X iv : f un ct - a n / 97 10 00 2 v 3 2 4 Ju l 2 00 7 Quantum mechanics and operator algebras on the Hilbert ball ( The revised )
نویسنده
چکیده
Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C-algebra is represented as a function algebra on the set of pure states with a noncommutative ∗-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space P(H) of a Hilbert space H. The space P(H) is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature c. The Hilbert ball BH is defined by the open unit ball in H and it is a Kähler manifold with c < 0. We introduce the quantum mechanics on BH. As an application, we show the structure of the noncommutative function algebra on BH. MSC: 81R15; 32Q15; 58B20; 46L65
منابع مشابه
ar X iv : f un ct - a n / 97 07 00 9 v 1 2 8 Ju l 1 99 7 One - parameter representations on C ∗ - algebras
Strongly continuous one-parameter representations on a C *-algebra A and their extension to the multiplier algebra are investigated. We also give a proof of the Stone theorem on Hilbert C *-modules and look into some related problems.
متن کاملar X iv : f un ct - a n / 94 10 00 1 v 2 5 F eb 2 00 4 RELATIVE CONVOLUTIONS . I PROPERTIES AND APPLICATIONS
To study operator algebras with symmetries in a wide sense we introduce a notion of relative convolution operators induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolution etc.) and their different combinations. Basic properties of relative convolutio...
متن کاملar X iv : f un ct - a n / 97 04 00 4 v 1 2 1 A pr 1 99 7 Examining the dual of an algebraic quantum group .
In the first part of this paper, we implement the multiplier algebra of the dual of an algebraic quantum group (A, ∆) (see [13]) as a subset of the space of linear functionals on A. In a second part, we construct the universal corepresentation and use it to prove a bijective corespondence between corepresentations of (A, ∆) and homomorphisms on its dual.
متن کاملar X iv : f un ct - a n / 94 06 00 5 v 2 2 4 Ju l 1 99 5 An Algebraic Spin and Statistics Theorem
A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebra...
متن کاملar X iv : f un ct - a n / 97 04 00 5 v 1 2 1 A pr 1 99 7 A natural extension of a left invariant lower semi - continuous weight
In this paper, we describe a natural method to extend left invariant weights on C *-algebraic quantum groups. This method is then used to improve the invariance property of a left invariant weight. We also prove some kind of uniqueness result for left Haar weights on C *-algebraic quantum groups arising from algebraic ones.
متن کامل