Mathematical Principles of Theoretical Physics

190 CHAPTER 4. UNIFIED FIELD THEORY

u. A tensoru 0 is called an extremum point ofFwith the divA-free constraint, ifu 0 satisfies
the equation

(4.1.31)

d

dλ

∣

∣

∣

λ= 0

F(u 0 +λX) =

∫

M

δF(u 0 )·X

√

−gdx= 0 ∀Xwith divAX= 0.

Principle 4.5(Principle of Interaction Dynamics).

1) For all physical interactions there are Lagrangian actions

(4.1.32) L(g,A,ψ) =

∫

M

L(gμ ν,A,ψ)

√

−gdx,

where g={gμ ν}is the Riemannian metric representing the gravitational potential, A

is a set of vector fields representing the gauge potentials, andψare the wave functions

of particles;

2) The action (4.1.32) satisfy the invariance of general relativity, Lorentz invariance,

gauge invariance and the gauge representation invariance;

3) The states(g,A,ψ)are the extremum points of (4.1.32) with the divA-free constraint

(4.1.31).

Based on PID and Theorems3.26and3.27, the field equations with respect to the action
(4.1.32) are given in the form

δ

δgμ ν

(4.1.33) L(g,A,ψ) = (∇μ+αbAbμ)Φν,

δ
δAaμ
(4.1.34) L(g,A,ψ) = (∇μ+βbaAbμ)φa,

δ

δ ψ

(4.1.35) L(g,A,ψ) = 0

whereAaμ= (Aa 0 ,Aa 1 ,Aa 2 ,Aa 3 )are the gauge vector fields for the electromagnetic, the weakand
strong interactions,Φν= (Φ 0 ,Φ 1 ,Φ 2 ,Φ 3 )in (4.1.33) is a vector field induced by gravita-
tional interaction,φais the scalar fields generated from the gauge fieldsAaμ, andαb,βbaare
coupling parameters.
Consider the action (4.1.32) as the natural combination of the actions for all four interac-
tions, as given in (4.1.17)-(4.1.20):

L=LEH+LEM+LW+LS+LD.

Then (4.1.33)-(4.1.35) provide the unified field equations coupling all interactions. Moreover,
we see from (4.1.33)-(4.1.35) that there are too many coupling parameters which need to
be determined. Fortunately, this problem can be satisfactorily resolved, leading also to the
discovery of PRI (Ma and Wang,2014h). Meanwhile, we remark that it is the gauge fields
Aaμappearing on the right-hand sides of (4.1.33) and (4.1.34) that break the gauge symmetry,
leading to the mass generation of the vector bosons for the weak interaction sector.