366 CHAPTER 6. QUANTUM PHYSICS
whereΩ⊂Rnis a bounded domain, and{A,B}=AB+BAis the anti-commutator.
Now, we derive the spectral equations from (6.4.32) for the massless bound states. Let
the solutionsψof (6.4.32) be in the form
ψ=e−iλt/ ̄hφ, φ=(
φ^1
φ^2)
,
Then equation (6.4.32) are reduced to the eigenvalue problem
−hc ̄(~σ·~D)^2 φ+
ig
2{(~σ·~D),A 0 }φ=iλ(~σ·~D)φ,and by (6.4.20) which can be rewritten as
[−hcD ̄^2 +g~σ·curl~A](
φ^1
φ^2)
+
ig
2{(~σ·~D),A 0 }(
φ^1
φ^2)
=iλ(~σ·~D)(
φ^1
φ^2)
(6.4.33) ,
(φ^1 ,φ^2 )|∂Ω= 0 ,where{(~σ·~D),A 0 }is the anti-commutator defined by
(6.4.34) {(~σ·~D),A 0 }= (~σ·~D)A 0 +A 0 (~σ·~D).
The eigenvalue equations (6.4.33)-(6.4.34) are taken as the model for the energy levels of
massless bound states. The mathematical theory (Theorem 2.42) established in Subsection
3.6.5laid a solid foundation for the energy level theory providedby (6.4.33).
Remark 6.18.In the equations (6.4.21)-(6.4.22) and (6.4.33) for bound states, we see that
there are terms
(6.4.35) ~μ·curl~A for massive particle systems,
(6.4.36) g~σ·curl~A for massless particle systems.
In (6.4.35), the physical quantity~μ= ̄hg~σ/ 2 mrepresents magnetic moment, and the term in
(6.4.36) is magnetic force generated by the spin coupled with ether weak or strong interaction.
In other words, in the same spirit as the electric chargeeproducing magnetism, the weak and
strong chargesgw,gscan also produce similar effects, which we also call magnetism.
Indeed, all three interactions: electromagnetic, weak, strong interactions, enjoy a com-
mon property that moving charges yield magnetism, mainly due to the fact that they are all
gauge fields.
6.4.3 Charged leptons and quarks
According to structure and interaction types, we shall discuss the energy levels for three
groups of particles. charged leptons and quarks, hadrons, mediators. In this subsection we
only consider the case of charged leptons and quarks.