Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces E. Huguet1 and J. Renaud2 1 - Universit´e Paris Diderot-Paris 7, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Batiment Condorcet, 10 rue Alice Domon et L´eonie Duquet, F-75205 Paris Cedex 13, France. 2 - Universit´e Paris-Est, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Batiment Condorcet, 10 rue Alice Domon et L´eonie Duquet, F-75205 Paris Cedex 13, France. ∗ (Dated: February 1, 2013) We show that the Laplace-Beltrami equation (cid:3) a= j in (R6,η), η := diag(+−−−−+), leads 6 under very moderate assumptions to both the Maxwell equations and the conformal Eastwood- Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. Thisresultisobtainedthroughageometricformalismwhichgives,asbyproduct,simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space. 3 1 PACSnumbers: 04.62.+v 0 2 I. INTRODUCTION coordinatesofR6 withthespecificpropertythattheirre- n striction to a particular Minkowski space of the covering a are the usual Minkowskian Cartesian coordinates. Each J This paper describes a geometrical framework for the space of the covering together with its corresponding set 1 study of conformally invariant fields in conformally flat ofMinkowskianCartesiancoordinatescanbeturnedinto 3 spaces in four dimensions. Applications to scalar and electromagnetic fields are made. In particular, we show a local chart of the cone modulo the dilations, which is qc] th+at),thweheeqreua(cid:3)tionis(cid:3)th6ae=Lajplianc(eR-B6e,lηt)r,aηm:i=opdeiaragt(o+r,−le−ad−s tohfutsheenkdeoywiendgrweidtihenatMs tionkpoewrfsokrimansaimtlapsl.eTcahlicsualatltaiosniss.onIne - −under very mo6derate assumptions to both the Maxwell particular, it allows us to use the local Weyl rescaling to r handle global problems. g equationsandtheconformalEastwood-Singergaugecon- [ ditions[1]on4Dconformallyflatspaces. Inaddition,our The study of conformal fields, as the electromagnetic proof of this result allows us to propose a new fiber bun- field, viewed as restrictions to four dimensional spaces of 1 v dleinwhich,broadlyspeaking,theMaxwellequationson fields on R6 can be traced back to the seminal paper of 6 a conformally flat space are converted into constrained Dirac [2]. There, he introduces the “six cone formalism” 4 scalar equations on Minkowski space. This drastically which sets the conditions, mainly the homogeneity of 6 simplifies the practical calculations. thefieldsonR6,toobtainconformalMinkowskianfields. 7 This approach takes into account the SO (2,4) symme- Thebasicgeometricalideaistobuildfour-dimensional 0 . 1 spaces as the intersection in (R6,η), of a surface and the try of the Minkowskian equations under consideration 0 fromthebeginning. Thisformalismhasbeenextendedin five-dimensionalnullcone(invariantunderthelinearcon- 3 group theoretical context by Mack and Salam [3] almost formal group SO (2,4)). The metric on such a space is 1 0 inducedfromthatof(R6,η). Inparticular, foranygiven 35 years later and in study of conformal generalizations : v Robertson-Walker (RW) metric, one can always find a of QED on Minkowski and Anti-de Sitter spaces in the i mid80’sbymanyAuthors(see[4]andreferencesherein). X surface such that the induced metric is the RW metric. The fact that the conformal group SO (2,4) includes as Now, to each point of a space obtained in this way cor- 0 r a responds a half line of the cone, hence all spaces can be subgroups, besides the Poincar´e and the anti-de Sitter groups, the de Sitter group was our starting point in the realized as subsets of the set of the half lines: the cone study of the relation between conformal scalar field in modulo the dilations. In addition, in the intersection of the de Sitter and Minkowski space [5],[6]. The general- two such subsets the spaces are related through a Weyl ization to the electromagnetic field in a conformal gauge rescaling. As a special case, the Minkowski space can onthedeSitterspacewastackledin[7]. Therefollowing be obtained in that scheme. This allows us to build a [4] we used a method of auxiliary fields to obtain a two- particular atlas of the cone modulo the dilations. In ef- pointfunctioninaconformalgaugewhichreducestothe fect, copies of that Minkowski space can be obtained by Eastwood-Singer gauge [1]. These works was concerned displacingthesurfaceintersectingtheconethankstothe byfreequantumfields,butpartofthemethodsusedalso action of O(2,4). This generates a covering of the set of appliedinthecaseofclassicalfieldswithsources. Inthat the half lines. One can go a step further by introducing context, the tools previously build, especially the use of Minkowskian charts on the set of half lines, allowed us to reproduce, with simplified calculations, the result ob- ∗Electronic address: [email protected], tained by Higuchi and Cheong [8] for the problem of two [email protected] charges in de Sitter space. The geometrical framework 2 we build up in the present work encompass and general- Forconvenienceareminderofdefinitionsandconventions izes the method we developed in these previous studies. is given in Appendix C. It extends the formalism to a class of conformally flat spaces which contains in particular the spaces endowed with a RW metric. It provides an atlas which permits 1. The general framework a global use of the Weyl relation to Minkowski space. Last but not least, it also provides a deeper view on the The main geometrical construction is pictured in Fig. geometrical nature of the objects involved. 