Mathematical Principles of Theoretical Physics

346 CHAPTER 6. QUANTUM PHYSICS

For the Klein-Gordon fields(Ψ,Φ)T, the Hamiltonian for a central force field is given by

(6.2.59) H=

1

2

∫[

|Φ|^2 +c^2 |∇Ψ|^2 +

1

̄h^2

(m^2 c^4 +V(r))|Ψ|^2

]

dx

The Hamiltonian energy operatorHˆof (6.2.59) is given by

(6.2.60) Hˆ=

(

HˆΨ 0

0 HˆΦ

)

, HˆΦ=Φ, HˆΨ=

[

−c^2 ∆+

1

h ̄^2

(m^2 c^4 +V)

]

Ψ.

The angular momentum operatorJˆis

(6.2.61) Jˆ=

(

Lˆ 0

0 Lˆ

)

+sh ̄σˆ, σˆ= (σ^1 ,σ^2 ,σ^3 ).

wheresis the spin of bosons, andLˆis as in (6.2.51).
For scalar bosons, spins=0 in (6.2.61) and the Hermitian operators in the conservation
law (6.2.45) are

Lˆ 11 =Lˆ 22 =Lˆ, Lˆ 12 =Lˆ 21 = 0 , HˆΦ,HˆΨas in (6.2.60)

Then by
∫
Φ†LˆΨdx=−

∫

Ψ∗Lˆ†Φdx,

∫

HˆΨ†LˆΦdx=−

∫

Φ†LˆHˆΨdx,

we derive the conservation law (6.2.45), i.e.
∫[
Φ†LˆΨ+Ψ†LˆΦ−HˆΨ†LˆΦ−Φ†LˆHˆΨ

]

dx= 0.

However, it is clear thatHˆandJˆin (6.2.60) and (6.2.61) don’t satisfy (6.2.45) for spins 6 =0.
Hence the quantum rule of angular momentum for bosons holds true.

Remark 6.15.The Angular Momentum Rule is very useful in the weakton modeland the
theory of mediator cloud structure of charged leptons and quarks, which explain why all
stable fermions with mediator clouds are at spinJ= 1 /2.

6.3 Solar Neutrino Problem

6.3.1 Discrepancy of the solar neutrinos

The solar neutrino problem is known as that the number of electron neutrinos arriving from
the Sun are between one third and one half of the number predicted by the Standard Solar
Model. This important discovery was made in 1968 by R. Davis,D. S. Harmer and K. C.
Hoffmann (Davis, Harmer and Hoffman, 1968 ).