Dear Dr. Ambjørn,

I'm a big fan of CDT, and fully agree with Lee Smolin that "now at least we know one way to do this",

http://www.god-does-not-play-dice.net/Loll.html

May I ask a question regarding your latest gr-qc/0607013 v1.

You and your colleagues acknowledged that in causal dynamical triangulations (CDT) you assume a global time-foliation, and then wrote:

"In two dimensions it is natural to study the proper-time "propagator", i.e. the amplitude for two space-like boundaries to be separated a proper time (or geodesic distance) T. While this is a somewhat special amplitude, it has the virtue that other amplitudes, like the disk amplitude or the cylinder amplitude, can be calculated if we know the proper-time propagator [10, 11, 12, 5]."

I wonder if you can formulate the proper-time propagator in three dimensions, and am curious to learn about the corresponding global time-foliation. My efforts can be read at

http://www.god-does-not-play-dice.net/Chrusciel.html#comment

Kindest regards,

Dimi Chakalov
--
http://www.god-does-not-play-dice.net
http://www.god-does-not-play-dice.net/download.html

Note: There is a well-known, and very simple idea about the dynamics of spacetime here, which I couldn't find in the two references below.

Read about the 'the art of building spacetime', from Renate Loll et al. [Ref. 1], and the 'pre-causality on appropriately defined chunks of space' from Daniele Oriti [Ref. 2]. See also Carlo Rovelli's contribution to the book "Approaches to Quantum Gravity" here. My contribution can be read here, only I wasn't invited.
 

Dimi Chakalov
July 10, 2006
Last update: July 12, 2006
 

[Ref. 1] J. Ambjorn, J. Jurkiewicz, R. Loll, Quantum Gravity, or The Art of Building Spacetime, hep-th/0604212 v1. Contribution to the book "Approaches to Quantum Gravity", ed. D. Oriti, Cambridge University Press.
http://arxiv.org/abs/hep-th/0604212

"What is more natural than constructing space from elementary geometric building blocks? It is not as easy as one might think, based on our intuition of playing with Lego blocks in three-dimensional space.
...

"In order for the sum over geometries to produce a quantum theory of gravity in which classical geometry is reproduced in a suitable limit, we therefore need a different principle for selecting the geometries to be included in this sum.
...

"Before discussing CDT in more detail let us comment on the nature of the geometries contributing to the path integral. It is important to emphasize that in a *quantum* theory of gravity a given spacetime geometry as such has no immediate physical meaning. The situation is really the same as in ordinary quantum field theory or even quantum mechanics, where individual field configurations [psi](x, t) or particle paths x(t) are not observable. Only certain expectation values related to the fields or paths can be observed in experiments. This does not mean there cannot exist limits in which it is appropriate to talk about a particular field configuration or the path of a particle in an approximate sense. In the case of our actual universe, down to the smallest distances that have been probed experimentally, it certainly does seem adequate to talk about a fixed classical spacetime geometry. Nevertheless, at sufficiently small distances it will no longer make sense to ask classical questions about spacetime, at least if we are to believe in the principles of conventional quantum theory.
...

p. 4: " A natural building block for a d-dimensional spacetime is a d-dimensional equilateral simplex with side-length a, and the path integral is approximated by performing the sum over all geometries (of fixed topology -footnote 4) which can be obtained by gluing such building blocks together, each geometry weighted appropriately (...).

"Afterwards we take the limit a --> 0."

Footnote 4: "There is an interesting and long-standing discussion about whether one should include topology changes in a quantum theory of gravity. ... "

 

[Ref. 2] Daniele Oriti, The group field theory approach to quantum gravity, gr-qc/0607032 v2. Expanded version of a contribution to "Approaches to Quantum Gravity - toward a new understanding of space, time, and matter", edited by D. Oriti, to be published by Cambridge University Press.
http://arxiv.org/abs/gr-qc/0607032

"There exist fundamental building blocks or atoms of space, which can be combined in all sorts of ways and can give rise to all sorts of geometry and topology of space. At the perturbative level spacetime is the (virtual) history of creation/annihilation of these fundamental atoms; it has no real existence, at least no more real existence in itself than each of the infinite possible interaction processes corresponding to individual Feynman graphs in any field theory; the interaction/evolution of these building blocks does not leave neither the geometry nor the topology of space fixed, but treats them on almost equal footing as dynamically evolving; the description of this evolution is necessarily background independent (from the point of view of spacetime) because spacetime itself is built from the bottom up and all of spacetime information has to be reconstructed from the information carried by the ‘atoms’ and thus by the Feynman graphs. (...) Spacetime information is thus necessarily encoded in structures that do not use per se a notion of spacetime.

