Mathematical Principles of Theoretical Physics

14 CHAPTER 1. GENERAL INTRODUCTION

• The field describing electromagnetic interaction is theU( 1 )gauge field{Aμ},repre-
  senting the electromagnetic potential, and the Lagrangianaction density is

(1.5.1) LEM=−

1

4

Aμ νAμ ν+ψe(iγμDμ−me)ψe,

in which the first term stands for the scalar curvature of the vector bundleM⊗pC^4.

The covariant derivative and the field strength are given by

Dμψe= (∂μ+ieAμ)ψe, Aμ ν=∂μAν−∂νAμ.

• For the weak interaction, theSU( 2 )gauge fields{Wμa|a= 1 , 2 , 3 }are the interacting
  fields, and theSU( 2 )Lagrangian action densityLWfor the weak interaction is the
  standard Yang-Mills action density as given by (1.3.2).


• TheSU( 3 )gauge action densityLSfor the strong interaction is also in the standard
  Yang-Mills form given by (1.3.2), and the strong interaction fields are theSU( 3 )gauge
  fields{Skμ| 1 ≤k≤ 8 },representing the 8 gluon fields.

It is clear that the action coupling the four fundamental interactions is the natural combi-
nation of the Einstein-Hilbert functional, the standardU( 1 ),SU( 2 ),SU( 3 )gauge actions for
the electromagnetic, weak and strong interactions:

(1.5.2) L

(

{gμ ν},Aμ,{Wμa},{Skμ}

)

=

∫

M

[LEH+LEM+LW+LS]

√

−gdx,

which obeys all the symmetric principles, including principle of general relativity, the Lorentz
invariance, theU( 1 )×SU( 2 )×SU( 3 )gauge invariance and PRI.

4 Unified Field Theory

With PID, the PRI covariant unified field equations are then given by:

Rμ ν−

1

2

gμ νR+

8 πG

c^4

Tμ ν=

[

∇μ+α^0 Aμ+αb^1 Wμb+αk^2 Skμ

]

(1.5.3) φνG,

∂μ(∂μAν−∂νAμ)−eJν=

[

∇ν+β^0 Aν+βb^1 Wνb+βk^2 Skν

]

(1.5.4) φe,

Gabw

[

∂μWμ νb −gwλcdbgα βWα νcWβd

]

(1.5.5) −gwJνa

=

[

∇ν+γ^0 Aν+γb^1 Wνb+γk^2 Skν−

1

4

m^2 wxν

]

φaw,

Gk js

[

∂μSμ νj −gsΛcdjgα βScα νSdβ

]

(1.5.6) −gsQνk

=

[

∇ν+δ^0 Aν+δb^1 Wνb+δk^2 Skν−

1

4

m^2 sxν

]

φks,

(1.5.7) (iγμDμ−m)Ψ= 0 ,

whereΨ= (ψe,ψw,ψs)Tstands for the wave functions for all fermions, participating respec-
tively the electromagnetic, the weak and the strong interactions, and the current densities are
defined by

(1.5.8) Jν=ψeγνψe, Jνa=ψwγνσaψw, Qνk=ψsγντkψs.