Friday, October 10, 2008

Observables in quantum gravity

Moshe Rozali has kindly initiated a blog discussion about this important topic. Let us join, too.


The goal of every quantum-mechanical theory is to predict the probabilities that particular physical quantities - "observables" - will take one value or another value after some evolution of the system, assuming certain initial conditions.

For example, we are using Schrödinger's equation to predict the probabilities that a particle appears at a certain point of the screen in a double-slit experiment. Alternatively, we are predicting the probability that the (decay products of a) Higgs boson and/or superpartners will appear inside a pixel of the LHC detector, and so forth.

Mathematics of quantum mechanics makes it inevitable that observables have to be identified with linear operators on the Hilbert space of allowed states. The allowed values of observables are the eigenvalues of the corresponding operators and the probabilities are squared absolute values of the appropriate complex amplitudes.

In the case of mechanics, the "fundamental" observables are usually x,p - position and momentum - but one may construct more complicated ones such as the angular momentum J (the generator of rotations) or the Hamiltonian (the generator of time evolution). Each particle typically carries its own x,p,J, terms contributing to H, and so forth.

In quantum field theory, the natural observables "look" different. While quantum field theory still implies the existence of particles with their momenta and (usually not quite localized) positions, every quantum field theory is a quantized version of a field theory, after all.

So the natural observables are fields such as phi(x,y,z,t) or E(x,y,z,t) - the latter is the electric field at a given point of spacetime. (Let's neglect additional dimensions of space.) More precisely, these objects are "operator-distributions" rather than operators and you should integrate them over some regions (with test functions) to get genuine operators whose commutators are functions rather than distributions.

But that's not a conceptually difficult technicality: every physicist who knows how to manipulate with distributions (such as the delta-function) may deal with the operator distributions, too. She can continue to call them "operators".

The operators phi(x,y,z,t) tell us something about the state of the system - the field phi - at a given point. So this operator has nothing to do with points in space that are separated: in fact, the (graded) commutator of phi(x,y,z,t) with an operator at a point separated by a space-like interval must vanish. That's why we call the operator "local". Special relativity implies that spatially separated regions can't influence each others so it is not surprising that such local operators exist.

Quantum field theory is naturally rewritten in terms of these local operators. The Hamiltonian is typically an integral over space. Other operators that can be measured may also be constructed as integrals of functions of the local operators (and their derivatives) over space. Even if we add Yang-Mills gauge invariance, it is still possible to construct local operators associated with a point in space that are gauge-invariant and whose eigenvalues are thus fully measurable.

(Gauge-non-invariant operators depend on the gauge i.e. on a convention: they cannot be directly measured. For example, the magnetic field strength in electromagnetism is gauge-invariant but the vector potential itself is not.)

Adding gravity

What happens if we add the metric tensor and gravity? Well, something does. In general relativity, we must also add the diffeomorphism group into the full package of gauge symmetries. Only gauge-invariant operators are independent of conventions. Only gauge-invariant operators can be directly measured.

Some laymen at Moshe's blog are deeply confused about the very basic questions here. Special relativity is not invariant under diffeomorphisms - unless we express the special-relativistic theory with additional, completely redundant, unphysical degrees of freedom (a metric tensor whose curvature must vanish and which is therefore "non-dynamical" in this case).

Without this useless extension, special relativity only allows the invariance under Poincaré transformations and puts inertial frames (but not other frames) on equal footing. States are not required to be annihilated by any generators of transformations of spacetime. The fact that the commenter named "iphigenia" is not able to see that the metric tensor is non-dynamical in special relativity and it (or diffeomorphisms) cannot therefore cause any dynamical problems in special relativity (unlike GR) doesn't mean that people with 60 IQ points above "iphigenia" are also unable to throw the unphysical degree of freedom out.

On the other hand, general relativity is invariant under diffeomorphisms. Both the metric tensor at each point and the diffeomorphism symmetry are necessary in every description of general relativity that keeps the important symmetries manifest. Because the spacetime is generically curved, there exists no natural subset of "inertial frames" and all coordinate systems are equally good to express the equations of general relativity.

This diffeomorphism symmetry turns out to be an important problem in general relativity where it cannot be thrown away. Before we added gravity, the quadruple of numbers (x,y,z,t) in phi(x,y,z,t) described a very physical point in spacetime. In a different reference frame, you would associate the point with a different quadruple of coordinates. But once you pick your coordinates, there is a one-to-one map between the quadruples of coordinates and the "objective" points in spacetime: this map is independent on the state of the system.

