Where we stand-Baez talk at Luminy

[Careful]

Most people, I think, go by the way it's defined, for example, in Wald: General covariance and no nondynamically defined objects (aether). That's how I understood Kea. But then of course GR is Background Independent.

But the entire subtlety in the argument is what you call ``kinematical´´ and ``dynamical´´ (is the topological/differentiable structure of the manifold ``kinematical´´ at the classical level (I would think it is) ? ); Kretchmann showed that even Newtonian physics can be given a covariant formulation ; any background frame or eather can be dynamically implemented as constraints on the equations of motion. A nice, recent example of this is how 't Hooft implements dissipation at the quantum level through constraints on the states. In this way, you can start out from a classical harmonic oscillator (using an unconvential Hamiltonian whose quantized version is the same as the classical one) and end up with a free quantum particle or a quantum harmonic oscillator *depending* upon the constraints you use.

If you really want to inforce the idea that you must find a quantum version of general covariance, then you have to solve the quantum constraint algebra. Otherwise, what becomes the true meaning of BI?

BTW if I follow your definition of BI literally, then isn't is justified to state that the construction of local observables by Dittrich and Rovelli is *not* BI ? Remember: you explicitely said that the ``reference frame´´ was KINEMATICAL.

Cheers,

Careful

 

[hellfire]

As a layman that tries to understand some basic ideas about quantum gravity I am wondering about the confusion with the concept of background independence and I would expect some clarification from this discussion. If general relativity is background independent, then it seams to me that string theory must be also background independent, because the Einstein field equations can be derived from it's classical action making some low energy approximations (Is this right? Anyway, it is still a mistery to me how this is possible since it is not a theory of spacetime). However, everywhere one reads that string theory is not background independent. I had rather expected the usual definition of background independence to be related to the fact that a background is used to quantize perturbatively around it. Comments, please.

 

[Careful]

**As a layman that tries to understand some basic ideas about quantum gravity I am wondering about the confusion with the concept of background independence and I would expect some clarification from this discussion. **

This is even still a point of confusion between specialists.


** If general relativity is background independent, then it seams to me that string theory must be also background independent, because the Einstein field equations can be derived from it's classical action making some low energy approximations (Is this right? Anyway, it is still a mistery to me how this is possible since it is not a theory of spacetime). **

Yep, that is right.

** However, everywhere one reads that string theory is not background independent. I had rather expected the usual definition of background independence to be related to the fact that a background is used to quantize perturbatively around it. **

Well not entirely true, there is a priori nothing wrong with picking out a background, splitting the action in a free and interacting part around it and try to quantize it perturbatively. It is just that this procedure should *not* depend upon the chosen ``background´´ spacetime. If you could find another spacetime (not necessary a solution to the vacuum equations) around which such procedure is well defined and you can show that minkowski (and perhaps some other highly symmetric spacetimes) are ``singular´´ points, then you are done. It is simply so that the *physics* should not ``substantially´´ depend upon this procedure. My question is if the LQG people are *really* doing something which differs from such procedure (and my guess is not) *now*.

In the early days, background independence was called *quantum covariance*, i.e. quantization of the constraint algebra (wave function of the universe stuff) and that was definitely different from the stringy strategy, here you were trying to obtain a quantum version of diffeomorphism invariance (while diffeomorphism invariance is also present in QFT - the measure is covariant - but then at the kinematical level). The philosophy behind this being that diffeo invariance is a *dynamical* statement in GR (note: this is a particular interpretation) instead of a kinematical one. That is, calculate the Poisson brackets of the dynamical phase space variables with the smeared out constraints, evaluate the result on shell (that is plug in the equations of motion), and you will see that the result corresponds to the Lie derivative of this quantity with respect to the associated vectorfield (the traditional gauge transformations). However, this programme seems to be largely abandonned because of some very persistent problems showing up - apart from major conceptual difficulties (LOCAL observables !). ADDENDUM : for sake of clarity, this distinction was only stressed AFTER perturbation theory on Minkowski failed (not to be in blatant conflict with post 149).

So, what is the correct point of view here?? Is covariance really a kinematical aspect of the game, or a dynamical one? At the classical level, this makes no difference whatsoever and that what appears to be a ``kinematical construct´´ can be turned into a ``dynamical statement´´ and vice versa. I admit that covariance was an important guideline for Einstein in constructing his theory of general relativity, but is it really a substantial, fundamental part of it QUANTUM MECHANICALLY?

This, f-h is also something which Kuchar adresses regularly : the classical is entirely different from the quantum. So, that is why I say that general covariance is not such a clear cut notion.

