise to an infinite number of conservation laws. This subgroup is obtained by a certain identification of future-acting to past-acting super-translations, α^+(z)=α^-(z). We refer the reader to <cit.> for more details.







CHAPTER: CANONICAL REALISATION OF BMS.


In this chapter we will  study a subset of the symmetries of the equations of motion of a relativistic free field ϕ of arbitrary spin in flat space-time. This object can be regarded as a classical field, or as a quantum operator; for definiteness, here we will only consider the first case, though our main conclusions will hold in the second case as well.

Let the equations of motion of ϕ:ℝ^d+1→ℂ^s  be

    𝒟ϕ=0

for 𝒟 a certain Poincaré invariant linear differential operator. By Poincaré invariance we mean that 𝒟 commutes with translations and Lorentz transformations:

    [𝒟,-i∂_μ]=[𝒟,ix_[ν∂_μ]+spin]=0

where “spin” corresponds to a finite-dimensional matrix that generates the representation of the Lorentz Group under which ϕ transforms. In simple terms, Poincaré invariance means that 𝒟 does not depend on x, and that its spin indices are contracted in a Lorentz-invariant way. Typical examples include the Klein-Gordon equation, the Dirac equation or the Maxwell equations.

What we want to do is to find the on-shell symmetries of 𝒟, that is, the differential operators Q such that

    [𝒟,Q]∝𝒟


It is clear that if -i∂_μ is a symmetry of 𝒟, then so is f(-i∂_μ) for any function f. Standard translations are recovered from f by taking it to be the identity. We will see that, for a certain family of functions f, the associated differential operators f(-i∂_μ) will correspond to super-translations, and they will therefore furnish a representation of the 𝔟𝔪𝔰 algebra. The point of this construction is that it is qualitatively independent of the number of space-time dimensions, so it provides a natural extension of the 𝔟𝔪𝔰_4 algebra to any d.

instead of considering ϕ to be a free field, we could take it to be an interacting field such that the interactions vanish at large times t→±∞. In fact, this is the usual setting considered in interacting quantum field theories, where fields are asymptotically free (e.g., the LSZ formalism <cit.> or the more technical Haag-Ruelle scattering theory <cit.>, the Gell-Mann and Low theorem <cit.>, etc.). This would lead to two groups of symmetries, at t→±∞, and one could perhaps draw an analogy with BMS^±, acting on ℐ^±.

The equation 𝒟ϕ=0, being free, can be solved most easily in momentum space. We will frame our discussion directly in terms of a,a^*, that is, the Fourier modes of ϕ. From now on, no explicit reference to the underlying field ϕ shall be made (we will come back to it in Chapter <ref>).



Let us therefore introduces a pair of canonical variables a_k⃗,σ,a_k⃗,σ^*, with algebra

    {a_k⃗,σ,a_k⃗',σ'}   =0
    
    {a_k⃗,σ,a^*_k⃗',σ'}   =δ_σσ'δ_k⃗k⃗'

where {·,·} denotes a Dirac bracket.

Here, k⃗∈ℝ^d is a continuous index – the momentum – and σ∈ℕ is a discrete index – the spin. On the other hand, δ_σσ' is the Kronecker delta, while δ_k⃗k⃗' is the Dirac delta on the mass hyperboloid,

    δ_k⃗k⃗' (2π)^d2√(k⃗^2+m^2) δ(k⃗-k⃗')




Given this pair of canonical variables, one may realise the Poincaré algebra through the standard construction,

    P_μ   ∑_σ∫ a^*_k⃗,σk_μ a_k⃗,σ ḳ
    
     M_0i    t P_i+∑_σ∫ a^*_k⃗,σℳ_0i a_k⃗,σ ḳ+spin
    
     M_ij   ∑_σ∫ a^*_k⃗,σℳ_ij a_k⃗,σ ḳ+spin

where ḳ is the invariant measure on the mass hyperboloid,

    ḳ1/(2π)^ddk⃗/2√(k⃗^2+m^2)

and ℳ_μν are the differential operators

    ℳ_0i    ik_0∂_i
    ℳ_ij    2ik_[i∂_j]


The differential operators ℳ_μν satisfy the Lorentz algebra,

    [ ℳ_αβ,ℳ^μν]=4iδ^[μ_[αℳ^ν]_β]

while the functions P_μ,M_μν satisfy the Poincaré algebra,

    { P^μ, P^ν}   =0
    { P_α, M^μν}   =2iδ^[μ_α P^ν]
    { M_αβ,  M^μν}   =4iδ^[μ_[α   M^ν]_β]


if we regard ℳ_μν as a vector field instead of a differential operator, then it is straightforward to check that it satisfies the Killing equation, that is, ℳ_μν generates the isometries of the mass hyperboloid which, as a manifold (Euclidean AdS_d), is maximally symmetric. This corresponds to the well-known isomorphism Iso(EAdS_d)=SO(1,d).

If we think of the functions a,a^* as the Fourier modes of a free field, then the objects P_μ, M_μν are actually the Noether charges corresponding to space-time symmetries. In particular, P_μ are the generators of space-time translations, and they are conserved. In fact, for free theories one may actually construct an infinite number of conserved charges,

    Q_f=∑_σ∫ a^*_k⃗,σf(k⃗) a_k⃗,σ ḳ

where f is an arbitrary function defined on the mass hyperboloid. For each f we will have a charge which, albeit perhaps non-local, is conserved Q̇_f=0.

The standard Poincaré algebra is obtained by taking f(k⃗)= k^μ, in which case the conserved charge Q_f coincides with the generators of translations, P^μ. The generalisation of the Poincaré algebra to the 𝔟𝔪𝔰 algebra is obtained by extending the functions k^μ into the larger set χ_ℓ^λ(k⃗),

    k^μ   → mχ^λ_ℓ(k⃗)
    
    P^μ   → P^λ_ℓ m∑_σ∫ a^*_k⃗,σχ_ℓ^λ(k⃗) a_k⃗,σ ḳ

for a certain set of (dimensionless) functions χ^λ_ℓ defined on the mass hyperboloid. Here, ℓ=(ℓ_1,ℓ_2,…,ℓ_d-1)∈ℤ^d-1 is a certain multi-index whose precise role will be clarified below; while λ∈ℝ is a real parameter. The set of indices (λ,ℓ) generalise the vector index μ.

We will call the symmetries generated by the conserved charges P_ℓ^λ super-translations. As long as the set of functions {χ^λ_ℓ} contains k^μ for certain values of (λ,ℓ), the corresponding conserved charges will agree with the generators of standard translations. Furthermore, if the functions {M_μν,P_ℓ^λ} satisfy a closed algebra, then we will have succeeded in finding a generalisation of the Poincaré algebra. The new algebra, the 𝔟𝔪𝔰 algebra, will be the sought-after infinite-dimensional generalisation of the Poincaré algebra. In what follows, we will describe how the functions χ^λ_ℓ are constructed.


The functions P^λ_ℓ clearly commute among themselves, while the functions M_μν satisfy the Lorentz algebra. The 𝔟𝔪𝔰 algebra is obtained by extending the relation { P_α, M^μν}=2iδ^[μ_α P^ν] into

    { P^λ_ℓ, M_μν}=∑_α∈ℤ^d-1c^α_μν  P^λ_ℓ+α

for a certain set of structure constants c^α_μν=c^α_μν(λ,ℓ). Equivalently, we can also write the relation above as

    ℳ_μνχ^λ_ℓ=∑_α∈ℤ^d-1c^α_μν χ^λ_ℓ+α




From this equation we see that what we want  is essentially to construct an infinite-dimensional representation of the Lorentz algebra. Therefore, we will define the functions χ^λ_ℓ as the eigenvectors of the Casimir operator of the Lorentz algebra,

    Δ1/2ℳ_μνℳ^μν

such that

    Δχ^λ_ℓ=λχ^λ_ℓ


For one thing, we note that the functions k^μ themselves satisfy Δ k^μ=d k^μ, which nicely fits in our picture: we are trying to generalise the functions k^μ into a larger set, so it is reassuring to see that they indeed satisfy the equation we are using to define χ^λ_ℓ.