The previous sentences fail to hold once you add gravity. Why? Because the numbers (x,y,z,t) are just coordinates that can be reparameterized in an arbitrary way, without changing the physics. So by saying what (x,y,z,t) are, you don't really identify any "objective" point in space. So you don't know which operator phi(x,y,z,t) you can possibly mean. The freedom to reparameterize the coordinate is large enough for (x,y,z,t) to mean anything you want or anything you don't want.

You might say that we faced the same problem in special relativity, too. There were also different acceptable coordinate systems over there. However, what's important is that we could agree upon a few conventions about our coordinates and then all quadruples (x,y,z,t) meant something specific. It's not the case in general relativity because you would need to reveal an infinite amount of information to determine what point is associated with any (x,y,z,t).

This infinite difference is a reason why the reparameterizations in special relativity are "global symmetries" while those in general relativity are "local symmetries". States can't be required to be invariant under global symmetries - because the energy etc. would have to be zero all the time - but they should be required to be invariant under local symmetries - e.g. because you wouldn't know how they evolve with time (the local, gauge transformations can always depend on time).

Moreover, the choices of coordinates in general relativity cannot be "canonical". In special relativity, the spacetime is flat so if you determine the coordinates of a few points, you may just assume that the coordinates are extrapolated linearly across the spacetime that is linear, too.

However, the spacetime is curved in general relativity: for general states (and their gravitational fields), the coordinates have to be "inherently non-linear". Moreover, the precise curvature of the spacetime does depend on the state of the physical system. For example, if an atom is found at one place, it has a different gravitational field - different profile of spacetime curvature - than if it is localized at another point. Dead and alive cats have different gravitational fields, too.

So the fact that you can't choose any "canonical" coordinates - and therefore "objective" gauge-invariant observables similar to phi(x,y,z,t) - depends on two complications:
  1. the physical observables must be gauge-invariant i.e. independent of diffeomorphisms; that means that they can't depend on arbitrary coordinates
  2. there are no "simple" non-arbitrary coordinates because the spacetime is curved according to the matter inside.
At Moshe Rozali's blog, various people have asked which of these circumstances actually explains that you can't define any simple gauge-invariant local observables in quantum gravity: in reality, it is only the union of both of them that makes things hard.

Now, I feel the urge to say that you could imagine that you can define some "objective" coordinates in a curved spacetime of general relativity, too. For example, choose a reference point P that is very far from all matter (somewhere near infinity). And then you can parameterize points in spacetime by specifying
  1. in which direction Omega (angular coordinates) from the point P the point is sitting (a small region around P is flat so Omega behaves just like in non-gravitational physics)
  2. what is the proper distance or proper time (the length of the shortest geodesic) from P to the point you want to describe
However, this particular convention is difficult to realize because geodesics in spacetime are complicated, especially for generic mass distributions. Moreover, the coordinates won't be unique in general: recall that in the case of gravitational lensing, two different light rays can get from a galaxy to your eye (along two different geodesics). Moreover, the definition assumes that the metric tensor is a good-behaving, nearly "classical" tensor that exists everywhere in space. In reality, the metric tensor is wildly fluctuating, especially at very short distances. If you wanted to follow the geodesics accurately, it would be much more messy than the classical ideas indicate.

For these reasons, attempts to define privileged coordinates in spacetime based on geodesics, proper distances, and extrapolations are not very well-defined, reliable, convergent, or convenient.

Incidentally, string theory gives us a better way to define privileged coordinates, the light cone gauge. In the light cone gauge, all fields or string fields are naturally interpreted as functions of a "light-cone time", x^+. The remaining coordinates, x^- and the transverse x^i coordinates, could a priori be redefined as well. But the Hamiltonian in the light cone gauge - the generator of translations in x^+ - automatically gives you a preferred value of all these coordinates.

The light-cone gauge coordinates are working well for the superselection sector of the flat space - all states that converge to a flat spacetime at infinity. You may talk about the evolution in the "bulk" or the evolution after "finite time". But let us assume that the reader doesn't like any gauge-fixed description of the physics because it obscures the "natural symmetries" and it could become problematic if the density of matter is high (and the fields are strong). Let's imagine that the reader wants the democracy between all the coordinates (and the Lorentz symmetry) to remain manifest.

Scattering and holography

Well, if this is her dream, the situation changes dramatically. In quantum field theory without gravity, we had the operators phi(x,y,z,t) and we could have computed their correlators - expectation values of their products in the vacuum state - for arbitrary values of (x,y,z,t) for each operator. By Fourier transform, these became the Green's functions of the external momenta and the external momenta (p_x,p_y,p_z,p_t) could have been any off-shell momenta.

