Mathematical Principles of Theoretical Physics

388 CHAPTER 6. QUANTUM PHYSICS

By (6.5.39)-(6.5.41), the equations (6.5.42)-(6.5.43) are written as

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν+

(

∇μ+

n 1 e

̄hc

Aμ+

n 2 e

̄hc

̃Aμ

)

(6.5.44) φνg,

Gab

[

∂νAbν μ−

n 1 e

hc ̄

λcdbgα βAcα μAdβ

]

(6.5.45) −n 1 eΨγμτaΨ

=

[

∂μ−

1

4

k 12 xμ+

n 1 e

̄hc

αAμ+

n 2 e

̄hc

α ̃A ̃μ

]

φa,

G ̃kl

[

∂νA ̃lν μ−

n 2 e

hc ̄

̃λijlgα βA ̃iα μA ̃j

β

]

+

i

2

n 2 e

[

(DμΦ)†( ̃τkΦ)−(τ ̃kΦ)†(DμΦ)

]

(6.5.46) ,

=

[

∂μ−

1

4

k 22 xμ+

n 1 e

̄hc

βAμ+

n 2 e

hc ̄

β ̃A ̃μ

]

φ ̃k,

iγμ

[

∂μ+

in 1 e

hc ̄

A^0 μ+

in 1 e

hc ̄

Aaμτa

]

Ψ−

c

̄h

(6.5.47) M 1 Ψ= 0 ,

gμ νDμDνΦ−

(c

h ̄

) 2

(6.5.48) M^22 Φ= 0 ,

where the energy-momentum tensorTμ νin (6.5.44) is

Tμ ν=−

1

2

gμ ν(LAN^1 +LAN^2 +hc ̄ LD+hc ̄ LKG)+

1

2

(6.5.49) (DμΦ)†(DνΦ)

−

1

4

Gabgα βAaμ αAbν β−

1

4

G ̃klgα βA ̃kμ αA ̃l

ν β.

The energy-momentum tensorTμ νcontains the massesM 1 ,M 2 , the kinetic energy and elec-
tromagnetic energy.

It is clear that both sides of the field equations (6.5.44)-(6.5.48) are all generated by the
fundamental principles. It is the view presented by Einstein and Nambu and shared by many
physicists that the Nature obeys simple beautiful laws based on a few first physical principles.
In other words, the energy-momentum tensorTμ νis now derived from first principles and is
geometrized as Einstein and Nambu hoped.

Systems with four interactions

The above systems with gravity and electromagnetism in general describe the bodies in
lower energy density. For the systems in higher energy density, we have to also consider
the weak and strong interactions. The interactions are layered as shown below, which were
derived in (Ma and Wang,2015b,2014g):