From Quantum Speed Limit to Mass and Velocity of Fundamental Particles*

    Inspired by recent developments of quantum speed limit (QSL), I prove the quantum velocity only depends on the mass of fundamental particles, and prove that there exists a uniqe topological dimensionless monodromic value:  QSL(max)=4. This research gets an explicit relation between dynamic mass (m) and velocity (v):

v= 4 cos²(θ)= 4x (1- m²) or m=sin(θ)= √(1 - v/4) where v = r_coh², r_coh is quantum coherent radius.

    In fact, recent result of QSL is the represent pseudo-speed |ψ₀|ψ_τ| ≈ λcos(Λ(ρ₀,ρ_τ)). Using Coxeter reflection of root system, we obtain that  v = 4 cos(θ) which is an isomorphic map with the pseudo-speed, which is proved by the T-Duality theory associated with mirror symmetry and quantum flop. I found that at special angle θ=0 or 2π, v_limit= 4 cos(θ) = 4 which just the wall in wall-crossing phenomena. This squar root of limit velocity r_coh=2 is radius of called large circle R=2, which divided the complex circle into two domain:

(i). the inner sheaf canonical retract deformation domain (r < 2) where there are a retract deformation m=0.5, v=4(1-0.5²) = 3 isometry with light speed c, as the Universe we leave; another is critical mass m=  √3 /2, v= 4(1- (√3/2)²)= 4x1/4 =1, which is an autonomous manifold, as the Milky Way.

(ii). the outer sheaf escape deformation domain (r > 2) where are an escape space with the Black-Hole. We found there exist two classes of the Black-Hole: Mass-Over Black-Hole and Energy-Over Black-Hole. For Mass-Over Black-Hole, m> m_crit=√3/2, θ > 65^o9, m=√5/2, v= 4(1-(√5/2)²) = 4x(-1/4)= -1 means that escape direction unit velocity. For Energy-Over Black-Hole, r_coh= √5 = 2.236 > 2 or v > 4 and mass m= √(1 - 5/4) = √-¼ = i/2 which is a complex number, i.e. called Tachyno particles.

    We give a geometric method described process from mass to velocity in quantum complex circle by wall-crosing formula. For example, from a spectrum point β=1/2 + i √3/2 to mirror symmetric point ω=-1/2 + i √3/2 by mirror reflection map. We found two methods of wall-crosing: (1). critical angle > θ₁= 65^o9, < θ₂=24^o1, (2). directly crossing the wall R=2. it is clear that the first method connected with hyperbolic quantum orbits; the second method connect with elliptic orbits which are standard Reeb orbits. 

     This study solves a long-time open problem about mass and velocity of fundamental particles in theoretical physics, the deformations of the Universes.