Tetsuyuki Yukawa

Lax form of the quantum mechanical eigenvalue problem

Abstract The quantum mechanical eigenvalue problem with the hamiltonian H = H 0 + tV is written as a set of dynamical equations for the eigenvalues x n ( t ) and the matrix elements V nm ( t ) regarding the parameter t as time. By appropriate changes of variables it can be expressed as a pair of matrix equations with the Lax form, hence we are able to write all the possible constants of the motion explicitly. Implications of these constants to the statistical properties of levels are discussed.

Simulation of 2D dynamical triangulation with higher order curvature terms

Abstract We present a numerical result of the Monte Carlo simulation of two dimensional random surface generated by dynamical triangulation under influence of higher order curvature terms. We measure the fractal dimension and related quantities such as the boundary length distribution for various lattice sizes up to 400 000 triangles. The boundary length distribution is found to scale nicely. We also find cross-over transition between fractal and flat surfaces through varying the strength of higher order terms.

Scaling Behavior in 4D Simplicial Quantum Gravity

Scaling relations in four-dimensional simplicial quantum gravity are proposed using the concept of the geodesic distance. Based on the analogy of a loop length distribution in the two-dimensional case, the scaling relations of the boundary volume distribution in four dimensions are discussed in three regions: the strong-coupling phase, the critical point and the weak-coupling phase. In each phase different scaling behavior is found.Scaling relations in four-dimensional simplicial quantum gravity are proposed using the concept of the geodesic distance. Based on the analogy of a loop length distribution in the two-dimensional case, the scaling relations of the boundary volume distribution in four dimensions are discussed in three regions: the strong-coupling phase, the critical point and the weak-coupling phase. In each phase a different scaling behavior is found.

Phases and fractal structures of three-dimensional simplicial gravity

Abstract We study phases and fractal structures of three-dimensional simplicial quantum gravity by a Monte Carlo calculation with a lattice size V =10 4 . After measuring the surface area distribution (SAD) which is the three-dimensional analog of the loop length distribution (LLD) in two-dimensional quantum gravity, we classify the fractal structures into three types: (i) In the strong coupling (hot) phase, strong gravity makes the space-time one crumpled mother universe with many fluctuating baby universes of small size around it. This is a crumpled phase with a large Hausdorff dimension d H =4.98±0.05. The topologies of the sections are extremely complicated. (ii) At the critical point, we observe that the space-time is a pseudo-fractal manifold which has one mother universe with many baby universes of small and middle size around it. The Hausdorff dimension d H is 3.93±0.05. We observe some scaling behaviors for the sections of the manifold. This manifold resembles the fractal surface observed in two-dimensional quantum gravity. (iii) In the weak coupling (cold) phase, the mother universe disappears completely and the space-time seems to be a branched-polymer with a small Hausdorff dimension d H =1.948±0.003. almost all of the sections have the spherical topology S 2 in the weak coupling phase.

2D QUANTUM GRAVITY: THREE STATES OF SURFACES

Two-dimensional random surfaces are studied numerically by the dynamical triangulation method. In order to generate various kinds of random surfaces, two higher derivative terms are added to the action. The phases of surfaces in the two-dimensional parameter space are classified into three states: flat, crumpled surface, and branched polymer. In addition, there exists a special point (pure gravity) corresponding to the universal fractal surface. A new probe to detect branched polymers is proposed, which makes use of the minbu(minimum neck baby universe) analysis. This method can clearly distinguish the branched polymer phase from another according to the sizes and arrangements of baby universes. The size distribution of baby universes changes drastically at the transition point between the branched polymer and other kind of surface. The phases of surfaces coupled with multi-Ising spins are studied in a similar manner.

Scaling structures in four-dimensional simplicial gravity☆

Abstract Four-dimensional(4D) spacetime structures are investigated using the concept of the geodesic distance in the simplicial quantum gravity. On the analogy of the loop length distribution in 2D case, the scaling relations of the boundary volume distribution in 4D are discussed in various coupling regions i.e. strong-coupling phase, critical point and weak-coupling phase. In each phase the different scaling relations are found.

Complex Structures Defined on Dynamically Triangulated Surfaces

A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as

Landau damping in the degenerate Fermi liquid

n

Complex structure of a DT surface with T2 topology

scalar fields (

Finite size effects for the ising model coupled to 2-D random surfaces

n=0,1