S= J d4x, (2-101)

where the integration is performed over the region between any two spacelike

hypersurfaces. With variations taken in the dynamical variables za'and Aa which

vanish on these hypersurfaces, the action suffers the variation


 6S = (-moga'b' + CeFab ) 6za'dr

 + i [- (47)-1 g12Fab;b + ] j Aad4x, (2-102)

provided that r is taken to be the proper time of the particle (and will henceforth

be assumed) such that


 ga'b' ab' = -1, (2-103)

 ga'b hab' = 0, (2-104)

 ga'b/ a'iZb' g= -a'b' 2. (2-105)

Application of this action principle yields the dynamical equations

 mnoxa' eFib' b', (2-106)

 gl/2 Fabb 47a. (2-107)

By the fact that


 1
 Fabba (Rbac Fb + RbabcF a) 0, (2-108)

one can show via Eq. (2-107) that the current density is conserved.

 Conservation of the stress-energy tensor.

 The stress-energy tensor of the system is given by


Tab tickle + TIabd,
 Particle field'


(2-109)