In gravity, we are forced to talk about the on-shell amplitudes only - those that are relevant for scattering. Why is it so? It is because the general correlators are hard because the nature of the operators phi(x,y,z,t) etc. is ill-defined. However, things simplify if you study scattering.

The initial and final states in the scattering process correspond to safely, spatially separated particles such as gravitons. Because they are so separated, the gravitational field around them is very weak and, in fact, universal. So it actually makes a perfect sense to define the initial or final state with a graviton that has a certain momentum (determined up to the accuracy of 1/X where X is an arbitrarily huge distance that separates the gravitons): the spacetime at infinity - where the incoming particles arrive or the outgoing particles leave - behaves just like in special relativity: you may forget about diffeomorphism and curvature over there. You can rightfully assume that there exist coordinates in which the space at infinity is flat. In these coordinates, things are as clear as in special relativity.

The graviton might still have a gravitational field around it, even when it is at infinity, but you don't need to know any details about it to define the external states. It is enough to say what the on-shell momenta are. By locality, which is a feature of the theory, you may also argue that there must exist multi-particle states in which the individual particles have independent momenta and are still safely separated. The further the particles are, the better approximation yours is.

For this reason, every meaningful quantum theory of gravity must be able to calculate the scattering amplitudes of gravitons (and perhaps other particles that are present) arbitrarily accurately if they scatter from infinity, assuming that the theory admits spaces where particles can flee to infinity.

The case of AdS/CFT is very clear in this respect. Let's talk about an AdS5/CFT4 pair, to be very specific. There is also a compact five-manifold (a sphere etc.) but let us neglect these extra dimensions because they're compact: all fields may be expanded into Kaluza-Klein spherical harmonics, anyway.

So the quantum gravitational theory must be able to compute the scattering of gravitons whose 5-momenta are on-shell i.e. light-like. With this constraint, there are only 4 independent parameters for each such momentum. It turns out that by the AdS/CFT dictionary, the scattering amplitudes are fully encoded in correlators of (off-shell) local operators (for gravitons, it's the stress-energy tensors) on the boundary.

Note that each such an operator on the boundary depends on four coordinates, e.g. (x,y,z,t), which is exactly the right number to parameterize light-like momenta of gravitons in five dimensions. The gravitational theory has five large dimensions but only four of them are accessible for the computation of exact correlators: that's why one dimension is effectively lost. In other words, it is a manifestation of holography.

On the other hand, the boundary theory allows you to access all Green's functions so it is not holographic. It is the very presence of gravity - and diffeomorphisms and/or black holes - that makes certain theories holographic. Recall that holography implies that the number of degrees of freedom (entropy) can't exceed the surface area in Planck units. But the Planck area is proportional to Newton's constant, "G", so if you turn off gravity, it goes to zero and the inequality becomes vacuous: "the entropy should be less than infinity".

Holography manifests itself in many other ways: the black hole is the final stage of a gravitational collapse of any localized system. Its entropy is only proportional to the area of its event horizon but by the second law of thermodynamics, it can't be lower than the entropy of any system that led to the birth of the black hole. It follows that all localized states in the given volume must have entropy lower than the area: the area in the Planck units tells you how many degrees of freedom you need to describe anything in the region. Whatever happens there is encoded in a "hologram" on the surface.

Holography: does it depend on string theory?

Holography emerges at many places of string theory. When you try to compute the amplitudes for strings, you can see that the results are only meaningful - conformally invariant on the worldsheet - if the external particles are on-shell (the dimension of the vertex operators must be (1,1) to keep them marginal and integrable over the worldsheet). The AdS/CFT is another manifestation of the holographic nature of string theory: whatever happens in the bulk (with gravity) may be described by a non-gravitational theory on the boundary.

So is it OK to say that holography only holds in string theory? I think it would be an incorrect conclusion. String theory is a very "specific", well-defined theory of quantum gravity where similar aspects appear to be crisp and clear. However, I think it is obvious that many arguments supporting holography have nothing to do with "strings" per se. Holography is a property of any consistent theory of quantum gravity (which is probably a term equivalent to "string theory" anyway, but even if it is not, the first part of this sentence should be valid).

So which observables do we have?

As we have suggested, in the flat Minkowski space or the AdS space, the scattering amplitudes at infinity are the observables to be studied. And they can be computed by stringy methods - from the worldsheet correlators or the boundary correlators. Does it mean that string theory (or any theory of quantum gravity) can't say anything about finite regions of the bulk?