"Finally, there would be a fundamental discreteness of spacetime and a key role played by causality, in the pre-geometric sense of ordering related to the notion of orientation (so that it would probably be better to talk about ‘pre-causality’)."

============

Subject: The continuum limit of the path integral for pure gravity
Date: Wed, 07 Sep 2005 19:03:42 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Renate Loll <loll@phys.uu.nl>
CC: Chris Isham <c.isham@imperial.ac.uk>,
     Claus Kiefer <kiefer@thp.uni-koeln.de>,
     Thomas Thiemann <Thomas.Thiemann@aei.mpg.de>,
     Szabados Laszlo <lbszab@rmki.kfki.hu>,
     Des Johnston <d.a.johnston@ma.hw.ac.uk>

Dear Professor Loll,

I have great respect for your work [Ref. 1], and agree with Lee Smolin that "now at least we know one way to do this",

http://focus.aps.org/story/v14/st13

May I share with you and your colleagues my thoughts on the issue in the subject line. Please correct me if I'm wrong.

In footnote 4, you said that the configurations contributing to the path integral are labeled by discrete *or* continuous parameters, and you have used the former option in your Causal Dynamical Triangulations (CDT) approach to quantum gravity [Ref. 1].

In the path-integral approach to QFT, the amplitude is not a complex
*number*, but some kind of density, in the sense that the "history" can wiggle up and down in the so-called time [tau] of the virtual paths [Ref. 1, Fig. 1], as explained by Roger Penrose,

http://www.God-does-not-play-dice.net/Barbara.html#note

The situation with this 'some kind of density' should change
dramatically in the absence of any background structure whatsoever [Ref. 1, p. 20], and I can't see how we could distinguish between discrete *or* continuous parameters (cf. above).

Perhaps we could obtain the correct continuum limit and explain the
buildup of the physical time  t  in the path integral for gravity [Ref.
1, p. 7] by "instances" of [tau] by adopting the idea of two modes of
spacetime,

http://www.God-does-not-play-dice.net/Kiefer.html

Last but not least, I believe the idea of two modes of spacetime can
supply *the only possible* mechanism for dynamic dark energy [Ref. 1, footnote 24, p. 21], which springs from the so-called empty space.

I will be happy to elaborate, in private, and to explain the whole story at ENRAGE's workshop next year in Edinburgh.

Best regards,

Dimi Chakalov
 

Reference

[Ref. 1] R. Loll, J. Ambjorn, J. Jurkiewicz, The Universe from Scratch, hep-th/0509010 v2.
http://arxiv.org/abs/hep-th/0509010

p. 6, Footnote 4: "... whether the configurations contributing to it are
labelled by discrete or continuous parameters. In CDT we will meet an example of the former."

p. 7: "It is important to realize that a single path is now no longer an
assignment of just three numbers (the coordinates xi of the particle) to every moment [tau] in time, but rather the assignment to every [tau] of a whole array of numbers (the space-space components gij(x) = gij(x, [tau]) of the metric tensor gµv(x)) for each spatial point x. This is simply a consequence of gravity being a field theory with infinitely many degrees of freedom. The path integral for gravity can thus be written as

        [XXX]                           (2)

where Sgrav now denotes the classical gravitational action associated
with a spacetime metric g_µv with initial and final boundary condition gi and gf , separated by a time distance t. Like in the particle case, the individual spacetime configurations interpolating between the initial and final spatial geometries have nothing a priori to do with classical spacetimes, and are much more general objects.

"Again, one would expect to be able to retrieve the full quantum
dynamics of spacetime from the path integral (2), which is a
superposition of all possible ways in which an empty spacetime can be curved[6]. In other words, the collective behaviour of the virtual
spacetimes contributing to the gravitational propagator (2) should tell
us what quantum spacetime *is*.

p. 11: "The short-distance cutoff  a  is an important part of our regularization of the spacetime geometries in the gravitational propagator. We will take the limit a --> 0 as part of the search for a
so-called continuum limit of the path integral over the regularized
geometries.

p. 20: "Talking about observables, one must keep in mind that an
innocent-looking question like "what is the value of the metric tensor
g_µv at point x?" is among the most difficult to answer in a
nonperturbative approach like ours. Firstly, although CDT histories come with a notion of proper time, they do not otherwise carry any natural coordinate system. Even if we introduced coordinate systems on the individual triangulated spacetimes, there is no way to mark "the same point" simultaneously in all of them. This is a consequence of the fact that individual points do not have any physical significance in empty space; in the absence of matter there is simply nothing we could "mark" the point  x  with."
 
