The Hagedorn temperature in theoretical physics is the temperature where hadronic matter (i.e. ordinary matter) is no longer stable, and must either "evaporate" or convert into quark matter; as such, it can be thought of as the "boiling point" of hadronic matter. The Hagedorn temperature exists, because the amount of energy available is high enough that matter particle (quark-antiquark) pairs can be spontaneously pulled from vacuum. Thus, naively considered, a system at Hagedorn temperature can accommodate as much energy as one can put in, because the formed quarks provide new degrees of freedom, and thus the Hagedorn temperature would be an impassable absolute hot. However, if this phase is viewed as quarks instead, it becomes apparent that the matter has transformed into quark matter, which can be further heated. The Hagedorn temperature was discovered by the German physicist Rolf Hagedorn.
It is about the same as the mass-energy of the lightest hadrons, the pion, at 130-140 MeV per particle or about 2 × 1012 K. This energy range can be routinely probed in particle accelerators such as CERN's Large Hadron Collider. Matter at Hagedorn temperature or above will spew out fireballs of new particles, which can again produce new fireballs, and the ejected particles can then be detected by particle detectors. This quark matter has been detected in heavy-ion collisions at SPS and LHC in CERN (France and Switzerland) and at RHIC in Brookhaven National Laboratory (USA).
In string theory, a separate Hagedorn temperature can be defined for strings rather than hadrons. This temperature is extremely high (1030 K) and thus of mainly theoretical interest.
Because of the divergence, many people come to the incorrect conclusion that it is impossible to have temperatures above the Hagedorn temperature, which would make it the absolute hot temperature, because it would require an infinite amount of energy. In equations:
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 rapidly, because it gets a finite contribution from energy levels that accumulate at the ionization energy. The states that cause the divergence are spatially big, since 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.
In string theory
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. Traditional grand-unified string models place this in the magnitude of 1030 kelvin, two orders of magnitude smaller than the Planck temperature.
- Peter Tyson (December 2007). "Absolute Hot: Is There an Opposite to Absolute Zero?". PBS NOVA. Retrieved 2008-12-21.
- Ericson, Torleif, Rafelski, Johann. The tale of the Hagedorn temperature. CERN Courier, Sep 4, 2003. 
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