Mathematical Principles of Theoretical Physics

186 CHAPTER 4. UNIFIED FIELD THEORY

are the Riemannian metric{gμ ν}and the gauge fields{Gaμ}. In addition, the symmet-

ric principles also determine the Lagrangian actions ofgμ νandGaμ;

2) PID determines the field equations governinggμ νandGaμ; and

3) The solutionsgμ νandGaμof the field equations determine the geometries ofMand

the vector bundles.

The geometry of the unified fields refers to the geometries ofM, determined by the
following known physical symmetry principles:

(4.1.16)

principle of general relativity,

principle of Lorentz invariance,

U( 1 )×SU( 2 )×SU( 3 )gauge invariance,

principle of representation invariance (PRI).

We shall introduce the two principles PID and PRI in Section4.1.5.
The fields determined by the symmetries in (4.1.16) are given by

• general relativity:gμ ν:M→T 20 M, the Riemannian metric,


• Lorentz invariance:ψ:M→M⊗p(C^4 )N, the Dirac spinor fields,


• U( 1 )gauge invariance:Aμ:M→T∗M, theU( 1 )gauge field,


• SU( 2 )gauge invariance:Wμa:M→(T∗M)^3 , theSU( 2 )gauge fields,


• SU( 3 )gauge invariance:Skμ:M→(T∗M)^8 , theSU( 3 )gauge fields.

The Lagrange action for the geometry of the unified fields is given by

(4.1.17) L=

∫

M

[LEH+LEM+LW+LS+LD]

√

−gdx

whereLEH,LEM,LWandLSare the Lagrangian actions for the four interactions defined
by (4.1.5)–(4.1.8), and the action for the Dirac spinor fields is given by

(4.1.18) LD=Ψ(iγμDμ−m)Ψ.

Here

Ψ= (ψe,ψw,ψs),

m= (me,mw,ms),

and

(4.1.19)

ψe:M→M⊗pC^4 1-component Dirac spinor,

ψw:M→M⊗p(C^4 )^2 2-component Dirac spinors,

ψs:M→M⊗p(C^4 )^3 3-component Dirac spinors.