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+igwWμaσa)ψw,

Dμψs= (∂μ+igsSkμτk)ψs.

The geometry of unified fields consists of 1) the field functions and 2) the Lagrangian
action (4.1.17), which are invariant under the following seven transformations:

1) the general linear transformationQp= (aμν):TpM→TpMwithQ−p^1 = (bμν)T, for

anyp∈M:

(4.1.21)

(g ̃μ ν) =Qp(gμ ν)QTp,

A ̃μ=aνμAν,

W ̃μa=aνμWνa for 1≤a≤ 3 ,

̃Skμ=aνμSkν for 1≤k≤ 8 ,

̃γμ=bνμγν, ̃∂μ=aνμ∂ν,

with no change on other fields, whereγμare the Dirac matrices;

2) the Lorentz transformation onTpM:

(4.1.22)

L= (lνμ):TpM→TpM, Lis independent ofp∈M,

(g ̃μ ν) =L(gμ ν)LT, A ̃μ=lμνAν,

W ̃μa=lνμWνa for 1≤a≤ 3 ,

S ̃kμ=lνμSνk for 1≤k≤ 8 ,

Ψ ̃=RLΨ, RLis the spinor transformation matrix,

∂ ̃μ=lνμ∂ν,

with no change on other fields;

3) theU( 1 )gauge transformation onM⊗pC^4 :

(4.1.23)

Ω=eiθ:C^4 →C^4 ∈U( 1 ),

(

ψ ̃e,A ̃μ

)

=

(

eiθψe,Aμ−

1

e

∂μθ

)

,

4) SU( 2 )gauge transformation onM⊗p(C^4 )^2 :

(4.1.24)

Ω=eiθ

aσa

:(C^4 )^2 →(C^4 )^2 ∈SU( 2 ),

(

ψ ̃w,W ̃μaσa,m ̃w

)

=

(

Ωψw,WμaΩσaΩ−^1 +

i

gw

∂μΩΩ−^1 ,ΩmwΩ−^1

)

.