This justifies the introduction of transversely and longitudinally-polarized photons, and of the related concept of transverse and longitudinal structure functions, when considering these virtual states as effective probes of the internal quark and gluon contents of the hadrons. From a mathematical point of view, one considers the SO(2,1) group instead of the usual SO(3) group encountered in the usual massive case discussed above. This explains the occurrence of two transverse polarization vectors ϵ T λ = 1 , 2 {\displaystyle ~\epsilon _{T}^{\lambda =1,2}~} and ϵ L {\displaystyle ~\epsilon _{L}~} which satisfy ϵ T 2 = − 1 {\displaystyle ~\epsilon _{T}^{2}=-1~} and ϵ L 2 = + 1 , {\displaystyle ~\epsilon _{L}^{2}=+1~,} to be compared with the usual case of a free Z 0 {\displaystyle ~Z_{0}~} boson which has three polarization vectors ϵ T λ for λ = 1 , 2 , 3 ; {\displaystyle ~\epsilon _{T}^{\lambda }{\text{ for }}\lambda =1,2,3~;} each of them satisfying ϵ T 2 = − 1 . {\displaystyle ~\epsilon _{T}^{2}=-1~.}