Josh Willis replies to comment on LQG and the diffeomorphism group

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marcus

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Originally posted by Urs

...I don't know how much longer I will have the energy arguing that there are problematic steps if the people working on LQG won't agree with this assessment. Well, my goal is not to fix what I (and many others who explicitly told me so) consider problems of the LQG approach.
I sympathize strongly because I understand from Eric F that you are in graduate school where one's own research is almost a survival issue!

I would certainly understand you focussing your obvious talent and energy on your own favorite topics and not on "fixing the problems of LQG!"

If I were to try to sum up, in the simplest possible way, how you see these problems which you mention, I suppose I would say this (Is this correct?)

I believe you think that diffeomorphism invariance should be implemented by operator equations on the kinematical state space----constraint equations. these operators will correspond to generators of the diffeomorphism group.

I believe you wouold reject the fairly elementary and transparent construction on page 4 of
http://arxiv.org/gr-qc/0403047 [Broken]
(the Fairbairn/Rovelli paper)
which defines a projection map from a preliminary hilbertspace
containing some unphysical redundancy down to the more concrete kinematical state space. This seems to be the thrust of your comment,
or perhaps you would describe it differently.

I am still trying to apply your reasoning to Rovelli's book or to this paper and to see clearly whether it amounts to anything *in that context*. I want to see what, if anything, your comments say about LQG proper not just about the Thiemann "Loop-String" paper, which (although it got quite some attention from those doing string research) seems somewhat periferal to the main body.
 
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marcus

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the conversation continued today

Thomas Larsson replied to Josh Willis, adding another voice to
the Josh/Urs conversation:

----quote from today's post by Larsson---

Josh Willis <jwillis@gravity.psu.edu> wrote:

> (3) When the symmetry group is infinite dimensional, as for instance
> Dif(S^1), any neighborhood of the identity contains group elements that
> cannot be obtained by exponentiating the Lie algebra. Thus, the algebra
> doesn't carry the full information about the group.
>


There are also exponented vector fields that are not
diffeomorphisms. My impression is that this distinction is a
mathematical subtlety without real physical significance. The
important thing is that the diffeomorphism group and the algebra
of vector fields share essentially the same representations; the
classical irreps are tensor densities and exact forms.


Most interesting Lie algebra cocycles can be integrated to group
cocycles, e.g., the Virasoro algebra gives the Schwarzian
derivative. Yuly Billig has recently written down the
analogous group cocycles in higher dimensions.


> > I believe that this won't be possible, because all the information
> > about the usual quantum effects have been eliminated in the
> > 'LQG-string' and they won't reappear in any limit.
>
> I don't understand this statement at all. See my other response (that I
> hope to write :) ) to your other reply to my post. But at various
> points you seem to me to have implied that the absence of anomalies
> means that one won't get the "usual quantum effects." But there are
> lots of quantum effects besides the existence of anomalies!


Urs Schreiber's arguments have some interesting logical
consequences. If canonical quantization of the string leads to
conformal anomalies (only as an intermediate step before
introducing ghosts, but anyway), canonical quantization of
gravity should lead to similar intermediate diffeo anomalies.
However, pure gravitational anomalies in field and string theory
can only arise in 4k+2 spacetime dimensions. Thus string theory
is also incapable of producing such diffeo anomalies in 3D and
4D.


Note the difference compared to anomaly cancellation: here no
anomalies can be written down at all. The reason is that the
relevant anomalies are functionals of the observer's trajectory,
which is simply not available unless you introduce it.


It thus seems that Urs Schreiber's argument, if correct, proves
that string theory is wrong.
-----------end quote----------
 
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Urs

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I must say that your reportage ("first person journalism") from Ulm was great!
Thanks. I have now also included a few photographs. See here (you have to scroll down a little).
 

marcus

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Originally posted by Urs
Thanks. I have now also included a few photographs. See here (you have to scroll down a little).
Whoah! The statues of Albert sticking his tongue out are terrible kitsch!

I like the stark machine-look of the Ulm university architecture, glad you included that snapshot of the building.

But who is the smiling guy who has intruded into your lecture
about field theory? He is getting in the way of the overhead projector.

If you keep on putting snaps of people like Ashtekar on line you will make Coffee Table famous and we will have to pay you an honorarium to get you to come here to PF.

thanks for the link in spite of it all
 

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