I What Are the Empirical Challenges Facing Quantum Gravity Theories?

[Nullstein]

Thanks @Nullstein! Do you also have a reference for more details?

You're welcome! You can check out the reference above, but the case of the harmonic oscillator is also discussed here.

 

[Fra]

Your paper is also from 2004.

You seem right, I focused on the content first. I don't know however, why a paper that looks published 2004, has the date 2018 written inside it? A typo? Look at the first page.

/Fredrik

 

[gentzen]

You seem right, I focused on the content first. I don't know however, why a paper that looks published 2004, has the date 2018 written inside it? A typo? Look at the first page.

No, not a typo, just a typical artifact of how arXiv compiles and caches tex. You can find similar artifact in thousands of other papers on arXiv. Try not to make any unwarranted conclusions from it.

 

[Demystifier]

only countable sums of spin network states are allowed in the LQG Hilbert space

Why? In ordinary QM a wave packet such as a Gaussian is a superposition of uncountably many plane waves, so why isn't something similar allowed in LQG?

If your reply is that plane waves are not in a Hilbert space, then I note that they are in a rigged Hilbert space. In that case, can a rigged Hilbert space be constructed for LQG and can it solve the problem you are referring to?

 

[Nullstein]

Why? In ordinary QM a wave packet such as a Gaussian is a superposition of uncountably many plane waves, so why isn't something similar allowed in LQG?

If your reply is that plane waves are not in a Hilbert space, then I note that they are in a rigged Hilbert space. In that case, can a rigged Hilbert space be constructed for LQG and can it solve the problem you are referring to?

Riggend Hilbert spaces are used in LQG as well, but this is a different situation. You can use plane waves in QM and they are elements of the rigged Hilbert space, but in order to form a physical state, you have to integrate them over all $k$, rather than take an uncountable sum over all $k$ (that's not possible). On the other hand, you have, e.g., the harmonic oscillator eigenstates, labeled by natural numbers $n$, which form a basis of the actual Hilbert space and in order to form physical states, you take (countable) sums over all $n$. In the case of harmonic oscillator eigenstates, it doesn't make sense to integrate over $n$. Now if you define a wave packet as an integral over uncountably many plane waves from the rigged Hilbert space, then the resulting state can still be expanded into countably many harmonic oscillator eigenstates.

The spin network states in LQG are like the harmonic oscillator states in QM. They are members of the actual Hilbert space (not a rigged Hilbert space) and labeled by discrete numbers. One has to sum them, rather than integrate. However, they form an uncountable basis, because the LQG Hilbert space is not separable. It's then just a fact of mathematics, unrelated to any physical theory, that sums of uncountably many terms only exist if at most countably many terms are non-zero. So the expansion of any state is necessarly an at most countable sum of spin network states. You can't expand a state into something like an integral of spin network states. Now even if you introduced a rigged Hilbert space on top of the LQG Hilbert space and defined states as integrals over an uncountable number of generalized states from this rigged Hilbert space, the resulting state would be in the actual Hilbert space and could be expanded in the spin network basis with at most countably many non-zero coefficients.

Objects	How many are there?	Where do they live?	How do we obtain physical states?
Harmonic oscillator eigenstates	Countably many	(separable) Hilbert space of QM	Sum of countably many terms
Plane waves	Uncountably many	Rigged Hilbert space on top of QM Hilbert space	Integral over k
Spin network states	Uncountably many	(non-separable) Hilbert space of LQG	Sum over countably many terms
Elements of a potential rigged Hilbert space on top of the LQG Hilbert space	Uncountably many	Rigged Hilbert space on top of LQG Hilbert space	Integral over something

 

[Fra]

I think topic also calls for reflection on what is distinguishable and measurable from the perspective of an observer/agent. This is a weak point in as far as I recall, most theoretical camps. The avoidance of these question is clear even when you read Rovellis RQM. Already there one can smell a suspicious application of QM. At least it's far from conceptually clear. Yet one continues to build onto a questionable principles.

Where is the physical "hilbert space" or the corresponding information, "encoded" in LQG?

/Fredrik

 

[Demystifier]

They are members of the actual Hilbert space (not a rigged Hilbert space) and labeled by discrete numbers. One has to sum them, rather than integrate. However, they form an uncountable basis, because the LQG Hilbert space is not separable.

I don't understand that on the level of set theory. How can an uncountable set be labeled by discrete numbers?

 

[Nullstein]

I think topic also calls for reflection on what is distinguishable and measurable from the perspective of an observer/agent. This is a weak point in as far as I recall, most theoretical camps. The avoidance of these question is clear even when you read Rovellis RQM. Already there one can smell a suspicious application of QM. At least it's far from conceptually clear. Yet one continues to build onto a questionable principles.

