# Viscosity/Entropy ratio and unitary evolution in quantum gravity

Calculations of the shear viscosity of the quark-gluon plasma (QGP), observed in heavy ions collisions, are performed via the AdS/CFT correspondence [ref1]. One finds that the shear viscosity of the CFT, on the boundary of an AdS spacetime with a black hole in its interior, is equal to the shear viscosity of the fluid theory living on the black hole horizon surface. This quantity in turn can be calculated by considering the cross-section for graviton absorption [ref2] by the horizon in the limit that the frequency $\omega \rightarrow 0$ (low-energy limit).

The end results suggest that the fluid on the horizon (and its dual fluid the QGP) is as close to being a "perfect" fluid as Nature allows. What is found is that the shear viscosity $\eta$ of the horizon is proportional to its surface entropy density $s$. It is conjectured that there is a lower bound:

$$\frac{\eta}{s} \ge \frac{1}{4\pi}$$

(in natural units) for the viscosity to entropy ratio for any hydrodynamical system.

These results would seem to imply that the evolution of the QGP cannot be given by a unitary theory. Perhaps this is just a naive misinterpretation of the physics on my part. If unitarity in quantum gravity is presumed to be sacrosanct then how does one reconcile the dissipative behavior of the field theory of a QGP and a black hole horizon with the demands of unitarity?

Edit: @Lubos' answer is fairly complete but I would like to see what others have to say. Thus the bounty. Have at it!

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These results would seem to imply that the evolution of the QGP cannot be given by a unitary theory.

Quite on the contrary. Both sides are equivalent so the AdS/CFT is a proof that the black hole dynamics is unitary - because its equivalent to another theory (the CFT) whose dynamics is given by a manifestly and exactly Hermitian Hamiltonian. This also implies that the information is not lost when black holes evaporate.

I am confused by your statement about "sacrosanct unitarity of quantum gravity". If it is sacrosanct, why do you claim it is not true? In reality, even if some people could call it sacrosanct today, the unitarity of quantum gravity is much more nontrivial - and has been much more controversial for decades - than the unitarity of the laws governing quark-gluon plasma. The latter are self-evidently unitary.

The AdS/CFT correspondence became the most well-known method - but not the only method - to resolve the much harder problem of unitarity of quantum gravity. It's the key method that has also convinced Hawking that he was wrong and the information is preserved. And the answer is Yes, any evolution in quantum gravity in an asymptotically AdS (or flat) spacetime is unitary even if it involves the birth and disappearance of black holes.

You may be confused by the term "dissipation". Dissipation doesn't mean that the theory fails to be unitary. Dissipation is a process in which macroscopic forms of energy are converted to the microscopic forms of energy - heat - which allow the entropy to increase. But the full microscopic theory is still unitary; if you measure the degrees of freedom describing the QGP accurately, they will be shown to evolve in a unitary way (even though, the effective theories we use in practice may display some disappearance of the non-uniformities and information).

But in the black hole case, people didn't know any reason (e.g. a description) that would indicate that the information was preserved, even in principle. This changed with the AdS/CFT correspondence, Matrix theory, and others. Since the late 1990s, the status of "dissipation" in quantum gravity has been on par with the status in field theories. At the fundamental level, the evolution is unitary and preserves the information even though approximate, macroscopic effective descriptions of the phenomena fail to see this fact.

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@Lubos thanks for your answer. You say that dynamics of the CFT (N=4 SUSY-YM) is "given by a manifestly and exactly Hermitian Hamiltonian." So my question is how does a macroscopic dissipative theory arise from a Hermitian Hamiltonian? Is this analogous to asking how an imperfect classical fluid with irreversible dynamics can arise from the reversible, time-symmetric microscopic laws of motion describing a gas of classical atoms ,the resolution to which is traditionally given by Boltzmann's H-Theorem? – user346 Feb 18 '11 at 10:20
Dear space_cadet, dissipative theories - with friction, viscosity, and other irreversible processes that create heat - arise simply because the macroscopic theory doesn't remember the detailed motion of the microscopic constituents, and it replaces it by a statistical description such as the local temperature. Most of the degrees of freedom are then obviously lost. – Luboš Motl Feb 18 '11 at 10:41
Boltzmann's theorem isn't - and hasn't ever been - any resolution to a paradox that existed. It is just a rigorous proof of a statement - the second law of thermodynamics - that was believed as a sacrosanct principle long before most physicists thought that thermal phenomena are due to chaotic motion of atoms. Instead, you seem to be asking about the resolution of a (fringe science) Loschmidt's paradox. Well, the entropy increases - behaves asymmetrically - in a Universe with T-invariant microscopic laws because the logic describing the Universe has an arrow, the logical arrow of time. – Luboš Motl Feb 18 '11 at 10:44
The thermodynamic arrow of time is simply inherited and proved to coincide with the logical arrow of time, and the H-theorem is just the first specific way to prove how this occurs in some quantitative detail. But of course, the H-theorem still has to assume the logical arrow of time - that the past evolves from the future, and not the other way around. No universe with time could exist without a logical arrow of time. The past and future are asymmetric because the past is the "assumptions" and future is its "logical consequences", and assumptions and their implications are not symmetric. – Luboš Motl Feb 18 '11 at 10:46
I wanted to say that in the effective macroscopic theories, the proposition that the "detailed microscopic degrees of freedom are lost" doesn't mean that they're lost in principle. It just means that by the very definition of the effective theory, these degrees of freedom are deliberately thrown away - denied - by the person who uses the effective theory. But this detailed information is needed to reverse the motion. When it's thrown away, the evolution is irreversible - and non-unitary. However, in principle, with all the knowledge, the evolution is unitary and reversible. – Luboš Motl Feb 18 '11 at 10:48