In order to better understand the equation above, it is useful to point out that Δ is actually the Laplace-Beltrami operator on the mass hyperboloid k^2+m^2=0. In other words, given the flat metric ds^2=-dk_0^2+d k^2 and the embedding k^0→ +√(k⃗^2+m^2), the induced metric is ds^2=g_ijdk^idk^j, where

    g_ij=δ_ij-k^ik^j/k⃗^2+m^2    inverse⟺     g^ij=δ_ij+k^i k^j/m^2

with determinant g=(g_ij)=m^2/k⃗^2+m^2.

With this, the induced Laplacian reads

    1/m^2Δ=1/√(g)∂_i(√(g)g^ij∂_j)=g^ij∂_i∂_j+d/m^2 k^i∂_i

which agrees with (<ref>). For completeness, we also mention that the Ricci scalar of this manifold is R=-d(d-1)/m^2 and the Christoffel symbols are Γ^k_ij=-g_ijk^k/m^2.


With this, we will define our functions χ^λ_ℓ to be the eigenvectors of Δ, that is, the solutions of

    Δχ^λ_ℓ=λχ^λ_ℓ


The operator Δ is formally self-adjoint, and has an empty point spectrum <cit.>. In what follows we will be interested in the continuous spectrum of Δ and, in particular, in the singular value λ=d, as we shall discuss later on. For this particular value of λ, the equation above becomes invariant under conformal transformations g_ij→Ω^2 g_ij, χ→Ω^(2-d)/2χ.

The equation (<ref>) does not determine χ^λ_ℓ uniquely, because the eigenvalues λ are degenerate; this is where the labels ℓ come into play. In order to fix the functions χ^λ_ℓ, we will simultaneously diagonalise the set of commuting operators {Δ, ℳ_12,Δ_S^n}, where Δ_S^n is the Laplacian of the unit n-sphere,

    Δ_S^n=∑_n>i>jℳ_ijℳ^ij

whose eigenvalues are of the form -ℓ_n(ℓ_n+n-1), with ℓ_n∈ℕ a non-negative integer (see the appendix <ref>).

With this, our functions χ^λ_ℓ are defined through

    {Δχ^λ_ℓ   =λχ^λ_ℓ
    ℳ_12χ^λ_ℓ   =ℓ_1χ^λ_ℓ
    Δ_S^nχ^λ_ℓ   =-ℓ_n(ℓ_n+n-1)χ^λ_ℓ     n=2,3,…,d-1
    .

with |ℓ_1|≤ℓ_2≤ℓ_3≤⋯≤ℓ_d-1. The last of these labels, ℓ_d-1, plays the role of “total angular momentum”, so we will sometimes denote it by Lℓ_d-1. 


The equations (<ref>), together with univaluedness and finiteness, are enough to uniquely determine the functions χ^λ_ℓ. We will solve them next. These equations are separable in spherical coordinates (ρ,θ):

    k^1   =ρsinθ_d-1sinθ_d-2⋯sinθ_2sinθ_1
    
    k^2   =ρsinθ_d-1sinθ_d-2⋯sinθ_2cosθ_1
    
    k^3   =ρsinθ_d-1sinθ_d-2⋯cosθ_2
       ⋯
    
    k^d-1   =ρsinθ_d-1cosθ_d-2
    
    k^d   =ρcosθ_d-1


In these coordinates, the Laplacian reads

    1/m^2Δ=(1+ρ^2/m^2)∂_ρ^2+(d-1/ρ+d ρ/m^2)∂_ρ+1/ρ^2Δ_S^d-1

where Δ_S^d-1 is the Laplacian on the (d-1)-sphere. The equations

    ℳ_12χ^λ_ℓ   =ℓ_1χ^λ_ℓ
    Δ_S^nχ^λ_ℓ   =-ℓ_n(ℓ_n+n-1)χ^λ_ℓ     n=2,3,…,d-1

determine the angular part of χ(ρ,θ) to be the spherical harmonics Y_ℓ,

    χ^λ_ℓ(ρ,θ)=f(ρ)Y_ℓ(θ),

while the radial equation, as given by

    (Δ-λ)χ(ρ,θ)=0

reads

    (1+z^2)f”+(d-1/z+d z)f'-(L(L+d-2)/z^2+λ)f=0

where z=ρ/m, and a prime denotes differentiation with respect to z.

Through  the change of variables f(z)=z^L F(-z^2), this equation adopts the standard hypergeometric form <cit.>

    z(zd/dz+α)(zd/dz+β)F=zd/dz(zd/dz+γ-1)F

where

    α   =1/4(+√((d+1)^2+4 (λ -d))+d-1+2 L)
    β   =1/4(-√((d+1)^2+4 (λ -d))+d-1+2 L)
    γ   =d/2+L





The solution to the radial equation thus reads

    f^λ_L(ρ)   =c_1 (ρ/m)^L  _2F_1[d+L+Λ-1/2,L-Λ/2;d/2+L;-ρ ^2/m^2]
       +c_2(m/ρ)^d+L-2_2F_1[Λ-L+1/2,2-d-Λ-L/2;2-d/2-L;-ρ ^2/m^2]

where _2F_1 is the Gaussian hypergeometric function, and

    Λ1/2 (√((d+1)^2+4 (λ-d))-d+1)    inverse⟺    λ=Λ (Λ+d-1)



As L≥ 0, the solution corresponding to c_2 is singular at ρ→ 0, so we will only keep the one corresponding to c_1 (which we set to c_1≡ 1, which we dub the canonical normalisation). For λ<0 the function f_L^λ(ρ) decays as ρ→∞, while for λ≥0 it does not. From now on we will focus on the second case.

Using _2F_1(α,β;γ;z)=1+𝒪(z), together with <cit.>

    _2F_1(α ,β ;γ ;-z)   =z^-αΓ (γ )  Γ (β -α )  /Γ (β ) Γ (γ -α ) _2F_1(α ,α -γ +1;α -β +1;-1/z)
       +z^-βΓ (γ )  Γ (α -β ) /Γ (α ) Γ (γ -β ) _2F_1(β ,β -γ +1;β-α +1;-1/z)

we see that the asymptotic behaviour of f_L^λ(ρ) is

    f^λ_L(ρ)=(ρ/m)^L+⋯   ρ≪ m
    (d+2 Λ-3)!/(d+2 Λ-4)!!(d+2 L-2)!!/(d+L+Λ-2)!(ρ/m)^Λ+⋯   ρ≫ m





From this expression we see that for λ<d the function f^λ_L(ρ) grows sub-linearly with ρ, while for λ>d the growth is super-linear. The edge case λ=d is of particular interest: for example, for λ=d, the lowest modes L=0 and L=1 reproduce the functions k^μ, because mf^d_0=√(ρ^2+m^2) (see <cit.>) and mf^d_1=ρ:

    mχ_00⋯0^d   ∼√( k^2+m^2)
    
    mχ_ℓ_1ℓ_2⋯1^d   ∼linear combinations of k^i

and therefore k^μ∈{χ_ℓ^λ}, as required. The higher-order modes L≥ 2 have a more complicated expression, but they are all 
linear in ρ for large ρ. This is an essential property of the radial function f_L^d(ρ), because it guarantees that the integral that defines P_ℓ^d converges as long as that that defines P^μ does. We note that if λ<d, the integral is still convergent, so the admissible range for λ is 0≤λ≤ d, which is equivalent to 0≤Λ≤ 1. If we wish to pick a larger value of λ, we must impose a faster decay rate for the Fourier modes a_k⃗ (for arbitrary λ, we need | k|^Λ |a|^2∈ L^1(ℝ^d,ḳ)).

the second argument of f_L^λ in the hypergeometric function (<ref>) is 1/2(L-Λ). Therefore, if Λ is an integer, then this argument vanishes for L=Λ, in which case the radial function becomes f=ρ^L. This in turns means that the mode χ_ℓ^λ, for L=Λ∈ℕ, is a (harmonic) polynomial in k of order L, that is, χ∼linear combinations of k^i_1k^i_2⋯ k^i_Λ.


