What's wrong with Canonical QG

[Klaus_Hoffmann]

ain't Quantum gravity problem solved? i mean if you have the ADM Hamiltonian

\mathcal{H}=\frac{1}{2}\gamma^{-1/2}(\gamma_{ik}\gamma_{jl}+\gamma_{il}\gamma_{jk}-\gamma_{ij}\gamma_{kl})\pi^{ij}\pi^{kl}-\gamma^{1/2}{}^{(3)}R

then you apply the usual quantization procedure with \pi_{ij} = -i \frac{\delta }{\delta \gamma_{ab}}

so you can calculate the Wave function (state) and the energies so the problem would be solved then what's happening

 

[Demystifier]

The main problem with quantum gravity is that it leads to nonrenormalizable infinities. The canonical approach above does not solve it.

 

[kharranger]

That's right, there's no problem quantizing gravity per se. The problem is that the resulting quantum field theory is not renormalizable so it is not valid at arbitrarily high energy scales.

 

[marcus]

I agree with Demy and with kharranger (kharranger welcome!)

but just want to mention that the Hamiltonian turns out to be a contraint on the state of geometry on a fixed spatial slice, leading people to wonder about time evolution and what observables they can define, but that did not turn out to be such a problem after all (so I just mention it as marginal remark)

the people who have been addressing that business lately are Bianca Dittrich and Johannes Tambornino, anyone interested should look their work up in arxiv. Also Dittrich has some video talks in the Perimeter PIRSA archive.

Her work with relational variables makes canonical Gen Rel much better and I feel sure will lead to considerable improvement in the canonical quantization (as by LQG and related approaches).

 

[john baez]

ain't Quantum gravity problem solved? i mean if you have the ADM Hamiltonian

\mathcal{H}=\frac{1}{2}\gamma^{-1/2}(\gamma_{ik}\gamma_{jl}+\gamma_{il}\gamma_{jk}-\gamma_{ij}\gamma_{kl})\pi^{ij}\pi^{kl}-\gamma^{1/2}{}^{(3)}R

then you apply the usual quantization procedure with \pi_{ij} = -i \frac{\delta }{\delta \gamma_{ab}}

so you can calculate the Wave function (state) and the energies so the problem would be solved then what's happening

The real problem is much more fundamental than the nonrenormalizability issue some others have mentioned.

The real problem is that the ADM Hamiltonian is a Hamiltonian constraint. In other words, it equals zero on all classical solutions of general relativity! So presumably it should also vanish on all quantum states when you quantize the theory. This is called "the problem of time"; it shows up in any http://math.ucr.edu/home/baez/background.html" , of which general relativity is the most famous.

So, you can't compute energies as you suggest. Energy makes sense for asymptotically Minkowskian solutions of general relativity, but then it's expressed as an integral over the sphere at spacelike infinity - the boundary term you get when deriving the above Hamiltonian constraint.

If you want to learn more about this, start with http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html" on canonical quantum gravity and the problem of time: the equation you wrote down is basically equations 3.3.20-21 there,but then go on to read equation 3.3.26 and the subsequent section called "The Role of the Constraints".

The appendix on the ADM formalism in Wald's book is also helpful.

 

[marcus]

JB.

You have said much better what I was trying to say in post #4 about the Hamiltonian constraint and the problem of time.

I also made a separate point there as well, where I referred to what Rovelli calls complete observables or relational obserables, these having been developed quite a bit by Bianca Dittrich recently. I'd be much interested to know if you have any reaction to Dittrich's work or are familiar with it---and if you think it has a bearing on QG?

I'll get a couple of Dittrich links, in case anyone else is interested.

Here's an earlier paper
http://arxiv.org/abs/gr-qc/0507106
Partial and Complete Observables for Canonical General Relativity
B. Dittrich
33 pages
(Submitted on 25 Jul 2005)

"In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture."

the thing that impresses me is that it isn't messy or kludgy. the mathematics looks like it was meant to be. She also has a video slideshow presentation online at Perimeter and a slide/audio on Jorge Pullin's ILQGS site, if I remember right. Maybe I'll fetch links to those media later. A nice thing is that her complete observable formulation (really just an improvement on Rovelli's earlier work) seems to have been *fruitful* as can be seen in these papers building on it.

