Mathematical Principles of Theoretical Physics

30 CHAPTER 1. GENERAL INTRODUCTION

Table 1.1: Weakton quantum numbers

Weakton Qe Qw Qs Qc Le Lμ Lτ B J m

w∗ 2 / 3 1 1 0 0 0 0 1 / 3 ± 1 / 2 0

w 1 − 1 / 3 1 0 1 0 0 0 0 ± 1 / 2 0

w 2 − 2 / 3 1 0 -1 0 0 0 0 ± 1 / 2 0

νe 0 1 0 0 1 0 0 0 − 1 / 2 0

νμ 0 1 0 0 0 1 0 0 − 1 / 2 0

ντ 0 1 0 0 0 0 1 0 − 1 / 2 0

numberB, lepton numbersLe,Lμ,Lτ,spinJ, and massm. The quantum numbers of weak-
tons are listed in Table1.1.

5.3.4 Weakton constituents and duality of mediators

The weakton constituents of charged leptons and quarks are given by

(1.12.2)

e=νew 1 ,w 2 , μ=νμw 1 w 2 , τ=ντw 1 w 2 ,

u=w∗w 1 w 1 , c=w∗w 2 w 2 , t=w∗w 2 w 2 ,

d=w∗w 1 w 2 , s=w∗w 1 w 2 , b=w∗w 1 w 2 ,

wherec,tandd,s,bare distinguished by the spin arrangements; see (5.3.20).
The weakton constituents of the mediators and their dual mediators are given by
(1.12.3)
γ=cosθww 1 w 1 −sinθww 2 w 2 (⇈,), γ 0 =cosθww 1 w 1 −sinθww 2 w 2 (↑↓,↓↑),
Z=sinθww 1 w 1 +cosθww 2 w 2 (⇈,), H^0 =sinθww 1 w 1 +cosθww 2 w 2 (↑↓,↓↑),
W−=w 1 w 2 (⇈,), H−=w 1 w 2 (↑↓,↓↑),
W+=w 1 w 2 (⇈,), H+=w 1 w 2 (↑↓,↓↑),
gk=w∗w∗(⇈,), gk 0 =w∗w∗(↑↓,↓↑).
Remarkably, both the spin-1 mediators and the spin-0 dual mediators in the unified field
theory have thesameweakton constituents, but with different spins. The spin arrangements
clearly demonstrate that there must be spin-0 particles with the same weakton constituents
as the mediators. Consequently, there must be dual mediators with spin-0. This observation
clearly supports the unified field model presented earlier. Conversely, the existence of the
dual mediators makes the weakton constituents perfectly fit.
Also, a careful examination of weakton constituents predicts the existence of an additional
mediator, which we call theν-mediator:

(1.12.4) φν^0 =∑
l

αlνlνl(↓↑), ∑

l

αl^2 = 1 ,

taking into consideration of neutrino oscillations. When examining decays and reactions of
sub-atomic particles, it is apparent for us to predict the existence of this mediator.

With the above weakton constituents of charged leptons, quarks and mediators, we can
then verify the four basic requirements as mentioned earlier, based in part on the weak inter-
action potentials.