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Quantum gravity in 2 + 1 dimensions . . .
 LIVING REVIEWS IN RELATIVITY
, 2005
"... In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body o ..."
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Cited by 137 (0 self)
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In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)dimensional vacuum gravity in the setting of a spatially closed universe.
Vicious walkers, friendly walkers and Young tableaux: II With a wall
 J. Phys. A: Math. Gen
"... Research supported by the Australian Research Council. ..."
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Cited by 49 (4 self)
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Research supported by the Australian Research Council.
Scaling in quantum gravity
 Nucl. Phys. B
, 1995
"... The 2point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2point function with geodesic distance determines the fractal dimension dH of spacetime. The integral of the 2point function determines the entropy exponent γ, i.e. the ..."
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Cited by 42 (10 self)
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The 2point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2point function with geodesic distance determines the fractal dimension dH of spacetime. The integral of the 2point function determines the entropy exponent γ, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2point function connects γ and dH by a quantum gravity version of Fisher’s scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation. 1 1
The Universe from Scratch
, 2005
"... A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19thcentury founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the n ..."
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Cited by 25 (3 self)
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A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19thcentury founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein’s theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of curvature. What we still lack is a definitive theory of quantum gravity to give us a detailed and quantitative description of the highly curved and quantumfluctuating geometry of spacetime at this socalled Planck scale. – This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from
(2+1)Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
, 2007
"... We perform a nonperturbative sum over geometries in a (2+1)dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometrie ..."
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Cited by 16 (6 self)
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We perform a nonperturbative sum over geometries in a (2+1)dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues.
A discrete history of the Lorentzian path integral
, 2008
"... In these lecture notes, I describe the motivation behind a recent formulation of a nonperturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) spacetimes, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solv ..."
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Cited by 12 (3 self)
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In these lecture notes, I describe the motivation behind a recent formulation of a nonperturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) spacetimes, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a welldefined Wick rotation, (ii) possessing a coordinateinvariant cutoff, and (iii) leading to convergent sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d = 2 and d = 3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a nonperturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.
Brownian motion, ChernSimons theory, and 2d YangMills,” Phys
 Lett. B
"... We point out a precise connection between Brownian motion, ChernSimons theory on S 3, and 2d YangMills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of ChernSimons theory on S 3 with gauge group U(N). The probability of starting wi ..."
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Cited by 12 (3 self)
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We point out a precise connection between Brownian motion, ChernSimons theory on S 3, and 2d YangMills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of ChernSimons theory on S 3 with gauge group U(N). The probability of starting with an equalspacing condition and ending up with a generic configuration of movers gives the expectation value of the unknot. The probability with arbitrary initial and final states corresponds to the expectation value of the Hopf link. We find that the matrix model calculation of the partition function is nothing but the additivity law of probabilities. We establish a correspondence between quantities in Brownian motion and the modular S and Tmatrices of the WZW model at finite k and N. Brownian motion probabilitites in the affine chamber of a Lie group are shown to be related to the partition function of 2d YangMills on the cylinder. Finally, the randomturns model of discrete random walks is related to Wilson’s plaquette model of 2d QCD, and the latter provides an exact twodimensional analog of the melting crystal corner. Brownian motion provides a useful unifying framework for understanding various lowdimensional gauge theories. 1 1
Gauge fixing in Causal Dynamical Triangulations
, 2008
"... We relax the definition of the AmbjørnLoll causal dynamical triangulation model in 1 + 1 dimensions to allow for a varying lapse. We show that, as long as the spatially averaged lapse is constant in time, the physical observables are unchanged in the continuum limit. This supports the claim that th ..."
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Cited by 6 (1 self)
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We relax the definition of the AmbjørnLoll causal dynamical triangulation model in 1 + 1 dimensions to allow for a varying lapse. We show that, as long as the spatially averaged lapse is constant in time, the physical observables are unchanged in the continuum limit. This supports the claim that the time slicing of the model is the result of a gauge fixing, rather than a physical preferred time slicing.