QFT and QSM in gen. cov. bndry form: ways it could fail?

[marcus]

This seems to be the most interesting development in QG currently. I want to know what you think might present serious obstacles to completing the program. Where could it go wrong?

The idea is that QFT and quantum statistical mechanics (QSM) need to be given a general covariant formulation. We have to avoid the idea of "transition applitude" between "initial" and "final" states because these ideas have no gen. cov. meaning. So what now seems to be the obvious thing to do is generalize the idea of "initial and final" information to the entire boundary of the spacetime region in which the process occurs.

Developing the general boundary approach has been due largely to Robert Oeckl. It can be thought of as alternative to Dirac canonical quantization where the Hamiltonian vanishes identically and process is frozen on a fixed hypersurface slice.

There are enough different people now at work in this approach that I don't feel comfortable labeling discussion by reference to specific papers or authors. So I'm starting a new thread to address the general topic.

 

[marcus]

Here are some links, and some earlier threads, for reference.
Tomita state-dependent (observer independent) time: https://www.physicsforums.com/showthread.php?t=660941
H&R's paper generalizing the concept of equilibrium: https://www.physicsforums.com/showthread.php?t=669658
Oeckl positive boundary form of QT: https://www.physicsforums.com/showthread.php?t=704934
Frank Hellmann's paper: http://arxiv.org/abs/1105.1334
Rennert Sloan paper with response see bottom p. 21 http://arxiv.org/abs/1308.0687

I should note that even if there is broad agreement on the goal of general covariant physics and that GBF (general boundary formulation) is the preferred way to go, there still could be problems with specific implementations. For instance is the EPRL version of spinfoam suitable, or the holonomy version, or the twistor version?
I'm not focusing on questions of implementation, but other people may wish to---or we might get around to that later on.

 

[Physics Monkey]

Without knowing too much about the specific proposals going in, here are some thoughts. I'll be interested to learn more as we go.

Possible objections I see:

1. The proposal isn't really different from specifying initial and final states and (spatial) boundary conditions.

2. Experiments are not done in a generally covariant way.

3. Statistical mechanics beyond conventional equilibrium is very difficult, even with a global time.

4. [With fluctuating gravity] Need to integrate over gravitational field on boundary, not clear how to do so.

5. Boundary assumes locality? Can't be a fundamental formulation.

Some of these possible objections are probably much more serious than others.

 

[atyy]

I think in Bianchi-Haggard-Rovelli "boundary is mixed" the boundary is where there is classical spacetime, because they invoke the equivalence principle.

But if the boundary is something like a generalization of AdS/CFT, then maybe the boundary wave function does not have a geometrical interpretation and spacetime only emerges in the bulk. If such a thing is possible, then maybe the boundary formulation doesn't have to assume locality.

 

[marcus]

Without knowing too much about the specific proposals going in, here are some thoughts. I'll be interested to learn more as we go.

Possible objections I see:

1. The proposal isn't really different from specifying initial and final states and (spatial) boundary conditions.

...

3. Statistical mechanics beyond conventional equilibrium is very difficult, even with a global time.

...

Interesting to get your thoughts on this!
I'll reply piecemeal, one or two possible objections at a time.
3. as you may have noticed they are working out a general covariant notion of equilibrium based on information flow between two processes placed in contact. http://arxiv.org/abs/1302.0724

The challenge to come up with and use an appropriate concept of equilibrium could be a productive one. It's nontrivial ("very difficult" as you say) and I find the results of tackling it in 1302.0724 fascinating.
Under GR two regions in equilibrium can have both different temperature and different rates of time flow. But the two types of difference can balance out. A new understanding of equilibrium (in general covariant context) seems to lead to a deeper understanding both of temperature and time.

1. Perhaps as you say it isn't "really" different but it is somewhat different since one cannot uniquely designate which part of the boundary is initial, and which is final, and which is side conditions.
Specifying what is to be regarded as initial etc. would require a metric or some other prior structure which one doesn't have. So it stikes me as *enough* different that it's intriguing to see how they handle it.
E.g. see http://arxiv.org/abs/1306.5206

 

[marcus]

5. Boundary assumes locality? Can't be a fundamental formulation.
...

Could you expand on this point? (Hopefully) in simple terms, why can't Oeckl's approach provide a fundamental formulation?

