On Inner Sheaf Space, Outer Sheaf Space and Determinativeness*

    Recent research results of the quantum speed limit (QSL) show that there exists an explicit relation between dynamic mass and velocity of fundament particles: v = λcos²θ, which imply that mass: m=sinθ. Using Coxeter reflection, I found that there exits a more exact relation: v=4 cos²θ, which means maximum topological quantum velocity is v_max=4 cos²(0)=4. This monodromic dimensionless value 4 shows it is just the radius of called large complex circle S¹, i.e. R=2 or the coherent radius r_coh=2.

    This paper recovers the most important phenomenon that the maximum topological quantum velocity v_max=4 or r_coh-max= 2  is a key determinativeness of inner sheaf space and outer sheaf.

(i). if r< r_coh-max there is an inner sheaf space: for example, in the solar system the mass= 0.5 (near electro-magnetic wave), then its velocity is: v=4(1- 0.5²)= 3, i.e. called light speed. At same time, in the Milky-way system the mass = √3/2 (critical mass), then its velocity is : v=4(1-(√3/2)²) = 1 without any deformation, which is autonomous system. It shows that in inner sheaf space the mass and velocity are real values.

(ii). if r > r_coh-max then there is a outer sheaf space or called Black-Holes. For the mass-over type Black-Hole, m= √5/2 > m_crit=√3/2, then its velocity. v=4(1- (√5/2)²)= 4x(-1/4)= -1 which means an counter-current. For the energy-over type Black-Hole, v=5 or r_coh= √5 ≈ 2.236 > 2, i.e. just in the outer sheaf space, its mass is m=√1-(√5)²/4) = √(1- 5/4)=√-1/4= i/2. In outer sheaf space, indeed there is Higgs particles with mass m_Higss=1.25 >> m_crit ≈ 0.866.

    The inner sheaf space and outer sheaf space have an essential different topological structure, by other words, there is not an isomorphism between the inner sheaf space and outer sheaf space. it shows that the noncommutative deformation without assumptions on the singularity and in arbitrary dimension is a ideal deformation-theoretic framework which can detect the non-isomorphism locus around a closed point.

    Using the classical hydrodynamics, we also find out the limit R=2. In 2013, I presented a model of two-phase flow in porous media with generalized oil-water viscousity ratio M, the ratio of chemistry and gravitational potential:

∂u/∂t = ▽{(Mu(1-u)/(1+(m-1)u)∇u}.

The quantization result is:

Mu² - u - 1 = 0. Its saturations are u = (1 ± √(1 - 4M))/2M. If M=1/4, u=2 which is a determinativeness.