Where is the physical "hilbert space" or the corresponding information, "encoded" in LQG?

LQG is just a regular quantum theory and you can interpret it according to your favorite interpretation, so new problems don't really arise regarding that in LQG. The new interpretational problem that arises in LQG (and in most other quantum gravity theories for that matter) is the problem of time.

I don't understand that on the level of set theory. How can an uncountable set be labeled by discrete numbers?

Well, in practice, it's really labeled in terms of a set of embedded graphs, equipped with the discrete topology (no two graphs are in any way similar) and angular momentum quantum numbers on the edges. That's a discrete, but uncountable set. If you want to stick to numbers, you could use the well-ordering theorem and instead label it by uncountable ordinal numbers (again equipped with the discrete topology). But anyway, the point is that the index set is discrete.

To put it completely in set theoretic terms: Any Hilbert space is isomorphic to a sequence Hilbert space $l^2(\kappa)$, where $\kappa$ is some cardinal number. A basis for this space is given by $e_\omega$, where $\omega <\kappa$ are the ordinal numbers smaller than $\kappa$. The most general vector in this space can be written as $v=\sum_{\omega<\kappa} c_\omega e_\omega$ with at most countably many non-zero $c_\omega$.

 

[jim mcnamara]

Fair warning. Sigh. We seem to be back on topic, which is Physics, not funding Physics. Please continue in that vein. Thank you.

 

[*now*]

..., because the LQG Hilbert space is not separable.

Was a reference provided specific to Rovelli when this was raised earlier in this thread?

 

[Nullstein]

Was a reference provided specific to Rovelli when this was raised earlier in this thread?

Yes, the references given in post #45 address Rovelli of course.

(If you wonder about the model given in Rovelli and Vidotto: This is a spin foam model that doesn't attempt to implement continuous spacetime symmetries in the first place. They are supposed to emerge only on macroscopic scales. Spin foam models start from a discretized version of classical GR and just quantize it, very similar to lattice QFT. The Hilbert space is modeled on fixed graph only, given by the choice of discretization of classical GR. Superpositions of different graphs are impossible.)

 

[*now*]

Thank you, those references also seem quite old and might not account for some recent developments.

This talk from a few years ago seems to expand on some of the issues in the OP paper in the first hour, although I could have a bad link that doesn’t let me watch all of it- hope it works ok here
Also, from what I have seen, at around 50 minutes or so, a list of some developments is displayed in a slide, introduced as to be taken with a smile.

 

[Nullstein]

Thank you, those references also seem quite old and might not account for some recent developments.

Well, the Hilbert spaces used in LQG/Spin foams are still the same today, so the arguments are still valid today. What developments are you thinking of? Can you give some references?

 

[martinbn]

Well, the Hilbert spaces used in LQG/Spin foams are still the same today, so the arguments are still valid today. What developments are you thinking of? Can you give some references?

In the Zakopane lectures, Rovelli constructs and explicitely says that the Hilbert space is seperable.

 

[andresB]

So they can work on whatever they want to work on, without any need for writing proposals for their research?
I am skeptical...

People get paid for doing research?! nobody told that to my 3rd world university.

Joke aside, the issue is not in the highest ranks, pretty much every professor in theoretical physics I know works in whatever he/she wants. The problem is at the Ph.D. and postdocs levels. People tend to go to where the money is, and, in my experience looking for postdocs, there is a disproportionate amount of money and jobs in superstrings and the like. Hence, people tend to flock to those topics resulting in more money being invested in recruiting even more people. It is a "positive" feedback loop.

 

[Nullstein]

In the Zakopane lectures, Rovelli constructs and explicitely says that the Hilbert space is seperable.

That's the same model as the one given in Rovelli and Vidotto. It's formulated on a lattice and doesn't implement any spacetime symmetry at all. There is only the internal Lorentz symmetry at the vertices.

There is no way out. All Hilbert spaces in LQG/Spin Foam models fall under one of the following two cases:

1. The Hilbert space contains uncountably many graphs and states on two different graphs are orthogonal. Then the Hilbert space is non-separable. Continuous symmetries may be implemented, but no nontrivial states can be invariant under a continuous group of symmetries.
2. The Hilbert space is modeled on a lattice, then it may be separable, but no continuous group of spacetime symmetries can be implemented.

The problem is basically that in LQG/Spin Foams, geometry is excited only on lattice-like structures like foams or graphs, i.e. subsets of the manifold that are nowhere-dense. By the Baire category theorem, no neighborhood of any point can be the countable union of nowhere-dense sets. And since states can at most be defined on a countable set of graphs, most points of the neighborhood are not equipped with any kind of geometry and thus no neighborhood can be locally isometric to a region of Minkowski spacetime. In order to circumvent this simple result, there is no other option than to allow geometry to be excited on neighborhoods and no LQG-type model does that.

