Unitary Wave Theory

This paper presents a unitary wave theory for the Macro-Universe to Meso-Nano to Micro-Fundamental particle in the Unitary Space-Time (UST). Using the nanoparticle diffraction X-ray peak profile analysis, specially according to Bragg's law, nλ = 2d Sin(θ), where θ - the Bragg angle, n- the order of diffraction, λ- the X-ray wavelength and d is the spacing between planes of given Miller indices. we construct the unitary wave theory, which can apply for all matters such as the Universe and Fundamental Particle. The Complex Circle Method (CCM) or Maximum Complex Torus in the Generalized Complex Structures is a strong quantum topological tool.

The reduced generalized complex structure (RGCS) in the complex circle is nothing but the quotient complex structure obtained from holomorphic quotient. Let ν be nanoparticle frequency, we have the Bragg's law: n V/ν = 2d Sin(θ). From the Generalized Newton's Laws (Zhji-An Luan, CAP 2019 Congress), we have V=4 Cos²(θ) and then the frequency 

ν = n 4 Cos²(θ) /(2d Sin(θ)) = 2n Cos²(θ)/(d Sin(θ)) = 2d/n Cot²(θ) Sin(θ),

which is just a Quotient Complex Structure of the complex circle. If (M, I) is a complex manifold, then any reduction of J_I by an extended action will result in a generalized complex structure and

J_I (T^*M) = T^*M

and the Courant Algebroid:

D= TM + T^*M

For a fixed mass M = Sin(θ), we finally obtain an Universal Frequency (UF):

v =  2βd/n Cot²(θ).

Here we obtain a monodromy law of all Matter- Wave, which means that the photon, electron, phonon and graviton havd a homotopy topological quantum structure, a Extended Dirac Instanton form:

Ω= (2n ± √3) + (2n±√3)^-1 = (2n ± √5) - (2n ± √5)^-1 |_n=1 = 4 (Maximum light speed > 3....km/s).

By the GNL theory:

GMνλ = 1 = Id, then ν = 1/(GMλ), where M- mass, G- generalized Newton's Constant Variety. If fixing a Standard Wavelength λ₀ = √3, we have the unitary wave form of

v = Ψ/(GM) ,where Ψ = 1/√3 is the Fu - Xi Universe Constant. 

This study provides a unique relation of Wave- Matter.