202 CHAPTER 4. UNIFIED FIELD THEORY
Remark 4.9.In the standard model, the wave functionsψwandψsinLWandLSare as
follows
(4.3.5)
ψw=(
l 1
l 2)
Lthe left-hand lepton pairs,ψs=
q 1
q 2
q 3
qi the quark with ith color.The reason whyψwandψsare taken in the form (4.3.5) is that the standard model, in partic-
ular the GWS electroweak theory, is oriented toward to computing the transition probability
for the decay and scattering. In the PID field theory, it is unnecessary to takeψwandψs
as in (4.3.5), because this model is oriented toward to interaction potentials and the basic
mechanism.
Remark 4.10.According to the standard model, the field particles corresponding to elec-
tromagnetic, weak, and strong interactions are described byU( 1 ),SU( 2 ),SU( 3 )gauge fields.
Hence we take theU( 1 )×SU( 2 )×SU( 3 )Yang-Mills action together withLEHas the action.
However, if only consider the field theory for anN-particle system withN 1 electric,N 2 weak,
N 3 strong charges, then the action sectors in (4.3.1) should be taken as
LEH=R,
LEM=SU(N 1 )Yang-Mills action,
LW=SU(N 2 )Yang-Mills action, and
LS=SU(N 3 )Yang-Mills action.In Section6.5, we shall discuss the unified field theory for multi-particlesystems.
We are now in position to establish unified field equations obeying PRI and PSB. By PID,
the unified field model (4.1.33)-(4.1.34) are derived as the variational equation of the action
(4.1.17) under the divA-constraint
〈δL,X〉=0 for anyX with divAX= 0.Here it is required that the gradient operator∇Acorresponding to divAare PRI covariant.
The gradient operators in different sectors are given as follows:
(4.3.6)
Dgμ=∇μ+α^0 Aμ+α^1 bWμb+α^2 kSkμ,
Dμe=∇μ+β^0 Aμ+βb^1 Wμb+βk^2 Skμ,Dwμ=∇μ+γ^0 Aμ+Γ^1 bWμb+γk^2 Skμ−1
4
m^2 wxμ,Dμs=∇μ+δ^0 Aμ+δb^1 Wμb+δk^2 Skμ−1
4
m^2 sxμ,where
(4.3.7)
mw,ms,α^0 ,β^0 ,γ^0 ,δ^0 are scalar parameters,
αa^1 ,βa^1 ,γa^1 ,δa^1 are first-orderSU( 2 )tensors,
αk^2 ,βk^2 ,γk^2 ,δk^2 are first-orderSU( 3 )tensors.