The Dreaded Yasha is coming on Tuesday

[marcus]

I confess to a fondness for troublemakers and heretics. The action functional of GR (as much as anything else) is supposed to be ℝeal.
But the dreaded Yasha has contrived to have it be ℂomplex
and he will be talking at the online LQG seminar in just a few days, on 7 May.
http://relativity.phys.lsu.edu/ilqgs/
The imaginary part of the GR action and the large-spin 4-simplex amplitude
Download the slides PDF ahead of time so you can scroll thru the slides as directed while listening to the online audio.

Here is a related video talk from earlier this year,to suggest what the talk may be about:
http://pirsa.org/13040106/
The imaginary part of the gravitational action and black hole entropy
Yasha Neiman
Abstract: I present a candidate for a new derivation of black hole entropy. The key observation is that the action of General Relativity in bounded regions has an imaginary part, arising from the boundary term. The formula for this imaginary part is closely related to the Bekenstein-Hawking entropy formula, and coincides with it for certain classes of regions. This remains true in the presence of matter, and generalizes appropriately to Lovelock gravity. The imaginary part of the action is a versatile notion, requiring neither stationarity nor any knowledge about asymptotic infinity. Thus, it may provide a handle on quantum gravity in finite and dynamical regions. I derive the above results, make connections with standard approaches to black hole entropy, and speculate on the meaning of it all. Implications for loop quantum gravity are also discussed.
18/04/2013 - 2:30 pm

Here is a related paper, to suggest ideas of what the talk may be about.
http://arxiv.org/abs/1303.4752
Imaginary action, spinfoam asymptotics and the 'transplanckian' regime of loop quantum gravity
Norbert Bodendorfer, Yasha Neiman
(Submitted on 19 Mar 2013)
It was recently noted that the on-shell Einstein-Hilbert action with York-Gibbons-Hawking boundary term has an imaginary part, proportional to the area of the codimension-2 surfaces on which the boundary normal becomes null. We extend this result to first-order formulations of gravity, by generalizing a previously proposed boundary term to closed boundaries. As a side effect, we settle the issue of the Holst modification vs. the Nieh-Yan density by demanding a well-defined variational principle. We then set out to find the imaginary action in the large-spin 4-simplex limit of the Lorentzian EPRL/FK spinfoam. It turns out that the spinfoam's effective action indeed has the correct imaginary part, but only if the Barbero-Immirzi parameter is set to +/- i after the quantum calculation. An interpretation and a connection to other recent results is discussed. In particular, we propose that the large-spin limit of loop quantum gravity can be viewed as a high-energy 'transplanckian' regime.
22 pages, 5 figures

 

[atyy]

I think it's also heretical that the large spin limit is transplanckian - isn't that usually guessed to semiclassical?

 

[MTd2]

Why Yasha is dreadful?

http://en.wikipedia.org/wiki/Yaksha

 

[atyy]

"We in this section will consider “the” large-spin limit in a more straightforward context - as the special subset of states in the fundamental theory that happen to have large spin labels. There is no coarse-graining implied. ... Instead, we see it as a consistency check on the quantization procedure itself."

"We conclude that the two classical GR’s are in fact two opposite putative limits of the quantum theory, in terms of the energies of the quanta involved. The observed continuum GR corresponds to a low-energy “IR regime” of subplanckian quanta (gravitons). The discrete classical GR of the large-spin limit corresponds to a high-energy “UV regime” of transplanckian quanta (spins and intertwiners). The situation is summarized in figure 5."

 

[MTd2]

Just one thing. If he is so Dreaded, why not going beyong complex numbers and go all out to include clifford algebras?

 

[marcus]

Yasha is not dreaded, of course. I was being ironical, and so were you just now. In my opinion it would have sounded better if you had said Quaternions. The Hindu/Buddhist statues link was beautiful. Now I know a new word:
Yaksha.

 

[MTd2]

In my opinion it would have sounded better if you had said Quaternions.

