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Background independence in AdS spaces

Moshe Rozali opened his and David Berenstein's blog to a discussion of background independence (BI) and emergent character of gravity, especially in the context of the anti de Sitter (AdS) space. As explained in that thread, the following facts are important:

  • states in a theory of quantum gravity with different asymptotic conditions than the AdS space are physically disconnected; they're in different superselection sectors
  • states within the same superselection sectors allow any physical configurations "in the bulk", i.e. inside the space defined by the boundary, to evolve; all physically conceivable configurations in the "bulk" are included in the gauge theory; the latter is therefore at least as background independent as general relativity
  • Riemannian geometry, strings, string fields, and/or branes are just approximate, emergent degrees of freedom in this context; they're no longer fundamental, universally valid, or leading to a known exact description of physics
  • gauge theory is the fundamental description of all physical phenomena in the superselection sector and it is defined beyond any perturbative expansion and beyond other limits
  • while this gauge theory can be found as a limit of another string theory, it can also be formulated completely independently of string theory, which is the philosophy relevant for the AdS/CFT correspondence
  • there is additional, virtually unquestionable evidence that string theory is more background-independent: it inevitably makes topology transitions legal, and leads to the exact equivalences of physical phenomena at geometrically vastly and qualitatively different backgrounds (dualities)
Let me discuss these points one by one.

Superselection sectors

In a theory of gravity, extending general relativity, the spacetime may have many different shapes. The space may get curved or less curved, approximate event horizons may be created for a while, the shape (and topology) of internal dimensions of space may change.




However, there are certain changes that can't really happen because they would take an infinite amount of effort or energy. If your spacetime is infinite, you can't change the spacetime in such a way that the asymptotic character of the spacetime at infinity qualitatively changes.

More concretely, in the flat or AdS space, the metric tensor at infinity approaches the metric tensor of the empty space plus small corrections, encoding the mass (and other moments of the energy distribution). If you imagine that such a space evolves into another space that has completely different asymptotics, it is similar to adding new objects into your spacetime whose energy is infinite. That can't be done in the real world where the energy always stays finite.

We say that the states in the Hilbert space are divided into many superselection sectors: the full Hilbert space is a direct sum. You have a chance to evolve from one state to another state in the superselection sector but you can be absolutely certain that the probability to evolve into a state in another superselection sector is exactly zero because these other states are "infinitely different". That's true both in classical physics as well as quantum physics. And it would be arguably true even in a new non-classical non-quantum framework that you may speculatively propose to replace quantum mechanics.

Superselection sectors also separate states according to different values of conserved quantities. So you often hear that states with a different total electric charge belong to different superselection sectors, too. Incidentally, different values of the electric charge also imply a different behavior of fields - namely the electric field - at infinity: the potential goes like Q/r. But we want to talk about superselection sectors that differ more substantially at infinity than by terms that decrease as a negative power law.

It's important to realize that the segregation of states into different superselection sectors is not a bug of one particular theory of quantum gravity or another. Instead, it is a demonstrable fact that every consistent theory must respect. While it may be a nice dream to have a description that treats all conceivable superselection sectors as objects on equal footing, something that we will call M2009 later, there is no logical necessity for such a description to exist. The differences between the superselection sectors are objectively huge. 

Freedom in the bulk

At the same moment, our inability to change the shape of space at infinity doesn't prevent us - and our theories - from changing the space in equally qualitative ways in finite regions of the "bulk". That's the case of the CFT description of physics in the AdS space, too.

Because the conformal field theory on the boundary is a complete theory, it allows one to evolve the initial state into any state in the same superselection sector. It follows that the whole superselection sector must be a part of the CFT Hilbert space. In the bulk description, we know that it is possible to create black holes, let them evaporate, and change the background in many other ways. It follows that all these transformations are automatically allowed by the boundary CFT, as can be explicitly checked in dozens of cases.

Geometry is no longer fundamental

The third point is that geometry encoded by the metric tensor used to be an exact, fundamental degree of freedom in classical general relativity. But assuming that this remains to be the case under all circumstances would be far too strong an assumption. In fact, we know this not to be the case. Using the intuition of perturbative string theory, we know that there are other, massive closed string modes besides the massless metric fields. There are also open string modes and D-branes. 

Moreover, holography tells us that the maximum entropy in a given region is much smaller than what the local field theory on a "real" geometry would lead us to believe.

But it is not just the geometry as imagined by Einstein in 1915 that becomes an approximate concept in the AdS/CFT. In fact, the string fields generated by strings from perturbative string theory are emergent, approximate notions, too. At a finite value of the string coupling, strings are interacting and they are comparably fundamental as other objects. It leads me to the fourth point.