He has been developing what he calls the GBF (general boundary formulation) for some 8-10 years now, as a format for quantum theories. It involves assuming that you can study a process by treating it as confined in a compact bounded spacetime region.

I gather that's what you mean by assuming locality---that it's valid to study what goes on in a bounded compact region, rather than being required to formulate a theory of the universe as a whole. Please correct me if that's a mistaken interpretation.

So a theory cannot be fundamental unless it comprises the entire universe?

Would that be because nothing is ever completely isolated and disentangled from the rest of existence, so a picture of anything in isolation ("in a box") is necessarily approximate, hence not fundamental? I could see the justice of objection along those lines, but it wouldn't be the most urgent concern.

What worries me is the thought that I may be missing something more subtle, so I'm hoping you will spell this point out.

 

[atyy]

In #4 I'd assumed that what Physics Monkey meant by "locality" was something like classical spacetime existing, so that an observer could set up a local proper frame that coincided with Minkowskian coordinates - basically some form of the EP is applicable. So that's why I thought Bianch-Haggard-Rovelli's mixed boundary assumed locality - at least at the boundary - since they use the EP to make their proposal.

The second part of #4 was wondering if the Oeckl formulation could still hold without the Bianchi-Haggard-Rovelli EP assumption. If such a thing is possible, it would seem to be a generalization of AdS/CFT.

 

[Physics Monkey]

Indeed, I am very worried about formulating the theory in terms of something like spacetime which is anyway supposed to be emergent.

This analogy has been made before, but it bears repeating: just as one wouldn't quantize a fluid made ultimately of atoms by quantizing the velocity field (except, perhaps, as an effective theory), it seems like putting too much emphasis on the useful semiclassical variables is not a good idea when one is seeking a "microscopic" definition of the theory.

So along these lines, yes, I worry that inside, outside, boundary, etc. have no completely general meaning.

This expectation seems to be born out by ads/cft, where although the boundary is non-fluctuating, bulk locality is definitely an emergent property of the "fluctuating geometry" and need not occur in every state of the theory or in every theory.

All that being said, this objection seems most relevant for gravity. When one has a given background and a conventional qft, then I might imagine that one doesn't have to worry as much. Although it still may be that interactions are non-local, etc.

 

[Physics Monkey]

So a theory cannot be fundamental unless it comprises the entire universe?

I would argue that in a sense this is correct.

Now I admittedly begin from the assumption that there is ultimately some kind of "conventional" quantum system underlying the theory we are trying build. It may be holographic, or look nothing like the eventual macroscopic variables, etc. but I want to assume it exists. I'm very aware that this may be an incorrect assumption.

That said, I do feel that a "fundamental theory" should ultimately define a sensible stand alone quantum system. We may couple it to something else, study the effective dynamics of subsystems, etc. But I like to think that any "fundamental theory", to be called that, should have the property that it COULD be the entire universe. Only question then is, is it our universe?

Example: lattice regulated standard model (no gravity) is a sensible quantum system. Neglecting gravity, we could be living as low energy beings in such a world. Its consistent and well-defined, it just happens to be wrong. And perhaps one can usefully formulate and apply a general boundary formalism to it, but I can define it independently.

 

[Haelfix]

A formalism that is entirely built on locality is already at considerable odds with the available lore on quantum gravity. It's been understood since the 60s that you can't write down local diffeormorphism invariant operators in a theory with gravity, so you have to, at best, live with boundary observables (eg correlators, or an Smatrix). These of course will only be defined in various superselection sectors, but that seems like the best we can do for now.

The Blackhole information paradox has further muddled the water, indicating that locality will need to be violated at a fundamental level in order to preserve unitarity, indicating that naive local field theories w gravity are probably wrong at some level (this includes naive attempts at quantizing gravity like LQG).