Here are slides to her talk at Loops '07:
http://www.matmor.unam.mx/eventos/loops07/talks/1B/Dittrich.pdf

ILQGS slides and audio for her 28 November 2006 talk are here:
http://relativity.phys.lsu.edu/ilqgs/

http://arxiv.org/abs/gr-qc/0610060
A perturbative approach to Dirac observables and their space-time algebra
Bianca Dittrich, Johannes Tambornino
23 pages
(Submitted on 11 Oct 2006)

"We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity...

http://arxiv.org/abs/gr-qc/0702093
Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
Bianca Dittrich, Johannes Tambornino
39 pages, 1 figure
(Submitted on 15 Feb 2007)
"We develop a gauge invariant canonical perturbation scheme for perturbations around symmetry reduced sectors in generally covariant theories, such as general relativity. The central objects of investigation are gauge invariant observables which encode the dynamics of the system..."

 

[Demystifier]

The real problem is much more fundamental than the nonrenormalizability issue some others have mentioned.

The real problem is that the ADM Hamiltonian is a Hamiltonian constraint. In other words, it equals zero on all classical solutions of general relativity! So presumably it should also vanish on all quantum states when you quantize the theory. This is called "the problem of time"; it shows up in any http://math.ucr.edu/home/baez/background.html" , of which general relativity is the most famous.

True, but many experts think that the problem of time is not really a problem.
For example, many think that the concept of relational time is the solution.

Another solution which some people (including myself) find acceptable is the Bohmian interpretation of quantum gravity. The main idea is that the quantum Wheeler-DeWitt equation is analogous to the classical Hamilton-Jacobi equation for gravity, which also suffers from the "problem" of time. In the classical case the problem solves by the fact that the Hamilton-Jacobi equation is only a part of the description of the system; there is also another equation that describes the time dependence of the metric. In the Bohmian interpretation of quantum gravity, an analogous equation is added to the Wheeler-DeWitt equation, solving the problem of time. This also solves the problem of "observer" in quantum cosmology.

 

[kharranger]

I would also argue that the problem of time is not really a problem to the same degree as non-renormalizability. The problem of time just reflects the fact that gravity is a gauge theory, and as in any gauge theory we use a gauge invariant description because we are too stupid to think up and/or use an action written only in terms of the true observables. The problem of time seems to be just the problem of finding the gauge invariant observables of GR. The observables in GR are presumably complicated and non-local, but the quantization in terms of these observables should in principle be straightforward.
Non-renormalizability, however, is a fundamentally more serious issue because it signals that the theory is incomplete and unpredictive beyond some regime. This problem may not be solvable by cleverness the way the problem of time presumably can be, because there may be many different renormalizable completions of gravity that reduce to GR at low energies, and then only experiment could single out the correct completion.

 

[meopemuk]

I would also argue that the problem of time is not really a problem to the same degree as non-renormalizability.

In my opinion, the problem of time is very simple and very fundamental. The essence of this problem is that time is not an observable. In my definition an observable is a numerical property that can be measured in the physical system, and whose measured value depends on the state of the system. Traditional observables, like position, momentum, mass, spin, are well covered by this definition. However, time is different. The value of time does not depend on the state of the observed system. It actually does not depend on whether we even have a physical system to observe. We read time by looking at the clock, which is rather a part of our reference frame than a part of the observed system.

So, there is a fundamental difference in definitions of time and position on a very basic level. Quantum mechanics correctly recognizes this difference and describes position as a Hermitian operator and time as a classical parameter. However, special (and general) relativity does not make this distinction. For this theory space and time coordinates are interchangeable. This is the problem. As long as there is such a difference in treatments of space and time by QM and SR (or GR), I don't see a chance to reconcile these two branches of physics.

Eugene.