Gauge theory is the only exact description of the physics

If we study this important superselection sector of string theory - and quantum gravity, the relevant gauge theory provides us with the exact description of all physical objects and phenomena. In the gauge theory, morally defined on the boundary of the AdS space at infinity, we can prove the existence of the metric tensor and other fields following the rules of supergravity in the right limit, excited strings with the right energies, branes, black holes, and other things expected from other approaches to string theory.

On the other hand, all these objects are "derived" in this context. The degrees of freedom of gauge theory are the only "exact" ones. While the particular gauge theory could look too special and you may be surprised that it is being claimed to contain "all of physics", the real situation actually makes sense. The gauge theory description is only convenient for states that can be naturally embedded into the AdS spacetime with the right, supersymmetric asymptotics.

While the CFT morally describes any type IIB stringy physical phenomena - and, by dualities, all of string theory - that can occur in the bulk, only the objects and processes that easily respect the superselection sector constraints have a simple enough description in terms of the CFT.

Other superselection sectors have different descriptions than the boundary CFT. For example, M-theory in the flat 11-dimensional space is another superselection sector of the "whole" string theory that has a nice description if you also compactify a light-like coordinate of radius R. Then the M-theory superselection sector splits into infinitely many additional superselection sectors with the light-like momentum component equal to N/R. Each of them is described by the U(N) BFSS matrix model. Again, it is a completely exact description of the superselection sector.

While the different boundary conditions at infinity lead us to very distinct exact descriptions of the corresponding superselection sectors, we can still verify that all these descriptions represent the same theory. In many of these pictures of string theory, you may often embed configurations with the same large, nearly flat region of space that contains strings and branes whose properties and interactions may be derived.

Whenever two definitions of string theory "overlap" in their predictions what can happen in the bulk, all of their predictions for the objects and processes in the bulk that we can calculate - and there are very many of them - agree for all these overlapping descriptions. So even though the AdS/CFT correspondence wants you to assume very different boundary conditions at infinity than the type IIB matrix model (I mean the superconformal limit of the 2+1-dimensional gauge theory whose status was simplified by the recent membrane minirevolution), both of them allow us to locate the same large type IIB space inside them and calculate the stringy S-matrix for the flat type IIB spacetime, at least in principle.

All the physical phenomena that have been verified exactly yet nontrivially agree and it is very unreasonable to expect that a real discrepancy emerges in the future. We are talking about the same theory - because the finite-volume phenomena always agree once we guarantee that we have reached the same points of the stringy configuration space. However, this uniqueness and universality of the stringy phenomena doesn't change anything about the fact that the full Hilbert space of quantum gravity is split into many completely decoupled superselection sectors whose known exact descriptions "look" very different.

Gauge theory requires no string theory

You may be surprised that all the complicated phenomena of string theory may be derived from something as ordinary as gauge theory. But it is a fact, supported by roughly 5,000 diverse and convincing papers. The richness of "emergent" phenomena that are hiding behind equations as simple as the gauge theory Lagrangian - especially in the "modern" regime when the number of colors and/or the coupling constant is large - is one of the most important conceptual insights of the last decade in theoretical physics.

So we know that gauge theory and string theory - in a particular background, i.e. a particular AdS superselection sector  - are really two different photographs of the same pretty structure. Or a pretty woman, if you find the word "structure" too boring. But that doesn't change anything about the definition of the "gauge theory". In fact, we mean the same thing by the gauge theory as the people did 30 years ago, long before it was seen to be a limit of D-brane physics. Whatever the people were able to properly calculate 30 years ago remained true. But in the last decade, physics has found many new "emergent phenomena" of this gauge theory that imply that the theory actually contains all the rich content of string theory.

A gauge theory itself emerges at many places of string theory as a limit. The low-energy limit of heterotic string theory, M-theory on ADE singularities, or D-brane Lagrangians are just three examples. In these contexts, there are many other physical phenomena - e.g. excited strings - that prevented the people from writing the full definition of the physical laws (because they didn't know what "string theory" exactly was).

However, this complication was circumvented in the case of many superselection sectors, especially the AdS-like ones. If the asymptotic spacetime literally looks like the AdS space - instead of a flat space with a complicated black brane with an AdS-like near-horizon geometry - then we are instructed to go to the very limit of the D-brane physics, indeed. We are left with the ordinary gauge theory without any additional stringy stuff. And this theory can be defined independently of string theory. Nevertheless, it is exactly equivalent to string theory in the AdS space.

If you care, the gauge theory supplemented with the massive open string fields etc. would be equivalent, at least at some qualitative level, to type IIB string theory on the background with a stack of D3-branes that includes both the AdS-like near-horizon geometry as well as the asymptotic Minkowski region. In this version, the duality would become a duality between two types of string descriptions - one of them based heavily on open strings and the other on closed strings. Neither of the two sides would be exactly defined and the content of the duality would be somewhat equivalent to the open-closed dualities that you can get from reinterpreting the cylinder diagram as an annulus.

