Is there any definitive answer on unitarity in quantum gravity

[petergreat]

Given that in GR there's no gauge invariant choice of "equal-time slices", how is unitarity formulated in quantum gravity? I guess the problem may be absent if spacetime is asymptotically flat. But what happens in other cases? In AdS/CFT, the notion of unitarity comes from the CFT side. Is this the only situation in quantum gravity where we have a well-defined notion of unitarity? Is there any definition (read: not proof) of unitarity on the bulk side, without resorting to the CFT dual?

Is there any consensus on this issue? Is it a settled question?

 

[petergreat]

Another issue is that there exists spacetime with no space-like Cauchy surface, so it's not possible to predict the future even if you have complete knowledge about the system on an initial "equal-time slice". Does this pose further problems to a definition of unitarity?

 

[Fra]

It doesn't answer your question, but conceptually the way I see the origin of unitarity is that it applies to expectations.

Ie. ALL expected evolutions are unitary, with respect to the observer encoding the expectations. But since expectations and the observers structures are constantly evolving as new information is arrives, sometimes causing the prior hilbert structure to be deformed then there is no objective or finite unitarity.

But this is not a problem since neither is ther in the genereal case an objective expectation. All expecations are observer dependent, and there is no external~objective description of the relation between these expectations.

That unitarity is still almost recovered as objective in many cases is just because we are observing subsystems of the universe.

The BH paradox I conceptually envision beeing solved this way: information is lost(or hidden), but it is also recovered(or more properly reGAINED, reconstructed) as the BH evaporates. The idea that BH radiation contains no information is then only with respect to the BH itself; not with respec to the environment - it akes a sufficiently complex observer to decode the BH radiation (in principle that is). And it's the external observer that concludes unitarity or not. If you ask the BH itsel, it just radiates random information.

The exact implementation of this, and how such view still be be example consistent with SM remains to lay out in detail, but at least conceptually this is not something I'm sleepless over.

I think the original idea in QM, seen as an extrinsic measurement theory where the unitary evolution is fundamental, it's hard to conceptually picture how non-unitariy can make sense. And how information ca be destroyed and created. I think it can't, if you inist on the extrinsic description. But then, is this extrinsic description sensible for an inside observer? I think not, so the paradox is due to confusion, and instead the "problem" is just to find the improved intrinsic measurement theory having these new traits.

/Fredrik

 

[tom.stoer]

Could you please define what you mean exactly be unitarity?

 

[petergreat]

Could you please define what you mean exactly be unitarity?

At time t the state of the system (universe) is given by a Hilbert space vector. At time t+dt the state of the system is obtained by left-multiplying the previous state vector by a Unitary matrix, usually given by exp (-i H dt). The Hamiltonian representation of the unitary evolution operator is important in my question. All I want to make sure is that it is indeed unitary.

The above description relies on the fact that we can choose "equal-time slices" of the universe at time t and t+dt. I think this is problematic in GR because there is no good way of choosing (global) equal-time slices.

 

[petergreat]

@Fra: Thanks for the answer. I sort of get your point. As long as information is never lost, and as long as expectations evolve in a manner that conserves probability, then we have unitarity, at least in the view of a local observer.

However, this is on a very conceptual level. A more mathematical answer is still desirable. Perhaps we should take AdS/CFT literally and assert that a global notion of unitarity is only possible in a holographic sense? Perhaps we can even assert that the basic structure of quantum mechanics must be modified, if one insist on finding a description of the universe in a direct manner, rather than resorting to the CFT dual? In string theory all the "scattering amplitudes" are S-matrices relating asymptotic states, and one can't calculate finite time evolution. This seems to make sense.

However, such an answer look very problematic for non-AdS universes.

 

[Fra]

However, this is on a very conceptual level. A more mathematical answer is still desirable.

Absolutely, that's why I wrote it doesn't answer your question. I personally consider the full answer to your question to be an open question.

But I think trying to understand what we are asking, and what form of the answer we expect is a strong guide.

