document: the @xmath0-invariance of the maxwell equation was discovered by cunningham and bateman a century ago . 
   however in order to quantize the maxwell field and due to gauge freedom , a gauge fixing condition is necessary . 
   the lorenz gauge is usually used , which breaks the @xmath0 invariance . 
   nonetheless since such a symmetry mights apear to lack physical meaning , its breaking does not bother many people @xcite . 
   the purpose of the present paper is to demonstrate the benefits of keeping this fundamental symmetry when quantizing the maxwell field in conformally flat spaces ( cfs ) . 
   the starting point is the following . 
   a classical @xmath0-invariant field can not , at least locally , distinguish between two cfss @xcite . 
   so why not maintain the @xmath0-invariance during the quantization process in a cfs ? 
   doing so , a free field living in a cfs might behave like in a flat space and the corresponding wightman two - point functions can be related to their minkowskian counterparts . 
   the work @xcite confirms this assertion in the special case of maxwell field in de sitter space . 
   indeed , a new and simple two - point wightman function @xmath1 was found and which has the same physical ( gauge independent ) content as the two - point function of allen and jacobson @xcite . 
   this is because the faraday propagator @xmath2 is the same . 
   the present work extends to general cfss and clarify the quantum structure of the formalism developed in @xcite . 
   we use dirac s six - cone formalism and realize all cfss as intersections of the null cone and a given surface in a six - dimensional lorentzian space . 
   the introduction of auxiliary fields and the use of the gubta - bleuler quantization scheme are necessary to deal with gauge freedom of the maxwell field . 
   another important ingredient is the use of a well - suited coordinate system . 
   this allows to @xmath0-invariant cfs formulas to get a minkowskian form . 
   the main result is a set of wightman two - point functions for maxwell and auxiliary fields . 
   this paper is organized as follows . 
   [ geom ] sets the coordinates systems and the geometrical construction of cfss . 
   [ fields ] defines the fields and gives their dynamical equations . in sec . 
   [ quantum - field ] , the dynamical system is solved , the quantum field is explicitly constructed and the two - point functions are written down . 
   some technical details are given in appdx . 
   [ details ] . 
   the infinitesimal @xmath0 action on the fields @xmath3 is expanded in appdx . 
   [ action ] and their @xmath0-invariant scalar product is given in appdx . 
   the six - dimensional lorentzian space @xmath4 is provided with the natural orthogonal coordinates @xmath5 and equipped with the metric @xmath6 . 
   quantities related to @xmath4 and its null cone @xmath7 are labeled with a tilde . 
   we define a second coordinate system @xmath8 , @xmath9 where the four components @xmath10 is the so - called polyspherical coordinate system @xcite and @xmath11 . a straightforward calculation yields @xmath12 which means that the component @xmath13 carries alone the homogeneity of the @xmath14 s . 
   using the system @xmath15 , the null cone reads @xmath16 a five - dimensional surface in @xmath4 is defined through @xmath17 where the real and smooth function @xmath18 depends only on @xmath10 and @xmath19 and is then homogeneous of degree @xmath20 . 
   the intersection of @xmath21 and @xmath22 is a four - dimensional space @xmath23 where the index @xmath24 in @xmath25 refers to @xmath26 ) . 
   regarding to its metric inherited from @xmath27 , precisely @xmath28 @xmath25 turns out to be a cfs . 
   a smooth move of the surface @xmath29 , which corresponds to changing the function @xmath30 , amounts to perform a weyl rescaling . 
   this locally relates all cfss and permits to go from one to another . 
   note that for @xmath31 , @xmath25 reduces to minkowski space @xmath32 and accordingly the four components system @xmath10 yields the usual cartesian system . 
   the gradients @xmath33 are extensivelly used in this article . 
   the function @xmath30 does not depend on @xmath13 and thus @xmath34 . 
   the choice of the function @xmath30 , including its @xmath19 dependence has to be done in such a way to ensure the invariance ( in @xmath4 ) of the surface @xmath22 under the action of the isometry group associated to the desired @xmath25 four - dimensional space . since the @xmath4 null - cone is @xmath0-invariant , the resulting @xmath25 will be invariant under its isometry group . 
   let us consider an example : @xmath35^{-1}$ ] , where @xmath36 is a constant . 
   the associated surface @xmath22 and thus the corresponding @xmath25 are left invariant under the action of de sitter group @xcite . 
   also , @xmath25 is a de sitter space . 