Well, it depends on the accuracy you want. If you want completely well-defined results and you are not ready to accept a huge set of conventions that define your privileged coordinates, such as the light-cone gauge or worse, it is only the scattering "from infinity" that is completely well-defined. However, if particles scatter from distances comparable to 10^{-18} meters, like those on the LHC, you should realize that this distance, while small for humans, is still gigantic in comparison with the string scale or the Planck scale.

It means that all collisions at the LHC may be viewed as on-shell collisions from infinity. Yes, there is also physics that can't be reduced to collisions, such as the analysis of the spectrum of hadrons. But for this physics, four-dimensional gravity is pretty irrelevant, up to a very tiny error. So the on-shell limitations of quantum gravity won't cripple your ability to "practically" calculate any realistic situation.

The higher energy scale you choose and the closer you approach the Planck scale, the more important the scattering experiments become for your observational tests. Near the Planck scale, the curvature of space and the fluctuations of the spatial geometry may become important but the scattering will become the only doable method to probe this physical regime. Even the tiniest microscopic black holes must be studied by scattering. (And the big ones are described well by classical general relativity, with the rest of the matter living on this curved classical background.)

I should emphasize that there are many more ways to determine non-scattering physics out of the full theory. For example, you may always derive a low-energy approximation of your theory and treat it classically or semiclassically. This can tell you a lot of things about "local" physics that doesn't obviously reduce to scattering. But in some sense, all of this physics is encoded in the scattering amplitudes.

De Sitter space

Another problem is that not all spaces have a region at infinity where it is easy to define the identity of particles with a certain momentum. For example, the energy density of our space seems to be dominated by the cosmological constant. Assuming that the cosmological constant is the right explanation of the observed "dark energy", this emptiness will get even worse and our Universe will be increasingly similar to de Sitter space. The spatial slices of this space look like sphere and your particles can never escape "quite" to infinity even though 13.7 light years is pretty far relatively to the Planck length.

Nevertheless, if you academically insist that your theory should calculate some quantities that are absolutely exact and, in principle, testable with an arbitrary accuracy, de Sitter space will show that you are far too immodest. A certain degree of uncertainty - such as the random thermal radiation coming from the cosmological horizon - seems to be an innate feature of de Sitter space. This uncertainty (and the thermal radiation) is too weak to matter for any conceivable experiment we can do today or in any foreseeable future but it seems to be there.

Once again, I must admit that the comments above are not a proof of a no-go theorem. There can exist a very specific Hilbert space with very well-defined observables whose evolution makes complete sense in de Sitter space. But because it seems clear that we can never measure things absolutely accurately in a de Sitter space anyway, it is questionable whether we really want and need a theory that can predict things absolutely accurately. Maybe we don't. If we don't want it, it still remains puzzling what it means to have a full theory that inherently predicts all probabilities inaccurately.

Preon degrees of freedom in the bulk

Some people could propose that the metric tensor is composed out of other objects or fields that are defined in the same "bulk". It can be a composite of gauge fields, superconducting stuff, preons, or any other buzzword of the same type. Well, I doubt it is the case. But more importantly, I don't think that any of these assumptions would solve the basic problem with the gravitational degrees of freedom, as explained at the beginning. When you try to define the observables like g_{mn}(x,y,z,t) in quantum gravity, the main problem is not the g_{mn} part but the (x,y,z,t) part.

Also, I find any idea that tries to present the metric tensor as a privileged degree of freedom to be misguided and obsolete. The metric tensor is clearly just a low-energy effective degree of freedom arising from a theory that contains and must contain much more stuff. In perturbative string theory, closed strings can carry infinitely many "Hagedorn" excitations that are in principle equally important as the graviton mode. The only way to make "graviton" look special is to go to long distances.

Beyond the perturbative series, there are always many black hole microstates. A pole (or branch cut) in a scattering amplitude corresponding to an intermediate black hole is as important as a pole arising from an intermediate graviton. At some level, when you think out of the box of low-energy approximations, black hole microstates must be equally important as excited string modes or the graviton mode itself. Well, black holes look like "composites" of gravitons (and their fields) but I feel that this proposition only holds - as a form of tautology - if you assume that the gravitational field is fundamental while the "black hole microstate fields" are not.

String field theory

What about some kind of string field theory? String field theory is a way to rewrite string theory that is as similar to an ordinary quantum field theory as you can get. However, it must contain infinitely many elementary fields, corresponding to all excited states of the string, and a correspondingly enhanced gauge symmetry principle. Some people like to say that string field theory is as "background-independent" (in the stringy sense) as general relativity in the first place.

