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[...]... replaced by a “time” parameter defined by deformation along spin boundaries as will now be explained Let the spin be located at the vertices of the triangulation A given spin configuration can be decomposed into spin clusters where spin either point up or down The spin boundaries can be viewed as closed loops passing through the centers of the triangles and crossing the links (i.e living on the lattice... dimension is best explained by considering the Ising model coupled to gravity in the context of dynamical triangulations The model has a critical point as a function of the Ising coupling Away from the critical point the geometric aspects of the combined theory coincide with those of pure gravity, but at the critical point, where the theory is believed to describe a conformal field theory of central...This page intentionally left blank THE STRUCTURE OF 2D QUANTUM SPACE-TIME Jan Ambjørn The Niels Bohr Institute Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark INTRODUCTION The free relativistic particle provides us with the simplest example of quantum geometry” The action of a free relativistic particle is just the length of its world line* in R d The classical path between two space-time points x... simulations indicate that we have a two-phase structure as a function of the bare gravitational coupling constant, the two phases being separated by a first order phase transition For large bare gravitational coupling constant we are in the phase with infinite dh (the limit of infinite bare gravitational coupling constant corresponds to not including the Einstein-Hilbert term in the action at all, i.e the situation... 129 J Ambjørn, J Jurkiewicz and Y Watabiki, Nucl Phys B454 (1995) 313 E Gava and K.S Narain (ICTP, Trieste), Phys.Lett B263 (1991) 213 N Ishibashi and H Kawai, Phys.Lett B322 (1994) 67 hep-th/9312047 R Nakayama and T Suzuki, Phys.Rev D54 (1996) 3985 hep-th/9604073; Phys.Lett.B354 (1995) 69 hep-th/9503190 J Ambjørn and Y Watabiki, Noncritical string field theory for two-D quantum gravity coupled to (p, ... metric Field- operators with specific partial gauge fixing g αβ = e ø α β, where scaling properties when the volume V → λV willn pick up anomalous scaling in Liouville gravity If the classical scaling is expected to be 〈Φ n 〉 λV = λ 〈Φ n 〉 V, the quantum Liouville scaling will be (89) where c is the central charge of the conformal field theory coupled to two-dimensional quantum gravity 15 Spin Boundaries... of the spin system The total magnetization at βc will be (95) In 2d the magnetization at the critical point is determined by the largest cluster of spin The clusters of spins are described by percolation theory and if p denotes probability that a site belongs to a largest cluster, it is known that (96) The largest spin clusters at the critical point will be fractal In peculation theory one defines the... solved since we know w(z, g) (for details see 10, 9 ) However, we are interested in the two-point function It is obtained from the two-loop function be closing the exit loop with a “(cap” (i.e the full disk amplitude w(l, µ)) and shrinking the entrance loop to a point The corresponding equation is (71) Since w (z, g) and G (z1 , z 2 ; r ) are known we can find G µ (r), µ → µ c , i.e in the continuum... 1/2 coupled to quantum gravity, the string susceptibility jumps from –1/2 (the value for pure gravity) to –1/3, the value for a c = 1/2 theory coupled to gravity according to the KPZ formula (80) At the same time the values of critical exponents related to magnetic properties of the spin system differ from the flat space values of the exponents, showing that gravity influences the critical properties... step out will include all the triangles in the loop One can view the step as if we “lasso” the loop and in this way create two new boundaries, each with a the same distance to the original link The spins at the two boundaries will be opposite The procedure is illustrated in fig 7 Following the above described procedure one can systematically proceed outwards from a given triangle where the three spins . Jurkiewicz Series B: Physics New Developments in Quantum Field Theory Edited by Poul Henrik Damgaard Niels Bohr Institute Copenhagen, Denmark and Jerzy Jurkiewicz Jagellonian. http://www.ebooks.kluweronline.com Print ISBN 0-3 0 6-4 581 6-0 eBook ISBN 0-3 0 6-4 707 5-6 PREFACE Quantum field theory is one of most central constructions in 20th century theo- retical