   in this section , we explain how to obtain the @xmath0-invariant maxwell field in @xmath25 from a six - dimensional one - form . following dirac @xcite 
   , we consider a one - form @xmath37 defined in @xmath4 homogeneous of degree @xmath38 and which decomposes on the @xmath39 basis as @xmath40 the components @xmath41 are homogeneous of degree @xmath42 and obey to the equation @xmath43 this equation is naturally invariant under the @xmath0 action since this group has a linear action when acting in @xmath4 . 
   we then decompose the one - form @xmath37 on the basis @xmath44 corresponding to the system @xmath45 ( [ coord+muc ] ) , with a slight but capital modification on the @xmath46 component . 
   there are two ways , the first decomposition reads @xmath47 the second is given by @xmath48 now , identifying ( [ eq-1 ] ) with ( [ eq-2 ] ) , one obtains the relation between the fields @xmath49 and @xmath50 through @xmath51 all the fields @xmath52 and @xmath53 are by construction homogeneous of degree @xmath54 . as a consequence , @xmath55 and @xmath56 
   this amounts to project the fields @xmath52 on @xmath57 and @xmath53 on @xmath58 . then projecting the fields on the null cone @xmath21 yields @xmath59 thus @xmath60 and @xmath61 are respectively @xmath25 and minkowski fields . 
   though in a slightly different maner , this relation was obtained in @xcite in the particular case of de sitter space and was called the `` extended weyl transformation '' . 
   the fields @xmath62 and @xmath63 are auxiliary fields and the field @xmath64 is , up to the condition @xmath65 the maxwell field . 
   this will become clear here after . 
   let us now turn to the dynamical equations . 
   our strategy is to transport minkowskian @xmath0-invariant equations to get @xmath0-invariant equations in the @xmath25 space . 
   the first step is thus to write down the minkowskian equations which are obtained using the equation ( [ equation - a ] ) and the relation ( [ a(a)-m ] ) . 
   this system reads @xmath66 the corresponding system in @xmath25 is obtained using ( [ extendedweyl - bis ] ) , @xmath67 where all contractions are performed using @xmath68 even though we are in the curved space @xmath25 . 
   the field @xmath69 obeying to the system above is not yet the maxwell one . 
   nevertheless , the constraint @xmath70 simplifies the system ( [ syst1m - h ] ) and leads to @xmath71 despite their minkowskian form , these equations are the maxwell equation and a conformal gauge condition on any conformally flat space . this is due to the use of the polyspherical coordinate system ( [ coord+muc ] ) , which makes apparent the flatness feature of the @xmath25 spaces . 
   the constraint @xmath72 reduces the extended weyl transformation ( [ extendedweyl - bis ] ) into the identity @xmath73 recovering the ordinary vanishing conformal weight of the maxwell field @xmath69 . after some algebra , the covariant form of ( [ maxwellh1 ] ) takes the form @xmath74 where @xmath75 . the first line ( resp . 
   the second one ) is the covariant maxwell ( resp . 
   the eastwood - singer gauge @xcite ) equation in an arbitrary @xmath25 space . 
   this conformal gauge was first derived by bayen and flato in minkowski space @xcite . 
   its extension to curved spaces ( even cfss ) is not trivial and can be performed using adapted tools like the weyl - gauging technique @xcite or the weyl - to - riemann method @xcite . 
   note that the system ( [ system - covariant ] ) is valid only if @xmath76 ( an @xmath0-invariant constraint ) . 
   but the latter has to be fixed at the end of the quantization process , not at the begining . 
   indeed , the auxiliary field @xmath77 acts as a faddeev popov ghost field and its retention during the quantization process is necessary . 
   the constraint @xmath76 will be applied on the quantum space to select an invariant subspace of physical states and the wightman functions thus include the whole big space . 
   this is related to the undecomposable group representation ( see appendix [ action ] ) . 
   we now apply the gupta - bleuler quantization scheme @xcite . 
   this can be summarized as follows . 
   we have seen that @xmath78 is interpreted as the maxwell field in the eastwood - singer gauge ( [ system - covariant ] ) on the space @xmath79 when the constraint @xmath80 is applied . 
   the problem is that pure gauge solutions ( @xmath81 , with @xmath82 and @xmath80 ) are orthogonal to all the solutions including themselves . as a consequence , 
   the space of solutions is degenerate and no wightman functions can be constructed . to fix this problem , we consider the system ( [ syst1m - h ] ) , instead of ( [ system - covariant ] ) , for which @xmath83 and thus a causal reproducing kernel can be found . 
