Mathematical Principles of Theoretical Physics

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 175

whereφ= (φ 1 ,φ 2 )T:Ω→C^2 is a complex-valued function with two components, called
the Weyl spinor,~σ= (σ 1 ,σ 2 ,σ 2 )is the Pauli matrix operator as given by (3.5.36),~D=
(D 1 ,D 2 ,D 3 )is the derivative operator given by

(3.6.41) Dk=∂k+igAk fork= 1 , 2 , 3 ,

and{(~σ·~D),A 0 }is the anti-commutator defined by

{(~σ·~D),A 0 }= (~σ·~D)A 0 +A 0 (~σ·~D).

Remark 3.41.The equation (3.6.40) is essentially an eigenvalue problem of the first order
differential operator:
i ̄hc(~σ·~D)+gA 0 ,

which is called the Weyl operator. In addition, the operator(~σ·~D)^2 in (3.6.40) is elliptic and
can be written as

(3.6.42) (~σ·~D)^2 =D^2 −
g
̄hc

~σ·curl~A,

and~A= (A 1 ,A 2 ,A 3 )as in (3.6.41). The ellipticity of (3.6.42)gin (3.6.41)-(3.6.42) represents
the weak or strong interaction charge, andAμ= (A 0 ,A 1 ,A 2 ,A 3 )the weak or strong interaction
potential.
Note also that~A= (A 1 ,A 2 ,A 3 )stands for the magnetic component of the weak or strong
interaction. Hence, in (3.6.42), the term

g~σ·curl~A

represents magnetic energy generated by the weak or strong interactions, which is an impor-
tant byproduct of the unified field theory based on PID and PRI.

Since (3.6.40) is essentially an eigenvalue problem of first-order differential equations,
its negative and positive eigenvalues are infinite. However, if we only consider the physically
meaningful eigenstates, then the number of negative eigenvalues of (3.6.40) is finite. We now
give introduce these physical meaningful eigenvalues and eigenfunctions for (3.6.40).

Definition 3.42.A real numberλand a two-component wave functionφ∈H 01 (Ω,C^2 )are
called the eigenvalue and eigenfunction of (3.6.40), if(λ,φ)satisfies (3.6.40) and

(3.6.43)

∫

Ω

φ†

[

i(~σ·~D)φ

]

dx> 0.

The physical significance of (3.6.43) is that the kinetic energyEof the eigenstateφis
positive:E>0.
The following theorem ensures the mathematical rationality of the eigenvalue problem of
the Weyl operators.

Theorem 3.43(Spectral Theorem of Weyl Operators).For the eigenvalue problem (3.6.40),
the following assertions hold true: