Friday, March 25, 2005

Raphael's and Nima's talks

A small comment: a clarifying statement by Prof. Gerard 't Hooft has been added to the report about Sidneyfest.

I will describe two talks in the same text: Raphael Bousso's duality seminar he gave yesterday, and Nima's lunch seminar today.

Raphael: good and bad backgrounds in quantum gravity

Yesterday, Raphael Bousso, one of the kings of holography, discussed what the right quantum gravity observables could be and should be in different cosmological backgrounds. String theory is always smart and it carefully avoids all potential problems - for example it gives us the S-matrix in the asymptotically flat spacetimes, and closely related variables in the AdS space. But it has told us very little about the more complicated cosmological solutions.

This is both good and bad. It's good because it always reassures us that string theory is a consistent quantum theory of gravity. It's bad because despite string theory's silence, we want some answers to these questions, and we run into many problems, especially in the case of de Sitter space:
• thermal radiation that introduces noise and eventually kills us (you may kill someone with a spoon if you're patient enough)... Note added later: Raphael corrected me - the problem is not whether you're patient enough, but whether your victim is patient enough
• the existence of event horizons that prevents us from measuring the final state
• a related problem, the non-uniqueness of the vacuum
• and so forth...
Different backgrounds have various types of problems, and people usually grade the most typical FRW cosmologies as follows:
• very good: flat space (and AdS spaces) - the S-matrix exists and the problems are gone
• very good: deccelerating Universe whose future is much like in flat space, and therefore people used to say that it is essentially as good as the flat space
• very bad: the accelerating Universe with the equation state "w > -1", which are usually grouped together with the following group because of the existence of horizons
• very bad: de Sitter space - the ultimate example of the problems and subtleties we encounter in quantum gravity
Raphael presented some arguments that encourage you to regroup the four classes differently:
• very good: flat space and AdS spaces
• average: deccelerating Universes. Raphael says that this group is worse than advertised because you would always need to know an infinite amount of information about an infinite amount of matter to describe a state of this Universe
• average: accelerating Universes. Raphael argues that this category is better than its reputation because the radius of curvature grows to infinity, the temperature drops to zero, and the integrated energy from the "spoon" is actually finite
• very bad: de Sitter space
Nima - inflation is anthropically necessary

One hour ago, Nima Arkani-Hamed explained that inflation is necessary because of a reasoning analogous to Weinberg's anthropic "derivation" of the existence (and approximate size) of the cosmological constant. Much like the cosmological constant should be small for anthropic reasons, Nima says that the curvature of the spatial slices should also be small. But it may be non-zero, as far as the anthropic reasoning goes. However, Nima is afraid to predict that it should actually be nonzero - and of course, Cumrun and me were peacefully questioning whether this kind of anthropic reasoning based on "things that sound reasonable" without any chance for a quantitative definition of the word "reasonable" is still science.

Nevertheless, Nima presented interesting observations, especially these two:
• If you initiate a new Universe (from the Big Bang) by tunneling from another Universe, then the new Universe will have negative curvature of the spatial slices. It's because the time is measured by the proper time from the origin, and the places with the same proper time from the origin are the hyperboloids whose curvature is negative. This contradicts observations (and it would also prevent structure formation, Nima argued). This fact either means that our Universe was not created by tunneling from a higher-Lambda Universe, which is the interpretation I would prefer, or that there had to be an inflationary era that guaranteed the flatness, which is Nima's preferred interpretation (one that unfortunately makes the tunneling event before the inflation irrelevant). Incidentally, one reason to explain why we obtained the hyperboloids is that they are the analytical continuation of the spheres in Coleman's instantons : these instantons are spherically symmetric...
• There exists an intrinsic problem to obtain inflation from string theory in a pretty way. If there are CMB-like gravity waves found (so far it's not the case), it means that the inflaton had to move by an amount that is (much) greater than m_{Planck} - assuming the canonical normalization of the inflaton kinetic terms. This case - the existence of gravity waves and the associated big changes of the scalar field - would invalidate the low-energy effective field theory and it would require the full string theory instead. However, it is difficult to find compactifications of string theory in which the scalar fields can be changed by a lot - an increment greater than the axion decay constant divided by M_{Planck}^2, and Nima discussed the model-independent axion (dual to the B-field) and the Wilson lines...
There exists a ratio
• "number of axions" times "m_{string}^2" over "m_{Planck four-dimensional}^2"
that should be large, and Nima argues that it cannot be made large by a parameteric dependence. It is only large because it is something of the order "8 pi cubed" or something like that. (I don't quite understand the argument why it can't be made large by a large value of the string coupling.) There are many assumptions that enter this kind of considerations, but in the context of Calabi-Yau spaces, Nima says that (once we forget about the string coupling issues, and keep "g_{string" fixed) the Calabi-Yau spaces preferred by his inflationary anthropic principle are those that, roughly speaking, maximize the following ratio
• "the number of two-cycles" (each of them must have an area greater than "l_{string}^2") divided by the quantum volume of the Calabi-Yau three-fold
This is an interesting mathematical task - try to find a Calabi-Yau space that maximizes the density of handles, so to say. I believe that in this or next century, a similar criterion will be found to pick the right compactification, and it will be a relatively simple one. This kind of mathematical problem reminds me of the following amusing observation I did some time ago:
• maximize the ratio "dim(G) / rank(G)^2" among all compact Lie groups "G"
You can see that the ratio approaches "1" for SU(N) with N large. It approaches "2" for SO(N) with N large. It equals "3" for SU(2). But I think that the winner is "E_8" because the ratio is almost four: more precisely, it is "31/8" if I remember well. ;-) Just like there exists the best Lie group, namely E_8 ;-), I believe that there exists the best compactification of string theory; a scientific argument analogous to this counting will be found; and this vacuum will be ours (or perhaps, our Universe will be in the "top ten").

