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(Re-)Constructing the Universe
  My research is concerned with constructing spacetime from the bottom up, that is, finding a consistent theory which describes the microscopic constituents of spacetime geometry and the quantum-dynamical laws governing their interaction. In order to be consistent with already established physics, this theory must be able to reproduce the known classical laws of general relativity at sufficiently large distances. Of particular physical interest are predictions of new physical phenomena at intermediate scales which - with some luck and ingenuity - one may be able to relate to current or future observations and experiments. These can take the form of computable quantum corrections to well-established physics, induced by spacetime's nontrivial microstructure. At a more fundamental level, such a theory of quantum gravity should help us to understand the quantum properties of black holes and of spacetime geometry near the big-bang singularity, and how all of the known interactions, gravitational and nongravitational, fit together at the quantum level. - The quantum gravity theory I am proposing is obtained with the method of Causal Dynamical Triangulations and has already passed some nontrivial classical consistency checks. Its further physical implications are the subject of active research by my group in Nijmegen and elsewhere.

Armed with last century's insights into the nature of both quantum theory and general relativity, physicists believe that probing the structure of space and time at distances far below those currently accessible by our most powerful accelerators would reveal a rich geometric fabric, where spacetime itself never stands still but instead quantum-fluctuates wildly. One of the biggest challenges of theoretical physics today is to identify the fundamental "atoms of spacetime geometry" and understand how their interactions give rise to the macroscopic spacetime we see around us and which serves as a backdrop for all known physical phenomena.

Two pillars of contemporary physics support the expectation that as we resolve the fabric of spacetime with an imaginary microscope at ever smaller scales, spacetime will turn from an immutable stage into the actor itself. First, due to Heisenberg's uncertainty relations, probing spacetime at very short distances is necessarily accompanied by large quantum fluctuations in energy and momentum - the shorter the distance, the larger the energy-momentum uncertainty. Second, according to Einstein's theory of general relativity, the presence of these energy fluctuations, like that of any form of energy, will deform the geometry of the spacetime in which it resides, imparting curvature which is detectable through the bending of light rays and particle trajectories. Taking these two things together leads to the prediction that the quantum structure of space and time at the so-called Planck scale must be highly curved and dynamical.

A long held ambition of theoretical physicists is to find a consistent description of this dynamical microstructure within a theory of quantum gravity, which unifies quantum theory and general relativity, and to determine its ramifications for high-energy physics and cosmology. Given the extraordinary smallness of the Planck length, how can we achieve progress in describing a physical situation that cannot be directly probed by experiment in the foreseeable future? The way this is usually done is by first postulating additional dynamical principles or fundamental symmetries at small distances, which are not accessible to direct experimental verification, second, verifying that these do not conflict with standard quantum physics or general relativity as one goes to larger scales, and third, predicting new physical phenomena that can (at least in principle) be tested, or confirmed indirectly by astrophysical observations. Examples of fundamental building principles are that the universe is made up of tiny vibrating strings, or that spacetime at the Planck scale is not a continuum, but consists of tiny discrete grains.

Research into quantum gravity falls broadly into two categories: string-theoretic approaches, where the quantization of gravity appears almost as a by-product of a unified higher-dimensional and supersymmetric theory of everything, whose fundamental objects are strings and (mem)branes, and nonperturbative approaches to quantum gravity, whose primary aim is to quantize the gravitational degrees of freedom per se, introducing little or no additional structure such as supersymmetry or extra dimensions.

My research deals with the investigation of causal nonperturbative quantum gravity and belongs in the second category. It centres on the new and intriguing physical insight that causality at the Planck scale may be responsible for the fact that our universe is four-dimensional [1]. It aims to explore the detailed dynamical reasons behind this result and its implications for a range of physical phenomena which are sensitive to how spacetime behaves at very small scales. To achieve its objectives, my research program draws on a well-developed set of analytical and numerical tools, subsumed under the name of dynamical triangulations, as well as nonperturbative canonical methods.

The new result of the emergence of a four-dimensional world from causal quantum gravity provides strong evidence that the requirement that a dynamical theory of quantum spacetime reproduce the correct classical limit (including the fact that our macroscopic spacetime has the correct one time and three space dimensions) implies stringent constraints on what happens at the Planck scale. This is good news, since it indicates that the absence of experimental guidance in quantum gravity does not open the door to pure speculation. Any quantum theory of gravity must demonstrate how the smooth and seemingly structureless four-dimensional spacetime continuum we observe macroscopically can arise from the collective dynamical behaviour of its microscopic geometric building blocks. The outcome of past research suggests that this condition is not easy to meet; it may even turn out that it makes the Planck scale dynamics essentially unique.

