Mathematical Principles of Theoretical Physics

192 CHAPTER 4. UNIFIED FIELD THEORY

with different symmetry groups are combined linearly into terms in the corresponding gauge
field equations. For example, in the Weinberg-Salam electroweak gauge equations with
U( 1 )×SU( 2 )symmetry breaking, there are such linearly combinations as

(4.1.38)

Zμ=cosθWWμ^3 +sinθWBμ,

Aμ=−sinθWWμ^3 +cosθWBμ,

Wμ±=

1

√

2

(Wμ^1 ±iWμ^2 ),

whereWμa( 1 ≤a≤ 3 )are theSU( 2 )gauge fields, andBμis theU( 1 )gauge field. It is

clear that these terms in (4.1.38) are not covariant under the generalSU( 2 )representation

transformations. Hence, the Higgs mechanism violates the PRI. Since the standard model is

based on the Higgs mechanism, it violates PRI as well.

4.2 Physical Supports to PID

The original motivation for PID was to explain the dark matter and dark energy phenomena.

We have demonstrated that the presence of dark matter and dark energy leads directly to PID

for gravity.

There are strong theoretical, experimental and observational evidence for PID. The need

of spontaneous symmetry-breaking for generating mass of the vector bosons for the weak

interaction is a physical evidence for PID for the weak interaction. The quark confinement

requires the introduction of the dual gluon fields demonstrates the necessity of PID; see Sec-

tion4.5. In this section, we address the physical evidence from the following viewpoints:

7.6.6 Nature of dark matter and dark energy.

2) the non-existence of solutions for the classical Einstein gravitational field equations in

general situations,

3) the principle of spontaneous gauge-symmetry breaking,

4) the Ginzburg-Landau superconductivity theory, and

5) the gauge-fixing problem.

4.2.1 Dark matter and dark energy

The presence of dark matter and dark energy provides a strongsupport for PID. We recall the

Einstein gravitational equations, which are expressed as

(4.2.1) Rμ ν−

1

2

gμ νR=−

4 πG

c^4

Tμ ν,

whereTμ νis the usual energy-momentum tensor of visible matter. By the Bianchi identity,

the left-hand side of (4.2.1) is divergence-free, i.e.

(4.2.2) ∇μ(Rμ ν−

1

2

gμ νR) = 0.