Mathematical Principles of Theoretical Physics

4.6. WEAK INTERACTION THEORY 243

which are derived by the following transformation





k^2 W

k^2 W

k^2 Z



=√gw

2 ̄hc





1 i 0

1 −i 0

0 0

√

2









γ 1

γ 2

γ 3



,

withγ 2 =0 in the Pauli matrix representation.
The equation (4.6.43)-(4.6.46) are the model to govern the behaviors of the weak interac-
tion field particles (4.6.40) and (4.6.41). Two observations are now in order.

First, we note that these equations are nonlinear, and consequently, no free weak interac-
tion field particles appear.

Second, there are two important solutions of (4.6.43)-(4.6.44), dictating two different
weak interaction procedures.
The first solution sets

(4.6.49) Wμ±= 0 , φ^0 = 1.

ThenZsatisfies the equation

(4.6.50) Zμ+k^2 zZμ=−gwJ^0 μ+

1

4

(m

Hc

̄h

) 2

xμ.

The second solution takes

(4.6.51) Zμ= 0 , φ±= 1.

ThenWμ±satisfy the equations

(4.6.52) Wμ±+kw^2 Wμ±=−gwJ±μ+

1

4

(m

Hc

h ̄

) 2

xμ.

We are now ready to obtain the following physical conclusions for the weak interaction
field particles.

1.Duality of field particles.The field equations (4.6.43)-(4.6.46) provide a natural duality
between the field particles:
W±↔H±, Z↔H^0.

2.Lack of freedom for field particles.Due to the nonlinearity of (4.6.43)-(4.6.46), the
weak interaction field particlesW±,Z,H±,H^0 are not free bosons.

3.PID mechanism of spontaneous symmetry breaking.In the field equations (4.6.50) and
(4.6.52), it is natural that the massesmWandmZofW±andZbosons appear at the ground
states (4.6.49) and (4.6.51), with

mW=

h ̄

c

kW, mZ=

h ̄

c

kZ,

andkWandkZare as in (4.6.48).