 

Note: Renate Loll stressed that the question "what is the value of the metric tensor g_µv at point x?" is among the most difficult to answer in the absence of any background [Ref. 1]. But we need some kind of "background", or else the question is meaningless.

Thus, we introduce a new "background" -- the Holon of the whole universe as 'potential reality' -- and fix the spectrum of potential values of the metric tensor g_µv at point x with respect to [everything else in the universe]. A toy model was explained here. See also the shoal of fish metaphor here, and recall the basic idea of 'think globally, act locally'. More here and here.

If all this 'quantum theory without observers' is too difficult, try the explanation of 'potential reality' intended to my 12-year old daughter here.

As to the question posed above: the value of the metric tensor g_µv at point x is flexible like the centipede's leg. The point x is an ideal point, which can take temporary "real" values only "after" the completion of the bi-directional negotiation with [the rest of the universe], in the global mode of spacetime. The Holon is the ultimate 'chooser', since it is ONE. The implications for the Causal Dynamical Triangulations (CDT) approach to quantum gravity are quite obvious, I believe.
 

D. Chakalov
September 8, 2005

=======

Subject: The physical effects of the Immirzi parameter
Date: Fri, 09 Sep 2005 14:09:07 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Chung-Hsien Chou <chouch@phys.sinica.edu.tw>,
     Roh-Suan Tung <tung@shnu.edu.cn>,
     Hoi-Lai Yu <hlyu@phys.sinica.edu.tw>
CC: Giorgio Immirzi <giorgio.immirzi@pg.infn.it>,
     Xiaoning Wu <wuxn@phy.ncu.edu.tw>,
     Chiang-Mei Chen <cmchen@phy.ncu.edu.tw>,
     Hwei-Jang Yo <hjyo@astro.physics.uiuc.edu>,
     James M Nester <nester@phy.ncu.edu.tw>,
     J Fernando Barbero G <fbarbero@iem.cfmac.csic.es>,
     Renate Loll <loll@phys.uu.nl>,
     Chris Isham <c.isham@imperial.ac.uk>,
     Claus Kiefer <kiefer@thp.uni-koeln.de>,
     Thomas Thiemann <Thomas.Thiemann@aei.mpg.de>,
     Szabados Laszlo <lbszab@rmki.kfki.hu>,
     Des Johnston <d.a.johnston@ma.hw.ac.uk>,
     Ezra Newman <newman@pitt.edu>

RE: Chung-Hsien Chou, Roh-Suan Tung, Hoi-Lai Yu, Origin of the Immirzi
Parameter, gr-qc/0509028 v1,
http://arxiv.org/abs/gr-qc/0509028

Dear Drs. Chou, Tung, and Yu,

Please excuse my bulk email.

I was never able to get through Ashtekar's jungle,

http://www.God-does-not-play-dice.net/Hossain.html#note_3

and read with great interest your recent gr-qc/0509028 v1. May I share with you and your colleagues my hunch about the physical effects of the Immirzi parameter. Please correct me if I'm wrong.

You suggested that the Immirzi parameter could be interpreted as "the ratio between scalar and pseudo-scalar contributions in the theory", hence can be interpreted as "a measure of how a generally formulated covariant theory differs from Einstein gravity", under the stipulation that "if the Immirzi parameter y' is a physical property of the gravity sector, then it should be observable without the introduction of other matter field."

I believe the Immirzi parameter is indeed a physical property of the gravity sector, which represent its unique holistic effects,

http://www.God-does-not-play-dice.net/Loll.html#note

If true, the physical effects of the Immirzi parameter could be observed only by some sort of quasi-local measurements of the gravitational field, employing two "ideal" observers,

http://www.God-does-not-play-dice.net/Landsman.html#ideal

Just a wild guess, prompted by the imaginary unit used in the Immirzi parameter.

Kindest regards,

Dimi Chakalov
--
http://www.God-does-not-play-dice.net
http://www.God-does-not-play-dice.net/download.html