1,913 research outputs found
Functorial quantization and the Guillemin-Sternberg conjecture
We propose that geometric quantization of symplectic manifolds is the arrow
part of a functor, whose object part is deformation quantization of Poisson
manifolds. The `quantization commutes with reduction' conjecture of Guillemin
and Sternberg then becomes a special case of the functoriality of quantization.
In fact, our formulation yields almost unlimited generalizations of the
Guillemin--Sternberg conjecture, extending it, for example, to arbitrary Lie
groups or even Lie groupoids. Technically, this involves symplectic reduction
and Weinstein's dual pairs on the classical side, and Kasparov's bivariant
K-theory for C*-algebras (KK-theory) on the quantum side.Comment: 15 pages. Proc. Bialowieza 200
When champions meet: Rethinking the Bohr--Einstein debate
Einstein's philosophy of physics (as clarified by Fine, Howard, and Held) was
predicated on his Trennungsprinzip, a combination of separability and locality,
without which he believed objectification, and thereby "physical thought" and
"physical laws", to be impossible. Bohr's philosophy (as elucidated by Hooker,
Scheibe, Folse, Howard, Held, and others), on the other hand, was grounded in a
seemingly different doctrine about the possibility of objective knowledge,
namely the necessity of classical concepts. In fact, it follows from Raggio's
Theorem in algebraic quantum theory that - within an appropriate class of
physical theories - suitable mathematical translations of the doctrines of Bohr
and Einstein are equivalent. Thus - upon our specific formalization - quantum
mechanics accommodates Einstein's Trennungsprinzip if and only if it is
interpreted a la Bohr through classical physics. Unfortunately, the
protagonists themselves failed to discuss their differences in this
constructive way, since their debate was dominated by Einstein's ingenious but
ultimately flawed attempts to establish the "incompleteness" of quantum
mechanics.
This aspect of their debate may still be understood and appreciated, however,
as reflecting a much deeper and insurmountable disagreement between Bohr and
Einstein on the knowability of Nature. Using the theological controversy on the
knowability of God as a analogy, Einstein was a Spinozist, whereas Bohr could
be said to be on the side of Maimonides. Thus Einstein's off-the-cuff
characterization of Bohr as a 'Talmudic philosopher' was spot-on.Comment: 22 pages. Argument sharpened and references update
Lie groupoid C*-algebras and Weyl quantization
For any Lie groupoid , the vector bundle dual to the associated Lie
algebroid is canonically a Poisson manifold. The (reduced) C*-algebra of
(as defined by A. Connes) is shown to be a strict quantization (in the
sense of M. Rieffel) of . This is proved using a generalization of Weyl's
quantization prescription on flat space. Many other known strict quantizations
are a special case of this procedure; on a Riemannian manifold, one recovers
Connes' tangent groupoid as well as a recent generalization of Weyl's
prescription. When is the gauge groupoid of a principal bundle one is led
to the Weyl quantization of a particle moving in an external Yang-Mills field.
In case that is a Lie group (with Lie algebra ) one recovers Rieffel's
quantization of the Lie-Poisson structure on . A transformation group
C*-algebra defined by a smooth action of a Lie group on a manifold turns
out to be the quantization of the semidirect product Poisson manifold
defined by this action.Comment: 14 page
Quantization and the tangent groupoid
This is a survey of the relationship between C*-algebraic deformation
quantization and the tangent groupoid in noncommutative geometry, emphasizing
the role of index theory. We first explain how C*-algebraic versions of
deformation quantization are related to the bivariant E-theory of Connes and
Higson. With this background, we review how Weyl--Moyal quantization may be
described using the tangent groupoid. Subsequently, we explain how the
Baum--Connes analytic assembly map in E-theory may be seen as an equivariant
version of Weyl--Moyal quantization. Finally, we expose Connes's tangent
groupoid proof of the Atiyah--Singer index theoremComment: 16 pages, Proc. Constanta 200
Poisson spaces with a transition probability
The common structure of the space of pure states of a classical or a
quantum mechanical system is that of a Poisson space with a transition
probability. This is a topological space equipped with a Poisson structure, as
well as with a function , with certain properties. The
Poisson structure is connected with the transition probabilities through
unitarity (in a specific formulation intrinsic to the given context).
In classical mechanics, where p(\rho,\sigma)=\dl_{\rho\sigma}, unitarity
poses no restriction on the Poisson structure. Quantum mechanics is
characterized by a specific (complex Hilbert space) form of , and by the
property that the irreducible components of as a transition probability
space coincide with the symplectic leaves of as a Poisson space. In
conjunction, these stipulations determine the Poisson structure of quantum
mechanics up to a multiplicative constant (identified with Planck's constant).
Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\em Acta Math.}
{\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 82}
(1982) 497-509), we give axioms guaranteeing that is the space of pure
states of a unital -algebra. We give an explicit construction of this
algebra from .Comment: 23 pages, LaTeX, many details adde
Compact Quantum Groupoids
Quantum groupoids are a joint generalization of groupoids and quantum groups.
We propose a definition of a compact quantum groupoid that is based on the
theory of C*-algebras and Hilbert bimodules. The essential point is that
whenever one has a tensor product over the complex numbers in the theory of
quantum groups, one now uses a certain tensor product over the base algebra of
the quantum groupoid.Comment: 8 pages, to appear in `Quantum Theory and Symmetries' (Goslar, 18-22
July 1999), eds. H.-D. Doebner et a
Deformation quantization and the Baum-Connes conjecture
Alternative titles of this paper would have been `Index theory without index'
or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an
analytic version of deformation quantization based on the use of continuous
fields of C*-algebras. We review how a wide variety of examples of such
quantizations can be understood on the basis of a single lemma involving
amenable groupoids. These include Weyl-Moyal quantization on manifolds,
C*-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the
Baum-Connes conjecture for smooth groupoids as described by Connes in his book
Noncommutative Geometry. Concerning the latter, we use a different semidirect
product construction from Connes. This enables one to formulate the Baum-Connes
conjecture in terms of twisted Weyl-Moyal quantization. The underlying
mechanical system is a noncommutative desingularization of a stratified Poisson
space, and the Baum-Connes conjecture actually suggests a strategy for
quantizing such singular spaces.Comment: 21 page
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