%Kiyoshi Shiraishi:Regge %
04/26/2002
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Regge calculus


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\title{Review: Regge calculus と Cosmology}
\author{
 }
\date{1986年07月09日}
\begin{document}
\maketitle
\begin{abstract}
主な話のタネ

1. T. Piran and R. M. Williams, Phys. Rev. {\bf D33} (1986) 1622.

2. H. W. Hamber, Les Houches lecture notes, 1984.

3. Review by 大川氏 at KEK
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{最近のRegge calculusの話題}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}
\item Higher-derivative simplicial gravity
$\Leftrightarrow$ Quantum or Nonperturbative effect
\item (Inflationary) Cosmology (非)一様 (非)等方
\end{itemize}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{座標によらない``重力''の定式化}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{smoothな$d$次元多様体をflatな$d$次元単体(simplex)でおおう}

\begin{center}
\begin{tabular}{ccc}
 & simplex & \\
$2$次元 & triangle & 三角形 \\
$3$次元 & tetrahedron & 四面体 \\
$\vdots$ & $\vdots$ & \\
$d$次元 & $d$-simplex & 
\end{tabular}
\end{center}

・$d$-simplex は$d+1$個の頂点(vertices)と$d(d+1)/2$本の辺(edges)をもつ。

・辺の長さで形が決まる。

\subsection{曲率等を作る}

$\sqrt{g}R$($R$はscalar curvature)や$R_{ab}$,$R_{abcd}$などに関連した量(に対応するもの)
をつくる/定義する。

もちろん正しい連続極限(continuum limit)をもつもの

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{How?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{in $2$ dimensions}

\[
\epsilon_h=2\pi-\sum_{triangles~meeting~on~h}\theta_d
\]

$h$: hinge

『$2$-dim ではhingeはvertex』

\bigskip

in $2$ dim, すべてのhinge(頂点)についての和

\[
\sum_h \epsilon_h\rightarrow \frac{1}{2}\int\sqrt{g}R
\]

$\epsilon$: ``deficit angle'' と呼ぶ。

\bigskip

{\it cf.}
測地線三角形
\[
\frac{\alpha+\beta+\gamma-\pi}{S}\stackrel{S\rightarrow 0}{\longrightarrow}\frac{1}{2}R
\]
(see later)

\subsection{in $d$ dimensions}

several $d$-simplices meet on a $(d-2)$ dim. \underline{hinge}.
\[
\epsilon_h=2\pi-\sum_{d-simplices~meeting~on~h}\theta_d
\]
\[
\sum_h \epsilon_hA_h\rightarrow \frac{1}{2}\int\sqrt{g}R
\]
$A_h$: ``volume'' of the hinge

\bigskip

four tetrahedra meet on an edge (hinge).

\bigskip

See MTW Box 42.1

\bigskip

hinge=``ちょうつがい''

\section{examples}

\subsection{example 1}

$S^2$ topology をもつ空間を\underline{正}三角形(\underline{regular} triangle)
でおおう

4, 8, 20 triangles (Pythagoras)

\begin{tabular}{c|cccc}
 & $\#$ of vertices & $\#$ of edges & $\#$ of triangles & ひとつのhingeに集まる面の数 \\
\hline
正四面体 & 4 & 6 & 4 & 3 \\
正八面体 & 6 & 12 & 8 & 4 \\
正二十面体 & 12 & 30 & 20 & 5 \\
\hline
 & $N_0$ & $N_1$ & $N_2$ & 
\end{tabular}

ちなみに $N_0-N_1+N_2=2$

\bigskip

正多面体の例・・・一様等方

一様非等方の例・・・三辺の長さが異なる三角形1種で作った正四面体。

一様:『全ての頂点がequivalentである。』  $d$次元でも同じ定義

\subsection{example 2}

$S^3$ topology をもつ空間を\underline{正}四面体(\underline{regular} tetrahedron)
でおおう

5, 16, 600 tetrahedra (Coxeter, 1948)

\begin{tabular}{c|ccccc}
 & \# of vertices & \# of edges & \# of faces & \# of tetrahedra & ひとつのhingeに集まる胞の数 \\
\hline
正五胞体 & 5 & 10 & 10 & 5 & 3 \\
正十六胞体 & 8 & 24 & 32 & 16 & 4 \\
正六百胞体 & 120 & 720 & 1200 & 600 & 5 \\
\hline
 & $N_0$ & $N_1$ & $N_2$ & $N_3$ & $\alpha$
\end{tabular}

\subsection{古典作用積分:$S^2$}

2次元 $S^2$

\[
\sum_h \epsilon_h\stackrel{?}{\rightarrow} \frac{1}{2}\int_{S^2}\sqrt{g}R=4\pi
\]
なんとなれば,半径$a$の$S^2$では$R=d(d-1)/a^2=2/a^2$

$\frac{1}{2}\int_{S^2}\sqrt{g}R=\frac{1}{2}\times (4\pi a^2)\times \frac{2}{a^2}=4\pi$

\begin{itemize}
\item 正四面体 $\epsilon=\pi$,
\[
\sum_h\epsilon_h=\epsilon\times N_0=\pi\times 4=4\pi
\]
\item 正八面体 $\epsilon=2/3 \pi$,
\[
\sum_h\epsilon_h=\epsilon\times N_0=\frac{2}{3}\pi\times 6=4\pi
\]
\item 正二十面体 $\epsilon=1/3 \pi$,
\[
\sum_h\epsilon_h=\epsilon\times N_0=\frac{1}{3}\pi\times 12=4\pi
\]
\end{itemize}

exact!

