Mathematical Principles of Theoretical Physics

204 CHAPTER 4. UNIFIED FIELD THEORY

under theU( 1 )×SU( 2 )×SU( 3 )gauge transformation as follows

(4.3.17)

(

ψ ̃e,A ̃μ

)

=

(

eiθψe,Aμ−

1

e

∂μθ

)

,

(

ψ ̃w,W ̃μaσa

)

=

(

Uψw,WμaUσaU−^1 +

i

gw

∂μUU−^1

)

, U=eiθ

aσa

,

(

ψ ̃s,S ̃kτk

)

=

(

eiφ

kτk

ψs,SkμΩτkΩ−^1 +

i

gs

∂μΩΩ−^1

)

, Ω=eiφ

kτk

,

m ̃l=eiθ

aσa

mle−iθ

aσa

,

m ̃q=eiφ

kτk

mqe−iφ

kτk

.

However, the equations (4.3.9)-(4.3.15) are not invariant under the transformation (4.3.17)
due to the termsDgμφνg,Deνφe,Dwνφaw,Dsνφkson the right-hand sides of (4.3.9)-(4.3.12) con-
taining the gauge fieldsAμ,WμaandSkμ.
Hence, the unified field model based on PID and PRI satisfies thespontaneous gauge-
symmetry breaking as stated in Principle4.4and PRI.

4.3.2 Coupling parameters and physical dimensions

There are a number of to-be-determined coupling parametersin the general form of the uni-
fied field equations (4.3.9)-(4.3.15), and theSU( 2 )andSU( 3 )generatorsσaandτkare taken
arbitrarily. With PRI we are able to substantially reduce the number of these to-be-determined
parameters in the unified model to twoSU( 2 )andSU( 3 )tensors

{αaw}= (α 1 w,α 2 w,α 3 w), {αks}= (α 1 s,···,α 8 s),

containing 11 parameters, representing the portions distributed to the gauge potentials by the
weak and strong charges.
Also, if we takeσa( 1 ≤a≤ 3 )as the Pauli matrices (3.5.36) andτk=λk( 1 ≤k≤ 8 )as
the Gell-Mann matrices (3.5.38), then the two metricsGabwandGklsare Euclidian:

Gabw=δab, Gkls=δkl.

Hence we usually take the Pauli matricesσaand the Gell-Mann matricesλkas theSU( 2 )and
SU( 3 )generators.
For convenience, we first introduce dimensions of related physical quantities. LetErep-
resent energy,Lbe the length andtbe the time. Then we have

(Aμ,Wμa,Skμ):

√

E/L, (e,gw,gs):

√

EL,

(Jμ,Jμa,Qμk): 1/L^3 , (φe,φaw,φks):

√

E

√

LL

,

(h ̄,c):(Et,L/t), mc/h ̄: 1/L.

In addition, for gravitational fields we have

(4.3.18)

gμ ν: dimensionless, R: 1/L^2 , Tμ ν:E/L^3 ,

φμg: 1/L, G:L^5 /Et^4.