Mathematical Principles of Theoretical Physics

4.3. UNIFIED FIELD MODEL BASED ON PID AND PRI 203

Thus, the PID equations (4.1.33)-(4.1.34) can be expressed as

(4.3.8)

δL

δgμ ν

=Dgμφνg,

δL

δAμ

=Deμφe,

δL

δWμa

=Dwμφaw,

δL

δSkμ

=Dsμφks,

whereφνgis a vector field, andφe,φw,φsare scalar fields.

With the PID equations (4.3.8), the PRI covariant unified field equations are then given
as follows:^1

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

(4.3.9) Tμ ν+Dgμφνg,

(4.3.10) ∂μ(∂μAν−∂νAμ)−eJν=Deνφe,

Gabw

[

∂μWμ νb−gwλcdbgα βWα νcWβd

]

(4.3.11) −gwJνa=Dνwφaw,

Gk js

[

∂μSμ νj −gsΛcdjgα βScα νSβd

]

(4.3.12) −gsQνk=Dνsφks,

(4.3.13) (iγμDμ−m)ψe= 0 ,

(4.3.14) (iγμDμ−ml)ψw= 0 ,

(4.3.15) (iγμDμ−mq)ψs= 0 ,

whereDgμ,Deν,Dwν,Dsνare given by (4.3.6), and

(4.3.16)

Jν=ψeγνψe,

Jνa=ψwγνσaψw,

Qνk=ψsγντkψs,

Tμ ν=

δS

δgμ ν

+

c^4

16 πG

gα β(GabwWα μaWβ νb +GklsSα μk Sβ νl +Aα μAβ ν)

−

c^4

16 πG

gμ ν(LEM+LW+LS).

Remark 4.11.It is clear that the action (4.3.1)-(4.3.4) for the unified field model is invariant

(^1) We ignore the Klein-Gordon fields.