\subsection{古典作用積分:$S^3$}

3次元 $S^3$

hinge (edge) の長さを$\ell_h$とする。
\[
\sum_h \epsilon_h\ell_h\stackrel{?}{\rightarrow} \frac{1}{2}\int_{S^3}\sqrt{g}R=6\pi^2 a
\]
なんとなれば,半径$a$の$S^3$では$R=d(d-1)/a^2=6/a^2$

$\frac{1}{2}\int_{S^3}\sqrt{g}R=\frac{1}{2}\times (2\pi^2 a^3)\times \frac{6}{a^2}=6\pi^2 a$

\bigskip

一辺の長さ$\ell$の regular tetrahedron

$\cos\theta=1/3$,   $(\cos\theta=1/d)$

$\theta=1.231\ldots {\rm rad}$

体積 $\frac{\sqrt{2}}{12}\ell^3$

\bigskip

\[
\frac{\sum_h \epsilon_h\ell_h}{V_{total}}\stackrel{?}{\rightarrow} 
\frac{6\pi^2}{(2\pi^2)^{1/3}}=21.91\ldots
\]

正多胞体では
\[
\frac{(2\pi-\alpha\theta)\times N_1}{\left(\frac{\sqrt{2}}{12}\times N_3\right)^{1/3}}
\]

この値は,正五胞体のとき30.90,正十六胞体のとき26.41,正六百胞体のとき22.35。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cosmology への応用I}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(Piran+Williams)

\bigskip

\subsection{``3+1 formulation'' }

continuous time

\bea
S&=&\int d^4x\sqrt{-g}R \nn
&=&\int dt d^3\vx \left\{ N\sqrt{{}^3g}\left[{}^3R+\Tr (K^2)-(\Tr K)^2\right]\right. \nn
& &\left. +2N_i\left[\sqrt{{}^3g}(K^{ij}-g^{ij}K)\right]_{|j}\right\} \nonumber
\eea
where
\[
K_{ij}=-\frac{1}{2N}\left({}^3\dot{g}_{ij}-N_{i|j}-N_{j|i}\right)
\]

$N$: lapse,  $N_i$: shift (see MTW, その他重力の教科書)

\bigskip

・一様な場合 $N_i$はいらない

\bigskip

\[
\int d^3\vx\sqrt{g}R\rightarrow 2\sum_h \epsilon_h \ell_h
\]

\bigskip

symmetric tensor のあつかい
\[
T_{ij}\rightarrow T_a=T_{ij}\ell_a^i\ell_a^j
\]
\[
g_{ij}\rightarrow g_a=g_{ij}\ell_a^i\ell_a^j=\ell_a^2
\]

\bigskip

対応づけをしていくと
\[
\left[\Tr (K)^2-(\Tr K)^2\right]_{\alpha}=g^{\alpha a b}K_aK_b
\]
\[
g^{\alpha a b}\equiv -\frac{1}{V_{\alpha}^2}\frac{\partial^2V_{\alpha}^2}{\partial g_a \partial g_b},
\qquad K_a\equiv -\frac{1}{2N}\dot{g}_a
\]
非一様だと いろいろめんどうくさいが。


\subsection{Inflationary Universe with a massive scalar field}

一様非等方

非一様非等方

\bigskip

freezing of anisotropy, inhomogeneity at \underline{large scale}

small scale $\rightarrow$ more detailed R. C.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cosmology への応用II}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Kasner Universe (Lewis 1982)

\bigskip

このように空間がflatなときはtorusの一部として``block''を考えればよい。


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weak limit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

long wave, weak field limit (flat background)

(continuum theory $g_{\mu\nu}\rightarrow\eta_{\mu\nu}+h_{\mu\nu}$に対応

\[
{\cal L}=-\frac{1}{2}\partial_{\lambda}h_{\alpha\beta}V_{\alpha\beta\mu\nu}
\partial_{\lambda}h_{\mu\nu}+\frac{1}{2}C_{\mu}^2
\]
where
\[
V_{\alpha\beta\mu\nu}\equiv\frac{1}{2}\delta_{\alpha\mu}\delta_{\beta\nu}-
\frac{1}{4}\delta_{\alpha\beta}\delta_{\mu\nu}
\]
and
\[
C_{\mu}=\partial_{\nu}h_{\mu\nu}-\frac{1}{2}\partial_{\mu}h_{\nu\nu}
\qquad (gauge~fixing)
\]

に対応したものがcontinuum limitでえられる。 (Rocek+Williams)

\bigskip

(図 略)・・・のようなlatticeの4次元版,波数展開する。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Higher derivative}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Even in the formulation of simlicial gravity,
$\int\sqrt{g}R$ action is not adequate for computer simulation,
because of its \underline{unboundedness} (from below).