   this means that for quantum fields @xmath84 acting on some hilbert ( or krein ) space @xmath85 , we can not impose the operator equation @xmath86 . 
   instead , we define the subspace of physical states @xmath87 which cancels the action of @xmath88 . then the maxwell equation and the eastwood - singer gauge hold in the mean on the space @xmath89 . 
   the task seems complicated at first sight , but thanks to the correspondence ( [ extendedweyl - bis ] ) we only need to solve the minkowskian system ( [ syst1 m ] ) , which is already done in @xcite . 
   indeed , using the weyl equivalence between cfss , the whole structure of an @xmath90-covariant free field theory can be transported from minkowski to another cfs . in the following , 
   we solve the dynamical equations , obtain the modes , determine the quantum field , the subspace of physical states and finally compute the two - point functions .      the solutions of the minkowskian system ( [ syst1 m ] ) can be obtained from @xcite and read @xmath91 where @xmath92 are polarization vectors whose components are given by @xmath93 and verifying @xmath94 with respect to the scalar product ( [ scalarps - a ] ) . 
   the matrix @xmath95 relates the fields @xmath96 and @xmath97 ( [ matrix - s ] ) . 
   the scalar modes @xmath98 are solutions of the minkowskian @xmath0-invariant ( or massless ) sclalar field equation @xmath99 , @xmath100 where @xmath101 denotes the usual hyperspherical harmonics . 
   the normalization constant @xmath102 is chosen in order to get @xmath103 with respect to the klein - gordon scalar product . as a consequence , the solutions ( [ modes - a ] ) are normalized with respect to ( [ scalarps - a ] ) 
   , @xmath104 thus the general solution of the system ( [ syst1 m ] ) reads @xmath105 where @xmath106 are real constants . 
   let us now turn to the modes of the system ( [ syst1m - h ] ) . 
   they are obtained thanks to the extended weyl transformation ( [ extendedweyl - bis ] ) applied on the minkowskian modes ( [ modes - a ] ) @xmath107 these modes are normalized like ( [ norm ] ) but according to the scalar product ( [ scalarps - a - k ] ) . 
   the general solution on @xmath79 reads @xmath108 where the @xmath109 are some real constants . 
   note that when @xmath110 the solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) solve the maxwell equation in the eastwood - singer gauge . 
   we can now define the quantum fields and construct the fock spaces as usual . 
   the quantum fields corresponding to ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) are respectively defined through @xmath111 @xmath112 where the operators @xmath113 and @xmath114 are respectively the annihilators and creators of the modes ( [ modes - a ] ) in @xmath32 and the modes ( [ modes - a - k ] ) in @xmath25 . 
   the use of the same annihilators and creators for all cfss is highly important for our purpose . 
   indeed , this allows to define the the same vaccuum state @xmath115 through @xmath116 for any annihilator . 
   the one - particle states are built by applying the creators on the vacuum state @xmath117 and the multiple particle states of the fock spaces are constructed as usual . 
   moreover , the annihilation and creation operators obey to the following algebra @xmath118 =    [   \hat a_{{\scriptscriptstyle l}m ( \alpha)}^{\dag } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ] = 0 \\ & [   \hat a_{{\scriptscriptstyle l}m ( \alpha ) } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ]   = -\tilde \eta_{{\scriptscriptstyle \alpha}\beta } \delta_{{\scriptscriptstyle l}l ' } \delta_{{\scriptscriptstyle m}m'}. \end{split}\ ] ]    the subset of physical states in both spaces is determined thanks to the classical physical solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) verifying ( [ a+=0 ] ) . in quantum language , @xmath119 is a physical state iff @xmath120 where @xmath121 is the annihilator part of @xmath122 . 
   this implies the equality @xmath123 for any physical states @xmath119 and @xmath124 . 
   also , the subspace of physical states is the same in all cfss , which allows to transport physical quantities from minkowski space into the @xmath25 space . as a consequence , one obtains @xmath125 in @xmath25 and the corresponding minkowskian system in @xmath32 . 
   the quantum fields fulfill the maxwell equation together with the eastwood - singer gauge in the mean on the physical states . 
   we show in this part how to get the wightman two - point functions on @xmath25 from their minkowskian counterparts . 
   the wightman functions related to the minkowskian fields @xmath126 are defined through @xmath127 their expressions are given in @xcite and read @xmath128 where @xmath129 with @xmath130 stands for the wightman two - point function related to the minkowskian massless scalar field . 
   the wightman two - point functions related to the field @xmath131 are given by @xmath132 now , using ( [ modes - a - k ] ) , ( [ maxwelds2ptdef ] ) and ( [ wightman - k ] ) , allows to write the following capital formula @xmath133 where the @xmath134 terms read @xmath135 the wightman two - point functions ( [ dk - dm ] ) read @xmath136 where @xmath137 , @xmath138 , @xmath139 and @xmath140 . 