For example:

Could you please define what you mean exactly be unitarity?.

At time t the state of the system (universe) is given by a Hilbert space vector. At time t+dt the state of the system is obtained by left-multiplying the previous state vector by a Unitary matrix, usually given by exp (-i H dt). The Hamiltonian representation of the unitary evolution operator is important in my question. All I want to make sure is that it is indeed unitary.

Unitary time operator, is how current QM implements "conservation of probability", so when were talking about "unitarity" in the general sense, possibly also applying to the yet not known QG-theory, then I take it to be synonymous to conservation of probability, or somehoe the idea that the observer systems state is ALWAYS and for all future konwn to be constrained to a fixed configuation space (hilber space).

I'm suggesting that the idea of a fixed timeless configuration space, can't be right - except as a special case. The reason is that no inside observer can in finite time infer that.

If you look exactly at how a state space is inferred, then it's either just assumes from preconceptions, or simply spanned by history. The former is not a proper infernece and has no place in the reconstruction IMO. The latter view is interesting but here one must note exactly how history is truncated and encoded inside observers. One also sees that the state space is constantly challanged, and there simply is no such thing as an eternal timeless statespace as it's not inferrable.

Instead these things are only recovered in special cases, asymptotically in infinite time given infinitly complex observers (meaning it ONLY apples to subsystems).

So current QM, really does correspond to an idealized special asymptotic state that is in neither way applicable to the general case.

But it's equally clear why and when thse special cases do make sense for FAPP - it's when an sufficinetly complex observer do studies small subsystems of it's environment. Then these theories are recovered as limits.

This insight, strongly guides is in the search for the full, mathematical reconstruction.

I don't wish to post any immature details, I think it will take me many years before I would be willing to present my contributions.

@Fra: Thanks for the answer. I sort of get your point. As long as information is never lost, and as long as expectations evolve in a manner that conserves probability, then we have unitarity, at least in the view of a local observer.

What I think is that to each local observer, the EXPECTED time evolution is always unitary in the differential sense. Meaning you should see the differential future as tangent plane defined on the current state. Unitarity must hold in the tangentplane, but not globally except in special cases.

Perhaps we should take AdS/CFT literally and assert that a global notion of unitarity is only possible in a holographic sense? Perhaps we can even assert that the basic structure of quantum mechanics must be modified, if one insist on finding a description of the universe in a direct manner, rather than resorting to the CFT dual? In string theory all the "scattering amplitudes" are S-matrices relating asymptotic states, and one can't calculate finite time evolution. This seems to make sense.

However, such an answer look very problematic for non-AdS universes.

I am convinced that thte structure of QM needs to be modified. That's not the question for me. The question is how. And I agree that it's a delicate problem. For me it's easy to see why the notion of fixed hilbert space of the entire universe just makes no sense, because it's not inferrable by an inside observer; thus it has no place in the inference description.

The main problem is how to recover some effective level of objectivity and global notions, from a picture which is both local and subjective. This is where the evolutionary ideas come in for me.

/Fredrik

 

[Fra]

Is there any definition (read: not proof) of unitarity

I'd like do distinguish between the actual future change, and the EXPECTED change.

I think the EXPECTED change is always conserving information, by construction. Exactly what the best mathemtical representation will be like I'm not sure yet but roughly the current state, encodes the expectation... and without further information the expected evolution must preserve this information. Any deviation will be explained as a revision of the prior (this is the measurement).

So I am seeking for

1. A representation for the general observer structure, and how this makeup is one-2-one with expectations of it's own environment. (currently it's looking like a system of non-commuting combinatorically generated structures? Here information is preserved by construction, since it would be irrational for the EXPECTATION to violate it's premise. So the EXPECTED evolution MUST be information preserving, or we have a logical contradiction. The important thing is IMHO to distinguish between a observer dependent EXPECTATION (where the notion of info preservation is defined) and some realize sense to actual evoltuion. THe latter is an irrational view that I think we have hard to release ourselves from.