Even more recent is the scattering amplitude program, where we can see that there are perfectly sensible ways to write down the world we know in a way that loses manifest locality (presumably to be generalized into a more complete theory where such a concept is merely emergent).

All of these arguments very much point to a program that requires a way to abandon spacetime as we know it like PhysicsMonkey argues, and the need for something more general. At the moment, we really only have AdS/CFT as a general toy model to work with, and this seems like it lacks in explanatory power in many of the crucial questions as to exactly how all of this is supposed to work.

 

[atyy]

Or is it possible that any quantum mechanical theory, from the naive "Copenhagen" point of view is always a subsystem? Even when applied to what we consider "cosmology"?

In AdS/CFT, when the bulk gravity is classical, are the non-gravitational bulk fields quantum mechanical?

 

[MTd2]

It could fail at event horizon, if not enough care is taken:

http://arxiv.org/abs/1308.2665
http://arxiv.org/abs/1308.3697

 

[fzero]

Or is it possible that any quantum mechanical theory, from the naive "Copenhagen" point of view is always a subsystem? Even when applied to what we consider "cosmology"?

In AdS/CFT, when the bulk gravity is classical, are the non-gravitational bulk fields quantum mechanical?

The bulk gravity is only ever classical in an extreme limit, but even so, the limit in which the rest of the theory is classical is actually different. Let's recall how parameters are related in the case of IIB on AdS5. The gauge theory $N$ is the 5-form flux and is related to the volume of the 5-sphere (which is in turn related to the AdS radius $R$), so

$$N \sim ( M_P R)^4.$$

The limit in which gravity is classical is one in which the radius of curvature is large in Planck units, so this is the limit of large $N$ in the gauge theory.

The gauge theory coupling on the other hand is directly related to the string coupling

$$ g^2 N \sim g_s N \sim ( M_s R)^4,$$

where the last expression uses the relationship between the string scale, string coupling, and Planck scale. Stringy corrections, which include bulk scalar and gauge interactions, involve the string scale, rather than the Planck scale. So there is a range of values for the string coupling, where gravity is classical, but quantum string interactions are important.

In the limit where both $$N, g^2 N$$ are large, everything in the bulk is classical.

 

[marcus]

Without knowing too much about the specific proposals going in, here are some thoughts. I'll be interested to learn more as we go.

Possible objections I see:

1. The proposal isn't really different from specifying initial and final states and (spatial) boundary conditions.

2. Experiments are not done in a generally covariant way.

3. Statistical mechanics beyond conventional equilibrium is very difficult, even with a global time.

4. [With fluctuating gravity] Need to integrate over gravitational field on boundary, not clear how to do so.

5. Boundary assumes locality? Can't be a fundamental formulation.

Some of these possible objections are probably much more serious than others.

P.M. thanks for these points re Oeckl boundary formulation. Atyy responded to point #5

Or is it possible that any quantum mechanical theory, from the naive "Copenhagen" point of view is always a subsystem? Even when applied to what we consider "cosmology"?

by suggesting that quantum theory (as we know it, with classical observer outside the box establishing conditions, keeping track of time and probabilities) is inherently a theory of subsystems. It does not apply to whole universe, because no outside observer.

Over the course of a number of years various people (including Hartle) have made a considerable effort to adapt or reformulate quantum theory to suit it for modeling the whole universe. Surely one can see the boundary formalism as limited because it has a boundary. Maybe that makes it more adapted to study quantum gravity in the sense of geometry at microscopic level. I suppose it could potentially be "fundamental" in one sense, but not in others.

I responded to your points #1 and #3.

Interesting to get your thoughts on this!
I'll reply piecemeal, one or two possible objections at a time.
3. as you may have noticed they are working out a general covariant notion of equilibrium based on information flow between two processes placed in contact. http://arxiv.org/abs/1302.0724

The challenge to come up with and use an appropriate concept of equilibrium could be a productive one. It's nontrivial ("very difficult" as you say) and I find the results of tackling it in 1302.0724 fascinating.
Under GR two regions in equilibrium can have both different temperature and different rates of time flow. But the two types of difference can balance out. A new understanding of equilibrium (in general covariant context) seems to lead to a deeper understanding both of temperature and time.

