Properties And Applications Of Four Dimensional Special Conformal Transformation

In general,conformal field theory is dominated by two-dimensional cases,because two-dimensional conformal groups have Virasoro recurrence relationships that do not have other dimensions.Four-dimensional quantum field theory has been widely used in particle physics or high-energy physics.Quantum field theory lays the foundation for calculating the decay and scattering between elementary particles,the interaction between particles,and the coupling constant of running.This paper focuses on the special conformal transformations in four-dimensional conformal groups.Firstly,since the action of the photon field remains unchanged after conformal transformation,special conformal transformations are mainly applied to photon fields.We derive two different metric transformations from two definitions of line elements by conformal groups.Next,I will study the specific differences between the two definitions.Thus,two definitions corresponding to the active viewpoint and the passive viewpoint in physics are obtained.Due to the nonlinear nature of special conformal transformations,there are differences between active and passive viewpoints.Then,comparing the results of these two viewpoints with the conformal invariance obtained by the variational method,it is found that the results obtained by the variational method include the effect of one of the viewpoints on the photon field,which is understood in the article as an active viewpoint of transformation.Because the variational method represents the transformation of the real coordinate after transformation,which is similar to the image from the active viewpoint,while the redundant terms exactly represent the changes in the real coordinate.From the above analysis,if a nonlinear transformation such as a special conformal transformation is applied to a photon field,the active and passive viewpoints of the transformation will be different.Finally,consider the special conformal transformation of local infinitesimal,and combine the above calculation results.It is found that both active and passive viewpoints can represent a local infinitesimal special conformal transformation as a combination of nontrivial dilation and rotation.Moreover,the effects of active and passive viewpoints on the photon field correspond to the rotation matrix of the Lorentz group.Of course,it should be emphasized that the above rotation and dilation are both coordinate related,and this combination is not a linear combination,so special conformal transformations should exist in conformal groups.