Understanding The Electro-Magnetic 4-Potential In The Orthonormal Frame Bundle

The separation of the spin and orbital angular momenta of the electromagnetic field has been discussed frequently in recent years.The definition of spin and orbital angu-lar momenta depends on the independent electromagnetic 4-potential A,and thus does not have gauge invariance.If we replace A in their definition with the transverse part A~?that satisfies the Coulomb gauge,the angular momenta will satisfy gauge invari-ance.However,the replacement of A to A~?destroys the Lorentz covariance of spin and orbital angular momenta.We use covariant language to construct various Noether flow with the vector field with a mass in 4-dimensional space-time from the perspective of quantum field theory and analyze their Lorentz transformation forms.We obtain the Noether flow of the vector field with a massless and its transformation by modeling the vector field with a mass.There is an essential difference between the Lorentz transfor-mation with mass and massless vector fields due to the difference between the standard4-vector and the small group.We believe that this difference is an important reason why the electromagnetic 4-potential of the fixed gauge cannot have Lorentz covariance.We redescribe these difficulties of the angular momenta from the perspective of quantum field theory and tried to fix the gauge as the Lorentz gauge,but found that this method is invalid.Based on the information obtained,the electromagnetic 4-potential should be fixed to satisfy the Coulomb gauge.The electromagnetic 4-potential should depend on the local reference frame instead of the global reference frame.The transformation of the local reference frame is the intrinsic degree of freedom of the electromagnetic field.Therefore,ignoring the difference of Lorentz transformation between the local refer-ence frame and the coordinate frame,the perspective is limited to space-time,which may result in a fixed gauge of the electromagnetic 4-potential does not have Lorentz covariance.Therefore,we must shift the perspective to the tetrad bundle and improve the electromagnetic 4-potential satisfying the Coulomb gauge into the electromagnetic vector in the orthogonal frame bundle,so that the electromagnetic vector has the gauge invariance.We prove that the definition of this electromagnetic vector does not depend on the reference frame and thus has Lorentz covariance.Accordingly,the spin and or-bital angular momentums of an electromagnetic field constructed with this electromag-netic vector can be simultaneously gauge invariant and Lorentz covariant.We analyze the variation of the Maxwell equation caused by the new electromagnetic vector.Using the pullback mapping e~*,which is induced by the local reference frame,we project the electromagnetic 4-potential and the momentum into space-time.It is found to be com-patible with the classical definition,so our theory can match the existing experiments and observations.In the last chapter,we summarize and forecast the application and extension of the theory.