Now that we have characterised the modes χ_ℓ^λ, we can go back to our initial intention of enlarging the Poincaré algebra into the 𝔟𝔪𝔰 algebra. To this end, let us note that ℳ_μν commutes with (Δ-λ), and therefore

    (Δ-λ)ℳ_μνχ_ℓ^λ=0

which implies that ℳ_μνχ_ℓ^λ must be a linear combination of the modes themselves:

    ℳ_μνχ^λ_ℓ=∑_α∈ℤ^d-1c^α_μν(λ,ℓ) χ^λ_ℓ+α

where c^α_μν are a set of constants. This is precisely the relation we were after, (<ref>).


we have now completed the construction of the modes χ, and showed that they satisfy a closed algebra of the form [ℳ,χ]=χ.
A crucial ingredient of this construction was the fact that  the differential operator (Δ-λ) commutes with ℳ_μν. In fact, it possible to prove that (Δ-λ) is the most general second order differential operator that commutes with ℳ_μν, so our construction is in a sense unique: if we were to insist that the modes χ  have to be given by a differential equation, then this equation has to be (Δ-λ)χ=0.



The coefficients c_μν^α(λ,ℓ) are the structure constants of the λ-extended 𝔟𝔪𝔰_d+1 algebra:

    { P_ℓ^λ, P_ℓ'^λ'}   =0
    { M_αβ,  M^μν}   =4iδ^[μ_[α   M^ν]_β]
    { P^λ_ℓ, M_μν}   =∑_α∈ℤ^d-1c^α_μν(λ,ℓ) P^λ_ℓ+α






We note that this algebra is uncountable infinite-dimensional, its elements being labelled by the integers ℓ∈ℤ^d-1 and the real number Λ∈[0,1] (or a larger interval if we impose a faster decay rate on the Fourier modes). It contains the standard (gravitational) algebra, corresponding to the value λ=d, which itself contains the Poincaré algebra, corresponding to  all the integer values of |ℓ_1|≤ℓ_2≤…ℓ_d-1=1:

    λ𝔟𝔪𝔰_d+1⊃𝔟𝔪𝔰_d+1⊃𝔦𝔰𝔬(1,d)

of dimensions 2^ℵ_0> ℵ_0>1/2 (d+1) (d+2).

The algebra that is relevant to the gravitational problem is the one corresponding to λ=d, since it contains a Poincaré sub-algebra generated by the super-translations with L=0,1 (see Fig. <ref>).

1.0 







A rather straightforward method to calculate the structure constants c^α_μν(λ,ℓ) is to use the orthogonality of the spherical harmonics:

    c_μν^α(λ,ℓ)=1/f^λ_ℓ_d-1+α_d-1(ρ)∫ Y^*_ℓ+α(θ) ℳ_μνχ^λ_ℓ(ρ,θ) dΩ_d-1

which is in fact independent of ρ, so we can take ρ→ 0 or any other convenient value. In the appendix <ref> we quote the coefficients c_μν^α(λ,ℓ) for arbitrary λ, for d=3,4,5.

As the differential operators ℳ_μν have one unit of angular momentum (à la Wigner-Eckart), the indices α_i take values in α_i∈{0,±1}. One should also note that, as the angular momentum generators only act on the angular part of the modes, the spatial coefficients c_ij^α are only non-zero if α_d-1=0 (so that L→ L). On the other hand, the boost operators include a radial derivative ∂_ρ, so they do change the value of L. This means that c^α_0i will be non-zero for α_d-1=±1 (see Fig. <ref>).








Another interesting fact about the coefficients c^α_μν is that, when α_d-1=+1, they are proportional to (ℓ_d-1(ℓ_d-1+d-1)-λ). Therefore, if λ=Λ (Λ+d-1) with Λ∈ℕ, then c^α_μν(λ,ℓ)|^α_d-1=1_ℓ_d-1=Λ vanishes, which means that we get a closed sub-algebra, of dimension

    (SO(1,d))+∑_ℓ_d-1=0^Λ ∑_ℓ_d-2=0^ℓ_d-1⋯∑_ℓ_2=0^ℓ_3 ∑_ℓ_1=-ℓ_2^ℓ_21
       =1/2 d (d+1)+(d+Λ-2)!/(d-1)! Λ!(d+2 Λ-1)


a simple argument to prove that when L=Λ∈ℕ the action of the Lorentz operators do not raise the value of L (that is, that the coefficient corresponding to L→ L+1 vanishes) is to realise that for L=Λ, the mode χ_ℓ^λ is given by a linear combination of monomials of the form k^i_1k^i_2⋯ k^i_Λ, and therefore the derivatives ℳ_0i=ik_0∂_i cannot raise the power of ρ=| k| (it can only lower it or leave it as is). Therefore, we cannot have a term with χ_L+1∼ρ^L+1 in the expansion of ℳ_0iχ_ℓ.

In the particular case λ=d, that is, Λ=1, the second term of (<ref>) becomes d+1, which is the number of space-time translations in a manifold of d spatial dimensions. This is the Poincaré sub-algebra. For Λ=0 we get 𝔰𝔬(1,d)×𝔲(1). For other integer values of Λ, we get other closed sub-algebras (see Fig. <ref>), whose relevance and physical interpretation is not clear yet. For non-integer values of Λ, there are no closed sub-algebras.





In any case, equation (<ref>) proves that our construction does indeed lead to an infinite-dimensional generalisation of the Poincaré algebra. We will next discuss how this construction is related to the gravitational 𝔟𝔪𝔰 algebra. The correspondence between the canonical 𝔟𝔪𝔰 algebra and the gravitational one is most clear in what we will call the massless limit of the former, to be discussed below. Let us therefore discuss this limit, together with the opposite limit, the non-relativistic limit. These two limits correspond to taking ρ≫ m and ρ≪ m, respectively (see Fig. <ref>).





Let us begin with the massless limit  m→ 0. In this limit, the mass hyperboloid becomes singular, and the Laplacian blows up. One may nevertheless expand the Laplacian, the zero-modes, and the 𝔟𝔪𝔰 algebra in power series in m, and keep the leading terms. The result is

    Δ̃   lim_m→ 0Δ=ρ^2∂_ρ^2+d ρ∂_ρ
    ℳ̃_0i   lim_m→ 0ℳ_0i=iρ∂_i
    ℳ̃_ij   lim_m→ 0ℳ_ij=2ik_[i∂_j]
    χ̃_ℓ^λ   lim_m→ 0m^Λχ_ℓ^λ=(d+2 Λ-3)!/(d+2 Λ-4)!!(d+2 L-2)!!/(d+L+Λ-2)! ρ^Λ Y_ℓ(θ)

where we have used the limit ρ≫ m as given by (<ref>). From here on, a tilde ·̃ over an object means that it corresponds to the massless limit of the same.

These modes satisfy the same algebra as the standard modes, with the exact same coefficients, c̃^α_μν=c^α_μν. As the radial function f^λ_L(ρ) is a power function instead of a hypergeometric one, the calculation of the structure constants c^α_μν(λ,ℓ) is particularly simple in this limit. 

If we take the particular case λ=d, the massless modes become

    χ̃_ℓ^d=(d-1)!/(d-2)!!(d+2 L-2)!!/(d+L-1)! ρ Y_ℓ(θ),

to be compared with the asymptotic Killing fields, ξ(ℐ^+), as given by (<ref>). The L-dependent prefactor accounts for the different normalisation between the canonical and gravitational normalisations.