Cheers,

Careful

 

[f-h]

But the entire subtlety in the argument is what you call ``kinematical´´ and ``dynamical´´ (is the topological/differentiable structure of the manifold ``kinematical´´ at the classical level (I would think it is) ? ); Kretchmann showed that even Newtonian physics can be given a covariant formulation ; any background frame or eather can be dynamically implemented as constraints on the equations of motion. A nice, recent example of this is how 't Hooft implements dissipation at the quantum level through constraints on the states. In this way, you can start out from a classical harmonic oscillator (using an unconvential Hamiltonian whose quantized version is the same as the classical one) and end up with a free quantum particle or a quantum harmonic oscillator *depending* upon the constraints you use.

If you really want to inforce the idea that you must find a quantum version of general covariance, then you have to solve the quantum constraint algebra. Otherwise, what becomes the true meaning of BI?

BTW if I follow your definition of BI literally, then isn't is justified to state that the construction of local observables by Dittrich and Rovelli is *not* BI ? Remember: you explicitely said that the ``reference frame´´ was KINEMATICAL.

Cheers,

Careful

In this sense, yes this is justified. Also, the differentiable structure is a background structure in the classical case (topology is more subtle). We start with a manifold and pick one of the infinitely many differentiable structures.
BI isn't a rigorous statement. Kretschmars objection is of course silly, one can write any theory in a more general way if one breaks down the generality of the language by introducing distinguished elements (non rotational invariant systems can be written as rotationally invariant + a distinguished vector for example).

I agree we need to implement the constraint algebra (or the part of it responsible for rendering a certain structure kinematical). Most of it is implements, the gauge and 3Diffeo constraints in particular.

And yes, the construction and interpretation of R/D involves kinematics, but only to supply interpretations, the resulting Observables are invariant under the full constraint algebra.
If there is nothing in your universe but the BI theory of the one field, then this localisation is of course of questionable physical validity. Luckily that's not the case.

 

[Careful]

**In this sense, yes this is justified. Also, the differentiable structure is a background structure in the classical case (topology is more subtle). We start with a manifold and pick one of the infinitely many differentiable structures.**

Also topology is important since it can have curvature ramifications - the Gromov - Bishop theorems and so on...

**
BI isn't a rigorous statement. **

We are getting somewhere.

**
Kretschmars objection is of course silly, one can write any theory in a more general way if one breaks down the generality of the language by introducing distinguished elements (non rotational invariant systems can be written as rotationally invariant + a distinguished vector for example). **

It is not that simple, for example I can dynamically pick out a preferred coordinate system and thereby *appearantly* violating general covariance (see K. Kuchar work on quantisation in the gaussian gauge). I can write down a fully covariant action pricinciple which gives me Minkowski as a preferred background. On the other hand, I can write down GR as a gauge theory on Minkowski space time (see the work of Dorian, Hestenes and company), without having to worry about general covariance at all.

**I agree we need to implement the constraint algebra (or the part of it responsible for rendering a certain structure kinematical). Most of it is implements, the gauge and 3Diffeo constraints in particular. **

Well, also here you need to be careful, it is not so that the diffeomorphism *algebra* is implemented (your algebra does not exist due to the lack of weak continuity). So, it seems a very difficult task to be able to speak about a suitable interpretation of *spacetime* covariance (with the correct classical limit) here.

**
And yes, the construction and interpretation of R/D involves kinematics, but only to supply interpretations, the resulting Observables are invariant under the full constraint algebra. **

Ok, but that is at the *classical* level no big deal at all. Moreover, a hardcore relativist would expect local observables to be defined without kinematical background structure (and at the quantum level you have troubles with your Hamiltonian constraint) - so again you use a rather personal interpretation of BI here.

**
If there is nothing in your universe but the BI theory of the one field, then this localisation is of course of questionable physical validity. Luckily that's not the case. **

Well, I am not sure what you mean here but in *any* case you need to make such identifications. If you include matter you have to color particle 1 red, particle 2 blue, particle 3 yellow and so on - so you will have many red spots in each universe and depending upon the questions you ask your ``consciousness´´ will be in different superpositions of universes - sorry I like to state this in a path integral language, it makes everything more ``visual´´. Now, you might say: well that works apart form the tiny facts that your number of orthogonal states blow up super exponentially and the small issue that you do not have properly understood the quantum constraint algebra yet. Moreover, in this way, you cannot setup a unification between matter and geometry (so again, this is not necessarily a virtue). In case you would be interested in such enterprise different point identifications will lead to different physics (different ``states´´ if you want to). In that case, it seems much more intelligent to start out from an interacting matter theory on Minkowski and to deduce your metric as an effective variable as EINSTEIN himself suggested (you see everyone uses some words of the old master in different ways and ignores with the same ease other equally important ideas ). I mean, good old Albert was never able to explain the measure stick (idealized and put on the tangent space), a theory of measurement physically originates from matter interactions and is weakly temperature dependent (in the old days they just counted the number of atoms in a stick, noted down the temperature, and used that as a reference). Really, even if this is not going to change anything for you, it is good to know of the difficulties (again with EM say) the view of the measurestick as fundamental variable carries in itself.

Cheers,

Careful