Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531-- 562 (1989)  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... classical mechanics and reduction [18,32] with the C # algebraic Supported by a fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW) formulation of quantum mechanics and induction [15] and also with non commutative geometry [2] Starting with Rieffel s fundamental paper [27], various C # algebraic definitions of quantization have been proposed [29,12,30,15,31] Definition 2 below is closely related to these proposals, and is particularly useful in the context of the class of examples studied in this paper. These examples come from the theory of Lie groupoids and ....

....function appearing in Example 2. For small enough h a function f C # PW (g # Q) is then quantized by : Exp(X) q) # # g # f(#,Exp( When G and Q has a G invariant measure, the map f # Q is equivalent to the deformation quantization considered by Rieffel [27], who already proved that it is strict (also cf. 15] Note added in proof.All results remain true when the groupoid C # algebras are replaced by reduced ones. This is clear both from the proof of Lemma 3 and from the argument at the end of Sect. 4 (which should be attributed to E. Blanchard) ....

Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531--562 (1989)

....constructions involving homotopic algebra and 1 structures that go under this name as well, and which are actually closely related to formal deformation quantization; see [55] for a representative paper. The C algebraic approach to deformation quantization was initiated in 1989 by Rieffel [49], who observed that a number of examples of quantization could be described by continuous fields of C algebras in a natural and attractive way. As indicated above, some of his examples involve quantization as physicists know and love it, like Weyl Moyal quantization and related constructions ....

....quantization and related constructions (see, in particular, 50] for a survey) while others relate to noncommutative geometry. In the latter category, Rieffel s discovery that the familiar noncommutative tori can be seen as deformation quantizations of ordinary symplectic tori stands out [49, 51]. Noncommutative tori actually do have potential physical relevance through string theory [15] We refer to [31, 50] for surveys of the starting period of C algebraic deformation 2 quantization, including references up to 1998. Later work that is relevant to noncommutative geometry ....

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M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys. 122, 531--562 (1989).

....only a small fraction of the desirable features listed above. Nevertheless some of the techniques developed for this problem, like upper and lower symbols, or certain operators connecting spin systems of different spin [LS] are close to the approach of this paper. G) Deformation quantization. Ri1,Ri2,Lan] In this approach the emphasis is indeed on the structure of products and Poisson brackets, and it is in many ways close to ours. With each classical phase space function (typically the Fourier transform of a finite measure) one associates a specific family of h dependent operators, ....

.... of compact support have been made the basis of a discussion of the classical limit by Emch [Em1,Em2] In his approach each classical observable F 0 thus has a unique h sequence of quantum observables F h associated with it, which is also typical for deformation quantization approaches [Ri1,Ri2,Ri3] In our approach this constraint becomes unnecessary, both from a technical and from a conceptual point of view. Emch s main emphasis is on defining the (weak) convergence of states with respect to this particular set of sequences. The intersection between his classical states , and our ....

M.A. Rieffel: "Deformation quantization of Heisenberg manifolds", Commun.Math.Phys. 122(1989) 531--562 32

.... than Planck s constant itself, that are variable in nature) Various categories of algebras A have been proposed in the literature; for the sake of analytical convenience, we follow Haag Kastler [12] in taking quantum algebras of observables to be C algebras (in the present context also cf. [32]) In any case, the relevant algebraic feature is that A should have two internal multiplications that can be compared with Delta and f ; g in A 0 . In an operator algebra these are (one half times) the anticommutator [ and (i= times) the commutator [ Gamma , respectively. The ....

....(f g) f exp 2i 0 p x Gamma x p 1 A g: 1.2) The meaning of the limits in (1.1) and of the exponential in (1.2) and, more generally, of notions of dependence and convergence, needs to be made precise. In [1] formal power series in are used, whereas in [32] a C algebraic framework is used, in which questions of convergence are handled using the norm topology. An interesting mathematical point here is that under suitable conditions norm convergence implies that the K theory of A 0 is related to that of A [4] Also, in case that f and g are ....

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M.A. Rieffel, "Deformation quantization of Heisenberg manifolds", Commun. Math. Phys. 122 (1989) 531-562.

....algebras, non commutative geometry. MSC nos. 46N50, 81S05, 81S30. PACS nos. 02.40. m, 03.65.Db. Supported by an S.E.R.C. Advanced Research Fellowship 1 1 Introduction Building upon previous leads [5, 3] M. Rieffel recently proposed a mathematically satisfactory framework of quantization [36, 37]. The main point of his approach is to make precise the intuitive idea, due to Dirac, that quantum commutators (times i=h) should converge to Poisson brackets in the classical limit h 0. We combine this with an analogous requirement that the anti commutator converge to the pointwise product, ....

M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys. 122, 531-562 (1989).

....of classical observables. The same is true of approaches based on Feynman integrals [AHK] and on the limits of coherent states [Hep, Hag] An approach to the classical limit emphasizing the limit of observables and their algebraic structure has recently been developed in [We2] compare also [Rie, Em1]) This approach makes rigorous the intuitive criterion for deciding which observables in quantum theory may effectively be treated classically: classical observables should not change too much under small position or momentum translations, where, due to the relation p = hk, a small momentum ....