1. The idea of Lubos about best compactification of string theory justifies my comments below. To me the basic question is what compactification and dualities really mean at the deeper, I believe geometric, level.

TGD approach has led to a new view about dualities and to an interpretation of the spontaneous compactification in terms of wave-particle duality in infinite-dimensional context. This vision generalizes also to the stringy context if the notion of configuration space of strings (more general than loop space) is introduced and if it really makes sense.

*The trivial observation is that the flat target space TH=M^8 is tangent space of "compactified" target space H=M^4xCP_2.

*This trivial observation lifts to something non-trivial at the level of configuration space CH of 3-surfaces. The configuration space CTH of 3-surfaces associated with tangent space TH=M^8 corresponds to the tangent space of configuration space CH associated with H=M^4xCP_2.

The NEW element is the representability of the points of tangent space of CH as 3-surfaces in TH. Momenta p associated with 3-surfaces (positions q in CH) are represented as 3-surfaces! Even more, same 3-surface (and thus 4-surface) but with metric and Kahler structure induced from H resp. TH represents q resp. p!

*This gives rise to the duality and allows to interpret spontaneous compactification in terms of infinite-dimensional wave-particle duality. In perturbative/ wave picture you have 3-surfaces in M^8 and in non-perturbative/ particle picture you have 3-surfaces in H=M^4xCP_2.

*Geometric quantization can be performed using any Lagrange manifold of cotangent bundle of CH defining the duality map q-->p(q) (generalization of Bohr rules). This means that canonical transforms of the dualities define new dualities. Symmetries probably however leave only very few practical choices.

This picture generalizes also to M-theory if one introduces the configuration space of strings. There are good reasons to expect that this space exists only for target spaces, which are loop groups associated with products of Lie groups and possibly also coset spaces. The reason is that Kähler geometry does not exist unless you have infinite-dimensional group of Kac-Moody symmetries as isometries. This might resolve landscape problem to large extend but might also mean that M-theory fails experimentally.

For more about this see Dualities as wave-particle dualities in infinite-dimensional context? at my blog and references therein.

Matti Pitkanen

2. Hi Lubos, can you tell us or point us to a reference where the statement

Incidentally, one reason to explain why we obtained the hyperboloids is that they are the analytical continuation of the spheres in Coleman's instantons

is explained?

3. Hi Anon,
the equation for a sphere is

x^2 + y^2 + z^2 + te^2 = r^2

where te is the Euclidean time. After an analytical continuation to te=i.t where t is the real time and i is the imaginary unit, you obtain

x^2 + y^2 + z^2 - t^2 = r^2

which is the hyperboloid. The fields in Coleman's instantons are constant over the sphere, and therefore they're constant on the hyperboloids which is its continuation.

More about vacuum decay instantons, see e.g. Coleman's lectures (I hope that you find it in Aspects of Symmetry, for example).

Best
Lubos

4. Lubos Said:
"x^2 + y^2 + z^2 + te^2 = r^2

where te is the Euclidean time. After an analytical continuation to te=i.t where t is the real time and i is the imaginary unit, you obtain

x^2 + y^2 + z^2 - t^2 = r^2

"
True, but having complexified t, your signature is now +++-, and that looks like a definition of de Sitter spacetime, not a hyperboloid!

5. Dear Fyodor,

yes, one must also continue to the imaginary values of the radius.

x^2+y^2+z^2-t^2 = r^2

is the continuation of the sphere which is indeed a de Sitter space of signature ++-, but

x^2+y^2+z^2-t^2 = -r^2

is a hyperboloid with a Euclidean metric +++ on it. These hyperboloids slice the time-like region of the +++- Minkowski spacetime while the de Sitter spaces slice the space-like region of the spacetime where nothing happens (by causality).

All I needed was to say that the fields only depend on the invariant r^2 which can be negative in the time-like region where the hyperboloids appear.

All the best
Lubos

6. Right! So spheres can end up as hyperboloids under [enough!] complexification. Interesting.
Thanks!