Not really. Why stop at quaternions? Why not generalize the whole way?

Something good here:

http://arxiv.org/abs/1005.4300/

 

[marcus]

Yasha's slides are already posted.
http://relativity.phys.lsu.edu/ilqgs/neiman050513.pdf
His talk should be finishing right about now.
The audio should be up in a few hours.
http://relativity.phys.lsu.edu/ilqgs/

EDIT: the audio is up. There is some noise on the conference-call phone line, which I find is not a problem. The talk got a substantial participation. Yasha is at Penn State, and PSU people including Abhay were there. Also participating at other locations were Carlo, Laurent, Eugenio, Hal, Simone and one or two others whose voices I didn't recognize. The presentation is organized and steady, slides are very clear and complete. Interesting topic. I would say check it out.

 

[space_cadet]

The action functional of GR (as much as anything else) is supposed to be ℝeal.
But the dreaded Yasha has contrived to have it be ℂomplex

Actually that is not such a new idea. Gibbons and Hawking wrote a paper in 1977 ("Action Integrals and partition functions in quantum gravity") where they showed that on analytically continuing the time component to imaginary values - and, thus, complexifying the metric - the resulting action, when evaluated on space-times with boundary, has an imaginary component which can then be identified with the entropy obtained via the path-integral partition function approach. It turns out that the entropy thus obtained is identical to the Bekenstein-Hawking expression for both Schwarzschild and deSitter spacetimes. In order to have a consistent path-integral one needs to add a boundary term to the Einstein-Hilbert action when the spacetime has boundaries. The boundary contribution to the action is proportional to the trace of the extrinsic curvature of the boundary.

Neiman does discuss the relation of his result with that of Gibbons and Hawking starting in the 3rd para of page 5 of his (Yasha's) paper:

... our results complement the analysis of Euclidean black holes [12]. There, one also encounters an imaginary on-shell action in GR. Roughly, this leads to an exponential damping of transition amplitudes e^iS ∼ e^{−Im S} , which can be ascribed to a multiplicity of states. More precisely, the imaginary action in [12] is interpreted as the logarithm ln Z of a partition function. From the latter, one extracts the entropy, after correcting for terms related to conserved charges. These terms are e.g. E/T for energy and JΩ/T for an- gular momentum (E, T, J, Ω are respectively the energy, temperature, angular momentum and angular velocity). For a Euclidean black hole, the correction terms can be viewed as contributions from the boundary at infinity, while the entropy itself is a contribution from the horizon [16, 17].

where [12] refers to the Gibbons-Hawking paper. Yasha goes on to say:

The derivation here and in [15] has a number of advantages over the Euclidean one. First, it traces the imaginary action directly to properties of Lorentzian geometry. The analytical continuation we employ is more subtle, with no Wick rotation of the metric. As a result, stationarity is not required: one can calculate the imaginary action for arbitrary solutions. Also, the calculation works with finite regions, with no reference to the asymptotic boundary. This is more physical, especially in a positive-Λ cosmology. Finally, we recover the entropy formula directly from the imaginary action, with no correction terms such as E/T or JΩ/T. This can be traced both to the use of finite regions (there are no “terms at infinity” to speak of) and to the absence of a stationarity requirement (without it, thermodynamic potentials like T and Ω are not well-defined).

A couple of comments are in order.

