Mathematical Principles of Theoretical Physics

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 187 The derivative operatorsDμare given by (4.1.20) Dμψe= (∂μ+ieAμ)ψe, Dμψw= (∂μ+igwWμaσ ...

188 CHAPTER 4. UNIFIED FIELD THEORY 5) SU( 3 )gauge transformation onM⊗p(C^4 )^3 : (4.1.25) Ω=eiθ kτk :(C^4 )^3 →(C^4 )^3 ∈SU( 3 ...

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 189 (2) the gauge invariance of the variational equations, (4.1.29), means that the part ...

190 CHAPTER 4. UNIFIED FIELD THEORY u. A tensoru 0 is called an extremum point ofFwith the divA-free constraint, ifu 0 satisfies ...

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 191 We are now in position to introduce the principle of representation invariance (PRI) ...

192 CHAPTER 4. UNIFIED FIELD THEORY with different symmetry groups are combined linearly into terms in the corresponding gauge f ...

4.2 Physical Supports to PID It implies that the usual energy-momentum tensor satisfies (4.2.3) ∇μTμ ν= 0. However, due to the p ...

194 CHAPTER 4. UNIFIED FIELD THEORY the 10 unknown functions become ( −1 0 0 gij ) , gij=gji for 1≤i,j≤ 3. This observation impl ...

4.2. PHYSICAL SUPPORTS TO PID 195 Consider the influence of cosmic microwave background (CMB)radiation, the energy- momentum ten ...

196 CHAPTER 4. UNIFIED FIELD THEORY 4.2.3 Higgs mechanism and mass generation Principle4.4of gauge symmetry breaking is also a m ...

4.2. PHYSICAL SUPPORTS TO PID 197 which are the variational equations of the Yang-Mills action (4.2.22) LYM= ∫ − 1 4 Fμ νaFμ νa+ ...

198 CHAPTER 4. UNIFIED FIELD THEORY whereρ 6 =0 is a constant. Obviously, the following action (4.2.28) L= ∫ (LYM+LH)dx, and its ...

4.2. PHYSICAL SUPPORTS TO PID 199 whereφis a scalar field. The term−^14 (mc ̄h ) 2 xμis the mass potential ofφ, and is also rega ...

200 CHAPTER 4. UNIFIED FIELD THEORY Let J= c 4 π curl^2 A, Js= e^2 s msc |ψ|^2 A−i he ̄ s ms (ψ∗∇ψ−ψ∇ψ∗). Physically,Jis the tot ...

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 ...

202 CHAPTER 4. UNIFIED FIELD THEORY Remark 4.9.In the standard model, the wave functionsψwandψsinLWandLSare as follows (4.3.5) ψ ...

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 ...

204 CHAPTER 4. UNIFIED FIELD THEORY under theU( 1 )×SU( 2 )×SU( 3 )gauge transformation as follows (4.3.17) ( ψ ̃e,A ̃μ ) = ( ei ...

4.3. UNIFIED FIELD MODEL BASED ON PID AND PRI 205 The dimensions of the parameters in (4.3.9)-(4.3.15) are as follows (4.3.19) ( ...

206 CHAPTER 4. UNIFIED FIELD THEORY From the field theoretical point of view instead of the field particle point of view, the co ...