The main theoretical tool of my research is a nonperturbative version of Feynman's famous path integral, a well-known procedure to determine the quantum behaviour of a physical system from its fundamental constituents [2,3]. In its simplest form, it provides a solution to the quantum dynamics of a particle moving in a potential, by superposing all possible paths a virtual particle could have followed. Analogously, a path integral for gravity is a superposition of all possible "paths" our universe could have followed. A recipe for finding the desired quantum dynamics of gravity is therefore to compute a superposition of all possible spacetime geometries, which differ by being curved in arbitrary ways. Such a quantum averaging should produce the true, nonperturbative ground state of quantum gravity, in other words, "the mother of all vacua". The geometric properties of this vacuum state will reflect the collective behaviour of the quantum ensemble of all virtual spacetime geometries, including those with large curvature fluctuations at very small scales.

The path integral I use is both causal and nonperturbative, and its construction therefore differs crucially from previous approaches. An appreciation of its novelty requires a brief retracing of its history. The use of (mainly perturbative) path integral methods in gravity was made popular in the late 70s by the influential work of S. Hawking and collaborators on black holes and quantum cosmology in Euclidean quantum gravity [4,5]. In their Euclidean path integral, instead of superposing the usual Lorentzian spacetimes with one time and three space dimensions, one uses so-called Euclidean spaces which have four space dimensions. This is done mainly to circumvent technical difficulties.

The work on Euclidean path integrals was taken one important step further to a nonperturbative formulation, with both Quantum Regge Calculus [6,7] and, more recently, the related Dynamical Triangulations research program [8,9]. These approaches follow a rather straightforward and geometrically appealing route. In order to make the integral over all spaces computable, the curved spaces are represented in a regularized form, namely, by assembling them from elementary building blocks, usually four-dimensional analogues of triangles [10]. Although the individual building blocks are geometrically flat and therefore carry no curvature, gluing them together in all possible ways results in spaces or spacetimes that are locally curved in all possible ways. One then studies by a combination of analytic and numerical methods how a superposition of such geometries behaves in the limit as the size of the individual building blocks shrinks to zero and their number goes to infinity. If everything has been done right, the dependence on the details of the regularization in terms of discrete building blocks will be completely washed out in the process, and a quantum theory of spacetime will emerge.

Unfortunately, the Euclidean version of this quantization program has run into trouble, because the quantum geometry that emerges as a ground state in such a limit is simply not a four-dimensional quantity, let alone a Euclidean version of flat Minkowski space [11,12,13].

How is this possible? How can one obtain an effective geometry that is not four-dimensional by superposing virtual spacetime geometries that individually are of dimension four? This nonperturbative quantum phenomenon, completely at odds with our classical intuition of spacetime as a fixed inert background structure, beautifully illustrates that no aspect of such a dynamically generated geometry can be taken for granted, not even its dimension. Since all local curvature degrees of freedom of the geometry undergo large quantum fluctuations, a four-dimensional geometry can either crumple up to generate a geometry of an effectively higher dimension (like crumpling up a two-dimensional sheet of paper into a three-dimensional ball), or curl up to give a geometry of an effectively lower dimension (like rolling up a piece of paper into a thin tube, which will appear effectively one-dimensional at a scale much larger than the circumference of the tube). This is exactly what happens in nonperturbative Euclidean quantum gravity. It produces ground states of geometry that are either maximally crumpled with an infinite(!) effective dimension [14] or polymerized into thin and branched threads, with an effective dimension of two [8], neither of them promising candidates for the ultimate vacuum.

String theories, although not currently formulated as nonperturbative path integrals, suffer from a similar problem; namely, to explain how the four-dimensional world we see around us arises dynamically from a theory whose natural habitat is ten- or eleven-dimensional. There are some nonperturbative attempts [15] to obtain dimension four without curling up the extra dimensions "by hand".

At a time when these developments seemed to spell the end of research into dynamically generating spacetime from a superposition principle, a beautiful new idea came along which changed the rules of the construction of the gravitational path integral and drastically altered the resulting quantum spacetimes. The idea, put forward and developed by my collaborators J. Ambjørn, J. Jurkiewicz and me [16-18], is to re-introduce into the path integral the notion of causality which was lost in the transition to the standard formulation in terms of Euclidean geometries. (This happens because the principle of 'cause precedes effect' can only be formulated on spacetimes and not on Euclidean spaces, which lack a 'time'.) It leads to the causal, Lorentzian path integral mentioned earlier, which is based on the postulate that quantum spacetime should obey microcausality, namely, that it should respect the sequence of cause and effect even at the Planck scale. This implies a considerable extension of the causality principle we know to hold at macroscopic scales. The main technical tool for constructing a causal nonperturbative path integral is the method of Causal Dynamical Triangulations, a Lorentzian version of dynamical triangulations [19], in which the new fundamental principle of microcausality is implemented by taking superpositions of only those geometries which have a well-defined causal structure down to the very smallest scales.