1. Yasha says: "The analytical continuation we employ is more subtle, with no Wick rotation of the metric." ... it is not clear to me that what Yasha accomplishes is any different from what Gibbons-Hawking did in their paper. GH first complexify the metric then evaluate the action on various spacetimes. This boils down to computing the integral of the extrinsic curvature along the boundary. They get an imaginary result because they initially complexified the bulk metric. Yasha instead wants to calculate this integral directly in the Lorentzian case, rather than going to a complex (or Euclidean) metric. But in order to do so he has to analytically continue the "boost parameter" in the integral to imaginary values. Well, when in Lorentzian space the boost is a real number. Analytically continuing the Lorentz metric gives us a Euclidean metric, where the "boost" are now rotations. From the Lorentzian perspective, the fact that the boost acquires an imaginary component would thus appear to correspond to Wick rotating from Lorentzian to Euclidean, even if it is "only on the boundary" in Yasha's case. Bottomline, I'm not sure there is any difference between first complexifying the bulk metric and then evaluating the boundary term, or in retaining the Lorentzian signature and analytically continuing the boost parameter in order to obtain a finite result for the boundary term. Is this a case of "tamato, tomato", or perhaps I am mistaken in equating Yasha's analytic continuation of the boost term on the boundary to performing a Wick rotation of the background metric.
2. Yasha says: "The analytical continuation we employ is more subtle, with no Wick rotation of the metric. As a result, stationarity is not required: one can calculate the imaginary action for arbitrary solutions." ... I would like to see that calculation done for a non-stationary case. Maybe Yasha is already working on it. If so we should see something in the near future.
3. (Once again) Yasha says: Also, the calculation works with finite regions, with no reference to the asymptotic boundary. This is more physical, especially in a positive-Λ cosmology. ... as far as I can tell, GH do not make any reference to asymptotic boundaries. In addition they perform the calculation for the Kerr-Newman and deSitter (positive-Λ cosmology) and obtain the correct result. So I'm not sure what Yasha is referring to here by the second part of that statement.
4. This is a purely aesthetic observation, but I feel Yasha should have mentioned the work of Gibbons-Hawking in the abstracts of both the present paper and the one preceding it (http://arxiv.org/abs/1212.2922), before summarizing how his work differs. But that's just me.

Yasha is an inventive researcher and has definitely tried to break new ground with his work, particularly with the paper on "Causal Cells" (http://arxiv.org/abs/1212.2916). The present paper (http://arxiv.org/abs/1301.7041) has been published in JHEP, whose referees are undoubtedly far more qualified than me. Moreover, I've only had a cursory glance at both (GH and Yasha's) papers. So I've probably missed something and/or am dead wrong at some juncture. Perhaps Yasha or someone equally competent can clarify my doubts.

 

[MTd2]

Isn't a complex metric even older, related to twistors?

 

[julian]

Isn't a complex metric even older, related to twistors?

Twistors are to do with self dualality. Ashtekar, a student of Penrose, took it further and formulated GR solely in terms of self-dual variables. This simplified the Hamiltoninan formulation but made the theory complex.

 

[marcus]

Actually that is not such a new idea. Gibbons and Hawking wrote a paper in 1977 ("Action Integrals and partition functions in quantum gravity") where they showed that on analytically continuing the time component to imaginary values - and, thus, complexifying the metric - the resulting action, when evaluated on space-times with boundary, has an imaginary component which can then be identified with the entropy obtained via the path-integral partition function approach. It turns out that the entropy thus obtained is identical to the Bekenstein-Hawking expression for both Schwarzschild and deSitter spacetimes. In order to have a consistent path-integral one needs to add a boundary term to the Einstein-Hilbert action when the spacetime has boundaries. The boundary contribution to the action is proportional to the trace of the extrinsic curvature of the boundary.

Neiman does discuss the relation of his result with that of Gibbons and Hawking starting in the 3rd para of page 5 of his (Yasha's) paper:
...

Space_cadet, welcome again! I remember your participating in the braid-matter discussion: your paper about Bilson-Thompson model http://arxiv.org/abs/1002.1462 .
https://www.physicsforums.com/showthread.php?p=2592176#post2592176

Hi,

Thanks for the comments on my paper. It is nice to know that someone actually took the time to read it. BTW I would appreciate any and all feedback - no matter how brutal, since that is the best way to see what needs work.

As for whether or not this line of thought can lead to a prediction for the ... number of... space-time dimensions ...
...
Cheers,

Deepak

As I recall you also collaborated with Bilson-Thompson, I think on a braid matter tutorial.