 

[MathematicalPhysicist]

People get paid for doing research?! nobody told that to my 3rd world university.

Joke aside, the issue is not in the highest ranks, pretty much every professor in theoretical physics I know works in whatever he/she wants. The problem is at the Ph.D. and postdocs levels. People tend to go to where the money is, and, in my experience looking for postdocs, there is a disproportionate amount of money and jobs in superstrings and the like. Hence, people tend to flock to those topics resulting in more money being invested in recruiting even more people. It is a "positive" feedback loop.

Loop that's the keyword here...
We are living in a loop.
Also regarding the ideas in this community.

 

[MathematicalPhysicist]

By community, I am referring to the wide community of BSM.

 

[bland]

Government. (Which gets money from taxpayers.)

Your research is untainted by funds from former 45.

 

[Demystifier]

Your research is untainted by funds from former 45.

What's former 45?

 

[Fra]

But there is no justification for enlarging the symmetry subgroup.

The general motivation they mention seems to be that the moduli space is too large, and unphysical. After all, what is the original justification for the continuous symmetry? This is why I asked "where the hilbert space is encoded" in the other post.

Separable Hilbert space in Loop Quantum Gravity
Winston Fairbairn, Carlo Rovelli, Oct 25, 2018

"However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. "

It seems well put to describe the continuum mathematics as a poor choice of fine tuning of the math.

If one add another constraint - that any observer for example have a finite perspective, or a finite information processing resources; set aside the details, this must necessarily make the set of possibilities smaller; as it must be encodable by the observer. I haven't seen a proper argumentation for this however in any of Rovellis papers before, but the general sense in what they say makes sense, and it's rather the original uncountable set of possible states that is what should be questioned.

But what seems be be going on to a certain extents in most approaches (not just LQG), is IMO starting with something that does not make sense (formal expressions which are pathological from the perspective of "inside agents"), then we are force to "make up" arguments to "tame it". Any arguments are likely flawed or ad hoc. The better way should IMO be to step back before we lost track of what we are doing and ended up with formal expressions that are extrapolated way outside their domain of corroboration. For example the whole continuum business, may well be an approximation of something more fundamental - rather than the other way around, which seems to be the more common attitude.

/Fredrik

 

[Nullstein]

The general motivation they mention seems to be that the moduli space is too large, and unphysical. After all, what is the original justification for the continuous symmetry? This is why I asked "where the hilbert space is encoded" in the other post.

What's unphysical is the singular structure of the theory. The non-separability of the Hilbert space is just a consequence of that. Usual QFTs with non-singular observables don't have this problem. The diffeomorphism symmetry is well motivated, because it's a consequence of classical GR. Any enlargement of this symmetry group is an ad-hoc assumption. If there wasn't the problem with the non-separable Hilbert space, nobody would bother to make it.

"However, the continuous moduli labeling these classes do not appear to affect the physics of the theory"

This claim is pretty unsubstantiated, given that not even a single physical observable (i.e. one that commutes with the constraints) is known in this version of LQG.

[...] it's rather the original uncountable set of possible states that is what should be questioned.

It is being questioned, but the solution is certainly not to just define the problem away. And if you check the literature, you will find that nobody in the LQG community actually uses the Hilbert space proposed in that paper in practice.

 

[*now*]

... Can you give some references?

I think the talk gave some developments and references, including those external to the theory like the BMV tests
Witness gravity’s quantum side in the lab | Nature
Phys. Rev. Lett. 119, 240401 (2017) - Spin Entanglement Witness for Quantum Gravity (aps.org)
Also concerning testing, this is a paper concerning time
Frontiers | On the Possibility of Experimental Detection of the Discreteness of Time | Physics (frontiersin.org)
There may be allowances for incorporation of possible quantum or related aspects, but does the literature considered here include papers for instance like these
[2007.12635] Edge modes of gravity. Part III. Corner simplicity constraints (arxiv.org)
https://arxiv.org/abs/2104.12881 ?

 

[Nullstein]

I think the talk gave some developments and references, including those external to the theory like the BMV tests
Witness gravity’s quantum side in the lab | Nature
Phys. Rev. Lett. 119, 240401 (2017) - Spin Entanglement Witness for Quantum Gravity (aps.org)
Also concerning testing, this is a paper concerning time
Frontiers | On the Possibility of Experimental Detection of the Discreteness of Time | Physics (frontiersin.org)
There may be allowances for incorporation of possible quantum or related aspects, but does the literature considered here include papers for instance like these
[2007.12635] Edge modes of gravity. Part III. Corner simplicity constraints (arxiv.org)
https://arxiv.org/abs/2104.12881 ?

