the German text by Kiefer can be found here
http://www3.interscience.wiley.com/cgi-bin/abstract/109860245/ABSTRACT
this is the HTML abstract, which has a link to the PDF file.
the Kiefer article can also be found by scanning the table of contents of the January issue of Physik Unserer Zeit
http://www3.interscience.wiley.com/cgi-bin/jissue/109860236
and also it is at Loll website
http://www.phys.uu.nl/~loll/Web/press/press.html
Because we have a shortage of general audience description of CDT in English, I translated the short Kiefer article:
--------transl. from Claus Kiefer january P.U.Z.-----
QUANTUM GRAVITY: The four-dimensional world
Classical spacetime has four macroscopic dimension. In a future theory of quantum gravity one must take into account that even dimensionality will be a dynamical variable, for which only an expectation value can be provided. For reasons of consistency the number four must arise independently in the the semiclassical limit. Jan Ambjørn (Kopenhagen), Jerzy Jurkiewicz (Krakau) and Renate Loll (Utrecht) were able to show that in the path-integral framework it actually works out that way.[1]
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One of the most fundamental open problems in modern physics is the consistent unification of the quantum and gravitation theory. The chief difficulty consists in the fact that General Relativity knows no fixed prior-given background spacetime, but rather a dynamic geometry.
One quantizes other interactions, for example electrodynamics, ON a given spacetime, but with General Relativity one must quantize the very spacetime itself.
The most ambitious attempt to arrive at quantum gravity has been on the part of string theory, whose point of departure is the assumption that this goal can only be attained in the context of a unification of all the interactions.
Alternative approaches attempt to directly quantize Einstein's theory. Among these approaches are quantum geometrodynamics and loop dynamics (LQG). Ambjørn and colleagues chose an approach via the Feynmanian path integral
In quantum mechanics the path integral is summed over all possible paths a particle can take from one point to another. Most of the paths are continuous but nowhere differentiable. The result is a transition amplitude, which satisfies the Schrödinger-equation. In quantum gravity, the job is to sum over all possible four-dimensional geometries ("spacetimes") which fit between two three-dimensional geometries ("spaces")
Formally, it's easy to write down a so-called saddle-point approximation which in the semiclassical limiting case is dominated by a spacetime satisfying the Einstein field equations. But to go beyond this approximation to a clean calculation, one must define the sum over all geometries. This regularization occurs by discretizing and a subsequent continuum limit. The lattice theories of strong and electroweak interactions serve as models. There, however, the geometry is fixed ahead of time, while in the case of gravity it is dynamic.
Up till now the path integral has mostly been used in the Euclidean context, where one integrates only over fourdimensional space, rather than spacetime, geometries. This approach was made popular above all by Stephen Hawking. However problems arise there, among other things with the dimension.
Consider the expectation value for the effective Hausdorff dimension H. This is defined by the relation V(r) ~ <r>
H where V(r) is the volume of a ball with radius r. For a three-dimensional space this should come right out H = 3.
The Hausdorff dimension is known from the theory of fractals, however as a classical quantity and not a quantum expectation value. Since in quantum gravity there is no background, H is
a priori NOT equal to the dimension d of the building blocks used in the path integral summation. Notably, in the Euclidean path integral the value of H turns out to be 2, even for d >2.
Because of these and other problems, the above-mentioned authors have introduced the alternative way of "Lorentzian dynamical triangulations". Here one actually sums over
spacetimes rather than spaces, which seems physically more reasonable [2]. The discretization is accomplished by choosing, at some fixed (discrete) time, spatial tetrahedra which are then joined by four dimensional simplices to like tetrahedra at the next timestep. Thus the simplices represent the (discretized) spacetime. Figure 1 shows a typical configuration which in the example shown here consists of 91,100 simplices. The sum over all configurations in the path integral is performed by Monte-Carlo simulation.
The authors consider the mean value of the spatial separation between two points in a spatial volume and find, for the Hausdorff dimension defined earlier, the value H = 3.10 ±0,15. this is good evidence of the three-dimensionality of space (and thus the four-dimensionality of spacetime). From this it certainly does not yet follow that at the smallest scale there is actually a smooth three-dimensional space.
But nevertheless this result offers a pointer towards the existence of a continuum theory. Moreover it is interesting that this method only works with a postive cosmological constant--in agreement with observations.
The numerical value has however not been determined. The resulting dynamically produced quantum geometry can then serve as a background for the quantum fluctuations of other degrees of freedom.
[1] J. Ambjørn, J. Jurkiewicz, R. Loll, Phys. Rev. Lett. 22000044, 93, 131301.
[2] R. Loll, in: Quantum Gravity (D. Giulini, C. Kiefer, C. Lämmerzahl Hrsg.), SpringerVerlag, Heidelberg 2003. Claus Kiefer, Köln
FIG. 1 SIMULATION: Typical configuration ("spacetime") as it appears in a Monte-Carlo simulation. The time steps (here 40 in all) are upwards, the other two axes are spatial dimensions.[1]
----end quote from the Claus Kiefer article---