1. A four dimensional manifold X is obtained as the This article is organized as follows. The Sec. II is f devoted to the geometrical framework. We first gives a global view and motivate the main definitions. Then C C weturntothegeometricalformulationandconsequences ontensorsfieldsoftheassumptionsoftransversalityand x homogeneity. This allows us to see how the Weyl rescal- f X f ing emerges in this context. The action of the SO (2,4) 0 group on tensor fields is then discussed. The construc- tion of the Minkowskian atlas, and the expression of some properties in it follows. This section ends by the P f proof that all the RW metrics can be obtained in the present formalism. In Sec. III, we discuss the confor- mal scalar fields, we show how both equations and fields on R6 and on a conformally flat space are related. The Sec. IV contains the proof of the proposition that the [x] Xσ Maxwell equations and the Eastwood-Singer gauge con- S dition are obtained from homogeneous one form fields satisfyingtheLaplace-BeltramiinR6 andsometransver- sality requirement. Although the result is purely geo- metrical, the proof makes use of calculations performed FIG. 1: The main construction: A space Xf (bold line) is in the Minkowskian atlas. This make apparent the prac- obtained as the intersection Pf ∩C of Pf a submanifold of tical calculation method inherited from this geometrical R6 defined by the equation f(x) = 1, x ∈ R6 and the five- dimensional null cone C. To each point of X corresponds a framework. Some generalizations are discussed in Sec. f uniquehalf-line[x],thatisapointofC(cid:48)whichcanbeidentified V.Somepropertiesofhomogeneoustensorsarereminded with a point of the intersection (bold line) X of C with the in Appendix A. Appendix B resembles a few additional σ sphere S. comments on some particular coordinate systems for the sake of completeness. Definitions and conventions are collected in Appendix C. For convenience, the conven- intersectionofthefivedimensionalnullcone ofR6 and tions for indices are repeated here: a surface P := y R6 :f(y)=1 , in whichCf is homo- f { ∈ } geneous of degree one. On a manifold X each point is α,β,γ,δ,... = 0,...,5, f intercepted by a single half line of . Conversely, thanks µ,ν,ρ,σ... = 0,...,3, C tothehomogeneityoff,eachhalflineinterceptsatmost i,j,k,l,... = 1,...,3, once the manifold X . As a consequence each manifold f I,J,... = c,0,...,3,+. Xf can be realized as a subset Xf(cid:48) of a single set: the set (cid:48) of the half lines of , namely the cone modulo the The space R6 is provided with the metric dilaCtions. This constructCion can be translated in rela- η = diag(+, , , , ,+). A point of R6 is de- tions between spaces which are collected in the following − − − − noted by x or y, the letter y refers most often to the diagram linear structure of R6. R(cid:79)(cid:79)6(cid:99)(cid:99) (1) l(cid:48) II. GEOMETRY f lf (cid:111)(cid:111) A. Overview of the formalism and definitions Xf λf Xf(cid:48) ⊂C(cid:48) This section is intended to give the Reader a global in which view of the formalism. We motivate the introduction of key structures as the cone modulo the dilations or l(cid:48) :X(cid:48) R6 (2) the bundle B . We give their definitions and their main f f → f [x] x :=[x] X , properties,leavingtheproofstotheforthcomingsections. f f (cid:55)→ ∩ 3 where[x]isthehalflineof (thatisapointof (cid:48))which of degree 2 satisfying C C − contains the point x, (cid:3) a=j  6 lf :Xf →R6 (3) (cid:93)ηa|C ∈T(C) (6) x(cid:55)→lf(x)=x, (cid:93)ηj|Xf ∈T(Xf), is the canonical injection, and the field Af :=l∗(a) and the current Jf :=l∗(j) defined f f on X satisfy the Maxwell equations and the Eastwood- f λf :Xf(cid:48) →Xf (4) Singer gauge condition [1] : [x] x :=[x] X , (cid:55)→ f ∩ f  (cid:0)(cid:3) δν ν +Rν (cid:1)Af =Jf  f µ−∇µ∇ µ ν µ which is a diffeomorphism. Note that, thanks to the ho- (cid:18) (cid:18) 1 (cid:19)(cid:19) (7) mogeneity of f one has f(x/f(x))=1, and thus  (cid:3)f∇ν −2∇µ Rµν − 3Rgµν Afν =0, x xf = f(x). (5) where (cid:3)f is the Laplace-Beltrami operator on Xf. Note that for maximally symmetric spaces the Eastwood- Singergaugeconditionreducestothemorefamiliarform Now,asetofhalflinesmaycrossseveralmanifoldsX , f in other words the realizations X(cid:48) on (cid:48) of these man- R f C ((cid:3) + ) Af =0, ifolds have a non-empty intersection, in that case one f 6 ∇· candefinesomecommoncoordinatesystems(atleastlo- with a constant Ricci scalar R. Note also that the two cally). This property allows us to drastically simplify conditions appearing in (6) mean that the vector fields many practical calculations, this can be explained as fol- (cid:93) a = aα∂ and (cid:93) j = jα∂ are tangent respectively to lows. Firstly, as we will prove in Sec. IIE, the cone η α η α and to X . modulo the dilations as a manifold can be covered by a f C collection of realizations X(cid:48) of