Mathematical Principles of Theoretical Physics

216 CHAPTER 4. UNIFIED FIELD THEORY

4.4.4 Strong interaction field equations

The decoupled strong interaction field model from (4.3.21)-(4.3.25) is given by

∂νSkν μ−

gs

hc ̄

fijkgα βSiα μSβj−gsQkμ=

[

∂μ+

gs

hc ̄

δlsSlμ−

1

4

(m

πc

̄h

) 2

xμ

]

(4.4.34) φsk,

iγμ

[

∂μ+i

gs

hc ̄

Sbμτb

]

ψ−

mc

h ̄

(4.4.35) ψ= 0 ,

for 1≤k≤8, whereτk=τkare the Gell-Mann matrices as in (3.5.38), and

(4.4.36)

Skμ ν=∂μSkν−∂νSkμ+

gs

̄hc

SiμSνj,

Qkμ=ψ γμτkψ.

Taking divergence on both sides of (4.4.34) and by

∂μ∂νSkμ ν= 0 for 1≤k≤ 8 ,

we deduce the following dual field equations for the strong interaction:

∂μ∂μφsk+∂μ

[(

gs

̄hc

δlkSlμ−

1

4

m^2 πc^2

̄h^2

xμ

)

φsk

]

=−gs∂μQkμ−

gs

hc ̄

(4.4.37) fijkgα β∂μ(Sα μi Sβj).

The equations (4.4.34)-(4.4.35) also need 8 additional gauge equations to compensate the
induced dual fieldsφsk:

(4.4.38) Fsk(Sμ,φs,ψ) = 0 , 1 ≤k≤ 8.

We have the following duality for the strong interaction.

1.Gluons and dual scalar gluons. Based on quantum chromodynamics (QCD), the
field particles for the strong interaction are the eight massless gluons with spinJ=1, which
are described by theSU( 3 )gauge fieldsSkμ( 1 ≤k≤ 8 ). By the duality (4.4.1), for the strong
interactions we have the field particle correspondence

Skμ ↔φsk for 1≤k≤ 8.

It implies that corresponding to the 8 gluonsSkμ( 1 ≤k≤ 8 )there should be 8 dual gluons

represented byφsk, called the scalar gluons due toφskbeing scalar fields. Namely we have the
following gluon correspondence

gluonsgk ↔ scalar gluonsgk 0 for 1≤k≤ 8.

Gluons and scalar gluons are described by equations (4.4.34) and (4.4.37) respectively,
which are nonlinear. In fact,gkandgk 0 are confined in hadrons.