M.A. Rieffel:"Deformation quantization of Heisenberg manifolds", Commun.Math.Phys. 122(1989) 531--562

....is a highly non unique procedure, and the correspondence principle is the only physical principle allowing one to decide whether a particular procedure is correct. To our taste, the most satisfying mathematical framework for quantization is that of strict deformation quantization proposed in [R1]. i.e. Quantization of the Anosov dynamics on the torus has been discussed before by a number of authors. The original reference is [HB] where a scheme is proposed using a group of unitary matrices on a finite dimensional Hilbert space. The generator of this group was determined from (i) the ....

....relations above has been studied extensively by both physicists and mathematicians, and we refer the reader to [R2] for an overview and extensive list of references. In particular, it has been established that smooth elements in this algebra obey a strong version of the correspondence principle [R1]. III.B. For our purposes, we consider the von Neumann algebra A , generated by U and V . Recall [D] that an algebra of bounded operators R on a Hilbert space H is called a von Neumann algebra, if (i) it is closed under taking the hermitian conjugate, and (ii) it is equal to its bicommutant, R ....

Rieffel, M.: Deformation quantization of Heisenberg manifolds, Comm. Math. Phys., 122, 531--562 (1989)

....objects. Quantization is a highly non unique procedure, and the correspondence principle is the only physical principle allowing one to decide whether a particular procedure is correct. A natural mathematical framework for quantization is strict deformation quantization proposed by Rieffel [17]. The key requirement is that lim 0 fl fl fl fl 1 i [Q (f) Q (g) Gamma Q (ff; gg) fl fl fl fl = 0; where f; g 2 C 1 (M ) The dynamic component of quantization consists in defining a time evolution on the quantized phase space. A natural way of doing this is to find a ....

....where is an arbitrary normalized element of H 2 (C ) It is well known that is, in fact, a faithful normal trace on A . As 0, this trace reproduces the classical ensemble average given by (1) It is characterized by the property that (U m V n ) ffi m0 ffi n0 : Theorem 4.1. [17], 18] This algebra satisfies the conditions of strict deformation quantization. We introduce the following notation: X = U i Gammai= p 2 j ; Y = U i 1= p 2 j ; and observe that [X; Y ] 0: The operators X and Y generate an action of the group Z 2 on H 2 (C ; d ) We also ....

Rieffel, M.: Deformation quantization of Heisenberg manifolds, Comm.Math.Phys., 122, 531 -- 562 (1989)

....Of course, Theorem 2.5 gives more than a purely formal product, since we have operator norm bounds on remainders. We can in particular show that the map f 7 T k (f) satisfies two important properties in Rieffel s definition of deformation quantization in the the category of C algebras [19]. This was proven in [4] Corollary 2.6. For f; g 2 C 1 (X) kT k (f)T k (g) Gamma T k (fg)k = O(k Gamma1 ) and kk[T k (f)T k (g) Gamma iT k (ff; gg)k = O(k Gamma1 ) To close this section, we consider the issue of orderings. Suppose we have a map R : C 1 (X) Psi 0 (P ) ....

M. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), 531--562.

....set of values dictated by geometric quantization. S. KLIMEK and A. LESNIEWSKI I. Introduction In a series of papers [1] 2] 3] we studied non perturbative deformation quantization of Riemann surfaces. Our approach is based on the ideas of [4] for related developments, see also [5] [6], 7] and references therein. A satisfactory picture of uniformization of exceptional quantum Riemann surfaces emerged from these investigations. In the case of higher genus (g 2) Riemann surfaces, the uniformization on the quantum level is a more complex issue. In fact, if M is a Riemann ....

Rieffel, M. A.: Deformation quantization of Heisenberg manifolds, Comm. Math. Phys., 122, 531--562 (1989).

....A and C 1 (P ) respectively (barring boundedness considerations) For our purpose it does not matter very much what one exactly means by a quantization; the induction procedure may be applied to any data H; A; B; H . Ideally, these data correspond to a strict deformation quantization [37] (as redefined in [17] of the symplectic data, as in some of our examples in section 4. We now take our cue from three directions (details to be given later on in this paper) 1. Take G a locally compact group and H ae G a closed subgroup. Let (H) be a unitary representation of H on H ; we ....

....and ae a moment map is considered in [51] as a classical analogue of induction. To complete the parallel, we recall Rieffel s discovery that the group algebra C (G) is a deformation quantization of the Poisson algebra C 1 (g ) 38] which in specific cases is even strict in the sense of [37]. 2. Let (P; Q; H; pr) be a principal fibre bundle with projection pr : P Q and a compact Lie group H acting on the total space P from the right. The symplectic leaves of the Poisson manifold (T P) H are in one to one correspondence with the co adjoint orbits O in h , and, as originally ....

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Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math.Phys. 122, 531-562 (1989)

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Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531-- 562 (1989)

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