In what sense do these articles provide any new insight into the separability of the LQG Hilbert space? The first three articles are concerned with experimental testing only. And the research by Freidel et al. on edge modes is an independent approach to developing a theory of quantum gravity and so far mostly classical analysis. Little is known yet about the potential quantum gravity theory that is supposed to arise from this. The fourth paper of the series, which is presumably supposed to be on Hilbert space aspects, has been announced, but not appeared yet, so even if you want to count this new approach towards the LQG family of theories, no conclusions about the separability of the Hilbert space can be drawn so far.

 

[mitchell porter]

allow geometry to be excited on neighborhoods

I find myself wondering what this dictum actually means. That the observables in question are associated with open sets? With submanifolds?

 

[Nullstein]

I find myself wondering what this dictum actually means. That the observables in question are associated with open sets? With submanifolds?

It means, as you have guessed, that the observables are associated (at least) with open sets. In QFT, operators can be smeared with test functions that are defined on extended, 4-dimensional regions and you can probe arbitrarily small regions with that. In loopy theories, operators are smeared along lower-dimensional objects such as edges or faces. These objects are nowhere dense subsets of spacetime, so no countable union of them can form an extended, 4-dimensional region. Since all states in the LQG Hilbert space can be obtained by repeatedly acting on the vacuum with these singularly smeared operators, no geometry can be excited on extended, 4-dimensional regions, because only countably many terms are allowed.

 

[Fra]

What's unphysical is the singular structure of the theory.

Yes, this is unphysical, but why? I think it's not because physicists computations crash? I think the biggest problem with singularities, is that the are related to infinite information, in a sense that isn't easily renormalized.

But there are some infinites in other theories, that coincidentally are easily renormalized, but this coincidental success induces an unfortunatey illusion of that it makes universal sense. But the problem for me is the conceptual one with how a finite agent can _relate_ to arbitrary amount of information - in finite time even. So technically, even the curable infinites, are conceptually suspicious. This is why I see it as "unphysical", from the perspective of an observer/agent, which itself is made of matter.

The diffeomorphism symmetry is well motivated, because it's a consequence of classical GR. Any enlargement of this symmetry group is an ad-hoc assumption.

At least from my perspective, it is not trivial to heuristically carry over classical symmetries to an inferential framework where the standards are different(and better). This heuristic reasoning works sometimes, and sometimes not. In GR, I am not convinced that it's well motivated in the context of measurment theory with inside observers. That is "ad-hoc" to me.

/Fredrik

 

[martinbn]

Coming back to this, I have some questions.

That's the same model as the one given in Rovelli and Vidotto. It's formulated on a lattice and doesn't implement any spacetime symmetry at all. There is only the internal Lorentz symmetry at the vertices.

There is no way out. All Hilbert spaces in LQG/Spin Foam models fall under one of the following two cases:

1. The Hilbert space contains uncountably many graphs and states on two different graphs are orthogonal. Then the Hilbert space is non-separable. Continuous symmetries may be implemented, but no nontrivial states can be invariant under a continuous group of symmetries.
2. The Hilbert space is modeled on a lattice, then it may be separable, but no continuous group of spacetime symmetries can be implemented.

The problem is basically that in LQG/Spin Foams, geometry is excited only on lattice-like structures like foams or graphs, i.e. subsets of the manifold that are nowhere-dense. By the Baire category theorem, no neighborhood of any point can be the countable union of nowhere-dense sets. And since states can at most be defined on a countable set of graphs, most points of the neighborhood are not equipped with any kind of geometry and thus no neighborhood can be locally isometric to a region of Minkowski spacetime. In order to circumvent this simple result, there is no other option than to allow geometry to be excited on neighborhoods and no LQG-type model does that.

About 1. I assume you have something specific in mind, because it cannot be true in general without anything additional. For example take any group you like and let it act trivially, then all states are invariant. Do you have a group and and action in mind?

About 2. I don't understand this one. Why would be a spacetime group of symmetries be implemented? The graphs are not a lattice in a apriori space-time. Isn't the space-time and the symmetries supposed to emerge somehow?

 

[Nullstein]

Yes, this is unphysical, but why? I think it's not because physicists computations crash? I think the biggest problem with singularities, is that the are related to infinite information, in a sense that isn't easily renormalized.

The use of the word "singular" here doesn't refer to something becoming infinite. It just means that the spacetime geometry is modeled on structures whose dimension is lower than the dimension of the spacetime manifold.

About 1. I assume you have something specific in mind, because it cannot be true in general without anything additional. For example take any group you like and let it act trivially, then all states are invariant. Do you have a group and and action in mind?

