Mathematical Principles of Theoretical Physics

206 CHAPTER 4. UNIFIED FIELD THEORY

From the field theoretical point of view instead of the field particle point of view, the
coefficients in (4.3.21)-(4.3.24) should be

(4.3.27)

(α 1 w,α 2 w,α 3 w) =αw(ω 1 ,ω 2 ,ω 3 ),

(β 1 w,β 2 w,β 3 w) =βw(ω 1 ,ω 2 ,ω 3 ),

(γ 1 w,γ 2 w,γ 3 w) =γw(ω 1 ,ω 2 ,ω 3 ),

(δ 1 w,δ 2 w,δ 3 w) =δw(ω 1 ,ω 2 ,ω 3 ),

and

(4.3.28)

(αs 1 ,···,αs 8 ) =αs(ρ 1 ,···,ρ 8 ),

(β 1 s,···,β 8 s) =βs(ρ 1 ,···,ρ 8 ),

(γ 1 s,···,γ 8 s) =γs(ρ 1 ,···,ρ 8 ),

(δ 1 s,···,δ 8 s) =γs(ρ 1 ,···,ρ 8 ),

with the unit modules:

|ω|=

√

ω 12 +ω 22 +ω 32 = 1 ,

|ρ|=

√

ρ^21 +···+ρ^28 = 1 ,

using the Pauli matricesσaand the Gell-Mann matricesλkas the generators forSU( 2 )and
SU( 3 )respectively.
The twoSU( 2 )andSU( 3 )tensors in (4.3.27) and (4.3.28),

(4.3.29) ωa= (ω 1 ,ω 2 ,ω 3 ), ρk= (ρ 1 ,···,ρ 8 ),

are very important, by which we can obtainSU( 2 )andSU( 3 )representation invariant gauge
fields:

(4.3.30) Wμ=ωaWμa, Sμ=ρkSkμ.

which represent respectively the weak and the strong interaction potentials.
In view of (4.3.27)-(4.3.30), the unified field equations for the four fundamental forces
are written as

Rμ ν−

1

2

gμ νR+

8 πG

c^4

Tμ ν=

[

∇μ+

eαe

̄hc

Aμ+

gwαw

hc ̄

Wμ+

gsαs

hc ̄

Sμ

]

(4.3.31) φνg,

∂νAν μ−eJμ=

[

∂μ+

eβe

hc ̄

Aμ+

gwβw

hc ̄

Wμ+

gsβs

̄hc

Sμ

]

(4.3.32) φe,

∂νWν μa −

gw

hc ̄

(4.3.33) εbcagα βWα μbWβc−gwJμa

=

[

∂μ−

1

4

k^2 wxμ+

eγe

̄hc

Aμ+

gwγw

hc ̄

Wμ+

gsγs

hc ̄

Sμ

]

φwa,

∂νSkν μ−

gs

̄hc

(4.3.34) fijkgα βSα μi Sβj−gsQkμ

=

[

∂μ−

1

4

k^2 sxμ+

eδe

hc ̄

Aμ+

gwδw

̄hc

Wμ+

gsδs

hc ̄

Sμ

]

φsk,

(4.3.35) (iγμDμ−m)Ψ= 0.