1. Perhaps as you say it isn't "really" different but it is somewhat different since one cannot uniquely designate which part of the boundary is initial, and which is final, and which is side conditions.
Specifying what is to be regarded as initial etc. would require a metric or some other prior structure which one doesn't have. So it stikes me as *enough* different that it's intriguing to see how they handle it.
E.g. see http://arxiv.org/abs/1306.5206

It's worth noting that Oeckl's formalism is based on topological manifolds---continuous maps, no metric or differential structure. So it does not assume a lot of structure at the outset, more or less shapeless blobs. But it does assume that much.

Indeed, I am very worried about formulating the theory in terms of something like spacetime which is anyway supposed to be emergent.

This analogy has been made before, but it bears repeating: just as one wouldn't quantize a fluid made ultimately of atoms by quantizing the velocity field (except, perhaps, as an effective theory), it seems like putting too much emphasis on the useful semiclassical variables is not a good idea when one is seeking a "microscopic" definition of the theory.

So along these lines, yes, I worry that inside, outside, boundary, etc. have no completely general meaning...

That's an interesting reservation. It's about something inherent in the approach. We want to study a process. What process? The one that happened or will happen HERE, inside this boundary.
The one happening in this bounded 4d region.

It seems hard to think of anything more "nondescript" than that, more minimalist, taking less structure to specify what process you're talking about.

Oeckl's approach apparently grew out of 1980s work by Segal, Witten, Attiyah, and others on topological quantum field theory.

 

[marcus]

...
2. Experiments are not done in a generally covariant way.
...

General covariance is a gauge symmetry.

A feature of theories therefore, rather than experiments. Not sure this qualifies as an obstacle to success of general boundary approach.

 

[marcus]

I think we've responded to four of the five possible objections offered by Physics Monkey, which I for one appreciated being provided because it gave something to discuss and got us thinking.

In the meanwhile I've thought of another potential obstacle:

Wouldn't Oeckl's GBF (general boundary formulation) become unusable if a black hole were to form in the bulk region?

Because doesn't the forming of a BH add a section of boundary?

Of course this new section of boundary is entirely to the FUTURE of where the fictional singularity (presumably somehow resolved) used to be. Whatever happens beyond the hole cannot backreact on us. Maybe for some purposes it can simply be ignored. But it worries me that we seem to have no access to or control over the full boundary in the case where a BH occurs in the bulk region being studied.

 

[Physics Monkey]

General covariance is a gauge symmetry.

A feature of theories therefore, rather than experiments. Not sure this qualifies as an obstacle to success of general boundary approach.

Perhaps it is an abuse of language, but this kind of trivial "general covariance" was not what was meant.

The physical point is that as far as gravitational experiments are concerned, we sit on our little ball in space and collect data essentially from "infinity", e.g. radiation from distant regions. There is a nice classical background and we are not, as far as I am aware, doing experiments that require a sophisticated specification of generalized boundary data, e.g. we count photons from time t1 to time t2 and analyze their spectra.

Hence although a generalized boundary formalism may be of interest conceptually, it doesn't seem to be necessary for the kinds of experiments we can achieve currently.

 

[Physics Monkey]

It seems hard to think of anything more "nondescript" than that, more minimalist, taking less structure to specify what process you're talking about.

I suppose I have a different conception. Living in a world of lattices where one can easily engineer non-local interactions, it seems straightforward to me that in general everything depends on everything else.

Even continuity assumes some dimension of space, and dimensionality is meaningful only so long as interactions have an appropriate structure, e.g. not on a complete graph.

Since we expect that the final formulation of gravity must not involve any preformed notion of spacetime, it must not be possible to build such a formulation using local concepts, even at the level of topology. It seems to me that morally we should always start from the complete graph and see locality emerge.