Yes, of course I'm thinking of non-trivial group actions. But more specifically, the action of the diffeomorphism group in this formalism is given by $(U_\phi\psi_\gamma)(\vec g) =\psi_{\phi(\gamma)}(\vec g)$, where $\phi$ is a diffeomorphism. And unless $\phi=\mathrm{id}$, this action has the peculiar feature that $U_\phi\psi_\gamma$ is orthogonal to $\psi_\gamma$.

About 2. I don't understand this one. Why would be a spacetime group of symmetries be implemented? The graphs are not a lattice in a apriori space-time. Isn't the space-time and the symmetries supposed to emerge somehow?

A spacetime is a manifold equipped with a metric. The graphs are embedded into the spacetime manifold. The metric becomes a quantum field instead of a classical field. (LQG gives meaning only to coarse functions of the metric, smeared along lower dimensional submanifolds.) The points that don't lie on the graph are still there in the spacetime manifold and diffeomorphisms still act on the whole manifold. Such points just don't carry any geometry due to the peculiar nature of the theory. Now the hope is to construct states such that e.g. $\left<\hat g_{\mu\nu}(x)\right>$ (or at least the smeared versions of it) resembles classical solutions to the EFE with some quantum corrections. You can then ask for example if there are diffeomorphisms $\phi$ such that something like $\left<(\phi^*\hat g)_{\mu\nu}(x)\right> = \left<\hat g_{\mu\nu}(x)\right>$ holds. (It is not clear what properties really are desirable, but this is one reasonable thing one could ask for). Then you could call $\phi$ a quantum spacetime isometry.

 

[Fra]

It just means that the spacetime geometry is modeled on structures whose dimension is lower than the dimension of the spacetime manifold.

Do you see this as a physical problem? How?

/Fredrik

 

[Nullstein]

Do you see this as a physical problem? How?

I explained it in my earlier posts. It leads to difficulties with the existence of semiclassical states and the implementation of continuous symmetries.

 

[Fra]

The graphs are embedded into the spacetime manifold. The metric becomes a quantum field instead of a classical field. (LQG gives meaning only to coarse functions of the metric, smeared along lower dimensional submanifolds.) The points that don't lie on the graph are still there in the spacetime manifold and diffeomorphisms still act on the whole manifold. Such points just don't carry any geometry due to the peculiar nature of the theory.

But I think the potential point is that this embedding, is not physically justified. It's an artifact of the mathematical formalism (that is well justified for OTHER domains).

This may be semantics, but I think this is not a physical problem, it's a problem of the choice of mathematics to describe the models, and it's an open question. Ie. there is not physical requirement, that we "must have" a continuum in the Planck domain. The embedding is a mathematical one, and problems with that, is not a physical problem.

/Fredrik

 

[Nullstein]

But I think the potential point is that this embedding, is not physically justified. It's an artifact of the mathematical formalism (that is well justified for OTHER domains).

This may be semantics, but I think this is not a physical problem, it's a problem of the choice of mathematics to describe the models, and it's an open question. Ie. there is not physical requirement, that we "must have" a continuum in the Planck domain. The embedding is a mathematical one, and problems with that, is not a physical problem.

Sure, we don't know what the physics at the Planck scale is like and it may well be discontinuous. But Rovelli's point in the article in the OP is that we should abandon Lorentz-violating theories. Also the non-uniqueness of the discretization is a problem and may render the theory non-predictive if there isn't some kind of universality.

 

[mitchell porter]

It seems like one issue here, is whether you can get "Lorentz symmetry from superposition", to coin a phrase.

Consider a two-qubit superposition. The underlying observables are discrete, both in number and in spectrum, yet the space of quantum states is a continuum (a Riemann sphere) and possesses continuous symmetries, because superpositions have complex coefficients and complex numbers are a continuum.

Do any LQG theorists (or other quantum gravity theorists) explicitly say that they can obtain Lorentz or Poincare symmetry in an analogous way? Or perhaps as the n->infinity limit of something like that?

Penrose's spin networks are supposed to give you full rotational symmetry in such a limit, and I think one dream for twistor theory was that it would do the same for relativistic boosts too?

 

[Nullstein]

It seems like one issue here, is whether you can get "Lorentz symmetry from superposition", to coin a phrase.

Consider a two-qubit superposition. The underlying observables are discrete, both in number and in spectrum, yet the space of quantum states is a continuum (a Riemann sphere) and possesses continuous symmetries, because superpositions have complex coefficients and complex numbers are a continuum.

Do any LQG theorists (or other quantum gravity theorists) explicitly say that they can obtain Lorentz or Poincare symmetry in an analogous way? Or perhaps as the n->infinity limit of something like that?

Penrose's spin networks are supposed to give you full rotational symmetry in such a limit, and I think one dream for twistor theory was that it would do the same for relativistic boosts too?