We can now consider the opposite limit, the non-relativistic limit. In the limit m→∞, the mass hyperboloid becomes flat, and the Laplace operator becomes

    Δ̂ m^2lim_m→∞1/m^2Δ=m^2∂_i^2

which is, once again the Casimir operator, but now of the homogeneous Galilei Group instead of the Lorentz Group. The Galilei Group is generated by

    ℳ̂_0i    mlim_m→ 01/mℳ_0i=im∂_i
    ℳ̂_ij   lim_m→ 0ℳ_ij=2ik_[i∂_j]

such that Δ̂=ℳ̂_0iℳ̂^0i. Here and in the remainder of this document, we will place a hat ·̂ over an object to indicate that it corresponds to the non-relativistic counterpart of the corresponding relativistic object.

In this limit, the modes χ̂_ℓ^λ become

    χ̂_ℓ(ρ,θ)   1/m^Llim_m→∞m^Lχ_ℓ^λ
       =(ρ/m)^LY_ℓ(θ)

as given by the ρ≪ m limit of our previous f^λ_L(ρ), cf. (<ref>). In this limit, the modes become independent of λ. This was to be expected, because

    1/m^2Δχ^λ_ℓ=λ/m^2χ^λ_ℓ0,

which is independent of λ, and therefore so are non-relativistic modes (which satisfy Δ̂χ̂=0, being the non-relativistic limit of Δχ=λχ). This is consistent with the fact that the functions k^i satisfy Δ̂k^i=0, where Δ̂ is the flat Laplacian. Therefore, our modes χ̂_ℓ contain the functions k^i for L=1 (the case L=0 plays the role of the central charge of the Bargmann group, χ̂_0=const., as we shall discuss in the following chapter).






Unlike in the massless case, and due to the factor of m in the non-relativistic limit of the boost generators (cf. (<ref>)), here the algebra is not the same as in the relativistic case. If we write

    ℳ̂_μνχ̂_ℓ=∑_αĉ_μν^α(ℓ)χ̂_ℓ+α

then the new structure constants ĉ_μν are given by

    ĉ^α_ij   =c^α_ij
    ĉ^α_0i   =
    c^α_0i   α_d-1=-1
    
    0    α_d-1=+1


We call this new algebra the 𝔫𝔯𝔟𝔪𝔰 algebra, as it corresponds to the non-relativistic limit of the canonical realisation of the 𝔟𝔪𝔰 algebra. For consistency, one should realise this non-relativistic algebra in terms of a scalar field with a Galilean symmetry instead of a relativistic field. We postpone this construction to the next chapter. There we will also consider the non-relativistic limit in more abstract terms, that is, as an İnönü-Wigner contraction of the abstract 𝔟𝔪𝔰 algebra.


















CHAPTER: NON-RELATIVISTIC BMS.


In this chapter we will address the problem of a non-relativistic BMS Group. To define this group, we could try to repeat the analysis of Chapter <ref> but in the context of Newtonian space-times, characterised by a contravariant degenerate spatial metric h^μν and 
covariant vector τ_μ; see for example <cit.> and references therein. We will not do this here; instead, we will construct a possible candidate for the 𝔫𝔯𝔟𝔪𝔰 algebra by performing an İnönü-Wigner contraction of the relativistic 𝔟𝔪𝔰 algebra (in the same spirit as the Bargmann algebra can be obtained by contracting the Poincaré one <cit.>); and we will then provide an explicit realisation of the contracted algebra, by means of a free field with Galilean space-time symmetries, mimicking the construction of Chapter <ref>. This chapter is essentially a generalisation of the results of <cit.> to an arbitrary number of space-time dimensions.



§ THE ALGEBRA 𝔫𝔯𝔟𝔪𝔰 AS A CONTRACTION OF 𝔟𝔪𝔰.


The 𝔟𝔪𝔰 algebra is the semi-direct sum of the Lorentz algebra with the generators of the super-translations, which form an infinite-dimensional abelian sub-algebra. In a similar fashion, the 𝔫𝔯𝔟𝔪𝔰 algebra will be given by the semi-direct sum of the Bargmann algebra with the generators of super-translations. 


The canonical realisation of the λ-extended 𝔟𝔪𝔰_d+1 algebra in terms of the Fourier modes of  relativistic free field leads to the  algebra (<ref>)

    { P_ℓ^λ, P_ℓ'^λ'}   =0
    { M_αβ,  M^μν}   =4iδ^[μ_[α   M^ν]_β]
    { P^λ_ℓ, M_μν}   =∑ c^α_μν(λ,ℓ) P^λ_ℓ+α

where M_μν  are the  generators of Lorentz transformations, and P_ℓ^λ are the generators of super-translations. To simplify the discussion, we shall henceforth set λ≡ d, and omit this label altogether.  As discussed in the previous chapter, for this value of λ the algebra above contains a Poincaré sub-algebra. This sub-algebra is spanned by M_μν,P_ℓ, with ℓ_d-1=0,1. These operators form a closed sub-algebra of 𝔟𝔪𝔰_d+1 because the structure constants c^α_μν(ℓ) vanish when α_d-1=+1 and ℓ_d-1≥ 1.

In the particular case d=3, the algebra above agrees with the algebra of the gravitational 𝔟𝔪𝔰 (<ref>) (modulo an ℓ_2-dependent prefactor, resulting from the different normalisation for the generators of super-translations). We have argued that the correspondence should hold for any number of space dimensions d.

We now proceed to perform the contraction of this algebra in order to obtain its non-relativistic analogue. In order to accommodate the central extension of the Galilei algebra, we consider the direct product of the BMS Group with U(1), with generator Z, and introduce the following transformation: 

    H   ω(P_0+ Z)
    Ẑ   1/ω(P_0- Z)
    M̂_0i   1/ωM_0i
    M̂_ij    M_ij
        inverse⟺    
    P_0   =1/2ωH+ωẐ
    
    Z   =1/2ωH-ωẐ
    
    M_0i   =ωM̂_0i
    
    M_ij   =M̂_ij

together with

    P̂_ℓω^f(L)P_ℓ    inverse⟺     P_ℓ=ω^-f(L)P̂_ℓ,    L≥ 1,

where L=ℓ_d-1 and f(L) is an unspecified function, and ω is a dimensionless parameter which we shall take ω→∞ at the end.



In the limit ω→∞, the centrally-extended relativistic algebra (<ref>) becomes

    [M̂_ij,M̂_mn]   =4iδ_[i[mĴ_n]j]
    
    [M̂_ij,M̂_0m]   =2iδ_m[iM̂_j]0
    
    [M̂_0i,M̂_0j]   =0
    
    [P̂_ℓ,H]   =0
    
    [M̂_ij,H]   =0
    
    [M̂_0i,H]   =iP̂_i
    
    [P̂_ℓ,P̂_ℓ']   =0
    
    [M̂_ij,P̂_ℓ]   =∑ĉ^α_ijP̂_ℓ+α
    
    [M̂_0i,P̂_ℓ]   =∑ĉ^α_0iP̂_ℓ+α

where ĉ^α_ij c^α_ij and

    ĉ^α_0ilim_ω→∞ω^f(L)-f(L')-1 c^α_0i


In order to have a non-trivial ω→∞ limit, we must have

    f(L)-f(L')-1≡0

for some L'. Moreover, the structure constants c^α_0i are non-zero only if |L'-L|=1. This means that the only non-trivial contractions are the ones that satisfy

    f(L)=f(0)± L


Furthermore, in order to obtain the Bargmann algebra as a sub-algebra,  for L=1 we should recover the standard contraction P̂_i=P_i, so that f(1)=0. With this,

    f(L)=±(L-1)


The conclusion of this discussion is that if we restrict ourselves to scalings of the form P̂_ℓ=ω^f(L)P_ℓ then there are only two non-trivial contractions of the 𝔟𝔪𝔰_d+1 algebra, corresponding to either sign in f(L)=±(L-1), which we will call 𝔫𝔯𝔟𝔪𝔰^(↓) (corresponding to the plus sign) and 𝔫𝔯𝔟𝔪𝔰^(↑) (corresponding to the negative sign). In either case, the structure constants are given by

    f^(↓)(L)=L-1 
    ĉ^α_ij   =c^α_ij
    ĉ^α_0i   =c^α_0i   α_d-1=-1
     0    α_d-1=+1                 f^(↑)(L)=1-L 
    ĉ^α_ij   =c^α_ij
    ĉ^α_0i   =0    α_d-1=-1
    c^α_0i   α_d-1=+1



the first of these algebras is the exact same algebra we obtained by the non-relativistic limit ρ≪ m of the modes χ_ℓ (cf. (<ref>)). This was actually to be expected, because the factors of ω in the abstract contraction (<ref>), (<ref>) are the same as the factors of m in the explicit limit (<ref>), (<ref>). The limits ω→∞ and m→∞ are formally identical. The second algebra is new.