Minkowski spaces, each Beside the property by itself its proof makes use of a f new geometrical structure: the fiber bundle endowed with a Cartesian system of coordinates. We will call Minkowskian atlas this covering together with (cid:91) B∗ := T∗(R6). (8) these coordinates, and Minkowskian charts the elements f x of this atlas. Secondly, we will prove in Sec. IIC that in x∈Xf theirintersectiontworealizationsX(cid:48) arerelatedthrough f It is obtained in restricting the base space of the cotan- a Weyl rescaling. Finally, due to the existence of the gentbundleT∗(R6)toX . Thisobjectallowsustosolve above covering all the manifolds X are Weyl related to f f some problems in a more simple way than by tackling a Minkowski space, that is are conformally flat. In ad- them directly on X . It also describes more accurately dition, and (this is the point for the simplifications) a f the transition between the fields on R6 and those on X . conformal equation reads under its Minkowskian form in f More precisely, although under the assumptions made a Minkowskian chart. in (6) the Maxwell field Af is directly obtained as the Amongst the spaces obtained within this scheme the pullback of a. The proof of this result involves an inter- space X , which is obtained as the intersection of the σ mediate step in which the equations (cid:3) a = j are, in a cone with the 5-sphere S, provides a realization of the 6 sensewhichwillbemadepreciseinSec. IV,restrictedto abstract manifold (cid:48) itself. Indeed, S is intercepted only C the six conformal scalar equations once by each half line of (Fig. 1). We will use this realization of (cid:48), which is iCncluded in the Einstein space, R C ((cid:3)(s)+ )af =jf, (9) for drawing conformal diagrams in Sec. IIE and some f 6 α α practical calculations (Sec. III). where(cid:3)(s) isthescalarLaplace-BeltramioperatoronX f f andinwhichboththeaf’sandthejf’sbelongstotheset 2. Geometry and the Maxwell field ∗ of the sections of B∗. These equations reduce (when Bf f the two constraints of (6) are applied) to the Maxwell Other important structures, more specifically related equations and the Eastwood-Singer condition (7). How- to the Maxwell field, have to be introduced. In the ever they are more easier to handle. Even more, when schemeweuse,whichisinspiredbytheDirac’s“sixcone quantizing the electromagnetic field, these equations ac- formalism” [2], the Maxwell field Af, a one form field on count directly for the gauge fixing: part of the af ’s X , is obtained from a one form field a on R6 homoge- components leads to the Maxwell field, the others{car}ry f neous of degree zero. We remind (see Appendix A) that constraints [4, 9, 10]. the Cartesian components of a are then homogeneous of The field af is obtained from the field a by restricting degree -1. In Sec. IV we will prove the following prop- the base of T∗(R6) to X and the field Af from the field f erty : for a homogeneous of degree zero, j homogeneous af byfurtherrestrictingthefiberT∗(R6)toT∗(X ). The x x f 4 relationsbetweena,af andAf (alsotrueforj,jf andJf) show that if t fulfills the transversality condition then, are collected in the following diagram on X(cid:48) X(cid:48), the following property holds: f ∩ h a∈Ω1(R6) (cid:0)T(cid:48)f(cid:1)([x])=(cid:16)Khf([x])(cid:17)d(t)(cid:0)T(cid:48)h(cid:1)([x]). (12) l∗ f l(cid:101)f (cid:15)(cid:15) (cid:39)(cid:39) where we defined (cid:47)(cid:47) h(x) af ∈Bf∗ rf Ω1(Xf)(cid:51)Af Khf([x]):= f(x). (13) in which Thus the function Kf appears as the conformal factor. h Forfurtherreferenceswealsodefinetherelatedoneform l(cid:101)f :Ω1(R6)→Bf∗ (10) on C(cid:48) a af :=l(cid:101)f(a) W(f,h) :=dln(Kf)2. (14) (cid:55)→ h such that: x X , V T (R6), af(x)[V]=a(x)[V], Note that, no reference to a metric structure is made in f x ∀ ∈ ∀ ∈ these two definitions. Inordertoprove(12),webeginwithshowingthat,for rf :Bf∗ →Ω1(Xf) (11) x∈C and V(cid:48) ∈T[x](C(cid:48)) one has af r (af)=Af f(x)l(cid:48) ([x])[V(cid:48)]=h(x)l(cid:48) ([x])[V(cid:48)]+N, (15) (cid:55)→ f f∗ h∗ where N is a null vector which belongs to a half-line [x]. such that: x X , U T (X ), Af(x)[U]=a(x)[U]. ∀ ∈ f ∀ ∈ x f We first remark that f(x) l(cid:48)([x])= l(cid:48)([x])=Kh([x])l(cid:48)([x]). B. The transversality condition h h(x) f f f Then,differentiatingtherightmostterm,oneobtains,for This is a condition on tensor fields which will appear any x and V(cid:48) T ( (cid:48)), [x] very often. Roughly speaking, for a tensor t fulfilling ∈C ∈ C this relation, the map t Tf has a good behavior with l(cid:48) ([x])[V(cid:48)]=l(cid:48) ([x])[V(cid:48)]l(cid:48)([x])+Kh([x])l(cid:48) ([x])[V(cid:48)]. respect to group invaria(cid:55)→nce, Weyl relation, equations... h∗ Kfh∗ f f f∗ Thisconditionreadsasfollows: Atensort 0(R6)ful- The first term of the r.h.s. of this expression is propor- fillsthetransversalityconditionifandonly∈if,Tfopranyx tionaltox,andasaconsequence,belongstothehalf-line and for any V ,...,V T∗( ), t(x)(V ,...,V ) =∈0 [x]. The expression (15) follows at once. Cas soon as one o1f the arpgu∈menxtsCVi is equ1al to thpe dila- Now, let us consider V1(cid:48),...,Vp(cid:48) ∈T[x](C(cid:48)). Then using