Well, that's exactly what we have been discussion so far. The singlet state $\frac{1}{\sqrt 2}\left(\left|\uparrow\downarrow\right>-\left|\downarrow\uparrow\right>\right)$ is rotationally invariant, because you can expand the rotated state again in the up/down basis and it turns out to be the same. But in LQG, if you have a spin-network state $\psi_\gamma$ modeled on a graph $\gamma$ and you apply a diffeomorphism $\phi$ to it, then the transformed state $\psi_{\phi(\gamma)}$ is orthogonal to the initial state $\psi_\gamma$ (unless $\phi=\mathrm{id}$) and hence has no chance to be equal to the initial state. Under no circumstances can two nonzero orthogonal vectors be equal. The next question is: Can a superposition of spin-network states, such as $\sum_n c_n\psi_{\gamma_n}$ be invariant? This can at most be true for a discrete group of transformations, because the sum can contain at most countably many terms and a continuous group will create new spin-network states that weren't present in the original sum and thus again be orthogonal.

 

[maline]

The graphs are embedded into the spacetime manifold. The metric becomes a quantum field instead of a classical field. (LQG gives meaning only to coarse functions of the metric, smeared along lower dimensional submanifolds.) The points that don't lie on the graph are still there in the spacetime manifold and diffeomorphisms still act on the whole manifold. Such points just don't carry any geometry due to the peculiar nature of the theory.

Why do you claim that the manifold "exists", and in what sense? That is not the way I usually hear people talking about LQG. They usually speak as if the amplitudes for the various graphs and their vertex/edge observables are the full content of the state; i.e. "all that is".

I also don't understand why we are discussing diffeomorphism invariance at all. I thought the graph observables were supposed to correspond to spacetime volumes as identified by measurable criteria, such as GPS readings. In that case, the diffeomorphism freedom (really just a redundancy, like all gauge freedom) has already been factored out.

 

[Nullstein]

Why do you claim that the manifold "exists", and in what sense?

Why do you claim the opposite? I claim it, because that's how it is and how it is written in the literature. Plenty of references have been given in this thread, not only by me but also by others. Spin networks / foams are embedded lower dimensional structures in the spacetime manifold. Quantum gravity is the theory of the quantized metric field $g_{\mu\nu}(x)$, just like quantum electrodynamics is the theory of the quantized vector field $A_\mu(x)$ and so on. All these fields live on a manifold, quantization doesn't change that.

I also don't understand why we are discussing diffeomorphism invariance at all. I thought the graph observables were supposed to correspond to spacetime volumes as identified by measurable criteria, such as GPS readings. In that case, the diffeomorphism freedom (really just a redundancy, like all gauge freedom) has already been factored out.

Diffeomorphism invariance is discussed in quantum gravity, just like gauge invariance is discussed in any other quantum gauge theory. By the way, no local observables exist in GR. All Dirac observables are necessarily non-local (see Torre, "Gravitational Observables and Local Symmetries"), so GPS readings can't constitute observables in QG. In fact, not a single Dirac observable is known so far, except in the very restricted setting of asymptotically flat spacetime.

 

[maline]

By the way, no local observables exist in GR. All Dirac observables are necessarily non-local (see Torre, "Gravitational Observables and Local Symmetries"), so GPS readings can't constitute observables in QG.

"Local observables" at some point in the manifold don't exist, precisely because you can do a diffeomorphism and move everything to somewhere else in the manifold. But the observable "curvature at the point identified by GPS readings (a,b,c,d)" is physically well defined. One could hope to have a theory where all of the quantum observables corresponded to invariants of this sort, and I was under the impression that LQG claims to do this. (After all, the whole idea of "loops" is based on the Wilson loop as an invariant operator in gauge theory...)

 

[Nullstein]

"Local observables" at some point in the manifold don't exist, precisely because you can do a diffeomorphism and move everything to somewhere else in the manifold. But the observable "curvature at the point identified by GPS readings (a,b,c,d)" is physically well defined.

No, that is not enough to define an observable in GR. Diffeomorphism invariance doesn't suffice. The observable also needs to commute with the Hamiltonian constraint.

One could hope to have a theory where all of the quantum observables corresponded to invariants of this sort, and I was under the impression that LQG claims to do this. (After all, the whole idea of "loops" is based on the Wilson loop as an invariant operator in gauge theory...)

Wilson loops in LQG are only invariant under the additional internal symmetry group that arises when the theory is formulated in terms of vielbein fields instead of the metric. They are not invariant under the Bergmann-Komar group, i.e. they don't commute with the other constraints of the theory.

 

[Fra]

Also the non-uniqueness of the discretization is a problem and may render the theory non-predictive if there isn't some kind of universality.

