Mathematical Principles of Theoretical Physics

166 CHAPTER 3. MATHEMATICAL FOUNDATIONS

In fact, the following are three terms in (3.5.44), which involve contractions ofSU(N)tensors:

GabFμ νaFμ νb, Aaμτa, λbcaAbμAcν.

Obviously, these terms are also Lorentz invariant.

Remark 3.33.The purely mathematical logic requires the introduction ofthe modified Yang-
Mills action (3.5.44) and the SU(N) tensors. There is a profound physical significance. This
invariance dictates that mixing different gauge potentials from different gauge groups will
often lead to the violation this simple principle. As we shall see, the new invariance theory is
very important and crucial in the unified field model presented in the next chapter, where this
principle is called the Principle of Representation Invariance (PRI).

3.6 Spectral Theory of Differential Operators

3.6.1 Physical background

Based on the Bohr atomic model, an atom consists of a proton and its orbital electron,
bounded by electromagnetic energy. Due to the quantum effect, the orbital electron is in
proper discrete energy levels:

(3.6.1) 0 <E 1 <···<EN,

which can be expressed as

(3.6.2) En=E 0 +λn (λn< 0 ),

whereλn( 1 ≤n≤N)are the negative eigenvalues of a symmetric elliptic operator. HereE 0
stands for the intrinsic energy, andλnstands for the bound energy of the atom, holding the
orbital electrons, due to the electromagnetism. Hence there are onlyNenergy levelsEnfor
the atom, which are certainly discrete.
To see this, letZbe the atomic number of an atom. Then the potential energy forelectrons
is given by

V(r) =−

Ze^2

r

.

With this potential, the wave functionψof an orbital electron satisfies the Schr ̈odinger equa-
tion

(3.6.3) ih ̄

∂ ψ

∂t

+

̄h^2

2 m 0

∇^2 ψ+

Ze^2

r

ψ= 0.

Letψtake the form
ψ=e−iλt/ ̄hφ(x),

whereλis the bound energy. Puttingψinto (3.6.3) leads to

−

̄h^2

2 m 0

∇^2 φ−

Ze^2

r

φ=λ φ.