insuccessionthedefinitions(2)-(4),thehomogeneityoft, tion field ξ :=yα∂ , i.e. V −O→x. α i Note that, when t Ωp(∝R6), one often finds the lit- theproperty(15)andfinallythetransversalitycondition ∈ for t (Sec. IIB), one has tle bit stronger condition i t = 0 which, for p = 1, re- ξ duces to yαtα =0 (from which the name “transversality T(cid:48)f([x])(cid:2)V1(cid:48),...,Vp(cid:48)(cid:3) condition” originates after Dirac’s paper [2], altough in =(cid:0)l(cid:48)∗t(cid:1)([x])(cid:2)V(cid:48),...,V(cid:48)(cid:3) the context of mechanics this kind of condition is usu- f 1 p (cid:18) (cid:19) alytermed“horizontality”),equivalentlyinanindex-free =t x (cid:2)l(cid:48) V(cid:48),...,l(cid:48) V(cid:48)(cid:3) notation: (cid:93)ηtC T( ). This condition will eventually f(x) f∗ 1 f∗ p | ∈ C be required for any field considered here. For quantum fields,asusualwhenagaugeconditionispresent,theim- =(cid:18) 1 (cid:19)d(t)−pt(x)(cid:2)l(cid:48) V(cid:48),...,l(cid:48) V(cid:48)(cid:3) plementation of the constraint at the quantum level (in f(x) f∗ 1 f∗ p aquGanutpiztaa-tBiolne.uler quantization scheme) is done after the =(cid:18) 1 (cid:19)d(t)t(x)(cid:2)f(x)l(cid:48) V(cid:48),...,f(x)l(cid:48) V(cid:48)(cid:3) f(x) f∗ 1 f∗ p =(cid:18) 1 (cid:19)d(t)t(x)(cid:2)h(x)l(cid:48) V(cid:48)+N ,...,h(x)l(cid:48) V(cid:48)+N (cid:3) C. Homogeneity and Weyl relations f(x) h∗ 1 1 h∗ p p In this section, we consider two manifolds, X and X =(cid:18) 1 (cid:19)d(t)t(x)(cid:2)h(x)l(cid:48) V(cid:48),...,h(x)l(cid:48) V(cid:48)(cid:3) f h f(x) h∗ 1 h∗ p defined, as in Sec. IIA, as the intersections of and the manifolds f(y) = 1, and h(y) = 1. Let t ∈ TCp0(R6) be =(cid:18)h(x)(cid:19)d(t)T(cid:48)h([x])(cid:2)V(cid:48),...,V(cid:48)(cid:3), an homogeneous tensor field of degree d(t), we want to f(x) 1 p compare the fields Tf := l∗t and Th := l∗t. For this f h purpose we realize both fields on (cid:48) and obtain a Weyl relation between T(cid:48)f := l(cid:48)∗t andCT(cid:48)h := l(cid:48)∗t. Let us which is the announced result. f h 5 Finally, thanks to the isomorphism λ between the whichisthewell-knownactionoftheconformalgroupon f X ’s and the X(cid:48) and using the notation Tf for the ten- conformal pseudo scalar fields of weight r. f f sorsontheX(cid:48)’sinsteadofT(cid:48)f’s,onecanrecasttheabove f relation (12) under the more familiar form 2. Action of the group on the one-forms (cid:16) (cid:17)d(t) Tf(x)= Kf(x) Th(x), (16) h The group acts on a 1-form a of R6 through in which Tf(x) stands for T(cid:48)f([x]) = Tf(xf). In partic- (L )∗a(x)=g a(g−1x), (21) g ular, between metrics this relation specializes to · where g a stands for the spinorial action on the 6 com- gf(x)=(cid:16)Kf(x)(cid:17)2gh(x), (17) ponents·of a. h We are going to define a representation (Lf)c of g whichmakesapparentthattheEq. (12)isinfactaWeyl SO (2,4) on the 1-forms of X as well as a represen- 0 f relation in the usual sense. tation L˜f on the sections of B∗. Moreover we will prove g f thatthemapr intertwinestheserepresentationsassoon f as the form a fulfills the transversality condition: D. Action of SO (2,4) 0 (Lf)crfaf =rfL˜faf as soon as i a=0, (22) g g ξ InthissectionwespecifytheactionofSO0(2,4)onthe which reads various objects defined above. In particular, we define L˜faf(x)[U]=(Lf)cAf(x)[U], x X , U T (X ), the action of SO0(2,4) on the sections af of the fiber g g ∀ ∈ f ∀ ∈ x f bundle B∗ and show that the transversality condition as soon as i a=0. f ξ ensures the SO0(2,4) invariance of the construction. We first build the representation L˜fg on the field af in The natural action x Lgx = gx of SO0(2,4) on R6 averysimilarwayasforthescalarfield. Wejustimpose yields an action Lfg on X(cid:55)→f defined through: that˜lf intertwinestherepresentationsL˜fg and(Lg)∗ and obtain straightforwardly: X x Lfx=(gx)/f(gx) X . f (cid:51) (cid:55)→ g ∈ f L˜faf(x)[V]=(ωf(g−1x))r−1g af(Lf x)[V], (23) Setting ωf(x)=f(x)/f(gx) for any x R6, one obtains g g · g−1 the actiongof the conformal group on th∈e space time: for any V T (R6), which is a fortiori true for any x ∈ U T (X ). Xf (cid:51)x(cid:55)→Lfgx=ωgf(x)gx∈Xf. (18) W∈exnowfdefine the representation (Lf)c of the group g on the forms Af. As a result, we cannot define a repre- 1. Action of the group on the scalar fields sentationontheAf inthesamewayasabove. Aswewill see, one must impose a condition on a. In place of this construction, we define directly the representation (Lf)c Let us recall that, for a scalar field φ homogeneous of g degree r the operator l∗ is defined through: through f (Lf)cAf(x)[U]=(ωf(g−1x))rAf(Lf x)[(Lf ) U], l∗φ(x)=φ(l (x))=:Φf(x). (19) g g g−1 g−1 ∗ f f (24) Also, the group acts on φ through the natural represen- for x X and U T (X ). We begin with calculating f x f taantdiowneLn∗gφow(x)de=finφe(gt−h1exr)e.pTrehseenfitealtdioΦnf(Lisfd)ecfionfedthoencoXnf- (Lfg)∗∈on any U ∈T∈x(Xf) using the Leibniz rule: (cid:18) (cid:19) g U ∂f(gx)[U] ifnotremrtawl ignreosu(pLofn)cita.ndWtehejunsattiumraplorseeptrheasetntthaetioonpeLra∗toonrφlf∗: (Lfg)∗(x)[U]= f(·gx) −gx f(gx)2 . (25) The crucial remark is that the second term of the r.h.s. belongs to a line of the cone. This term will be denoted (Lf)cl∗ =l∗(L )∗. (20) g f f g as F in the following. At this time we can set down That is to say, using that f(g−1x) = ωf(g−1x) for any the representation (Lf)c, putting ω := ωf(g−1x) for the g g g x X , readability: f ∈ (cid:20) (cid:21) ((Lfg)cΦf)(x)=(Lfg)clf∗φ(x) (Lfg)cAf(x)[U]=ωrAf(Lfg−1x) ω1g−1·U +F =l∗(L )∗φ(x) f g (cid:20) (cid:21) 1 =φ(g−1x) =ωra(Lf x) g−1 U +F g−1 ω · (26) (cid:18) g−1x (cid:19) (cid:20) (cid:21) =φ f(g−1x)f(g−1x) =ωra(Lfg−1x) ω1g−1·U =(cid:0)ωgf(g−1x)(cid:1)rΦf(Lfg−1x), =(ωgf(g−1x))r−1g·af(Lfg−1x)[U], 6 the before last equality being due to the transversality siderations of the Sec. IIC a space X is related to X f σ condition. Theresult(22)followsimmediatelyfrom(23) through a Weyl rescaling, the subsets X(cid:48) can be con- f and (26). As a consequence, the map l∗ intertwines formally pictured in the plane (α,β). In the sequel we f the representations (Lf)c and L∗ as soon as a fulfills will call “(α,β)-diagram” this conformal mapping (see g g the transversality condition. Note that the identifica- Fig. (2) for an example.). One may note that the metric tion Xf = Xf(cid:48) allows to realize the group action on Xf(cid:48) element on Xσ in the {α,β,θ,ϕ} system is as well. ds2 =dβ2 dα2 sin2αdω2. (28) − − E. Minkowskian atlas and conformal diagrams This form shows that Xσ is indeed included (β belongs to a subset of R) in the static Einstein space of positive curvature. AsexplainedinSec. IIAallthespacesX arerealized assubsetsXf(cid:48) ofC(cid:48)throughtheonetoonemfapλf. Then, y4)N/o2w=, l1et. uAsscsohnoswidnerinth[5e] stphaiscespXacfeN,isfaNM:=ink(oyw5sk+i one can use the Weyl relation (16) in order to simplify space. The coordinate system xI , I = c,µ,+ defined practical calculations in problems involving conformally { N} by invariant equations. As well known, these calculations can be much simpler than those performed on Xf espe-  yαy cUianlfloyrtwuhneanteltyh,eosneecocnadnnspotacienXgehniesraalMreicnokvoewrstkhieanspoancee. xcN = 4fN2(y4α,y5) Xf with only one Minkowskian space. However, we will xµ = yµ (29) fporuorveMtihnaktowthskeicaonvsepriancgeso.fTthheeswechaonlebCe(cid:48)einsdpoowsesdibwleituhsicnog- x+N =ffN((yy44,,yy55)), ordinates systems in order to form an atlas. In addition, N N thesesystemscanbechosentobeCartesianMinkowskian (whose inverse is given for convenience in Appendix B) coordinates to makes calculations simpler. Now, the for- provides on X(cid:48) (cid:48) a chart, called N-chart in the malismwedevelopuseR6 asframework,inpracticalcal- sequel. TheresftNrict⊂ionCtotheconeisobtainedforxc =0 N culations the coordinates basis on which fields can be andthatforP forx+ =1. ThemetricinducedfromR6 expandedarechosenonR6. Asaconsequence,theCarte- on X(cid:48) is, inftNhe xµN -coordinate basis, η , the usual sian Minkowskian coordinates of the atlas have to be re- formfoNf the Minkow{skNi}metric. µν lated to coordinates in R6. In the sequel, we will call The points where the above system becomes singu- Minkowskian systems both the systems in R6 and those lar, that is on the subset y5 +y4 = 0 , correspond to deducedfromthemintheatlasof (cid:48). Thisatlaswillalso points at infinity in the Mi{nkowski space}X . They are benamedMinkowskianaswellascChartsthatcomposeit. mapped to the boundaries of the N-chart ofnNthe confor- We will prove that one Minkowskian plane PfN together mal (α,β)-diagram in Fig. (2). Similar considerations withoneMinkowskiancoordinatesystem,theN-system, apply to the space X defined by f :=(y5 y4)/2=1 are sufficient to generate the whole Minkowskian atlas. and the coordinate syfSstem xI , I =S c,µ,+−is obtained Note that, some general comments in relation with such by replacing N by S in (29){. S} kind of coordinates are made in Appendix B. A set of Minkowskian charts covering (cid:48) is finally Firstletusintroduceausefulgraphicalrepresentation obtained by moving the two surfaces P C, P with of (cid:48) as conformal diagram. Remind that (cid:48) can be re- some elements of SO (2,4). More precfiNsely, fwSe ob- C C 0 alized as the intersection Xσ of the 5-sphere, obtained tain new Minkowski spaces through the action of the (cid:112) Ftathhigervo.emur1yg)ah.pcPoPλnoσσvi,n.etnσsTieohn:f=itsXgsσlpoabaδcαnaeβldyXcCαoσ(cid:48)yoβaridr/sei2nn,taahwtteeuintrshaiyldlstyethneeemtnificdeoodαnwe,teβhdC,roθw(u,siϕgtehhe oXXn5feβ0S:)p=adreyafi5mn∂ee0dt−erthyr0so∂uu5bggh=rou∂pβ. oTfhSeOm0a(2n,if4o)ldsgeXnefrβaNte(dresbpy. { } ocobotardininedatebysyssettetmingrr11,r=2,αr2,β=,θ1,ϕindthefienheydptehrr-sopuhgehrical fβN(y):=fN(e−βX50y)= 21(cid:0)cosβy5+sinβy0+y4(cid:1)=1, { } yy05 ==rr1scionsββ fβS(y):=fS(e−βX50y)= 21(cid:0)cosβy5+sinβy0−y4(cid:1)=1. 