Geometric Quantization and Extreme Deformation over the Universes

I. Part: Geometric quantization

The heat equation is a kernel: u = exp(-x²/2t) /√2πt, where x-Space, t-Time. We rewrite the heat kernel to a complexified form: u= [exp(i x/√2t)]²/√2πt. Using Euler formula: 2 cos(x/√2t)= exp(i x/√2t) + exp(-i x/√2t) and 2i sin(x/√2t)= exp(i x/√2t) - exp(-i x/√2t), then the heat kernel becomes a triangle complex function u = f(cosθ, i sinθ) where θ=x/√2t, which lead the complexification of unifying Space-Time. Let t=1/(2π) be a standard variable unit, and let it be realized on a complex circle S¹. We obtain an explicit complex representation of geometric quantization: u = w(cosθ + i sinθ) and its conjugation u = w(cosθ - i sinθ), and also there exist an important reflection u → - 1/u. As a result, we finished total geometric quantization processes. Now we can prove that the Mass = sinθ, the Velocity V= (r_coh)², after unifying, we give a theoretical framework: G ○M ○V = Id =1, which constructs main frame of the Generalized Newton's Laws (GNL), which presented in the CAP(Canada Physics Associate) congress 2019 by Zhi-An Luan. Most important results of the GNL contain that M=√(1- V/4) , V=4(1-M²). Using these formula, we can explicitly give the Mass and Velocity of fundamental particles.

II. Part: Complexified Courant algebroids

It well known that the exact Courant algebroid: TM = TM + T^*M, which is direct product of tangent bundle and cotangent bundle, which realized in complex torus as tan(θ) + cot(θ), as √3 + 1/√3. The Courant courants play an important role in Mathematics and Theoretical Physics. I propose a new theory of CA - Complexified Courant algebroid (CCA). Using compexification of Courant algebroid, we have a T-Duality symmetric difference TG= TG + i T^*G, and its conjugation TG^' = TG - i T*G . In a complex plane we obtain realizations on complex circle (torus) as √5 - 1/√5, which has a periodic form: u=2n+√5- 1/(2n+√5) and v=2n-√5 -1/(2n-√5). When n=1, then u=v= 4. Same time, 2+√3 + 1/(2+√3) = 2-√3 + 1/(2-√3) = 4.

III. Part: Applications

Three quantum orbits in the Universes:

i. Original orbit M=√3 /2, V=4(1-3/4)=1, Abelian orbit without any deformation;

ii. Normal stable deformed orbit M=1/2, V=4(1-1/4)=3, which is measurable at Earth;

iii. Extreme orbit r_coh= √V= √5 =2.236 > radius of large circle 2, mass M=√(1-5/4)= i/2 - a complex number. It is a Black-Hole.

Black-Hole can escape, and can return.