Uncertainty Principles on Clifford Modules  

作　　者：Pan LIAN 

机构地区：[1]School of Mathematical Sciences,Tianjin Normal University,Tianjin 300387,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2024年第10期2537-2570,共34页数学学报（英文版）

基　　金：Supported by NSFC(Grant No.12101451);Tianjin Municipal Science and Technology Commission(Grant No.22JCQNJC00470)。

摘　　要：In this paper,we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra,which heavily depend on the Clifford algebraic structure.The obtained inequalities further lead to very general uncertainty inequalities on these modules.Some new phenomena arise,due to the non-commutative nature,the Clifford-valued inner products and the Krein geometry.Taking into account applications,special attention is given to the Dirac operator and the Howe dual pair Pin(m)×osp(1|2).Moreover,it is surprisingly to find that the recent highly nontrivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy–Schwarz inequality.This new observation leads to refined uncertainty relations in terms of the Wigner–Yanase–Dyson skew information for mixed states and other generalizations.These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.

关 键 词：Clifford algebra uncertainty principle Krein space Fischer decomposition Wigner-Yanase-Dyson skew information 

分 类 号：O413.1[理学—理论物理]