1 (27) are Minkowski spaces. Corresponding to each case, one yy4i ==rr2csionsαα,ωi(θ,ϕ) rceapnladceifinngefacboyorfdinatines(y2s9t)eman{dxsIβiNm}ilaanrlaylofgooruSstion{pxlINac}e, 2 N βN of N. As a consequence, we obtain an atlas of (cid:48), in the C inwhichβ [ π,π[, α,θ [0,π],ϕ [0,2π]andωi(θ,ϕ) usualsense,inwhichthesystemsofcoordinatesmakethe correspond∈to−theusualsp∈hericalcoo∈rdinatesonS2. The Weyl relations very transparent. Now every X(cid:48) space is f condition r =r =:r gives a system on the cone, r =1 an open subset of (cid:48) and can be endowed with the above 1 2 C gives a system on X . Now, since as a result of the con- atlas. σ 7 β The last term is for convenience divided by x+ in order N that tN be of the same degree of homogeneity (zero) as + the other components. One has tN =yαt =i t. (31) + α ξ In particular, the transversality condition reads tN = 0. + The same result applies in others Minkowskian coordi- nates. F. Robertson-Walker metric Here we prove that for any Robertson-Walker (RW) metric gRW one can find a function f such that the RW metric induced from R6 on the conformally flat space X :=P betheRobertson-WalkermetricgRW. fRW fRW ∩C In other words, each Robertson Walker space can be re- alized at least locally as a space X . fRW FollowingIbison[11]thesetcalled conformal(r,t) of α { } conformally flat metrics defined through : FIG. 2: The Minkowskian chart N (dotted region) and S ds2 =A2(x0, x )dx2; dx2 =η dxµdxν, (32) (crossed region) obtained respectively from the Minkowski µν (cid:107) (cid:107) spacesX ,f := 1(y5+y4)=1andX ,f := 1(y5−y4)= 1. ThesefNtwoNchart2s can be translatedfaSlongS the2the coordi- A2 being a conformal factor and xµ the Cartesian { } nate β leading to a complete covering of C(cid:48). This translation Minkowskiancoordinates, includestheRWmetrics. The belongs to SO (2,4) and corresponds to moving the planes notation (x0, x ) for the arguments of the function A 0 PfN, PfS in R6. The dashed lines are boundaries of a moved indicates that(cid:107)it (cid:107)depends separately of the time coordi- plane, the interior of that chart is on the l.h.s. (moved N- nates and of the space coordinates and that the space plane) or the r.h.s. (moved S-plane). coordinates appears only through the radius x . (cid:107) (cid:107) Let us assume now that A for some RW metric is given in a Minkowskian coordinate system, say the sys- tem xI (29). The function As a final remark, the space PfS can be obtained from { N} the space P through the transformation y4 y4 fN (cid:55)→ − x+ whichisalsoanelementofO(2,4). Asaconsequence,all f (x):= N , (33) RW A(x0, x ) the elements of our Minkowskian atlas can be obtained N (cid:107) N(cid:107) from P through a transformation of O(2,4) that is to say thrfoNugh an (inoffensive) Cartesian isometric change is invariant under the subgroup SO0(3) of SO0(2,4) gen- erated by y ∂ y ∂ . The equation f (x) = 1 defines of variables in R6. Thus, the whole atlas is generated i j − j i RW a SO (3) invariant space endowed with a metric which from P and the N-coordinates as announced in the 0 fN satisfies (32), thus a RW space. beginning of this section. III. SCALAR FIELDS Some explicit conditions in the N-charts In this section, we consider scalar fields on R6, homo- We note that, for x X , ∈ f geneous of degree r. These fields yield scalar fields on (cid:18) (cid:19) X . Weshowthatthesefieldsfulfillthescalarconformal x f x+(x)=f (x)=f =Kf (x). (30) equation as soon as the original field fulfill the Laplace- N N N f(x) fN Beltrami equation on R6. Let φ and j two scalar fields on R6 homogeneous of Also, in N-coordinates the dilation field reads ξ :=yα∂ =x+∂ . degree -1 and -3 respectively, and suppose that α N x+ N Now, a one-form t Ωp(R6) homogeneous of degree (cid:3) φ=j. (34) zero,isexpandedinthe∈coordinatebasis dxc,dxµ,dx+ 6 { N N N} as We claim that tN =tNcdxcN +tNµdxµN + xtN++dx+N. ((cid:3)f + R6)Φf =Jf, (35) N 8 where (cid:3) is the Laplace-Beltrami operator and R the Maxwell field on de Sitter space using a Gupta-Bleuler f scalar curvature for the space X . We consider X the schemewheretheconditionoftransversalityistranslated f σ realization of the cone up to the dilation defined Sec. to a condition on states after quantization. In second, IIE, on which all X spaces realize as subsets. Setting they can be used to solve classical propagation problems f u=r /r andv =1/r inthehypersphericalcoordinates for the Maxwell field: For instance, we considered in [7] 1 2 2 (27) and using the homogeneity of φ, the equation (34) the two-charges problem for the de Sitter space. reads (cid:18) (cid:19) 1 1 v2 (1−u2)∂u2+ u(1−u2)∂u+1+ u2∂β2 −∆S3 φ=j. B. Maxwell equation on Xf (36) In this section, we implicitly identify the spaces X The restriction to X (u=v =1) then gives f σ and X(cid:48), writing all the objects on the common manifold f (∂2 ∆ +1)Φσ =Jσ, (37) (cid:48). β − S3 C We now prove the statement of Sec. IIA: for a ho- which is the desired result. We obtain the similar re- mogeneous of degree zero, j homogeneous of degree 2 − sult for the other hyper surfaces X by using the Weyl satisfying f correspondence between the spaces X . Note that al- f (cid:3) a=j though straightforward the change of variables leading  6 to the equation (36) is rather cumbersome. (cid:93) a T( ) (39) η C | ∈ C (cid:93) j T(X ) η |Xf ∈ f IV. THE MAXWELL FIELD the field Af := l∗(a) and the current Jf = l∗(j) defined f f on X satisfy the Maxwell equations and the Eastwood- f In this section we deal with one-forms a and j of R6 Singer gauge condition [1] : satisfying (cid:3) a = j and such that a is homogeneous of 6 degree 0 and j homogeneous of degree 2 (remind that  (cid:0)(cid:3) δν ν +Rν (cid:1)Af =Jf