The two possible algebras above contain time and space translations, rotations, boosts, super-translations, and a central charge, and they both contain a Bargmann sub-algebra (see Fig.<ref>).







1.0 




It is important to note that in the case of the 𝔫𝔯𝔟𝔪𝔰^(↓) contraction, the boost operators lower the value of L to L-1. This is in stark contrast with the relativistic case, where we have simultaneous contributions from L-1 and L+1. Therefore, unlike in the relativistic case, here the algebra resulting from the `↓' contraction contains an infinite number of finite-dimensional sub-algebras, obtained by considering all the super-translation generators with 1≤ L≤Λ for given Λ∈ℕ; the dimension of these sub-algebras is given by (<ref>) (plus one, due to the central charge). All these sub-algebras contain a Bargmann sub-algebra, and the associated matrices
ĉ^α_ij,ĉ^α_0i provide an infinite number of finite-dimensional representations of the homogeneous Galilei group (finite-dimensional indecomposable representations  of homegeneous Galilei have been studied in <cit.>).


On the other hand, in the case of the 𝔫𝔯𝔟𝔪𝔰^(↑) algebra, the boost operators raise the value of L to L+1, which means that the only  finite-dimensional sub-algebra of 𝔫𝔯𝔟𝔪𝔰^(↑) is the Bargmann algebra (see Fig.<ref>).


1.0 




In the following section we will construct an explicit realisation of the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra corresponding to the plus sign contraction, and we will also discuss the possibility of adding dilatations and expansions.








§ CANONICAL REALISATION OF THE 𝔫𝔯𝔟𝔪𝔰 ALGEBRA.
 


We now proceed to construct  the canonical realisation of the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra. The construction will be analogous to the canonical realisation of 𝔟𝔪𝔰_d+1 from the previous chapter: we want to study the symmetries of galilean equations of motion of a free fields. As before, we will frame our discussion directly in terms of the Fourier modes of the non-relativistic field, a,a^*, in terms of which one may realise the Bargmann algebra through

    Ẑ   ∫ a^*( k)m  a( k) d k/(2π)^d
    
    H   ∫ a^*( k) k^2/2m a( k) d k/(2π)^d
    P̂_i   ∫ a^*( k)k_i a( k) d k/(2π)^d
    M̂_0i    tP̂_i+∫ a^*( k)ℳ̂_0i a( k) d k/(2π)^d
    M̂_ij   ∫ a^*_k⃗ℳ̂_ij a_k⃗ d k/(2π)^d

where Ẑ is the central charge which generates U(1) rotations (here, and due to a lack of Coleman-Mandula <cit.>, external symmetries and internal symmetries mix in a non-trivial way). The differential operators ℳ̂_0i,ℳ̂_ij are given by (<ref>):

    ℳ̂_0i   =im∂_i
    ℳ̂_ij   =2ik_[i∂_j]

and they satisfy the (homogeneous) Galilei algebra

    [ℳ̂_ij,ℳ̂_mn]   =4iδ_[i[mℳ̂_n]j]
    
    [ℳ̂_ij,ℳ̂_0m]   =2iδ_m[iℳ̂_j]0
    
    [ℳ̂_0i,ℳ̂_0j]   =0





On the other hand, the Noether charges satisfy the the Bargmann algebra,

    {M̂_ij,M̂_mn}   =4iδ_[i[mM̂_n]j]
    {M̂_ij,M̂_0m}   =2iδ_m[iM̂_j]0
    {M̂_0i,M̂_0j}   =0
    {P̂_ℓ,H}   =0
    {M̂_ij,H}   =0
    {M̂_0i,H}   =iP̂_i
    {P̂_i,P̂_j}   =0
    {M̂_ij,P̂_m}   =2iδ_m[iP̂_j]
    {M̂_0i,P̂_m}   =iδ_imẐ



What we want to do is to generalise the last column of this algebra, by introducing a set of functions χ̂_ℓ in place of the standard momentum,

    k^i   → mχ̂_ℓ(k⃗)
    P̂^i   →P̂_ℓ m∫ a^*_k⃗χ̂_ℓ(k⃗) a_k⃗ d k/(2π)^d

such that k_i∈{χ̂_ℓ} for some values of ℓ, thus extending the Bargmann algebra into the 𝔫𝔯𝔟𝔪𝔰 algebra.

As before, let us consider the quadratic Casimir operator of the homogeneous Galilei group, 

    Δ̂ℳ̂_0iℳ̂^0i=m^2∂_i^2

which coincides with the Laplace-Beltrami operator on ℝ^d (flat space). With this, we will define our modes as the zero-modes of Δ̂:

    Δ̂χ̂_ℓ( k)=0

together with the same angular dependence as in the relativistic case (cf. (<ref>)). As a consistency check, we note that, once again, the momentum k^i satisfies Δ̂k^i=0, so the family of zero-modes {χ̂} will contain the functions k^i as a subset.


The general solution to the equation above is

    χ̂_ℓ(ρ,θ)=f_L(ρ)Y_ℓ(θ)

where ρ,θ are the spherical components of k_i, and Y_ℓ are the spherical harmonics. On the other hand, f_L is given by the solution to

    f”+d-1/zf'-L(L+d-2)/z^2f=0

where z=ρ/m, and a prime denotes differentiation with respect to z. The solution to this equation is

    f_L(r)=c_1(ρ/m)^L+c_2(m/ρ)^L+d-2


As before, we set c_2≡0 to avoid the singular behaviour at ρ→ 0, and set c_1≡ 1. With this,

    χ̂_ℓ(ρ,θ)=(ρ/m)^L Y_ℓ(θ)


We see that the L=0 mode corresponds to the central charge (being momentum-independent), while the L=1 modes agree with the spherical components of k, so that, as expected, the family {χ̂_ℓ} contains the functions k_i as a special subcase. Comparing (<ref>) with the equation (<ref>), we see that the non-relativistic modes agree with the non-relativistic limit of the relativistic modes from the previous chapter, as the notation suggests. Therefore, they satisfy the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra

    ℳ̂_μνχ̂_ℓ=∑_αĉ_μν^α(ℓ)χ̂_ℓ+α

where the structure constants are given by (<ref>):

    ĉ^α_ij   =c^α_ij
    ĉ^α_0i   =
    c^α_0i   α_d-1=-1
    
    0    α_d-1=+1



With this, we define the generators of super-translations  as

    P̂_ℓ m∫ a^*_ kχ̂_ℓ( k) a_ k d k/(2π)^d



One should note that, unlike the relativistic case, here the existence of P̂_ℓ is not guaranteed by the existence of P̂_i, because the non-relativistic modes χ̂ scale as ρ^L for large ρ instead of linearly with ρ. Therefore, a_ k being square-integrable is not enough for the integral defining P̂_ℓ to converge; we must impose the stronger condition that | k|^L |a_ k|^2 is integrable for all L∈ℕ. If we only require for a finite number of super-translations to exist, those corresponding to 0≤ L≤Λ for a certain integer Λ, then we must impose that | k|^L |a_ k|^2 is integrable for all 0≤ L≤Λ. This doesn't interfere with the closedness of the sub-algebra, because the structure constants ĉ_μν^α only connect super-translations of order L among themselves and with super-translations of order L-1 (see the left diagram of Fig. <ref>).