   + to end this paper , let us consider an important particular case , that corresponding to de sitter space . 
   this case is obtained by specifying @xmath141 where @xmath142 is related to the de sitterian ricci scalar through @xmath143 . 
   the gradients ( [ upsilon ] ) read @xmath144 in this case we obtain simple expressions for the two - point functions related to the fields @xmath145 on de sitter space . 
   the three more relevant yield @xmath146 where we have used ( de sitter is a maximally symmetric space ) the standard unit tangent vectors @xmath147 and @xmath148 , the parallel propagator along the geodesic @xmath149 and the usual function @xmath150 of the geodesic distance @xmath151 relating @xmath152 and @xmath153 , @xmath154 . 
   see @xcite for a more precise statement . 
   note that the two - point function @xmath155 has the same physical content with the allen and jacobson two - point function @xcite . 
   an @xmath0-covariant quantization of the maxwell field in an arbitrary conformally flat space was presented . following dirac s six - cone formalism , all conformally flat spaces @xmath25 
   are realized as intersections of the null cone and a given surface @xmath29 . 
   the quantum field was explicitly constructed using the gupta - bleuler canonical quantization scheme and the wightman two - point functions were given . 
   the price to pay for this simplicity and the maintaining of the @xmath0 invariance during the whole quantization process was the introduction of two auxiliary fields @xmath156 and @xmath77 . 
   as a consequence , the maxwell field @xmath69 does not propagate `` alone '' but together with its two auxiliary fields . 
   the propagation must use all the wightman functions ( [ d - x - k - relevent ] ) and not only the `` purely '' maxwell one @xmath157 . 
   nonetheless , in a recent work @xcite , we have used the functions ( [ d - ds ] ) to propagate the maxwell field generated by two charges of opposite sign placed at the two poles of a de sitter space . 
   the calculations showed that only @xmath157 is involved , which trivialize the problem . 
   one can consider to use the two - point functions ( [ d - x - k - relevent ] ) to propagate the electromagnetic field for some charge distribution given in other cfss , like flrw spaces for instance . 
   one concludes that is much worth to maintain the @xmath0 symmetry during the whole quantization process when dealing with maxwell field in a conformally flat space . 
   the problem then goes back to minkowski and the calculations become much easier . 
   in fact , the classical and quantum structures of the free maxwell field are locally the same in all conformally flat spaces . 
   the remained question is to know if this is true for other free fields and how to deal with @xmath0-invariant interactions ? 
   i would like to thank m. novello , j. renaud and e. huguet for illuminating discussions and the cnpq for financial support . 
   considering ( [ field - a ] ) and ( [ eq-1 ] ) , expressing the basis @xmath159 in terms of @xmath44 and then identifying both sides , one obtains the expression of @xmath160 in terms of @xmath161 . 
   we find , after using the homogeneity properties , @xmath162 which reads @xmath163 this system can be inverted in @xmath164    following the same steps as above , one obtain the matrix linking the @xmath165 to the @xmath166 @xmath167     + a^{{\scriptscriptstyle k}}_4 [ \upsilon_c(1+x^2)-1 ]     \\ & & \qquad \qquad \qquad    + \upsilon_c a^{{\scriptscriptstyle k}}.x \biggr\ } \\     { \displaystyle a_\mu^{{\scriptscriptstyle k } } }     & = &   k   \biggl\ { a^{{\scriptscriptstyle k}}_5 \left ( ( 1 -x^2 ) \upsilon_{\mu } - \frac{1}{2}x_{\mu } \right )    \\ & &        +   a^{{\scriptscriptstyle k}}_4 \left ( ( 1 + x^2 ) \upsilon_{\mu } + \frac{1}{2 } x_{\mu } \right )   + a^{{\scriptscriptstyle k}}_\nu \left ( \upsilon_{\mu } x^\nu + \delta_{\mu}^{\nu } \right ) \biggr\ } \\   { \displaystyle   a_+^{{\scriptscriptstyle k } } }    & = & { \displaystyle k \biggl\{a^{{\scriptscriptstyle k}}_5 ( 1- x^2 ) } + a^{{\scriptscriptstyle k}}_4 ( 1 + x^2 ) + a^{{\scriptscriptstyle k}}.x \biggr\}.     \end{array } \right .\ ] ] this system can be obtained using the minkowskian system ( [ a(a)-m ] ) , the relation @xmath168 ( which comes out from the homogeneity properties of the fields ) and the extended weyl transformations ( [ extendedweyl - bis ] ) . 