tafohnridmsi−aml3ipsmlrieesstpthehecatteivqaeuαlayat)ni.odnWjα(cid:3)eafiarrest=hoshmjoowlgeeahndoes−wotuoisnoatfhdseeetgprroeefese−snix1t  (cid:18)(cid:3)ff∇µν−−∇2µ∇∇µ(cid:18)Rµνµ− 31Rνgµν(cid:19)µ(cid:19)Afν =0. (40) 6 copiesoftheequationoftheconformalscalarfield. Then, WefirstprovetheresultontheMinkowskianchartX , wegivetheproofthatAf T∗(X )satisfiestheMaxwell fN ∈ f after what, using the O(2,4) invariance of the hypoth- equationsassoonasafulfillsthetransversalitycondition, esis and of the conclusion, and the properties of the andthatanadditionalconditiononj allowsustorecover Minkowskian charts, we obtain this result on all the a conformal gauge condition. Minkowskian charts. In the following, for readability, we note aN,AN, for afN and AfN. We first consider the equation (cid:3) a = j, and apply (38) on the Minkowski A. Equation on B∗ 6 f space X defined by f (used to build the N-chart), fN N this yields the system Suppose that B∗ is endowed with a coordinate system f (dxyIα,dayαC)awrtheseiraenxsIysisteamnyofcocoorodridniantaetessysotenmTo∗f(RX6f).anInd (cid:3)N(s)aNα =jαN, (41) x such a coordinate system the equation on af reads where (cid:3)(s) is the scalar Laplace-Beltrami operator on N ((cid:3)(s)+ R)af =jf, (38) XstefNp w(cid:39)eXexf(cid:48)pNr,eassNin:=thl(cid:101)Ne(Na)-cahnadrtjoNf:=(cid:48) (l(cid:101)N29(j))t.heAsabaovseecsoynsd- f 6 α α C tem of equations (41). This leads to the system already where (cid:3)(s) is the scalar Laplace-Beltrami on X and R obtained in [10] (with slightly different notations) f f the Ricci scalar. This is a straightforward consequence of the fact that, on R6 in Cartesian coordinates one has ∂2aNµ +∂µaNc =jµN ((cid:3) a) =(cid:3)(s)a : sinceinaCartesiancoordinatesystem ∂2aN 2∂ aN 2aN =jN (42) yα6 oαf R6 6theαequation (cid:3)6a = j reduces to six copies ∂2a+N −=jN.· − c + { } c c (one per components of a and j) of the scalar equation (34), each components aα and jα being homogeneous of Then,weexpressthetwoconstraintsappearingin(39) degree d(aα) = 1 and d(jα) = 3, one can thus apply in the N-chart. The first one is the transversality condi- − − the result (35) of the Sec. III. tion applied to the field a, using the formula of ( 31) one Twocommentsareinorderconcerningequations(38). has In first, they do not present any gauge ambiguity and we proved in [9] that they allow a quantization of the aN =0. (43) + 9 The second condition (cid:93) j T(X ) implies the where the dots now refer to the metric gf and where we η |Xf ∈ f transversality of j since T(X ) T( ), thus havetakenintoaccountthehomogeneityofboththeelec- f ⊂ C tromagnetic and the current one-form fields. They have jN =0. (44) respectively a degree of 1 and of 3 which correspond + − − to conformal weights of zero and 2 for the components iInntahdedNiti-ocno,o(cid:93)rηdjin|Xaftes∈oTn(tXhfe)corenwer(ixtecs=(cid:93)ηj0[)fr]e=ads0, which of the fields : Afµ = ANµ, Jµf = (K−ffN)−2JµN. Finally, the second equation simplifies and we obtain the announced N result on the N-chart. Then, using the above remark ∂f 2 ∂f jN + ηµνjN =0. on O(2,4) invariance, we obtain the result on the whole c ∂x+ x+ µ ∂xν space. N N N Note that the Maxwell equations are obtained inde- Thanks to the homogeneity of f this equation rewrites pendently of the condition (cid:93) j T(X ), which is η |Xf ∈ f ∂ f used only to obtain the Eastwood-Singer gauge condi- jcN +2 ηµνjµN fν =0. tion. Note also that, in absence of source (j = 0), the Eastwood-Singer gauge condition is automatically ful- Then, using the definition (14), it becomes filled as soon as the transversality condition on a is ful- filled. jN =jN W(f,fN), (45) It is important to point out that from a physical per- c · spective the initial conditions and currents are given where the dot refer to the Minkowskian metric ηf. in the space Xf. In particular, the Minkowskian cur- Using the constraint (43-45), the system (42) becomes rents appearing in (47) are defined through the equa- tion Jf = (Kf )−2JN in order to satisfy the Weyl rela- ∂2aN ∂ ∂ aN =jN tion. µThey hafNve in general no physical meaning in the  µ − µ · µ Minkowski space. ∂2∂ aN =jN W(f,fN) (46) aN =· ∂ aN. · c · Owning to the map r the above system leads to V. COMMENTS ON SOME APPLICATIONS fN (cid:40)∂2ANµ −∂µ∂·AN =JµN (47) Besides the results presented in the previous sections, ∂2∂ AN =JN W(f,fN) theformalismdepictedinthepresentpaperexplicitsthe · · geometricalnatureofthevariousobjectsusedinourpre- on the Minkowski space X . Note that the map r viousworks. Thisallowsustoconsidersomestraightfor- becomes obvious in the N-cfoNordinates (29): for all xfN ward generalizations, let us comment briefly about the X one has Af(x)=af(x)=a (x). ∈ classical and quantum situations. f µ µ µ For the sake of argument, let us introduce a self ex- Themethodusedfortheclassicalpropagationproblem planatory symbolical notation for the two operators ap- consideredin[7]extendsnaturallytotheconformallyflat pearing in the l.h.s. of the Maxwell equations and the spacesXf. Tosummarize,givenasetofinitialconditions Eastwood-Singer gauge condition. With them the above ancurrentsinXf theproblemoffindingtheAf solution system reads: of the Maxwell equations in the Eastwood-Singer gauge (orinagaugecontainedinit)inX amountstouseinthe f (cid:40) (M [AN]) =JN Minkowskian atlas the propagation formula established N µ µ in [7], which uses only the Minkowskian scalar Green’s ES [AN]=JN W(f,fN), N · function. The quantization scheme used in [9] for the free elec- This result is available for the N-chart. Nevertheless, tromagnetic field on de Sitter space can be transposed the O(2,4) invariance of hypothesis and conclusion and here for the most part. The definition of an SO (2,4)- thefactthatanyMinkowskianchartcanbededucedfrom 0 invariant scalar product on the space of the solutions of theN-chartbymeanofanO(2,4)transformation,proves (38) and the obtention (from the known mode solutions that this result is true for any Minkowskian chart of our of (38)) of a reproducing kernel for the af’s with respect atlas. α tothatproductcanbereproducedalmostverbatim. The Now, using the same symbolical notation for the two result is: f (x,x(cid:48)) = η D+(x,x(cid:48)), where D+(x,x(cid:48)) operators appearing in the l.h.s of (40), and following Wαβ − αβ f f is the scalar two-point function on X . The construc- [1] to apply a usual (local) Weyl transformation between f tion of the Fock space, including the determination of X andX toeachEqs. ofthesystem(47),oneobtains fN f the physical subspace is formally identical. The general (cid:40) (cid:0)M [Af](cid:1) =Jf form of a covariant two-point function, a bi-tensor on a f µ µ specificX ,requiresmoredevelopments(notethat,such f ES [Af]+W(f,fN) (cid:0)M [Af](cid:1)=Jf W(f,fN), a function has been proposed recently in Minkowskian f f · · 10 coordinates [12]). This could be the object of future in- Now, moving to the more general coordinate basis vestigations. xI(y) on has { } T (λy)=Jα1(λy)...Jαp(λy)T (λy) I1,...,Ip I1 Ip α1,...,αp Acknowledgements (cid:18)∂yα(cid:19) whereJα(u):= ,u R6,ishomogeneousofde- I ∂xI ∈ TheauthorsthankC.Cachotforvaluablesdiscussions u gree 1 d(I). Taking homogeneity into account in the and suggestions about the Secs.(IIE) and (IIF). − r.h.s. of the above expression leads to T (λy)=λ((cid:80)k(1−d(Ik))+(d(T)−p)) Appendix A: Homogeneous tensors I1,...,Ip Jα1(y)...Jαp(y)T (y), × I1 Ip α1,...,αp For reference, we recall here some properties of an ho- from which the result follows. mogeneous (cid:0)0(cid:1)-tensors T of R6. p ThesetR6,togetherwiththescalarproduct<y ,y > 1 2 :=η yαyβ,isnaturallyendowedwithastructureoflin- αβ ear space. A point in R6 is located by a vector y whose Appendix B: Notes on Minkowskian systems components in Cartesian coordinates are yα . The lin- { } ear structure allows us to identify the tangent spaces at The system appearing in Sec. IIE has already been any point through ∂ (y) = ∂ (0). This makes sense for used in the literature (see for instance [10]), it is remi- α α expressionsasT(y)=T(λy)sincethetangentspaceaty niscent of systems called polyspherical systems (see for isidentifiedwiththetangentspaceatλy. Letusconsider instance [13]). One may note that they are framed in the dilatation map in R6 such a way that the “extremal”coordinates xc,x+ are { } functions of the constraints (defining the cone and ρλ :R6 →R6 the space PfN respectively) whereas the “centraCl” co- y λy, ordinates xµ are those on the background space Xf. (cid:55)→ Cartesian{Min}kowskian coordinates can be obtained on λ being a positive real number. Since Ty(R6) R6 this Xf setting x+ := f instead of x+ := fN, with this is also the result of a push-forward on the vec≡tor y. In choice conformally invariant equations will appears on other words ρλ∗∂α = λ∂α. For a (cid:0)0p(cid:1)-tensor field T of their Minkowskian form on Xf. the pullback reads (ρ∗T)(y)=λpT(λy). Now, an homo- For the de Sitter space, one may verify that the N geneous (cid:0)0(cid:1)-tensor fieλld T of degree d(T) = r is defined and S Minkowski charts correspond to the stereographic p through the relation projections from the North and South Poles from which their names originates. (ρ∗T)(y)=λrT(y). Finally, we note for convenience that the inverse of λ system (29) reads Thus, homogeneous (cid:0)0(cid:1)-tensor field T of degree r satisfy p  1 T(λy)=λ(r−p)T(y). (A1) y5 =x+N(1+xcN − 4ηµνxµNxνN) 1 y4 =x+(1 xc + η xµxν) ofTithsechoommpoogneennetistyinofsaomteenscoorofirdelidnactaenbbaesirselwatheodsetoctohoart- yµ =x+Nxµ−. N 4 µν N N N N dinates xI(y) I = 0,...,5 are each one homogeneous { } functions of degree d(I). Precisely, let us show that: for T 0(R6) a (cid:0)0(cid:1)-tensor field of R6, homogeneous of Appendix C: Conventions and definitions ∈ Tp p degree d(T) one has We summarize here the conventions, main structures p (cid:0) (cid:1) (cid:88) and maps used in this paper. They are discussed in Sec. d T =d(T) d(I ), I1,...,Ip − k IIA. Here are the conventions for indices: k=1 α,β,γ,δ,... = 0,...,5, in particular for x(y)=y µ,ν,ρ,σ... = 0,...,3, (cid:0) (cid:1) d T =d(T) p. i,j,k,l,... = 1,...,3, α1,...,αp − I,J,... = c,0,...,3,+. This result is obvious in the particular case x(y) = y since the Eq. (A1) reads in components The coefficients of the metric diag(+, , , , ,+) of R6 are denoted η . The definitions of−spa−ces−an−d maps αβ T(λy) =λ(r−p)T(y) . reads : α1,...,αp α1,...,αp