However, in the exact AdS limit, the open string description does simplify. The symmetry increases to the full superconformal symmetry of the AdS space and the theory becomes so simple that a full definition is known - it is the gauge theory that was known long before the duality revolution in string theory. The holographic duality becomes much more powerful because one side, the CFT, is completely well-defined, while the other, the type IIB string theory on the AdS background, is not quite independently well-defined (except by the gauge theory) but it is still extremely rich and admits lots of qualitative predictions as well as quantitative methodologies to expand physical quantities.

This AdS/CFT duality is useful in both directions. The gauge theory is useful for a complete definition of type IIB string theory on this background - and becomes very useful in extreme circumstances, e.g. in the high-curvature limit that was completely inaccessible e.g. by the supergravity approximation. And the type IIB string theory is extremely useful for the gauge theory because it gives us many new approximate tools and kinds of expansions - based on higher-dimensional gravity and its stringy extensions - that can be used to determine the physics of gauge theory i.e. a theory qualitatively analogous to QCD.

So the gauge theory is not a "natural" description of everything - of all superselection sectors - but it actually is a natural description of everything in quantum gravity that is only constrained by the AdS-like asymptotic conditions at infinity. Everything is allowed to happen in the bulk. The AdS/CFT correspondence forces us to make no assumptions about the character of the "bulk" except that it must satisfy the laws of physics.

Background independence in string theory goes beyond general relativity

In general relativity, the space could suddenly bend and deform. All these things are physically true in string theory because the long-distance limit of string theory is physically equivalent to an extension of general relativity. However, general relativity had some limitations.

For example, when the curvature radius or another radius of a portion of geometry went to zero, you would approach a classically singular configuration and general relativity couldn't tell you what's going on. You couldn't say whether dragons should appear whenever the sphere shrinks. Can the topology or the number of dimensions change at this mysterious point?

However, the classically singular configurations were always guaranteed to be physically inaccurate because at short enough distances, new physics linked to quantum mechanics had to take over. The idea of classical general relativity that the curvature could literally become infinite near the source and that you should still believe the simple geometric description is of course incorrect once you take the quantum phenomena into account.

The main goal of considering quantum gravity is to figure out what actually happens in these extreme circumstances. But that was a goal that was known for decades. What was actually not known were many of the shocking answers - what happens - that were only found recently. Some people only seem to care about the ambitious goals to learn things but they are actually not too interested in the answers. But they're extremely interesting!

For example, it was not known whether a sphere in higher-dimensional spacetime of general relativity could shrink to zero volume and change the topology so that another sphere expands. Now we know the answer. If you embed this thought experiment into a well-defined stringy framework - e.g. type IIB string theory on a Calabi-Yau manifold - you can exactly see that certain types of topology change are inevitably allowed (in type IIA and type IIB, the detailed phenomena that would happen are different). In fact, the transition can be seen to be physically smooth, using another equivalent description.

Topology change is possible. In the same way, it is possible to see that two very different shapes of spacetime can lead to exactly equivalent - i.e. experimentally indistinguishable - physical phenomena. Also, new dimensions can appear in space when a coupling constant becomes large. Internal electric and similar charges can continuously transform into momenta in new directions or winding/wrapping numbers describing the topology of various strings/branes. 

There are usually many ways to look at the same physical situation. And the full quantum theory - that we continue to call "string theory" despite the newly found trans-stringy richness of its content - seems to have very many classical limits.

So the concepts, objects, and phenomena are much more interconnected according to the cutting-edge picture of reality - string theory - than they have ever been. On the other hand, there exist certain "gaps" that will always lead to a segregation - especially the separation of the Hilbert space into superselection sectors. This will never go away.

A description independent of the boundary conditions?

You might be annoyed by the different descriptions of string theory that become more useful - or exact - for particular boundary conditions of spacetime only. Isn't there a universal description that covers everything? Well, that's surely an attractive idea but there are many ways how to specify what we exactly mean that are demonstrably nonsensical. Not all questions are open.

Imagine that the universal description of all of string theory is called M2009.

First of all, it seems almost certain that if M2009 exists, it must be possible to exactly derive things like the N=4 gauge theory in four dimensions as the right description of a particular superselection sector. There must be other ways to derive the BFSS matrix model as a description of a light-like compactified M-theory in 11 dimensions.

It sounds very ambitious to unify the AdS/CFT correspondence and the BFSS matrix theory - because the details why these gauge theories describe quantum gravity are very different at the technical level (even though some people, in my opinion very incorrectly, like to say that Matrix theory is a special case of AdS/CFT) - but it is conceivable that such a unified description exists. There are many other languages that must be derivable from M2009. But maybe, it exists. The possibility to derive so many diverse stringy phenomena from the gauge theory could also look stunning just a decade ago.

