Wave function in 2+1 dimensional causal dynamical triangulation - PowerPoint PPT Presentation

Wave function in 2+1 dimensional causal dynamical triangulation JGRG22 (2012 11.12) Takayuki HIKICHI (Nagoya Univ.) collaborator Yasusada NAMBU (Nagoya Univ.) Hiromi SAIDA (Daido Univ) 2012 年 11 月 12 日月曜日

Contents 3D Quantum Gravity defined by CDT (Causal dynamical triangulation) We consider space-time with a spatial boundary. We measure the dynamics of spatial volume and whether spatial geometry is homogeneous. 2012 年 11 月 12 日月曜日

Path integral and Wave function trajectory （ quantum mechanics ） → metric （ quantum gravity ） path integral representation of a wave function metric of a spatial boundary 2012 年 11 月 12 日月曜日

The ground state wave function is given by a path integral over all compact Euclidean geometries which have a boundary. Hartle&Hawking(1983) Wick rotation Euclidean path integral 2012 年 11 月 12 日月曜日

Causal dynamical triangulation → a discrete regularization method for gravitational path integral The space-time is discretized by simplices. Advantage ・ functional integral finite summation ・ avoid conformal divergence 2012 年 11 月 12 日月曜日

t+1 t ・ ・ ・ ・ ・ ・ Discretization of Lorentzian space-time & Wick rotation in CDT discretized 3D Lorentzian space-time squared length of space-like links squared length of time-like links Wick rotation (We choose ) 2012 年 11 月 12 日月曜日

正曲率（５枚） 平坦（６枚） 負曲率（８枚） Curvature of discretized space-time e. g. 2D positive(5) flat(6) negative(8) coordination number (number of simplices around a hinge(2d vertex, 3d link) ) curvature flat space coordination number 6(2d), 5.104...(3d) 2012 年 11 月 12 日月曜日

Wave function in CDT discretized wave function sum over all discretized space-time gravitational constant total number of vertices cosmological constant total number of simplices coupling constant of a total number of triangles on boundary term a boundary Definition of boundary term in 3d DT: S. Warner, S. Catterall, R. Renken, Phase diagram of three- dimensional dynamical triangulations with a boundary, Phys. Lett. B 442 (1998) 266–272, hep-lat/9808006. 2012 年 11 月 12 日月曜日

Simulation set-up ・ topology spatial slice ・ pure gravity ・ fix volume of the spatial boundary ・ fix space-time volume for technical reason ・ We set ・ We set gravitational constant We perform Markov chain Monte Carlo simulation. 2012 年 11 月 12 日月曜日

Results 2012 年 11 月 12 日月曜日

Typical configuration time spacial volume 2012 年 11 月 12 日月曜日

Spatial volume dynamics In DT classical space-time didn’t emerge. In CDT the averaged spacial volume at Euclidean time can be described by de Sitter instanton. constant 2012 年 11 月 12 日月曜日

de Sitter 600 measured value standard deviation 500 400 300 200 boundary 100 14 16 18 20 22 time 2012 年 11 月 12 日月曜日

de Sitter de Sitter 1000 400 measured value measured value standard deviation standard deviation 800 300 600 200 400 100 200 0 12 14 16 18 20 22 17 18 19 20 21 22 time time boundary boundary 2012 年 11 月 12 日月曜日

Homogeneity of spacial slice number of hinges which have same curvature 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 15 20 25 30 coordination number(local curvature) 2012 年 11 月 12 日月曜日

Homogeneity of spacial slice number of hinges which have same curvature number of hinges which have same curvature 3.5 60 3.0 50 2.5 40 2.0 30 1.5 20 1.0 10 0.5 0 0.0 1 2 3 4 5 6 7 8 9 10 15 20 25 30 1 2 3 4 5 6 7 8 9 10 15 20 25 30 coordination number(local curvature) coordination number(local curvature) 2012 年 11 月 12 日月曜日

Summary ・ CDT is a discrete regularization method for gravitational path integral. ・ The emergent space-time in corresponds to de Sitter instanon and the spatial slices are homogenous. 2012 年 11 月 12 日月曜日

Typical configuration time spacial volume 2012 年 11 月 12 日月曜日

Homogeneity of space-time number 8000 6000 4000 2000 0 curvature 1 2 3 4 5 6 7 8 9 10 15 20 2012 年 11 月 12 日月曜日