In any case, using (<ref>) we see that the functions P̂_ℓ satisfy the algebra

    {P̂_ℓ,M̂_μν}=∑_α∈ℤ^d-1ĉ^α_μν(ℓ) P̂_ℓ+α

which, together with the first two columns of (<ref>), agrees with the 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1 algebra (<ref>). In principle, it may possible to construct a realisation of 𝔫𝔯𝔟𝔪𝔰_4^(↑) by using the second solution to the radial equation, f_L∼ρ^2-L-d, but the fact that these functions are singular at the origin implies that the Fourier modes a_ k must go to zero faster than any polynomial if we want the integral that defines P̂_ℓ to converge. We will not consider this possibility any further here.

we have chosen the definition of χ̂ to be Δ̂χ̂≡ 0 in order to match the non-relativistic limit of the relativistic modes (cf. (<ref>)). We could have worked with a more general equation (Δ̂-λ̂)χ̂=0 for an arbitrary real constant λ̂>0. The solutions {χ̂^λ̂_ℓ} satisfy a more general algebra than 𝔫𝔯𝔟𝔪𝔰^(↓), but such that it reduces to the it when we take λ̂≡ 0. In order to obtain this generalised algebra by contracting the λ𝔟𝔪𝔰 algebra, one must take the ω,λ→∞ limit of the latter, while keeping λ̂≡λ/ω fixed. We state without proof that the radial equation for general λ̂ reads f^λ̂_L(z)=z^L   _0F_1(;d/2+L;-λ̂/4 z^2  ) where _0F_1 is the confluent hypergeometric function, and that the structure constants for general λ̂ are ĉ^α_μν(λ̂,ℓ)=c^α_μν(λ̂+ℓ_d-1(ℓ_d-1+d-1),ℓ). We will not study this extension of 𝔫𝔯𝔟𝔪𝔰 here. In what follows, we will restrict ourselves to the case λ̂≡0, that is, the regular 𝔫𝔯𝔟𝔪𝔰^(↓) algebra, where the modes are given by χ̂_ℓ( k)=(ρ/m)^LY_ℓ(θ).

The algebra 𝔫𝔯𝔟𝔪𝔰_d+1^(↓) has a very important property, one that is not present in the relativistic λ𝔟𝔪𝔰_d+1 case: here, the modes χ̂( k) are actually homogeneous polynomials in k^i, of degree L. This means that the symmetries generated by P̂_ℓ are local when acting on the field ϕ, meaning that 

    δ_ℓϕ(x)={P̂_ℓ,ϕ(x)}=χ̂_ℓ(-i∂)ϕ(x)

where χ̂_ℓ is a harmonic polynomial of degree L. In other words, χ̂_ℓ(-i∂) is nothing but a polynomial in ∂_i:

    χ̂_ℓ(-i∂)=∑_|α|=L𝒞^α∂_α

for a certain set of coefficients 𝒞^α. Differential operators of this form (and generalisations thereof), in the context of symmetries of partial differential equations, have been studied extensively in the literature; see for example <cit.>.  


Furthermore, the fact that the polynomials χ̂_ℓ( k) are homogeneous and of degree L implies that they satisfy

    𝒟χ̂_ℓ=Lχ̂_ℓ

where 𝒟 is the homogeneity operator, 𝒟 k^i∂_i=ρ∂_ρ. This means that if we define the dilatation operator as

    D 2tH+i∫ a^*_ k𝒟 a_ k d k/(2π)^d

then the generators of super-translations satisfy

    {D,P̂_ℓ}=iL P̂_ℓ

which extends the 𝔫𝔯𝔟𝔪𝔰 algebra to include dilatations, giving rise to a “Weyl-𝔫𝔯𝔟𝔪𝔰”. We note that the main obstruction of this extension to the relativistic algebra is the presence of a non-zero mass m, which introduces a length scale to the theory (so that it is not invariant under dilatations). From a more pragmatic point of view, the energy k_0=√( k^2+m^2) is not a homogeneous polynomial, so it is not an eigenvector of 𝒟 (unless we take m≡ 0).

Once we realise that the algebra admits dilatations, it becomes natural to ask ourselves about the action of the expansion operator (or Schrödinger conformal transformations), given by

    C-t^2H+tD+m/2∫ a^*_ kΔ̂a_ k d k/(2π)^d

and which satisfies <cit.>

    {D,C}=2iC,  {H,C}=iD,  {C,P̂_i}=iM̂_0i, etc.


Using

    [Δ̂,χ̂_ℓ]   =Δ̂χ̂_ℓ+2(∂_iχ̂_ℓ)∂/∂ k^i
       =2/im[∑_αĉ_0i^α χ̂_ℓ+α]∂/∂ k^i

we obtain

    {C,P̂_ℓ}= itL P̂_ℓ-im∫ a^*_ k[∑_αĉ_0i^α χ̂_ℓ+α]∂/∂ k^i a_ k d k/(2π)^d


The r.h.s. of (<ref>) is not an element of 𝔫𝔯𝔟𝔪𝔰^(↓)_d+1. This means that if we attempt to extend the algebra to include C, the resulting algebra is not closed, which seems to preclude a possible “Schrödinger-𝔫𝔯𝔟𝔪𝔰”. 

In any case, the r.h.s. (<ref>) is the bracket of two conserved quantities, which means that it is conserved as well. Indeed, a straightforward calculation confirms that

    [∂/∂ t+{·,H}]{C,P̂_ℓ}=0


This means that {C,P̂_ℓ}, despite not being an element of 𝔫𝔯𝔟𝔪𝔰, is a conserved operator, i.e., it generates a symmetry of the equations of motion. In the particular case L=1, this commutator agrees with the generators of boosts, {C,P̂_i}=iM̂_0i. It is tempting to let {C,P̂_ℓ} define a new kind of generator of symmetries, which would generalise the standard generators of boosts; we could dub these objects super-boosts. It will be interesting to explore their relation with the relativistic super-rotations.




CHAPTER: CONCLUSIONS AND OUTLOOK.
tocchapterConclusions and Outlook.


This completes our study of the BMS Group. Here we summarise the essential points and highlight those topics that require a further analysis.



  §.§.§ The BMS Group.


In Chapter <ref> we reviewed this group in its original context, the problem of characterising the asymptotic isometries of asymptotically flat space-times. We saw that, while in the strictly flat case the isometries are generated by a finite number of vector fields, once we relax the notion of isometry into an asymptotic isometry, the number of generators becomes infinite. In the first case, these vector fields generate the Poincaré Group, and in the second case they generate the BMS Group, which contains Poincaré as a sub-group. Moreover, we saw that one may further extend the symmetry algebra by including the infinite-dimensional set of super-rotations.

Finally, we mentioned that the generalisation of this procedure to higher dimensional manifolds is non-trivial, and there are several opposing views in the literature. The very existence of super-translations depends on the particular falloff rate for the component of the metric, and it is not clear how strong these falloffs should be in arbitrary dimensions. Even more so in the case of super-rotations, whose existence depends on whether the conformal Killing equation on the sphere admits or not an infinite number of solutions; and, as we have mentioned, that only happens in d=2,3. Therefore, unless we alter the definition of super-rotations in higher dimensions, one would not expect to find them in d>3.




  §.§.§ Canonical realisation of BMS.