Does M2009 contain some "universal" degrees of freedom, a universal Lagrangian? What would it mean? Would it be just another specific Lagrangian with some specific degrees of freedom that just happens to be special? How would you exactly "insert" the information about the right superselection sectors if your plan were to derive the known descriptions of these sectors? What would it mean for the degrees of freedom to be labeled by universal coordinates and indices in a universal spacetime? How would you connect them with the normal degrees of freedom of gauge theory or the worldvolume coordinates?

And if M2009 contains no "specific" degrees of freedom, what are its "primordial" pre-degrees of freedom? What does it mean to have degrees of freedom whose number, identity, or organization is not defined at the beginning? What does it mean for M2009 to organize its own degrees of freedom?

These are very tough, almost religious questions. At least they look so at this moment. But it is not quite impossible that someone will find very crisp answers to all of them soon.

What M2009 would be good for would be to derive all the known useful descriptions we have, to "map" the landscape from a new "satellite" perspective, and to suggest ways how the early cosmology may make portions of the landscape more relevant than others. All of the superselection sectors have a notion of time and usually also space and they respect some kinds of causality and locality - in spacetime or world sheet. 

I personally find it likely that physics only becomes doable once it is possible to derive the existence of "time" (much greater than the fundamental scale) of some type. Space is usually a part of the story, too. When the Universe is compact and finite, at the level of a "cosmology", all possible configurations in string theory are a part of the game and there is no separation to the superselection sectors. But I suspect that in this "compact mushed potatoes", there are no exact physical quantities replacing the S-matrix, either (recall the problems to define observables in de Sitter spaces). I think that a large chunk of time (or spacetime) is needed for accurate questions about physics to become meaningful

So all the interesting superselection sectors of M2009 are connected with a method to derive the existence of a time or spacetime from M2009. Each superselection sector gives you a new type of time. Still, there are way too many superselection sectors - with very chaotic and prescribed boundary conditions at infinity.

The situation becomes much more manageable when you only ask what are the superselection sectors and their vacua (ground states) that respect certain symmetries - for example, the 10-dimensional super Poincaré symmetry. The number of solutions is very small, roughly four one-parameter or two-parameter families (type I/HO, type IIA/M, type IIB, HE/HM). 

Is there a well-defined mathematical problem, M2009 combined with the assumptions about the symmetry, that allows you to derive these four 10-dimensional vacua? A similar mathematical problem (with a reduced number of the spacetime dimensions controlling the Poincaré group) would allow you to derive the exceptional duality symmetry group of supergravity (actually M-theory) on tori.

I still feel that it is useful to imagine M2009 as a strongly coupled generalization of the world sheet conformal field theory. In this sense, M2009 must be a meta-theory that dictates the rules that your rules for model building have to satisfy to be called quantum gravity. One class of the specific solutions to M2009, which is only valid perturbatively, is the whole framework of two-dimensional CFTs (with marginal vertex operators) coupled to gravity (i.e. with Riemann surfaces as world sheets) that can be used as a perturbative description of string theory.

Another solution of M2009 is a class of theories that are defined by their holographic boundary CFTs. Add the BFSS-like matrix descriptions of flat superselection sectors. And there can exist solutions of completely new types. What can M2009 be so that it has so closely physically related, but so philosophically different frameworks to approach physics? I am not sure but I am pretty certain that everything that is related to the basic and universal features of background independence but that is less ambitious than M2009 has already been answered and clarified and the new insights about string theory are critical for these known answers.

And that's the memo.

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reader Doug said...

Hi Lubos,

I can understand your skepticism.

Russian Academy of Science [RAS] physicists are applying mathematical dynamics, particularly in thermodynamics related to plasma physics, dynamos, magnetohydrodynamics and fluids including blood

Springer books for reference, searchable on Amazon:
1 - VV Kozlov, Dynamical Systems X: General Theory of Vortices

2 - SV Alekseenko, PA Kuibin, VL Okulov, Theory of Concentrated Vortices: An Introduction

#2 is especially good, dealing with vorticity [spin], flows [Beltrami ~ helical], sheets, perturbations and filament dynamics.

From the perspective of this physiologist Beltrami flow [Bf] is comparable to Zittwerbewegung [Z] discussed by David Hestenes. Z is about relativistic speeds at quantum distances while Bf is about slower speeds from millimeters likely to parsecs with planets about stars about galactic nuclei among neutrinos and cosmic rays.

Bf also seems applicable to nucleic acids. There is electron flow along the phosphate backbone which likely generates a magnetic field as well as transcription flow.