2. How this strucuture makes a random walk, according to the expectations.

3. How these structures are revised in the light of new information.

4. How selection takes place when such structures interact, and what kind of "nash equilibrium" points we can expect. If the SM model should pop out as a nash equilibrium (or a generalisation thereof) it would be bingo of course.

/Fredrik

 

[Fra]

I've often insisted on the distinction between a descriptive view and the decision problem.

One doesn't need to know economic theory to apprecaite this, I just referred to Nash equilibrium to hint the reasoning using.

If someone isn't sure what that is, just see how wiki describes it:

"The Nash equilibrium concept is used to analyze the outcome of the strategic interaction of several decision makers. In other words, it is a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation. Instead, we must ask what each player would do, taking into account the decision-making of the others."
-- http://en.wikipedia.org/wiki/Nash_equilibrium

How this relates to PHYSICS and physical interactions, as opposed to economical systems is what I've tried to convey in plenty of past posts.

The economical players here are of course; physical subsytems/observers, observing other subsystems. The decision making process and action (placing bets) correspond to the inferences systems makes about each other ie physical interaction.

The ACTION and microstructure of an atom, is one-2-one with a game strategy in an evolving game.

The point where the analogy fails is that there is not objective description of the came and now given statespace of strategies. But the association to game theory really IMHO help thinking in the right terms, rather than outdated geometrical terms, like physicists tend to do.

/Fredrik

 

[tom.stoer]

At time t the state of the system (universe) is given by a Hilbert space vector. At time t+dt the state of the system is obtained by left-multiplying the previous state vector by a Unitary matrix, usually given by exp (-i H dt). The Hamiltonian representation of the unitary evolution operator is important in my question. All I want to make sure is that it is indeed unitary.

The above description relies on the fact that we can choose "equal-time slices" of the universe at time t and t+dt. I think this is problematic in GR because there is no good way of choosing (global) equal-time slices.

Assuming that space-time allows for a global topology like R*M³ there are infinitly many spacelike foliations. You can chose any spacelike foliation you like. In the canonical formalism the coordinate time t is given by a direction which is locally orthogonal to the spacelike M³.

But there is no time-evolution operator. The reason is that t is a coordinate time only and that chosing the foliation is physically irrelevant. Therefore the operator H which acts as a tanslation operator in t-direction is a constraint operator that vanishes on physical states, ie. H|phys> = 0. Therefore a time translation U=exp(-iHt) is weakly equal to the the identity.

That means there is no problem with unitarity but with time itself. One has to reconstruct a "physical time" which is of course not given by the coordinate time (which is arbitrary as we have seen above).

 

[crackjack]

Just like for the boundary CFT, one can also talk of unitarity for sigma-model CFT for the bulk string theories. I would also like to know if unitarity has been proven for pure-spinor formalism.

 

[tom.stoer]

I guess the problem with my explanation is that AdS/CFT and canonical quantum gravity are too far away from each other. So my ideas regarding time and time-evolution cannot be applied to AdS/CFT.

 

[Fra]

But there is no time-evolution operator. The reason is that t is a coordinate time only and that chosing the foliation is physically irrelevant.

Yes, but is'nt it also in this statement that there is some good amount of fog?

I think all agree that simply changing lables makes no differece (coordinate system).

But I think we should distinguish between choosing coordinate system, with choosing the observer. If the coordinate system can be thought as one-2-one with an inside observer view, then this are less trivial.

Some people seemt to be happy to also think that choosing a particular observer is physically irrelevant; and instead only relations between observers is physical. But this does not make sense to me; because a measurement theory cannot be sensibly formulated without this choice.

The question we should ask to clarify is this: Is the choice of observer physically irrelevant or not? Let's first note: WHOSE choice is this?

Usually we think of it as a gedanken choice made by the theorist... and the theorist is external to the system and thus this choice IS irrelevant, but...

...how does this choice look like, if we consider it beeing made by an inside observer?

/Fredrik