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 furth...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.展开更多
Let （Zll, . . . , Z1N,... , Zml,. . . , ZmN, Wll,. . . , Wmm） be the coordinates in C^mN＋m2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as foll...Let （Zll, . . . , Z1N,... , Zml,. . . , ZmN, Wll,. . . , Wmm） be the coordinates in C^mN＋m2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows W=zz-t＋O（3）,where W ={wij}1≤ i,j≤ m and Z ={Zij}1≤i≤ m,1≤j≤N We prove that M is biholomorphically equiva-lent to the model W =zz-tif and only if is formally equivalent to it.展开更多
基金Supported by NSFC(Grant No.12101451)Tianjin Municipal Science and Technology Commission(Grant No.22JCQNJC00470)。
文摘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.
文摘Let （Zll, . . . , Z1N,... , Zml,. . . , ZmN, Wll,. . . , Wmm） be the coordinates in C^mN＋m2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows W=zz-t＋O（3）,where W ={wij}1≤ i,j≤ m and Z ={Zij}1≤i≤ m,1≤j≤N We prove that M is biholomorphically equiva-lent to the model W =zz-tif and only if is formally equivalent to it.