In Chapter <ref> we generalised the canonical construction of the 𝔟𝔪𝔰_4 algebra to an arbitrary number of space-time dimensions, and we argued that this result suggests a non-trivial gravitational 𝔟𝔪𝔰_d+1 for all d. We also generalised this algebra to include an additional label, λ, which makes the resulting algebra uncountable infinite-dimensional. For some particular values of λ, the corresponding algebra contains finite-dimensional sub-algebras, such as Poincaré for λ=d.

We have not addressed the problem of super-rotations in the canonical construction. One may presume that, like in the gravitational case, these symmetries are only present in d=2,3. These dimensions require a case-by-case analysis.

Another unexplored aspect of the canonical construction is the fact that it admits the extension in terms of the parameter λ. The particular value λ=d was relevant to the gravitational problem, but it would be nice to find a physical system whose group of symmetries is λ𝔟𝔪𝔰, with arbitrary λ.


Finally, we would like to mention that in massless theories the group of external symmetries usually gets enhanced to the Conformal Group, which includes conformal transformations and dilatations together with the standard Poincaré transformations. In our canonical construction, it may be possible to do the same in the case of massless super-translations, that is, to consider a conformal BMS, perhaps related to the gravitational conformal BMS constructed in <cit.>.





  §.§.§ Non-relativistic BMS.


In Chapter <ref> we studied some possible analogues of 𝔟𝔪𝔰 for non-relativistic systems, both by means of a contraction of the abstract relativistic algebra and by the canonical method. We saw that, in a sense, there are only two admissible contractions, and that one of them explicitly arises in the study of the symmetries of Galilei-invariant equations of motion for free fields.

Unlike in the relativistic case, the 𝔫𝔯𝔟𝔪𝔰 super-translations are local: they act as polynomials in the differential operator ∂_i. Being homogeneous, these polynomials have a simple behaviour under the action of the dilatation operator, which means that the 𝔫𝔯𝔟𝔪𝔰 algebra admits an extension that includes dilatations. However, it does not seem to admit conformal transformations, which means that we cannot use the non-relativistic super-translations to extend the Schrödinger algebra. The existence of super-rotations in the non-relativistic setting might prove essential in the construction of a possible non-relativistic conformal NRBMS.

Finally, it will be interesting to study the asymptotic symmetries of asymptotically flat Newtonian space-times and to check whether the algebra coincides with the one of the two 𝔫𝔯𝔟𝔪𝔰 algebras we have constructed. In other words, it will be nice to have an explicit verification of the correspondence between the canonical NRBMS Group and the gravitational one, provided the latter actually exists.


















CHAPTER: SPHERICAL HARMONICS.
 

Here we quote the form of the spherical harmonics on S^d-1. A much more detailed discussion can be found in <cit.>.

The metric on the sphere can be defined recursively through

    ds_1^2   =dθ_1^2
    ds_n^2   =dθ_n^2+sin^2θ_nds_n-1^2

where θ_1∈[0,2π) and θ_n∈[0,π). The volume element is

    dΩ_1   =dθ_1
    dΩ_n   
    =sin^n-1θ_n dθ_n dΩ_n-1


On the other hand, the Laplacian reads

    Δ_S^1   =∂^2/∂θ_1^2
    Δ_S^n   =1/sin^n-1θ_n∂/∂θ_n[sin^n-1θ_n∂/∂θ_n]+1/sin^2θ_nΔ_S^n-1


We define the spherical harmonics through

    [∂/∂θ_1-iℓ_1]Y_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1)   =0
    [Δ_S^n+ℓ_n(ℓ_n+n-1)]Y_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1)   =0

where n=1,2,…,d-1.



The first equation is solved by

    Y_ℓ_1ℓ_2⋯ℓ_d-1(θ_1,θ_2,…,θ_d-1)∝e^iθ_1ℓ_1

where the constant of proportionality is an arbitrary function of θ_2,…,θ_d-1. Univaluedness of Y implies that ℓ_1∈ℤ is an integer. The rest of equations are solved by

    Y_ℓ_1ℓ_2⋯ℓ_n(θ_1,θ_2,…,θ_n)=Y_ℓ_1ℓ_2⋯ℓ_n-1(θ_1,θ_2,…,θ_n-1)𝒥(θ_n)

where 𝒥(θ) satisfies

    [1/sin^n-1θ∂/∂θ[sin^n-1θ∂/∂θ]-ℓ_n-1(ℓ_n-1+n-2)/sin^2θ
    +ℓ_n(ℓ_n+n-1)]𝒥(θ)=0



If we set 𝒥(θ)=sin^ℓ_n-1θ y(cosθ), and make the change of variables x≡cosθ, this equation becomes

    (1-x^2) y”(x)-(2 μ+1)x y'(x)+ν (ν+2 μ) y(x)=0

where μℓ_n-1+(n-1)/2 and νℓ_n-ℓ_n-1. The solution to this differential equation is

    y(x)= c_1 C_ν^(μ)(x)+c_2 (1-x^2)^1/4 (1-2 μ) Q_ν+μ-1/2^1/2-μ(x)

where C_ν^(μ),Q_ν^μ are the Gegenbauer function and the Legendre function of the second kind <cit.>: 

    C_ν^(μ)(x)   Γ(ν+2μ)/Γ(2μ)Γ(ν+1)_2F_1(-ν,ν+2μ;μ+1/2;1-z/2)
    
    Q_ν^μ(x)   e^iμππ^1/2Γ(μ+ν+1)(x^2-1)^μ/2/2^ν+1x^μ+ν+1 ·
       ·_2F̃_1(μ+ν+2/2,μ+ν+1/2;ν+3/2;x^-2)

where F̃ denotes the regularised hypergeometric function. 

Finiteness at x=±1 requires c_2≡ 0, and ν to be a non-negative integer. By induction, ℓ_n∈ℕ, with |ℓ_1|≤ℓ_2≤ℓ_3…≤ℓ_n. The series for C_μ^(μ) now terminates; the Gegenbauer functions become polynomials:

    C_ν^(μ)(x)= ∑_j=0^⌊ν/2⌋(-1)^jΓ(ν+μ+j)/Γ(μ)j!(ν-2j)!(2x)^ν-2j



With this,

    y_ℓ_n,ℓ_n-1^(n)(x)∝ C_ℓ_n-ℓ_n-1^(ℓ_n-1+(n-1)/2)(x)∝(d/dx)^ℓ_n-1C_ℓ_n^((n-1)/2)(x)



If we normalise the functions to

    ∫_-1^+1 (1-x^2)^n/2+ℓ_n-1-1[y_ℓ_n,ℓ_n-1^(n)(x)]^2 dx=1

we get

    y_ℓ_n,ℓ_n-1^(n)(x)   (-1)^ℓ_n-1 2^ℓ_n-1+n/2-1Γ(n-12+ℓ_n-1) ·
       ·√((ℓ_n+n-1/2) Γ (ℓ_n-ℓ_n-1+1)π Γ (ℓ_n+ℓ_n-1+n-1)) C_ℓ_n-ℓ_n-1^(ℓ_n-1+(n-1)/2)(x)

where we have included the conventional Condon-Shortley phase (-1)^ℓ_n-1.

With this, our spherical harmonics are given by the recursive formula

    Y_ℓ_1(θ_1)   =1/√(2π) e^iθ_1ℓ_1
    
    Y_ℓ_1ℓ_2…ℓ_n(θ_1,θ_2,⋯,θ_n)   =sin^ℓ_n-1θ_n y_ℓ_n,ℓ_n-1^(n)(cosθ_n)Y_ℓ_1ℓ_2⋯ℓ_n-1(θ_1,θ_2,…,θ_n-1)

and they have been normalised to

    ∫ Y^*_ℓ_1ℓ_2⋯ℓ_n(θ)Y_ℓ'_1ℓ'_2⋯ℓ'_n(θ) dΩ_n=δ_ℓ_1ℓ'_1δ_ℓ_2ℓ'_2⋯δ_ℓ_nℓ'_n






left=2cm,right=2cm


CHAPTER: STRUCTURE CONSTANTS.


