### Objects in Your Phase Space May Be Smaller Than They Appear

Lubos Motl has a very interesting post on a new ArXiv paper (0807.4556) by Jan de Boer, Sheer El-Showk, Ilies Messamah, Dieter Van den Bleeken.

Because I don’t understand the technical details, let me just quote Lumo on what seem to me the most interesting parts:

What is the general lesson? The general lesson is that in quantum gravity, the number of degrees of freedom is often much lower than what you would need to realize all of your fantasies based on classical physics. The entropy bounds and the holographic principle were the old moral examples why it is so.

The new Benelux paper gives you a new and, in some optics, more concrete picture why not all of your classical fantasies are allowed. Why is it so? Simply because you often don't have enough quantum phase space (not even one Planck volume, up to powers of (2 pi), necessary for one quantum microstate) to realize them.

A priori, this comment could sound crazy to you. If you have large objects, the corresponding phase spaces - parameterizing things like the distances between the components of a bound state - are also large. And if they are large, these phase spaces will have a large enough volume in all of their regions to represent all the classical geometries rather faithfully.

But the vague argument above is actually incorrect. Long distances on the phase space actually don't imply large volumes. ;-) Picky mathematicians would always know that they don't but most physicists would suspect that the mathematicians' counter-examples are inevitably pathological. But they can't really be that pathological because they appear in the description of some of the most canonical black hole solutions in supergravity and string theory.

. . .

Once people look at them carefully, they could also understand the origin of holography (and the Bekenstein-Hawking entropy for general backgrounds) somewhat more constructively. What do I mean? The surprising feature of holography is "how do all those numerous degrees of freedom - that a priori seem to be associated with the large black hole interior - disappear?" And the answer could be that they don't really disappear but if you construct the corresponding phase space (whose cells form a basis of the Hilbert space), you find out that the volume of the phase space is much lower than you expected and the calculation reduces to those simple phase spaces that lead to the finite "S=A/4G" entropy.

The final paragraph is very similar to some points in the Carlip paper of my previous post. After some talk about 2 + 1 dimensional spacetime black holes Carlip says:

A typical black hole is neither two-dimensional nor conformally invariant, of course, so this result may at first seem irrelevant. But there is a sense in which black holes become approximately two-dimensional and conformal near the horizon. For

fields in a black hole background, for instance, excitations in the r–t plane become so blue shifted relative to transverse excitations and dimensionful quantities that an effective two-dimensional conformal description becomes possible [147–149].

Indeed, as noted in section 2.5.3, the full Hawking radiation spectrum can be derived from such an effective description [50,51].Martin,Medved, and Visser have further shown that a generic near-horizon region has a conformal symmetry, in the form of an approximate conformal Killing vector [150, 151].

…

As a count of microscopic degrees of freedom, the Bekenstein-Hawking entropy (2) has a peculiar feature: the number of degrees of freedomis determined by the area of a surface rather than the volume it encloses. This is very different from conventional thermodynamics, in which entropy is an extensive quantity, and it implies that the number of degrees of freedom grows much more slowly with size than one would expect in an ordinary thermodynamic system. This holographic” behavior [175, 176] seems fundamental to black hole statistical mechanics, and it has been conjectured that it is a general property of quantum gravity. It may be that the generalized second law of thermodynamics requires a similar bound for any matter that can be dropped into a black hole; a nice review of such entropy bounds can be found in [177]. The AdS/CFT correspondence discussed in section 3.1.3 is perhaps the cleanest realization of holography in quantum gravity, but it requires specific boundary conditions. A more general formulation proposed by Bousso [178] is supported by classical computations [179], and is currently a very active subject of research, extending far beyond its birthplace in black hole physics to cosmology, string theory, and quantum gravity.

Another fascinating subject that I wish I understood.

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