Mathematical Principles of Theoretical Physics

234 CHAPTER 4. UNIFIED FIELD THEORY

4.6 Weak Interaction Theory

4.6.1 Dual equations of weak interaction potentials

According to the standard model, weak interaction is described by theSU( 2 )gauge theory.
In Section4.3.4, we have demonstrated that the weak interaction potential is given by the
following PRI representation invariant

(4.6.1) Wμ=ωaWμa= (W 0 ,W 1 ,W 2 ,W 3 ),

where{ωa| 1 ≤a≤ 3 }is theSU( 2 )tensor as in (4.3.29).
Also, the weak charge potential and weak force are as

(4.6.2)

Φw=W 0 the time component ofWμ,

Fw=−gw(ρ)∇Φw,

wheregw(ρ)is the weak charge of a particle with radiusρ.
In this subsection, we establish the field equations for the dual potentialΦwandφwof the
weak interaction from the field equations (4.4.48)-(4.4.50).
Taking inner products of (4.4.48) and (4.4.49) withωarespectively, we derive the field

4.6.1 Dual equations of weak interaction potentials.

∂νWν μ−

gw

hc ̄

κabgα βWα μaWβb−gwJμw= (∂μ−

1

4

k^2 xμ+

gw

hc ̄

(4.6.3) γWμ)φw,
[
1
c^2

∂^2

∂t^2

−∆

]

(4.6.4) φw+k^20 φw−gw∂μJwμ

=

gw

̄hc

∂μ

[

κabgα βWα μaWβb−γWμφw

]

−

1

4

k^20 xμ∂μφw,

whereκab=εabcωc,Wμis as in (4.6.1),φw=ωaφwa, and

(4.6.5) Jwμ=ωaψγμσaψ, Wμ ν=∂μWν−∂νWμ+

gw

hc ̄

κabWμaWνb.

Experiments showed that theSU( 2 )gauge fieldsWμafor weak interaction field particles
possess masses. In addition, the dual Higgs fieldsφwaofWμaalso have masses. In (4.6.3),
there is no massive term ofWμ. However, we see that on the right-hand side of (4.6.3) there
is a term

(4.6.6)

gw

̄hc

γ φwWμ,

which is spontaneously generated by PID, and breaks the gauge-symmetry. Namely, (4.6.6)
will vary under the followingSU( 2 )gauge transformation

Wμa→Wμa−εbcaθbWμc−

1

gw

∂μθa

We shall show that it is the spontaneous symmetry breaking term (4.6.6) that generates mass
from the ground state ofφw.