Mathematical Principles of Theoretical Physics

4.3. UNIFIED FIELD MODEL BASED ON PID AND PRI 201

In Section4.1.3, we showed that the action functional obeys all the symmetric principles,
including principle of general relativity, the Lorentz invariance, theU( 1 )×SU( 2 )×SU( 3 )
gauge invariance and PRI, is the natural combination of the Einstein-Hilbert functional, the
U( 1 ),SU( 2 ),SU( 3 )Yang-Mills actions for the electromagnetic, weak and strong interac-
tions:

(4.3.1) L=

∫

M

[LEH+LEM+LW+LS]

√

−gdx.

Here

(4.3.2)

LEH=R+

8 πG

c^4

S,

LEM=−

1

4

Aμ νAμ ν+ψe(iγμDμ−m)ψe,

LW=−

1

4

GabwWμ νaWbμ ν+ψw(iγμDμ−ml)ψw,

LS=−

1

4

GklsSkμ νSμ νl+ψs(iγμDμ−mq)ψs,

whereRis the scalar curvature of the space-time Riemannian manifold(M,gμ ν)with Minkowski

type metric,Sis the energy-momentum density,GabwandGklsare theSU( 2 )andSU( 3 )metrics

as defined by (3.5.28),ψe,ψwandψsare the Dirac spinors for fermions participating in the

electromagnetic, weak, strong interactions, and

(4.3.3)

Aμ ν=∂μAν−∂νAμ,

Wμ νa =∂μWνa−∂νWμa+gwλbcaWμbWνc,

Skμ ν=∂μSkν−∂νSkμ+gsΛklrSlμSrν.

HereAμis the electromagnetic potential,Wμa( 1 ≤a≤ 3 )are theSU( 2 )gauge fields for the

weak interaction,Skμ( 1 ≤k≤ 8 )are theSU( 3 )gauge fields for the strong interaction,gwand

gsare the weak and strong charges, and

(4.3.4)

Dμψe= (∂μ+ieAμ)ψe,

Dμψw= (∂μ+igwWμaσa)ψw,

Dμψs= (∂μ+igsSkμτk)ψs,

whereσa( 1 ≤a≤ 3 )andτk( 1 ≤k≤ 8 )are the generators ofSU( 2 )andSU( 3 ).

Remark 4.8.For a vector fieldXμand an antisymmetric tensor fieldFμ ν, we have

∇μXν−∇νXμ=∂μXν−∂νXμ,

∇μFμ ν=∂μFμ ν,

where∇μis the Levi-Civita covariant derivative. Hence, the tensorfields in (4.3.3) and the

action (4.3.1) obey both the Einstein general relativity and the Lorentz invariance simultane-

ously under the transformations (4.1.21)-(4.1.22).