The exciting news is that this idea seems to work! Computer-generated superpositions of causal geometries show the emergence of an extended quantum ground state. Recent work which analyzes some of its geometric properties provides strong evidence that this quantum spacetime indeed behaves macroscopically like a four-dimensional universe [1,20,21]. These very encouraging findings imply that an essential aspect of the classical limit, the dimension of spacetime, emerges correctly, a consistency check other approaches to nonperturbative quantum gravity have yet to pass. This nontrivial result makes the model of causal dynamical triangulations a prime candidate for a theory of quantum gravity, although much work remains to be done to prove that it indeed is the correct theory. Further research is under way to determine other classical and semiclassical properties of the "reconstructed universe", as well as its true quantum structure.

Literature references

[1] J. Ambjørn, J. Jurkiewicz and R. Loll: "Emergence of a 4D World from Causal Quantum Gravity", Phys. Rev. Lett. 93 (2004) 131301 [arXiv: hep-th/0404156].
[2] R.P. Feynman and A.R. Hibbs: Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).
[3] H. Kleinert: Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, 2nd ed., World Scientific, Singapore (1994).
[4] G.W. Gibbons and S.W. Hawking (eds.): Euclidean Quantum Gravity, World Scientific, Singapore (1993).
[5] J.J. Halliwell: "Introductory Lectures on Quantum Cosmology", in Jerusalem Winter School 1989, 159-243.
[6] R.M. Williams: "Recent Progress in Regge Calculus", Nucl. Phys. B (Proc. Suppl.) 57 (1997) 73-81 [arXiv: gr-qc/9702006].
[7] H.W. Hamber: "On the Gravitational Scaling Dimensions", Phys. Rev. D61 (2000) 124008 [arXiv: hepth/ 9912246].
[8] J. Ambjørn and J. Jurkiewicz: "Four-Dimensional Simplicial Quantum Gravity", Phys. Lett. B 278 (1992) 42-50.
[9] M.E. Agishtein and A.A. Migdal: "Critical Behavior of Dynamically Triangulated Quantum Gravity in Four Dimensions", Nucl. Phys. B 385 (1992) 395-412 [arXiv: hep-lat/9204004].
[10] T. Regge: "General Relativity Without Coordinates", Nuovo Cim. A 19 (1961) 558-571.
[11] P. Bialas, Z. Burda, A. Krzywicki and B. Petersson: "Focusing on the Fixed Point of 4d Simplicial Gravity", Nucl. Phys. B 472 (1996) 293-308 [arXiv: hep-lat/9601024].
[12] B.V. de Bakker: "Further Evidence That the Transition of 4D Dynamical Triangulation is 1st Order", Phys. Lett. B 389 (1996) 238-242 [arXiv: hep-lat/9603024].
[13] R. Loll: "Discrete Approaches to Quantum Gravity in Four Dimensions", Living Rev. Rel. 13 (1998) http://www.livingreviews.org/lrr-1998-13.
[14] P. Bialas, Z. Burda, B. Petersson and J. Tabaczek: "Appearance of Mother Universe and Singular Vertices in Random Geometries", Nucl. Phys. B 495 (1997) 463-476 [arXiv: hep-lat/9608030].
[15] H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo and S. Shinohara: "Mean field approximation of IIB matrix model and emergence of four dimensional space-time", Nucl. Phys. B 647 (2002) 153-189 [arXiv: hepth/ 0204240].
[16] J. Ambjørn and R. Loll: "Non-Perturbative Lorentzian Quantum Gravity, Causality and Topology Change", Nucl. Phys. B 536 (1998) 407-434 [arXiv: hep-th/9805108].
[17] J. Ambjørn, J. Jurkiewicz and R. Loll: "A Non-Perturbative Lorentzian Path Integral for Gravity", Phys. Rev. Lett. 85 (2000) 924-927 [arXiv: hep-th/0002050].
[18] J. Ambjørn, J. Jurkiewicz and R. Loll: "Dynamically Triangulating Lorentzian Quantum Gravity", Nucl. Phys. B 610 (2001) 347-382 [arXiv: hep-th/0105267].
[19] R. Loll: "A Discrete History of the Lorentzian Path Integral", Lecture Notes in Physics 631 (2003) 137-171 [arXiv: hep-th/0212340].
[20] J. Ambjørn, J. Jurkiewicz and R. Loll: "Semiclassical Universe From First Principles", Phys. Lett. B 607 (2005) 205-213 [arXiv: hep-th/0411152].
[21] J. Ambjørn, J. Jurkiewicz and R. Loll: "Reconstructing the Universe", Phys. Rev. D 72 (2005) 064014.


R. Loll, © 2004 A. Müller