Mathematical Principles of Theoretical Physics

246 CHAPTER 4. UNIFIED FIELD THEORY

• U( 1 )gauge transformation:

L→e

i

2 βL,

φ→e−

2 iβ

φ,

R→eiβR,

Wμa→Wμa−

2

g 2

∂μβ,

Bμ→Bμ+

2

g 2

∂μβ.

4.4.2 Gravitational field equations derived by PID.

(4.6.58)

δLWS

δWμa

= 0 ,

δLWS

δBμ

= 0 ,

δLWS

δ φ

= 0 ,

δLWS

δL

= 0 ,

δL

δR

= 0.

We remark here that the Higgs field in this setting is includedin the Lagrangian action,
drastically different from the mechanism based on PID developed earlier.
In addition, the particleφ+represents a massless boson with a positive electric charge.
However, in reality no such particles exist. Hence we have totake the Higgs scalar field as

(4.6.59) φ=

(

0

φ

)

.

In fact, in the WS theory, the Higgs fieldφis essentially taken as the form (4.6.59). Under
the condition (4.6.59), the variational equations (4.6.58) of the SW action (4.6.55)-(4.6.57)
are expressed as follows:

Gauge field equations (massless):

(4.6.60)

∂νWν μ^1 −g 1 gα α(Wα μ^2 Wα^3 −Wα μ^3 Wα^2 )+

g 1

2

Jμ^1 −

g^21

2

φ^2 Wμ^1 = 0 ,

∂νWν μ^2 −g 1 gα α(Wα μ^3 Wα^1 −Wα μ^1 Wα^3 )+

g 1

2

Jμ^2 −

g^21

2

φ^2 Wμ^2 = 0 ,

∂νWν μ^3 −g 1 gα α(Wα μ^1 Wα^2 −Wα μ^2 Wα^1 )+

g 1

2

Jμ^3 −

g 1

2

φ^2 (g 1 Wμ^3 −g 2 Bμ) = 0 ,

∂νBν μ−

g 2

2

JμL−g 2 JRμ−

g 2

2

φ^2 (g 2 Bμ−g 1 Wμ^3 ) = 0.

Higgs field equations:

∂μ∂μφ−

1

4

(4.6.61) φ(g 12 WμaWμa+g^22 BμBμ− 2 g 1 g 2 Wμ^3 Bμ)

−λ φ(φ^2 −a^2 )+Gl(lLR+RlL) = 0.

Dirac equations:

(4.6.62)

iγμ(∂μ+ig 2 Bμ)R−GlφlL= 0 ,

iγμ(∂μ+i

g 2

2

Bμ−i

g 1

2

Wμaσa)

(

νL

lL

)

−GlR

(

0

φ

)

= 0.