   this is inverted in @xmath169 
   the @xmath0 infinitesimal action on the field @xmath3 is given by commutators of the group generators and the field . 
   first , we write down the infinitesimal transformations of the minkowskian fields @xmath96 which can be found in @xcite then we transport the resulting representation into @xmath25 . for any element @xmath170 
   , the related generator is denoted by @xmath171 and whose action on the field @xmath96 reads @xmath172 \\ &   = x_{e } \ 
   a_{{\scriptscriptstyle i}}^{{\scriptscriptstyle m } }   + \left ( \sigma_{e }   \right)_{{\scriptscriptstyle i}}^{{\scriptscriptstyle j } } \ 
   a_{{\scriptscriptstyle j}}^{{\scriptscriptstyle m } } \end{split}\ ] ] where the first part represents the scalar action and the second the spinorial action . setting @xmath173 
   the minkowskian infinitessimal action reads @xmath174 } \ 
   a^{{\scriptscriptstyle m}}_\tau   \\ &   \left ( x_{\mu\nu}^{{\scriptscriptstyle m } } \ 
   a^{{\scriptscriptstyle m}}\right)_+ =   x_{\mu\nu } a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the rotations , @xmath175 for the translations ; @xmath176}^\lambda   + x^\lambda \eta_{\mu\nu } ) a^{{\scriptscriptstyle m}}_\lambda - 2\eta_{\mu\nu } a^{{\scriptscriptstyle m}}_+     \\ & \left(k_\mu^{{\scriptscriptstyle m } } \ 
   a^{{\scriptscriptstyle m}}\right)_+ = k_\mu a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the special conformal transformations ( sct ) . 
   finally , we have @xmath177 for the dilations . 
   the undecomposable structure of the fields @xmath96 is made clear . under the @xmath0 action 
   , the component @xmath178 overlaps @xmath179 which in turn overlaps @xmath180 . 
   so we have the scheme @xmath181    the second step is to trasport the group action from minkowski to the @xmath79 space using the extended weyl transformation ( [ extendedweyl - bis ] ) @xmath182 a_{+}^{{\scriptscriptstyle k } } \end{split}\ ] ] where we have used @xmath183 for all @xmath184 . 
   also only the second part of the last line has to be computed . 
   note that the constraint @xmath189 ( [ a+=0 ] ) is @xmath0-invariant . 
   this is important since this constraint defines the subset of physical states . 
   the @xmath0-invariant scalar product for the minkowskian field @xmath96 reads @xmath190 where @xmath191 is some cauchy surface in @xmath32 and @xmath192 is a surface element . 
   an important point is that this cauchy surface is common to all the spaces @xmath25 since they are all conformally equivalent @xcite .    using ( [ lien - weyl ] ) and 
   ( [ extendedweyl - bis ] ) , the scalar product for the field @xmath166 is obtained from ( [ scalarps - a ] ) and reads @xmath193 where the @xmath25 surface element is related to its minkoskian counterpart by @xmath194 .
summary: we present an @xmath0-covariant quantization of the free electromagnetic field in conformally flat spaces ( cfs ) . 
   a cfs is realized in a six - dimensional space as an intersection of the null cone with a given surface . 
   the smooth move of the latter is equivalent to perform a weyl rescaling . 
   this allows to transport the @xmath0-invariant quantum structure of the maxwell field from minkowski space to any cfs . 
   calculations are simplified and the cfs wightman two - point functions are given in terms of their minkowskian counterparts . 
   the difficulty due to gauge freedom is surpassed by introducing two auxiliary fields and using the gupta - bleuler quantization scheme . 
   the quantum structure is given by a vacuum state and creators / annihilators acting on some hilbert space . 
   in practice , only the hilbert space changes under weyl rescalings . also the quantum @xmath0-invariant free maxwell field 
   does not distinguish between two cfss .