西郷 甲矢人
(さいごう・はやと)
Hayato Saigo
略歴
- 京都大学理学研究科(数学・数理解析専攻)博士課程修了
- プリンストン高等研究所(Interdisciplinary Studies, 2010-2011期)滞在後本学へ
電子のゆらぎなどにおける偶然性を「無知のせいにできない」という事実は、物理量の積について「AB=BA とは限らない」という「代数的」な構造に根ざしていることが知られています。私は、自然の根底に流れるこのような数理を探るために、代数的な観点からゆらぎを研究する理論=「代数的確率論」を研究しています。
量子古典対応の数理
新しい「独立性」概念
圏論による組織化
- 研究の応用領域
- 代数学・組み合わせ論・確率論から自然科学における数理モデルの研究まで多岐にわたる。
- 産官学連携で求めるパートナー
- 自然に満ち溢れるゆらぎを無理矢理に押さえ込もうとするのではなく、ゆらぎと巧みに付き合い、むしろそれを活かしていくような文化・技術を志向する方々。
Algebraic probability is such an algebraic framework extending probabilistic notions that its scope is to cover various areas of mathematics and sciences related to quantum theory.
While usual probability theory mainly treats random variables commuting with each other, in quantum theory it is essential to deal with noncommutative random variables which we call observables. Then states are nothing but the expectation functionals for noncommutative random variables.
Conceptually, an algebra of observables describes the system to be observed and a state an interface between observed and observing sides (a view point due to I. Ojima). By taking this point of view, we can obtain a deeper understanding of the fundamental problems in Physics. For instance, we have discovered the crucial role of the “Arcsine Law”, a famous probability law, in “Quantum-Classical Correspondence”.
One of the central issues of interest in algebraic probability as a mathematical theory is the extension of the notion of “independence”.
Among many kinds of generalizations “monotone independence” (Muraki) is a unique one as an “asymmetric” kind characterized by certain naturality conditions, in the sense that monotone independence of “X from Y” does not imply that of “Y from X”. The monotone version of Central Limit Theorem (CLT) is first obtained by Muraki, with its “limit distribution” not being the Gaussian but the Arcsine law.
We have succeeded in clarifying the intrinsic combinatorial structure of monotone limit theorems for the first time.
What is the most relevant to the unification of such types of the limit theorems is the notion of “cumulants”, which is, however, incompatible with the above sort of asymmetry inherent in the monotone independence. This was the most essential obstruction which prevented the transparent explanation for the logical relation between monotone CLT and the definition of monotone independence. In collaboration with T. Hasebe, we have overcome this difficulty by our discovery of “generalized cumulants” By the use of this, we have obtained quite simple proofs for the limit theorems in the monotone case.
Moreover, the combinatorial representation of the relation between moments and generalized cumulants are obtained by developing the arguments before due to myself. Now we are investigating the category-theoretic meaning of generalized cumulants.
I. Ojima, K. Okamura and H. Saigo, Derivation of Born Rule from Algebraic and Statistical Axioms, Open Systems and Information Dynamics, 21, 1450005 (2014).
T. Hasebe and H. Saigo, On operator-valued monotone independence, Nagoya Mathematical Journal, 215, 151-167 (2014).
H. Saigo, A new look at the Arcsine law and “Quantum-Classical Correspondence”, Inf inite Dimensional Analysis, Quantum Probability and Related Topics, 13,1250021 (2012).
I. Ojima and H. Saigo, Who has seen a free photon?, Open Systems and Information Dynamics 19, 1250008 (2012).
T. Hasebe and H. Saigo, The monotone cumulants, Annales de l’Institut Henri Poincaré;(B) Probabilités et Statistiques 47, No. 4, 1160-1170 (2011).