There are problems with this assertion. First, when we talk about the background, we should allow all fields - including the scalar fields - to take any values they can take, without making the description much less appropriate. The dilaton is perhaps the most important scalar field in string theory that is special in the perturbative series. But string field theory is so heavily based on "strings" that it is only good for a weak coupling (the dilaton goes to minus infinity). It even seems likely that string field theory doesn't tell us anything new about nonperturbative physics of string theory than other perturbative approaches. For example, attempts to write type IIA string theory as string field theory don't seem to imply that the strong-coupling limit has the 11-dimensional Lorentz invariance.

That's bad and the "special role" of the weak string coupling certainly shows that string field theory is not "quite" background-independent. Even if you solved the difficult technical problems of closed string field theory, the background independence wouldn't quite hold.

Matrix theory or matrix string theory are radically opposite in this sense: they are equally well-defined for any value of the string coupling, including the infinite value (in type IIA, you get back to M-theory captured by the original BFSS model). In this sense, it is "background-independent" because it deals equally well with any value of the coupling. On the other hand, you need a different model for each value and only models for superselection sectors that are asymptotically flat (or some pp-waves) are known: that cripples the "normal", geometric part of the background independence.

But both AdS/CFT and Matrix theory show that space and positions in space are not fundamental concepts. They're emergent. Systems that don't look anything like AdS5 or the 11-dimensional Minkowski space M11 end up behaving exactly as AdS5 or M11. That's really amazing. The people outside string theory who like to talk about emergent space or background independence or any of these nice words usually end up with one of these two outcomes:
  • they have a model that already has an underlying space where objects are "localized": whenever it is so, they can't really say anything about the question "what are the diffeomorphism, gauge-invariant operators"
  • they have a model in which no space exists at the beginning but they cannot show that a smooth space emerges, either: well, it's probably because it doesn't emerge in which case they have thrown the baby with the bath water
Which of these two problems occurs sometimes depends on what you are still ready to call "space". At any rate, if you end up with one of the results above, your theory is useless for the conceptual questions of quantum gravity. It is only interesting to describe a theory of emergent space if the space is (apparently) not there at the beginning but if it is (demonstrably) there at the end. ;-)

When you look at the successful descriptions of quantum gravity - and all of those that are known as of today were written by string theorists - they roughly fall into two groups:
  • descriptions that make unitarity manifest and allow you to calculate exact amplitudes in principle for any values of moduli etc.
  • descriptions that make the geometrical interpretation (and Lorentz symmetries) manifest but that suffer from various limitations (e.g. assumptions of weak coupling or low energies).
There seems to be a trade-off going on here. The more manifest the geometrical interpretation of your description is, the more limited the description is in its ability to deal with situations with high energy, strong coupling, high density, or rapid time dependence. That shouldn't be surprising because a geometrical interpretation of a spacetime is often known to emerge. But it often emerges at low energies and (for compact dimensions) in decompactification corners of the moduli space only.

You shouldn't be shocked that if you try to study more extreme situations, simple, geometrically manifest descriptions disappear. Well, they really disappear: it is not a bug of your description but rather a true fact about Nature. The "local" degrees of freedom associated with a large manifold only occur if you can show that the size of the manifold is large in string/Planck/brane units and if all physical quantities are changing much more slowly than by 100% per Planck length.

If you want to find a description - a choice of degrees of freedom - that are equally good in every situation of quantum gravity, it must be an amazing meta-geometric description that secretly knows about all the dualities and about all the ways how degrees of freedom can emerge from the interactions of others and how some of them can become light in various limits.

Frankly speaking, I am convinced that such a description, if it exists, cannot be based on any "predetermined set of degrees of freedom" that just happen to exhibit a rich variety of different behaviors. I think that the very composition and organization of the degrees of freedom that you should start with must be a solution to a set of consistency criteria. I am convinced that people should spend much more time by trying to understand which consistency criteria actually imply that things must go in one way and not another. At some moment, they will have to get back to the bootstrap program.

And yes, I am convinced that the best, most universal description of the degrees of freedom in quantum gravity will have no space - and probably no time - to start with. But because space - and especially time - is so essential to design any system of physics that qualitatively resembles what we know, we will learn some rules that allow you to define what you mean by space and especially time in the pre-geometric structure.

There is no guarantee that such a structure exists or will be found in five years or fifty years. But people simply have to keep on trying because such a setup would become an extremely robust pillar of any research of quantum gravity in particular and theoretical physics in general.

And that's the memo.

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