 

[marcus]

...
Even continuity assumes some dimension of space, and dimensionality is meaningful only so long as interactions have an appropriate structure,...
...

Certainly in this case we have dimensionality because Oeckl says up front his regions are topological manifolds (e.g. of dimension 4).
That means locally homeomorphic to neighborhood in R^d (e.g. R⁴).

So no metric, no differential structure, no geometry. Bi-continuous maps to patches of R⁴ but otherwise shapeless and smooth-less.

I recall in 2005 Lee Smolin came to the Loops conference at Potsdam AEI talking about how the world could have started out as a COMPLETE GRAPH (every vertex connected to every other) and by a process of severing connections it kind of relaxed into more familiar latticework (but with some disordered locality, or maybe that idea came later).

I can appreciate that from your latticeworld perspective the topological manifold seems like a lot of structure already---dimensionality even!

I suppose I have a different conception. Living in a world of lattices where one can easily engineer non-local interactions, it seems straightforward to me that in general everything depends on everything else.

Even continuity assumes some dimension of space, and dimensionality is meaningful only so long as interactions have an appropriate structure, e.g. not on a complete graph.

Since we expect that the final formulation of gravity must not involve any preformed notion of spacetime, it must not be possible to build such a formulation using local concepts, even at the level of topology. It seems to me that morally we should always start from the complete graph and see locality emerge​

Thats without doubt a philosophically interesting and (I suspect) mathematically fertile approach.

 

[marcus]

"General covariance is a gauge symmetry.

A feature of theories therefore, rather than experiments. Not sure this qualifies as an obstacle to success of general boundary approach."
​

Perhaps it is an abuse of language, but this kind of trivial "general covariance" was not what was meant.

The physical point is that as far as gravitational experiments are concerned, we sit on our little ball in space and collect data essentially from "infinity", e.g. radiation from distant regions. There is a nice classical background and we are not, as far as I am aware, doing experiments that require a sophisticated specification of generalized boundary data, e.g. we count photons from time t1 to time t2 and analyze their spectra.

Hence although a generalized boundary formalism may be of interest conceptually, it doesn't seem to be necessary for the kinds of experiments we can achieve currently.

Let me know if I'm wrong about this. For sure when you do experiments there's a choice of coordinates and all. What I meant by "general covariance" is that there is the group of diffeomorphisms and a theory can have the feature that its equations, its action etc, is symmetric under action of that group.
It's a feature that a theory can have, just like it can have Lorentz symmetry.

So two different representations of the world are the SAME if they can be mapped one to the other by an element of the group. Their difference is physically meaningless. So I was calling it "gauge". (That could be abuse of language, but you know what I mean.)

I don't think of diffeomorphism invariance (or general covariance) as trivial though.
================
I've been thinking about your objection and have come to a better understanding. You are asking what is the point of the GBF?

When do you need a gen cov formulation? When your thermodynamics involves the energy of the geometry and the entropy of the geometry (as well as the energy and entropy of other things.) You know about the Tolman temperature of the geometry. Time flow and temperature varying in different parts of gravitational field.

Or when you want to include the geometry as a general covariant quantum field theory. The geometry itself should be a field interacting with other fields. And there is no one preferred idea of time (because the spacetime geometry is one of the players).

So you point out that we don't do experiments in which the geometry is dynamic and interactive, taking part in the statistical mechanics.

We don't do experiments where QFT has to be general covariant (to include geometry). It is always sufficient to have Lorentz covariance on fixed background.

So why should one ever want to include dynamic interactive geometry in one's theory?

Well I think one motive is to be able to satisfactorily resolve the cosmological singularity and the one at the pit of a black hole. As you probably know there are thermodynamic problems about the start of expansion and the early universe. One wants to be able to include geometry as well as matter fields.

My feeling is that IF Oeckl's approach fails to prove useful it will be because it turns out not to be applicable to a very few stubborn problems like this.

I don't think it is making a bid for general use. You may have reflected on this already and could have come to similar conclusions. Anyway that's a very good question: "what's it good for?"