In this appendix we quote the value of the structure constants c^α_μν(λ,ℓ) for d=3,4,5.

For d=3,

    c_01^±+   =1/2 (ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_01^±-   =1/2√(2 ℓ_2+1/2 ℓ_2-1 (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    
    
    
    c_02^±+   =±i/2  (ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±-   =±i/2√(2 ℓ_2+1/2 ℓ_2-1(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    
    c_03^0+   =-i (ℓ_2 (ℓ_2+2)-λ )√(1/(2 ℓ_2+1) (2 ℓ_2+3)^3(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0-   =i √(2 ℓ_2+1/2 ℓ_2-1(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_12^00   =ℓ_1
    
    c_13^±0   =-1/2√((ℓ_2∓ℓ_1) (ℓ_2±ℓ_1+1))
    
    c_23^±0   =∓i/2√((ℓ_2∓ℓ_1) (ℓ_2±ℓ_1+1))



For d=4,

    c_01^±++   =-1/8 (ℓ_3 (ℓ_3+3)-λ )·
       ·√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2) )
    
    c_01^±-+   =-1/8 (ℓ_3 (ℓ_3+3)-λ )·
       ·√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )
    
    c_01^±+-   =-1/2√(ℓ_3+1/ℓ_3 (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_01^±–   =-1/2√(ℓ_3+1/ℓ_3  (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    
    
    
    
    
    
    
    c_02^±++   =∓i/8 (ℓ_3 (ℓ_3+3)-λ )·
       ·√(1/(ℓ_3+1) (ℓ_3+2)^3 (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±-+   =∓i/8 (ℓ_3 (ℓ_3+3)-λ )·
       ·√(1/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+1) (ℓ_3-ℓ_2+2)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )
    
    c_02^±+-   =∓i/2√(ℓ_3+1/ℓ_3  (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±–   =∓i/2√(ℓ_3+1/ℓ_3  (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))



    c_03^0++   =+i/4 (ℓ_3 (ℓ_3+3)-λ )·
       ·√(1/ (ℓ_3+1) (ℓ_3+2)^3 (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0-+   =-i/4 (ℓ_3 (ℓ_3+3)-λ )·
       √(1/(ℓ_3+1) (ℓ_3+2)^3 (ℓ_3-ℓ_2+1)(ℓ_3-ℓ_2+2) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_03^0+-   =+i √(ℓ_3+1/ℓ_3  (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0–   =-i √(ℓ_3+1/ℓ_3  (ℓ_2+ℓ_3) (ℓ_2+ℓ_3+1)/(1-2 ℓ_2) (2 ℓ_2+2-1)(ℓ_1-ℓ_2) (ℓ_1+ℓ_2))
    
    c_04^00+   =-i/4 (ℓ_3 (ℓ_3+3)-λ )√(/(ℓ_3+1) (ℓ_3+2)^3(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))
    
    c_04^00-   =i √(ℓ_3+1/ℓ_3(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))
    
    
    
    c_14^±+0   =1/2√( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_14^±-0   =1/2√( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_24^±+0   =±i/2√( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_24^±-0   =±i/2√( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_34^0+0   =-i √( (ℓ_3-ℓ_2) (ℓ_3+ℓ_2+2)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_34^0-0   =+i √( (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))



For d=5,

    c^±+++_01   =1/4 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3)(2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2)  (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c^±+-+_01   =1/4 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4-ℓ_3+1) (ℓ_4-ℓ_3+2)/ℓ_3 (ℓ_3+1)  (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1)(ℓ_2±ℓ_1+2))
    
    c^±-++_01   =1/4 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c^±–+_01   =1/4 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c^±+–_01   =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c^±++-_01   =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c^±-+-_01   =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2±ℓ_1) (ℓ_2±ℓ_1-1))
    
    c^±—_01   =1/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)  (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))



    c_02^±+++   =±i/4 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±-++   =±i/4(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_02^±–+   =±i/4(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/ (2 ℓ_4+3)(2 ℓ_4+5)^3(ℓ_4-ℓ_3+1)(ℓ_4-ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)  /(2 ℓ_2-1) (2 ℓ_2+1)   (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_02^±+-+   =±i/4(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3)(2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)   (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±—   =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_02^±+–   =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±++-   =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_02^±-+-   =±i/4√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))





    c_03^0+++   =-i/2(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/ (2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0-++   =+i/2 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/ (ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2+1) (2 ℓ_2-1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_03^0+-+   =-i/2(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0–+   =+i/2 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/ (2 ℓ_4+3)(2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_03^0++-   =-i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)(ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0-+-   =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2+1) (2 ℓ_2-1)(ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_03^0+–   =-i/2√(2 ℓ_4+3/2 ℓ_4+1 (ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_03^0—   =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))




    c_04^00++   =+i/2 (ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4+ℓ_3+3) (ℓ_4+ℓ_3+4)/(ℓ_3+1) (ℓ_3+2)  (ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))
    
    c_04^00-+   =-i/2(ℓ_4 (ℓ_4+4)-λ )·
       ·√(1/ (2 ℓ_4+3) (2 ℓ_4+5)^3 (ℓ_4-ℓ_3+2) (ℓ_4-ℓ_3+1)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))
    
    c_04^00–   =-i/2√(2 ℓ_4+3/2 ℓ_4+1 (ℓ_4+ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))
    
    c_04^00+-   =+i/2√(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4-ℓ_3-1)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))
    
    c_05^000+   =-i (ℓ_4 (ℓ_4+4)-λ )√(1/(2 ℓ_4+3) (2 ℓ_4+5)^3(ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+3))
    
    c_05^000-   =+i √(2 ℓ_4+3/2 ℓ_4+1(ℓ_4-ℓ_3) (ℓ_4+ℓ_3+2))
    
    c_15^±++0   =-1/4√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3) /(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_15^±–0   =-1/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_15^±-+0   =-1/4√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1) )
    
    c_15^±+-0   =-1/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))



    c_25^±++0   =∓i/4√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_25^±–0   =∓i/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_25^±-+0   =∓i/4√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2∓ℓ_1) (ℓ_2∓ℓ_1-1))
    
    c_25^±+-0   =∓i/4√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2±ℓ_1+1) (ℓ_2±ℓ_1+2))
    
    c_35^0++0   =+i/2√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2) (ℓ_3+ℓ_2+2) (ℓ_3+ℓ_2+3)/(2 ℓ_2+1) (2 ℓ_2+3) (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_35^0-+0   =-i/2√((ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/(ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+2) (ℓ_3-ℓ_2+1) /(2 ℓ_2-1) (2 ℓ_2+1) (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_35^0+-0   =+i/2√((ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3-ℓ_2) (ℓ_3-ℓ_2-1) /(2 ℓ_2+1) (2 ℓ_2+3)  (ℓ_2-ℓ_1+1) (ℓ_2+ℓ_1+1))
    
    c_35^0–0   =-i/2√( (ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1) (ℓ_3+ℓ_2) (ℓ_3+ℓ_2+1)/(2 ℓ_2-1) (2 ℓ_2+1)  (ℓ_2-ℓ_1) (ℓ_2+ℓ_1))
    
    c_45^00+0   =-i/2√( (ℓ_4-ℓ_3) (ℓ_4+ℓ_3+3)/ (ℓ_3+1) (ℓ_3+2)(ℓ_3-ℓ_2+1) (ℓ_3+ℓ_2+2))
    
    c_45^00-0   =+i/2√( (ℓ_4-ℓ_3+1) (ℓ_4+ℓ_3+2)/ℓ_3 (ℓ_3+1)(ℓ_3-ℓ_2) (ℓ_3+ℓ_2+1))





































































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