TITLE:
Heron’s Triangles, Golden Section and Quantization of Decays of Scalar, Strange Mesons and Δ, N Baryons in the Hyperbolic Lobachevsky Velocity Space
AUTHORS:
Valeriy Pavlovich Khеn, Aleksey Valerevich Khen
KEYWORDS:
Lobachevsky Velocity Space, Resonance Decay Triangles, Oricyclic Cotangent Triangle, Heron’s Hyperbolic Triangle, Golden Section, Stewart’s Theorem, Bretschneider’s Theorem, Quantization Resonance Decays
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.10,
October
28,
2025
ABSTRACT: The ends of the velocity vectors of the decay particles of resonance represent material points-velocities in the hyperbolic Lobachevsky velocity space of negative curvature k = −1/C2 (C = 1 is the speed of light, the rest masses of the decay particles are assigned to the points-velocities). Two points-velocities of the decay particles can be connected by a line segment and an arc of a line of constant curvature 0, called the oricycle. Archimedes’ leverage laws define a 3rd point on the arc of the oricycle to which an additive mass (sum of rest masses of particles) is assigned. Connecting 3 points-velocities by line segments, we obtain isosceles triangles of decays of resonances in the Beltrami model of the Lobachevsky velocity space. In the decay triangles of resonances, the golden section is found and the Stewart, Brettschneider theorems on oricyclic arcs are satisfied. Near the decay triangles of scalar, strange mesons and Δ, N baryons, isosceles triangles-satellites with integer values of their characteristics were found. On the satellite triangles, the Lorentz invariant function—the product of the length of the arc of the oricycle subtending the base and the cotangent of half the angle at the vertex opposite the base—takes integer values. The function is called the oricyclic cotangent of a triangle (OCT). In addition to the integer values of OCT, these satellite triangles also have the sum of the hyperbolic cosines of the lengths of the lateral sides and the hyperbolic cosines of the base lengths equal to integers. These satellite triangles are called Heron triangles. On Heron triangles, the generalized cosines of the angles between the tangent to the oricycle at the point-velocity of the additive mass and the tangent at the point-velocity of the base of the triangle take multiples of 1/2 values.