 

[ccdantas]

Dear Eugene,

Thank you for your (very clear) comment. I tend to agree with your point.

Christine

 

[Fra]

It actually does not depend on whether we even have a physical system to observe. We read time by looking at the clock, which is rather a part of our reference frame than a part of the observed system.

But doesn't your reference frame (and thus clock in your terminology) evolve? How can you possibly justify taking the clock outside of your system? In my thinking this is equivalent to ignore that our reference is also subject to change. "Reading a clock" or "reading a ruler" what is the fundamental difference? It's all just information anyway.

There is only one universe, and in it is the observer, the clock and the ruler. The first task of the observer before introducing the notion of space and time is to distinguish the subsystem or "patterns in information" that he can distinguish to be clocks and rulers. There must I think always be a certain undertainty in the clocks and the rulers. In this information view, it's not hard to imagine how the observer has a certain chance of mixing up the clock device and his ruler. This is why I think that poincare invaraiance is not deeply fundamental, because if no clocks and rules are distinguished to 100% certainty the notion of spacetime itself is fuzzy. I think it's a matter of relative order and disorder, when spacetime is formed. Classical rigid spacetimes are one extremes, the other extreme is probably the yet unclear QG domain.

So it all seems to be self references, and self organisation and I think that's the key to understanding it. That's my personal opinion at least.

/Fredrik

 

[Fra]

But doesn't your reference frame (and thus clock in your terminology) evolve? How can you possibly justify taking the clock outside of your system? In my thinking this is equivalent to ignore that our reference is also subject to change. "Reading a clock" or "reading a ruler" what is the fundamental difference? It's all just information anyway.

The most natural idea is that all interactions or observations you make is exactly what is the input to the update of your very reference. Analogous to an bayesian update (but a generalisation thereof). To in a certain sense one can say that spacetime is part of the observers expectations about the environment, and thus relative to the observer, and not just the observers "position or movement in spacetime" but to the prior of the observer. And this expectation is updated along with it's interactions. It's learning about the environment, and it's "equilibrating" with the environment. This I think has the potential to make things more easy to understand. To turn it into a formalism may not be trivial but I'm sure there is a way.

/Fredrik

 

[f-h]

Nonrenormalizable? We don't even know what renormalization (in the physical, Wilsonian, non perturbative sense) is supposed to be for a theory like GR. Which is basically due to energy being a problematic concept in GR as JB said. That in return is tied to the problem of time, and that again should in principle be solvable by relational methods, but while Bianca Dittrichs work shows that that is indeed the case for classical systems the situation is far from clear on the Quantum Mechanical side, as it is unclear what a relational probability interpretation should look like (or whether or not it is necessary).

OTOH if you can quantize the constraint canonically without running into infinitely many ambiguities then speaking about nonrenormalizability doesn't make sense except in the sense that the effective field theory you get from the fundamental theory is nr, but that's not a problem.

 

[meopemuk]

There must I think always be a certain undertainty in the clocks and the rulers. In this information view, it's not hard to imagine how the observer has a certain chance of mixing up the clock device and his ruler. This is why I think that poincare invaraiance is not deeply fundamental, because if no clocks and rules are distinguished to 100% certainty the notion of spacetime itself is fuzzy. I think it's a matter of relative order and disorder, when spacetime is formed. Classical rigid spacetimes are one extremes, the other extreme is probably the yet unclear QG domain.

I see your point. Clock, rulers, observers, reference frames are, in fact, quantum objects whose positions and velocities cannot be determined simultaneously. So, everything becomes fuzzy: results of measurements, the Poincare group, etc. I have no idea how one can build a theory on such shaky grounds. Maybe I haven't thought hard enough.

A different question worries me. We have established that measurements of space and time are fundamentally different, and that time even cannot be called an observable. Nevertheless special (and general) relativity assumes complete equivalence and interchangeability of space and time. Is this conclusion of SR really inevitable? or we can keep space and time separate while respecting such predictions of SR as relativity of simultaneity, time dilation, etc? My guess is that yes, we can. We probably went too far by declaring the space-time unification in the 4D Minkowski continuum.

