The thermodynamics of the Hagedorn mass spectrum

No bootstrap assumption is needed to derive the exponential growth of the Hagedorn hadron mass spectrum: It is a consequence of the second law applied to a relativistic gas, and the relativistic equivalence between inertial mass and its heat content. The Hagedorn temperature occurs in the limit as the number of particles and their internal energy diverge such that their ratio remains constant. The divergences in the $N$ particle entropy, energy, and free energy result when this condition is imposed upon a mixture of ideal gases, one conserving particle number and the other not. The analogy with a droplet in the presence of vapor explains why the pressure of the droplet continues to increase as the temperature rises finally leading to its break up when the Hagedorn temperature is reached. The adiabatic condition relating the particle volume to the Hagedorn temperature is asymptotic. Since it is a limiting temperature, and not a critical one, there can be no phase transition of whatever kind, and the original density of states used to derive such a phase transition is not thermodynamically admissible because its partition function does not exist

Noneuclidean Tessellations and their relation to Reggie Trajectories

The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the angular momentum states. Bound states are associated with real angular momenta while resonances are characterized by complex angular momenta. With a centrifugal potential, the half-plane is tessellated by crescents. The addition of an electrostatic potential converts it into a hydrogen atom, and the crescents into triangles which may have complex conjugate angles; the angle through which a rotation takes place is accompanied by a stretching. Rather than studying the properties of the wave functions themselves, we study their symmetry groups. A complex angle indicates that the group contains loxodromic elements. Since the domain of such groups is not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of the isometric circle is adapted since it treats all groups symmetrically. The pairing of circles and their inverses is likened to pairing particles with their antiparticles which then go one to produce nested circles, or a proliferation of particles. A corollary to Laguerre's theorem, which states that the euclidean angle is represented by a pure imaginary projective invariant, represents the imaginary angle in the form of a real projective invariant.Comment: 27 pages, 4 figure