I agreee this is a key problem, but I find it conceptually a more tractable and rational quest than the similar uniqueness problem of ambigous ways to cure poorly defined formal expressions which are similarly unpredictive and conceptually lost.

/Fredrik

 

[martinbn]

Yes, of course I'm thinking of non-trivial group actions. But more specifically, the action of the diffeomorphism group in this formalism is given by $(U_\phi\psi_\gamma)(\vec g) =\psi_{\phi(\gamma)}(\vec g)$, where $\phi$ is a diffeomorphism. And unless $\phi=\mathrm{id}$, this action has the peculiar feature that $U_\phi\psi_\gamma$ is orthogonal to $\psi_\gamma$.

A spacetime is a manifold equipped with a metric. The graphs are embedded into the spacetime manifold. The metric becomes a quantum field instead of a classical field. (LQG gives meaning only to coarse functions of the metric, smeared along lower dimensional submanifolds.) The points that don't lie on the graph are still there in the spacetime manifold and diffeomorphisms still act on the whole manifold. Such points just don't carry any geometry due to the peculiar nature of the theory. Now the hope is to construct states such that e.g. $\left<\hat g_{\mu\nu}(x)\right>$ (or at least the smeared versions of it) resembles classical solutions to the EFE with some quantum corrections. You can then ask for example if there are diffeomorphisms $\phi$ such that something like $\left<(\phi^*\hat g)_{\mu\nu}(x)\right> = \left<\hat g_{\mu\nu}(x)\right>$ holds. (It is not clear what properties really are desirable, but this is one reasonable thing one could ask for). Then you could call $\phi$ a quantum spacetime isometry.

I might be wrong, but this is not how I read Rovelli. For him the graphs are completely abstract combinatorial objects without an embedding in an apriori manifold. The space-time is a consequence. Even if you start with a manifold, representing space at a time or space-time, and consider graphs on them shouldn't you identify ones obtained after a diffeomorphisms? In other words for the state space the $\gamma$ and $\phi(\gamma)$ should be in the same equivalence class.

 

[Nullstein]

I might be wrong, but this is not how I read Rovelli. For him the graphs are completely abstract combinatorial objects without an embedding in an apriori manifold. The space-time is a consequence. Even if you start with a manifold, representing space at a time or space-time, and consider graphs on them shouldn't you identify ones obtained after a diffeomorphisms? In other words for the state space the $\gamma$ and $\phi(\gamma)$ should be in the same equivalence class.

No, there is definitely a manifold, upon which the theory is formulated. Of course, the diffeomorphism group is a gauge group, so it has to be modded out, but the equivalence classes depend on the background manifold. For example, in $\mathbb R^4$, all circles are equivalent, while on a torus, two circles may not be transformed into one another by a diffeo. The equivalence classes depend on the manifold-dependend knot classes. And the fact that the geometry is singular survives the modding out of the diffeos as well. If one geometry is singular, then it remains singular after the application of a diffeo. It's perfectly valid to choose a representative of the equivalence class and demonstrate the singular nature of the state on this particular representative.

 

[martinbn]

No, there is definitely a manifold, upon which the theory is formulated.

Again, that is not how I read him. In the Zakopane lectures, he explicitly says that this is a purely combinatorial graph. Then there is a comment, where he says that there are constructions with graphs on manifolds and what he likes and what he doesn't about them.

Of course, the diffeomorphism group is a gauge group, so it has to be modded out, but the equivalence classes depend on the background manifold. For example, in $\mathbb R^4$, all circles are equivalent, while on a torus, two circles may not be transformed into one another by a diffeo. The equivalence classes depend on the manifold-dependend knot classes. And the fact that the geometry is singular survives the modding out of the diffeos as well. If one geometry is singular, then it remains singular after the application of a diffeo. It's perfectly valid to choose a representative of the equivalence class and demonstrate the singular nature of the state on this particular representative.

Of course the equivalent classes depend on the manifold, the diff. group itself does. I am not sure I understand the relevance of the rest. If $\gamma$ is a graph, and $\phi$ a diffeomorphism, then $\phi(\gamma)$ is in the same class.

 

[Nullstein]

Again, that is not how I read him. In the Zakopane lectures, he explicitly says that this is a purely combinatorial graph. Then there is a comment, where he says that there are constructions with graphs on manifolds and what he likes and what he doesn't about them.

The Zakopane lectures are a heavily abbreviated, pedagogical version of what's explained elaborately in a proper textbook like Thiemann or even his own books. You can read the full construction there, which starts from the classical manifold and performs a discretization of classical GR on embedded graphs. Then, after many pages of calculations, you can derive some of the results contained in the Zakopane lectures. The Zakopane summer school was a one week event directed at beginner students. The abstract says: "The theory is presented in self-contained form, without emphasis on its derivation from classical general relativity." Of course, a lecture for a summer school is heavily condensed and cannot contain all the details. Notice that he never explicitely defines the set $\Gamma$ of "combinatorial graphs" in these lecture notes. That's because a proper definition requires the manifold. The graphs are not really just combinatorical objects, that's just a good enough description for a one week introductory course. But if you don't believe me and don't want to look it up in a textbook either, then I'm afraid there's nothing I can do to convince you.