 

[Fra]

I see your point. Clock, rulers, observers, reference frames are, in fact, quantum objects whose positions and velocities cannot be determined simultaneously. So, everything becomes fuzzy: results of measurements, the Poincare group, etc. I have no idea how one can build a theory on such shaky grounds. Maybe I haven't thought hard enough.

Something like that, yes. And I agree the ground is shaky, but if life could evolve from chaos, I think we can build a theory on shaky ground. It's just that we need to build the reference too. This isn't just making it all more complex and shaky, it's also easier to see the potential of unification once everything is shaken up a bit. My personal thinking around this is that this relational information thinking isn't consistent unless it's combined with a deep evolutionary mechanism. Somehow it's obvious that we need a reference, but it's also obvious that this reference is "alive". The consistent solution I have as a vision is to this seemingly circular reasoning is that our reference is evolving. So our reality is not a static thing, it's probably more like a steady state, which appear static at certain scales. I've been thinking about this in circles for a few months and it seems that it's hard to find even a starting point because formalism, representation and dynamics are deeply entangled. But I am convinced there is a solution to this, it seems intuitively obvious althought hard to nail.

A different question worries me. We have established that measurements of space and time are fundamentally different, and that time even cannot be called an observable. Nevertheless special (and general) relativity assumes complete equivalence and interchangeability of space and time. Is this conclusion of SR really inevitable? or we can keep space and time separate while respecting such predictions of SR as relativity of simultaneity, time dilation, etc? My guess is that yes, we can. We probably went too far by declaring the space-time unification in the 4D Minkowski continuum.

I haven't worked out the formal details yet, but I envision a similarly symmetry as local lorentz invariance, that implies an upper bound of information propagation, at least in the differential sense. It should I suspect follow from the nature of the relational information. There is a relation between the information of the clock and the ruler (or any other configuration parameter for that matter). I picture time to simply be a parametrisation of the excepted future (to be defined as the pak of a bayesian-analog of a generalised entropy), relative to a sub-future (pure clock device). But this is probably again only an expectation, in the sense that the analog of lorentz invariance is likely to be observed. Classicaly this expectation should peak to be effectively a "rule", in the QG domain OTOH I think the notion of spacetime and lorentz invariance gets blurred enough to not make sense. The statement of lorentz invariance is probably blurred, thus the answer gets even more ambigous.

At least conceptually and intuitively I don't there is a problem for a radical approach to reduce to GR and ordinary QM in the respective limiting case.

I think this is quite a challange, I've been thinking about this off and on in circles for a few months and I try to make infinitesimal progress and come up with a working formalism.

I suspect the formalism will to start with, look something like a generalization of thermodynamics. But in a bayesian outfit, and a more realistic treatment of the ensembles and the measurement problems. I think dynamics can be integrated into this. The action principles should be something like minimizing information divergences. Once I've toyed on a little I will compare it to the standard expression and see what terms are missing. Right now it looks like the assumption of complete knowledge of probabiliy distributions may have something to do with it. What I am trying to find is a generalized entropy that isn't just a "state function" it rather contains the expected dynamics itself, so that minimizing the action will show to be nothing but a ME method. There should I think not be a need to "quantize it" because it would take these effects into account from construction. Classical mechanics should be limiting cases. One challange is that I hope (eventually) to see some of the basic CM stuff pop out by itself, for example the basic inertia and Newtons gravity as a limiting case. That would serve as a checkpoint. The standard model phenomenology would have to be ontop of that, by evolving the model to higher levels. I don't know exactly how I should do it yet, but something suggest that there is a way.

I like Ariel Catichas thinking. He has some good thinking in his papers.
http://www.albany.edu/physics/ariel_caticha.htm But except that I havn't had much luck finding ideas elsewhere. Most approachers are taking off at a level of abstraction that has broken a clean line of reasoning long time ago. I'm too much of a philosopher to find any motivation in that. I think you need to have another mindset to appreciate it, it's not for me anyway.

/Fredrik