# Does the lack of modular nuclearity in string theory mean anything?

Nuclearity is a postulate in algebraic quantum field theory (AQFT). Basically, it says thermal states at any temperature always have a thermodynamic limit with extensive quantities. This is violated by string theory at the Hagedorn temperature. Does this mean anything?

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It means that the postulate is incorrect in general.

Whenever the number of degrees of freedom is large, there is some "extensivity". For example, a single string stores energy and entropy in many modes labeled by $n$, the Fourier mode.

However, it is surely incorrect that the entropy and other quantities is proportional to volume in general. In particular, quantum gravity obeys the holographic principle which implies that the entropy only scales as a surface area, not the volume. In that context, the disagreements are not just exceptions: they're generic and they show that "modular nuclearity" is much more wrong than the modest example of the Hagedorn temperature shows.

There are many other wrong assumptions underlying AQFT, too. The closer one studies renormalized field theory; string theory; quantum gravity, the more visible the invalidity of the AQFT prejudices becomes.

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You don't even need quantum gravity to see that extensivity breaks down when gravity is important. Solve the Tolman-Oppenheimer-Volkov equation for a self-gravitating perfect fluid with equation of state $p=\kappa\rho$. This gives an energy density $\rho \sim r^{-2}$. Cut off the solution at some radial size $R$ and join it on to a solution that goes to zero at a finite distance. Relate the entropy density to the energy density using the EOS, integrate, and you get entropy $\sim R^{\frac{1+3\kappa}{1+\kappa}}$. The entropy of the self-gravitating object never grows faster than the area. – Robert McNees Aug 12 '11 at 18:34

To add a little to Lubos's answer: Even classical gravity has a problem with a thermodynamic limit, because a constant energy density leads to a collapse in a finite time in the future or past, so thermodynamics in the presence of gravity is not trivial. You can think of this as a classical residue of holography. I believe the proper analog to a thermal state in gravity (or in string theory) is an empty dS space, since this is a maximal entropy configuration semi-classically with a finite temperature. de Sitter space is hard to describe in quantum gravity, because of the various paradoxes related to the finite surface area bounding horizon.

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