Of course the equivalent classes depend on the manifold, the diff. group itself does. I am not sure I understand the relevance of the rest. If $\gamma$ is a graph, and $\phi$ a diffeomorphism, then $\phi(\gamma)$ is in the same class.

The relevance is to explain how the graphs in LQG are not just combinatorical objects, but equivalence classes of embedded graphs (two circles viewed as combinatorical objects are equivalent, but viewed as embedded graphs can be inequivalent). The metric isn't an observable in GR/QG, because it doesn't commute with the constraints. But (a smeared version of) it exists on the kinematical Hilbert space and you can use it to show that the states in LQG cannot locally look like Minkowski spacetime. You just pick a representative from the equivalence class, which is just one particular embedded graph with excitations on the edges, and calculate expectation values of geometric operators. This feature is preserved by diffeomorphisms, so it's really a property of the whole equivalence class.

 

[martinbn]

I will have to be satisfied by this and take your word for it, because I will not find the time to attempt to read the books.

 

[Nullstein]

I will have to be satisfied by this and take your word for it, because I will not find the time to attempt to read the books.

Well, you don't need to read a whole book, you can just skim to the page where the state space gets defined. This article is a preliminary version of Thiemanns full book and differs mostly in some introductory remarks and the appendix. You could just go to section I.2 and read a few pages.

 

[haushofer]

Maybe it has been said before, and maybe it's not even relevant in this topic. But somehow LQG reminds me a bit about how people tried to quantize Fermi's theory of weak interactions. The solution there wasn't changing the rules of quantizing, but adjusting the field theory by softening the amplitudes via the vector bosons. Somehow I feel LQG is analogous to attempts to quantize an incomplete field theory, i.e. GR.

 

[Fra]

The solution there wasn't changing the rules of quantizing, but adjusting the field theory by softening the amplitudes via the vector bosons. Somehow I feel LQG is analogous to attempts to quantize an incomplete field theory, i.e. GR.

I share this view on LQG. When I started to look into Rovellis book years ago, I was lead to read up on his interpretation of QM (RQM), and there was IMO a weak spot. The big problem is that the nice relational standards, that Rovellis holds high, makes sense only at classical level. The way he conceptually connects it to QM essentially witout modicitation, seems unsatisfactory and somewhat conceptually inconsistent to me.

In Relativity, SR or GR. An observer is associated with a coordinate frame of reference (from which "observations" are made). In QM, the "observer" is the CONTEXT of the whole inference process.

In QFT whe almost get away with merging the external passive observer with unlimited information processing resources at infinity, and asymptotically flat spacetime. But this seems like a conincidental success that still is conceptually incomplete.

When entertaining the generalized observer equivalence in the quantum size, conceptual consistency suggets that considering ONLY the diffeomorphism constraint is missing out the observers internal complexity. This is an unsolved problem. ST does offer such internal complexity in the moduli spaces of the generalized background, but they are OTOH lost in it. It's for this reason I think that ant "extended diff" symmetry in itself is likely to be required in some way - because the moduli space of observers defined only by diff is bound to be larger than what is physically motivated, because the contraints of information processing is not accounted for.

/Fredrik

 

[Nullstein]

Maybe it has been said before, and maybe it's not even relevant in this topic. But somehow LQG reminds me a bit about how people tried to quantize Fermi's theory of weak interactions. The solution there wasn't changing the rules of quantizing, but adjusting the field theory by softening the amplitudes via the vector bosons. Somehow I feel LQG is analogous to attempts to quantize an incomplete field theory, i.e. GR.

I don't even think there is anything wrong with quantizing GR. In fact, discretizing it on a lattice and then quantizing it with standard methods can't be completely off. Even if you may not obtain the theory of everything this way, it's a reasonable intermediate step. What's problematic, however, is the rather odd choice of canonical variables and the missing investigation of the continuum limit.

 

[haushofer]

I don't even think there is anything wrong with quantizing GR. In fact, discretizing it on a lattice and then quantizing it with standard methods can't be completely off. Even if you may not obtain the theory of everything this way, it's a reasonable intermediate step. What's problematic, however, is the rather odd choice of canonical variables and the missing investigation of the continuum limit.

An intermediate step, like quantizing Fermi's theory of the weak interactions :P

Can you e laborate on these canonical variables and what's odd about them?