Hagedorn temperature

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In theoretical physics, the Hagedorn temperature is the temperature above which the partition sum diverges in a system with exponential growth in the density of states.

$\lim_{T\rightarrow T_H^-} Tr[e^{-\beta H}]=\infty$

Because of the divergence, many people come to the incorrect conclusion that it is impossible to have temperatures above the Hagedorn temperature because it would require an infinite amount of energy. In equations:

$\lim_{T\rightarrow T_H^-}E=\lim_{T\rightarrow T_H^-}\frac{Tr[H e^{-\beta H}]}{Tr[e^{-\beta H}]}=\infty$

This line of reasoning was well known to be false even to Hagedorn. The partition function for creation of Hydrogen-antiHydrogen pairs diverges even more badly, because it gets a finite contribution from energy levels which accumulate at the ionization energy. The states which cause the divergence are spatially big--- the electrons are very far from the protons. The divergence indicates that at a low temperature Hydrogen-antiHydrogen will not be produced, rather proton/antiproton and electron/antielectron. The Hagedorn temperature is only a maximum temperature in the physically unrealistic case of exponentially many species with energy E and finite size.

The concept of exponential growth in the number of states was originally proposed in the context of condensed matter physics. It was incorporated into high energy physics in the early 1970s by Steven Frautschi and Hagedorn. In hadronic physics, the Hagedorn temperature is the deconfinement temperature, while in string theory, it indicates a phase transition--- the transition at which very long strings are copiously produced. It is controlled by the size of the string tension which is smaller than the Planck scale by the some power of the coupling constant. By adjusting the tension to be small compared to the Planck scale, the Hagedorn transition can be much less than the Planck temperature.^[1] Traditional grand-unified string models place this in the magnitude of 10³⁰K, two orders of magnitude smaller than the Planck temperature.