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Matrices and algebras in the canonical tensor model

Obster, Dennis 京都大学 DOI:10.14989/doctor.k24168

2022.09.26

概要

まず第一章は導入である.重力の量子化の必要性や,期待される物理的成果などについて説明がなされる.次に,量子重力への様々なアプローチが紹介され,その困難などについても簡単に解説が行われる.また,空間と代数との双対性を意味するGelfandの定理が紹介され,3階テンソルが代数を定義できることを根拠として,3階テンソルが空間を記述する可能性について簡単に触れられる.そして,正準テンソル模型が3階テンソルを力学変数とする正準量子化によるアプローチであることが説明され,先の空間との双対性から,時空間の量子論となることが期待されることが説明される.

第二章では,まず,結合則を満たす可換代数と位相空間との双対性の数学的定式化が詳しく紹介される.更にラプラシアンを導入することによりその位相空間に距離計量が導入され,リーマン多様体の代数的定式化が説明される.

第三章は,Obster氏による独自な部分となる.正準テンソル模型の力学変数である3階テンソルはその添字についての交換対称性以外の構造を持たないため,可換代数ではあるものの結合則を満たさず,第二章で行われたようなリーマン空間との対応を直接持たない.また,正準テンソル模型のテンソルは一般に有限次元において定式化されており,無限自由度がある連続的なリーマン多様体との関係も不明である.Obster氏のオリジナルな発想は,そのような正準テンソル模型の3階テンソルとリーマン多様体を結びつけることにある.まず3階テンソルから作られる代数を考え,その中にある結合則を部分的に満たす部分を拡張することにより結合則を満たすような必要かつ最大の代数系を構成する.更に代数に単位元が存在することを要求することにより,その構成された代数に対してある種のスケール変換を実行し,このスケール変換の生成子としてラプラシアンを導入する.これらの操作により,リーマン多様体との関係が付けられる.更にこれらの操作において,データ解析の分野で知られているテンソルランク分解との密接な関連が指摘される.

第四章では,正準テンソル模型においてこれまでに知られた諸結果についての簡単な解説が行われる.一般相対論との古典的関係や,量子化とその結果得られる厳密な波動関数についての説明が行われる.また,データ解析において知られているテンソルのランク分解やパーシステントホモロジーの応用により,正準テンソル模型のダイナミクスを幾何学的位相的ダイナミクスに結びつけた結果なども紹介される.

第五章は共同研究の紹介である.原理的には上記の波動関数を調べれば正準テンソル模型のダイナミクスを導くことができる.しかし,波動関数の変数は3階テンソルであり,その自由度はNの3乗のオーダーとなり,調べるには巨大すぎ,なんらかの単純化・粗視化が必要となる.最も簡単なものとして,その波動関数の絶対値自乗を指数関数でregularizeし積分したものがある.これを実行するとある行列模型が得られる.しかし,その行列模型は通常の行列模型と異なり,片方の添字は通常通りペアで縮約されるが,もう一方は3重の形で縮約がなされるため,通常の行列模型における対角化の取り扱いができず,解くのが難しい.この共同研究では,主に数値計算を行うことによりその行列模型を調べた,その主な結果は,この行列模型には連続的な相転移を生じる点が存在し,正準テンソル模型がのNのleadingorderで丁度その上にあるということである.通常離散的理論と連続理論とが連続的な相転移点上で関連づくことを考えると,正準テンソル模型の連続極限の可能性を探る上で好ましい結果と考えられる.

第六章では,テンソルランク分解の体積という量の厳密な関数形が導かれる.テンソルランク分解はデータ解析において,テンソルから有益な情報を引き出す上で大事な技術となっている.行列の特異値解析のテンソルへの拡張の一種と考えられるが,行列の場合と違って確立された方法が存在せず,その性質については分からないことが多い.第五章で述べた行列模型の研究で得られた知見を活かすとテンソルランク分解の体積に相当する量が超幾何関数により厳密に表されることが導かれる.

第七章では,論文のまとめと将来解決すべき問題などが説明される.

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