Sci-Am:"Information in the Holographic Universe"

["Tim Gwinn" <***>]

In the Aug 2003 issue of Scientific American, there is an article entitled
"Information in the Holographic Universe" by physicist Jacob Bekenstein,
subtitled "Theoretical results about black holes suggest that the universe
could be like a giant hologram". Sounds slightly provocative, eh? :) I found
this article rather......unconvincing, so I decided to do a little
background research. Although some of it definitely exceeds me, I think I
have followed the crucial points, below. I hope someone more astute in math
and quantum mechanics can correct or confirm my observations.

In the true spirit of reductionism, the first paragraph in the article ends
with the sentence:"Indeed, a current trend, initiated by John A. Wheeler of
Princeton University, is to regard the physical world as made of
information, with energy and matter as incidentals."

On an equally optimistic note, the article ends: "What is the fundamental
theory like? The chain of reasoning involving holography suggests to
some...that such a final theory must be concerned not with fields, nor even
spacetime, but rather with information exchange between physical processes.
If so, the vision of information as the stuff the world is made of will have
found a worthy embodiment."

The world 'information' in this article seems to be used in different ways
at different points, but centrally it takes the form of Shannon information:
roughly, it concerns syntactic "bits" without regard for their meaning.
Shannon entropy is about how many "bits" can be stored in a given medium.
This is closely related to thermodynamic entropy. Ultimately, these are
related to possible configurations, and to "degrees-of-freedom", of the
constituents in a system.

The upshot of the article is that for a given volume of space, the maximum
entropy (and therefore, maximum information entropy) is finite and
determined not by the volume, but by the surface area of the volume. So, if
the volume is a spherical region of space, the maximum entropy is limited by
the size of the surface area of the sphere. This is remarkable in two ways:

First, it says that all the information content of the inside of the volume
can be mapped to the surface of the sphere. In effect, the 3-D volume is
perhaps just a "holographic" projection of a 2-D reality.

Secondly, it contends that the maximum entropy in the volume is finite. This
is perhaps the more interesting conclusion, because it effectively says that
there are only a finite number of degrees-of-freedom possible for the
constituents of a given volume of space. (The nicety in this is the
implication that these finite degrees of freedom would then be amenable to
complete description in some fundamental theory.)

But, if this were so, it would seem to have harsh consequences for Rosennean
complexity: a limited (finite) number of degrees of freedom would seem to
mean that the physical universe was incapable of the necessary richness of
configuration possibilities necessary to build most (if not all) complex
systems. Maybe the universe is approximately Newtonian after all?

The "holographic principle", on which this article rests, comes from Gerald
't'Hooft and is described in a paper of his entitled "Dimensional Reduction
in Quantum Gravity". (see http://arxiv.org/abs/gr-qc/9310026 )  The basic
idea is that by using a gedanken experiment with black holes, one can arrive
at the solution that the maximum amount of information regarding
degrees-of-freedom of the constituents of a spherical volume of space (in
this case, inside a black hole) is **finite** and a function of the
**surface area** of the volume, not the volume itself! This leads to the
conclusion or speculation that perhaps 3-D space is "really" only 2-D, since
all the necessary information for describing degrees-of-freedom of 3-D
(volume) is apparently encodable in only 2-D (a surface). Hence, the name
"holographic principle", since it is akin to how a 'flat' hologram encodes
3-D information. (His "holographic principle" also has a more general
meaning for relating physical theories, but that is another story.)

Another paper by physicist Raphael Bousso (who is mentioned in the Sci-Am
article) entitled "The Holographic Principle" (see
http://arxiv.org/PS_cache/hep-th/pdf/0203/0203101.pdf ) presents an even
more robust version (the "covariant entropy bound") of this notion that does
not rely entirely on black holes. Bousso describes numerous scenarios
demonstrating that this principle seems to apply without exception. As he
notes, though: "The origin of the bound remains mysterious. As discussed in
the introduction, this puzzle forms the basis of the holographic principle,
which asserts that the covariant entropy bound betrays the number of degrees
of freedom of quantum gravity." (p. 19)

On a quantum mechanical level - which is the level at which these papers
discuss these topics - entropy is related to the degrees of freedom present
in the quantum mechanical description. This description takes place in a
mathematical construct called "Hilbert space". The degrees of freedom of a
quantum mechanical system is measured by the dimensionality of the Hilbert
space. Specifically, entropy is the natural log (ln) of the number of
dimensions of Hilbert space.

Using a number of physical considerations (Planck limit, energy density
limits, etc.) , these various authors argue for reasons why the
dimensionality must be finite, and therefore the degrees of freedom are
limited. If we allow (for the sake of argument) that macroscopic physics can
be reduced to quantum mechanics, and we take the author's arguments
seriously, then it seems to follow quite directly that the aforementioned
limits on entropy must apply.

This is crucial: if the dimensionality of the Hilbert space were infinite,
then all of this goes out the window. There would be no "covariant entropy
bound", and no issue for Rosennean complexity.

Now (as best I understand it) Hilbert space provides for quantum mechanics
what phase space does for Newtonian mechanics. In Hilbert space, quantum
mechanical system states are described using coordinates and momentum in
ways analogous to Newtonian systems. This tells us where they are and how
they are moving. But if this is so, then these Hilbert space descriptions
suffer from the same flaws described by Rosen in "Life Itself" ch. 4: 'The
Concept of State'. In short, the attempt to describe such system using
state-based descriptions and disregarding all the (infinitely many)
higher-order temporal differentials of position, leads to an incomplete
description of the dynamics. As Rosen says of Newtonian and quantum
mechanics, "they remain different species of the same genus" [LI 105].

So it seems to me that the plausibility of these holographic notions, and of
the Sci-Am article, rests on an an artifact of Newtonian physics that has
trickled down to quantum mechanics. This artifact allows the Hilbert space
to appear to be potentially of finite dimensions. Removing that artifact
would lead to an infinite state description, and therefore nullify all the
holographic considerations above.

Contrary to the picture painted in the Sci-Am article, I think Bousso puts
it wisely in the introduction to his paper:
"Indeed, a broader caveat is called for. The covariant entropy bound is a
compelling pattern, but it may still prove incorrect or merely accidental,
signfiying no deeper origin. If the bound does stem from a fundamental
theory, that relation could be indirect or peripheral, in which case the
holographic principle would be unlikely to guide us to the core ideas of the
theory." (p. 2)

Regards,
Tim