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)
(mw,ms): 1/L, (α^0 ,β^0 ,γ^0 ,δ^0 ): 1/√
EL,
(αa^1 ,βa^1 ,γ^1 a,δa^1 ): 1/√
EL, (αk^2 ,βk^2 ,γk^2 ,δk^2 ): 1/√
EL.
Thus the parameters in (4.3.7) can be rewritten as(4.3.20)
(mw,ms) =(m
Hc
̄h,
mπc
h ̄)
,
(α^0 ,β^0 ,γ^0 ,δ^0 ) =e
̄hc(αe,βe,γe,δe),(αa^1 ,βa^1 ,γa^1 ,δa^1 ) =gw
hc ̄(αaw,βaw,γaw,δaw),(αk^2 ,βk^2 ,γk^2 ,δk^2 ) =gs
̄hc(αks,βks,γks,δks),wheremHandmπ represent the masses ofφwandφs, and all the parameters(α,β,γ,δ)
on the right hand side of (4.3.20) with different super and sub indices are dimensionless
constants.
4.3.3 Standard form of unified field equations
With (4.3.20) at our disposal, the unified field equations (4.3.9)-(4.3.15) can be simplified in
the form
Rμ ν−1
2
gμ νR=−8 πG
c^4Tμ ν+[
∇μ+eαe
̄hcAμ+gwαaw
hc ̄Wμa+gsαks
hc ̄Skμ]
(4.3.21) φνg,
∂νAν μ−eJν=[
∂μ+e
hc ̄βeAμ+gw
hc ̄βawWμa+gs
hc ̄βksSkμ]
(4.3.22) φe,
∂νWν μa −gw
̄hc(4.3.23) εbcagα βWα μbWβc−gwJμa
=
[
∂μ+e
̄hcγeAμ+gw
hc ̄γbwWμb+gs
hc ̄γksSkμ−1
4
(mHc
h ̄) 2
xμ]
φwa,∂νSkν μ−gs
̄hc(4.3.24) fijkgα βSα μi Sβj−gsQkμ
=
[
∂μ+e
hc ̄δeAμ+gw
̄hcδbwWμb+gs
hc ̄δlsSlμ−1
4
(mπc
h ̄) 2
xμ]
φsk,(4.3.25) (iγμDμ−m)Ψ= 0 ,
whereΨ= (ψe,ψw,ψs), and
(4.3.26)
Aμ ν=∂μAν−∂νAμ,
Wμ νa =∂μWνa−∂νWμa+
gw
hc ̄εabcWμbWνc,Skμ ν=∂μSkν−∂νSkμ+
gs
hc ̄fijkSiμSνj.Equations (4.3.21)-(4.3.25) need to be supplemented with coupled gauge equations to
compensate the new dual fields(φe,φwa,φsk). In different physical situations, the coupled
gauge equations may be different.