I What Are the Empirical Challenges Facing Quantum Gravity Theories?

[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?

 

[Nullstein]

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

It might as well become a success story, like quantizing classical electrodynamics. Nobody can tell at this point in time. At least, it's a reasonable approach, no less reasonable than quantizing classical electrodynamics. But it has to be done well. Sure it may fail, but it's worth a try.

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

They have no immediate physical meaning. What you do in LQG is to start from the vielbein formulation of GR, where the basic variables are the frame field and the spin connection. This is reasonable, one needs to do it anyway to allow for the inclusion of spinor fields. Then you make a 3+1 split, which is also reasonable if you want to obtain a Hamiltonian formulation. But then you go ahead and form new variables by adding the spin connection to the extrinsic curvature of the spatial slices. How is this a reasonable physical quantity? It's like adding apples and oranges and only accidentally works in 3 dimensions (because the adjoint representation of $SO(3)$ is equivalent to the defining representation). Morover, one introduces a new parameter (the Immirzi parameter), which classically cancels out, but remains important in the quantum theory. With these new variables, many equations simplify or become more elegant. The theory then looks like a Yang-Mills theory with additional constraints, but at the cost of having had to add apples to oranges in an early step of the calculation.

 

[*now*]

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.

Yes, I think I’ve previously linked e.g. a Lorentzian description in LQC somewhere in a thread here. There are varied alternatives and some crossovers and a description of Freidel’s recent work in a talk might interest Introduction to local holography - Laurent Freidel - Bing video . The tests linked for a start might add more weight towards distinguishing between differing descriptions, which might be related to papers such as this-Carlo Rovelli (Dated: February 8, 2018) [Written for the volume “Beyond Spacetime: The Philosophical Foundations of Quantum Gravity” edited by Baptiste Le Biha, Keizo Matsubara and Christian Wuthrich.]

Space and Time in Loop Quantum Gravity
Quantum gravity is expected to require modifications of the notions of space and time. I discuss and clarify how this happens in Loop Quantum Gravity.

https://arxiv.org/pdf/1802.02382.pdf

 

[physika]

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

mends and mends botches...

 

[Nullstein]

Yes, I think I’ve previously linked e.g. a Lorentzian description in LQC somewhere in a thread here. There are varied alternatives and some crossovers and a description of Freidel’s recent work in a talk might interest Introduction to local holography - Laurent Freidel - Bing video . The tests linked for a start might add more weight towards distinguishing between differing descriptions, which might be related to papers such as this-Carlo Rovelli (Dated: February 8, 2018) [Written for the volume “Beyond Spacetime: The Philosophical Foundations of Quantum Gravity” edited by Baptiste Le Biha, Keizo Matsubara and Christian Wuthrich.]

Space and Time in Loop Quantum Gravity
Quantum gravity is expected to require modifications of the notions of space and time. I discuss and clarify how this happens in Loop Quantum Gravity.

https://arxiv.org/pdf/1802.02382.pdf

I think I've lost track of what we're talking about here. Are we still discussing Lorentz invariance? Then I don't see how these references support your point.

 

[*now*]

I’d been thinking of issues generally that along with possible narrowing of alternatives discussed in the OP source there is breadth of other possible directions and emphases towards open questions, and those raised may be examples of, but replying has been problematic and on second thoughts I think that absent citing the author’s express words attempting to speak of the author’s possible opinions or intuitions or variations seems very problematic.

 

[arivero]

I was looking for discussion on a old paper by Rovelli and Connes and I wonder we do we keep closing topics...

https://www.physicsforums.com/threa...r-on-time-in-gen-cov-quantum-theories.392819/

I guess it is a general "feature" of forums, but well, this field of science is not fast-paced.