\bigskip

order $R^2$ terms should be considered.

\bigskip

仮に$\int\sqrt{g}\rightarrow\sum_h V_h$とすると

($V_h$: 各hingeごとにわりふった体積,see later)

\bigskip

\[
\frac{1}{2}\int\sqrt{g}R\rightarrow\sum_h A_h\epsilon_h=
\sum_h V_h\frac{A_h\epsilon_h}{V_h}
\]
したがって
\[
\frac{1}{2}R\Leftrightarrow \frac{A_h\epsilon_h}{V_h}
\]
の対応が考えられる。

すなわち
\[
\frac{1}{4}\int\sqrt{g}R^2\rightarrow=
\sum_h V_h\left(\frac{A_h\epsilon_h}{V_h}\right)^2=
\sum_h \frac{A_h^2\epsilon_h^2}{V_h}
\]

\bigskip

$V_h$を定義する

\[
V_h=\sum_{d-simplices~meeting~on~h}V_d
\]

2-dim

外心を結ぶVoronoi分割


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{loopによる定義}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
``Area~of~loop''\equiv A_{\Gamma_h}\propto
\frac{V_h}{A_h^{(d-2)}}
\]
これは何次元でもホントにArea。

ここで$A_h^{(d-2)}$はhinge自体の「体積」。

\bigskip


\[
I=\sum_h A_h^{(d-2)}\epsilon_h=
\sum_h V_h\frac{A_h^{(d-2)}\epsilon_h}{V_h}=
\sum_h V_h\frac{\epsilon_h}{A_{\Gamma_h}}
\]

\[
\frac{\epsilon_h}{A_{\Gamma_h}}=\frac{角度欠損}{面積}\rightarrow R
\]

前を見よ。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computer simulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

一般にむずかしい。・・・lattice自身がdynamical

\subsection{Monte Carlo}

Path integral の形をつくるとき,measureが複雑になってしまう。

\bigskip

例えば

三角不等式を満たすとき $F=1$,みたさないとき $F=0$のような$F$等が必要



\subsection{Stochastic}

(Hamber+Williams)

\bigskip

Langevin eq.
\[
\frac{1}{\ell_p(t)}\frac{d\ell_p(t)}{dt}=-\ell_p(t)\frac{\delta I[\ell]}{\delta \ell_p(t)}+
\sqrt{2}\eta_p(t)
\]

$t$: fictious ``5th time''

\bigskip

$\eta$: Gaussian White Noise

\[
\langle\eta\rangle=0,\qquad\langle\eta\eta'\rangle=\delta
\]

以下reference参照。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{Regge}
the Original work\\
T. Regge, Nuovo Cimento {\bf 19} (1961) 558.
\bibitem{MTW}
For a review\\
C. W. Misner, K. Thorne and J. A. Wheeler, {\it Gravitation}
(Freeman, San Francisco, 1973), Chap. 42.
\bibitem{leshouches}
H. W. Hamber, Les Houches lecture notes, 1984,
``Simplicial Quantum Gravity''.\\
H. W. Hamber and R. M. Williams, Phys. Lett. {\bf B157} (1985) 368;
Nucl. Phys. {\bf B248} (1984) 392; {\it ibid.} {\bf B267} (1986) 482;
{\it ibid.} {\bf B269} (1986) 712.
\bibitem{Piran}
T. Piran and R. M. Williams, Phys. Rev. {\bf D33} (1986) 1622;
Phys. Lett. {\bf B163} (1985) 331.\\
T. Piran and A. Strominger, Class. Quantum Grav. {\bf 3} (1986) 97.\\
R. M. Williams, Gen. Rel. Grav. {\bf 17} (1985) 559.\\
P. A. Collins and R. M. Williams, Phys. Rev. {\bf D7} (1973) 965.\\
S. M. Lewis, Phys. Rev. {\bf D25} (1982) 306.\\
\bibitem{Rocek}
M. Rocek and R. M. Williams, Phys. Lett. {\bf B104} (1981) 31;
Z. Phys. {\bf C21} (1984) 371; Class. Quantum Grav. {\bf 2} (1985) 701;
M. J. Duff and C. J. Isham eds. {\it Quantum Structure of Space and Time}
(CUP) 105.
\bibitem{Hartle}
J. B. Hartle, in {\it Quantum Gravity 3}, Moscow?
``Simplicial Quantum Gravity'';
J. Math. Phys. {\bf 26} (1985) 804; {\it ibid.} {\bf 27} (1985) 287;
Class. Quantum Grav. {\bf 2} (1985) 707.
\bibitem{FL}
R. Friedberg and T. D. Lee, Nucl. Phys. {\bf 242} (1984) 145.
\bibitem{JN}
A. Jevicki and M. Ninomiya, Phys. Lett. {\bf B150} (1985) 115;
Phys. Rev. {\bf D33} (1986) 1634.
\bibitem{LNN}
M. Lehto, H. B. Nielsen and M. Ninomiya, Nucl. Phys. {\bf 272} (1986) 213, 228.
\end{thebibliography}

\end{document} 

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