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184
Differentiable and algebroid cohomology, van Est . . . classes
, 2000
"... In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and ..."
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Cited by 93 (20 self)
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In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by WeinsteinXu [47]). As a second application we extend van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives a slight improvement of HectorDazord’s integrability criterion [12]. In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the van Est map. This extends EvensLuWeinstein’s characteristic class θL [17] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
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Cited by 57 (7 self)
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Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
Characterizing quantum theory in terms of informationtheoretic constraints
 Foundations of Physics
, 2003
"... We show that three fundamental informationtheoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibil ..."
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Cited by 56 (2 self)
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We show that three fundamental informationtheoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment—suffice to entail that the observables and state space of a physical theory are quantummechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about nonlocality and bit commitment. KEY WORDS: quantum theory; informationtheoretic constraints. Of John Wheeler’s ‘‘Really Big Questions,’ ’ the one on which most progress has been made is It from Bit?—does information play a significant role at the foundations of physics? It is perhaps less ambitious than some of the other Questions, such as How Come Existence?, because it does not necessarily require a metaphysical answer. And unlike, say, Why the Quantum?, it does not require the discovery of new laws of nature: there was room for hope that it might be answered through a better understanding of the laws as we currently know them, particularly those of quantum physics. And this is what has happened: the better understanding is the quantum theory of information and computation. 1
Deformation quantization of compact Kähler manifolds by BerezinToeplitz quantization
, 1999
"... For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via BerezinToeplitz operators. Results on their semiclassical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are use ..."
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Cited by 51 (6 self)
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For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via BerezinToeplitz operators. Results on their semiclassical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 44 (10 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type
, 2001
"... Let K be a connected Lie group of compact type and let T ∗ (K) be its cotangent bundle. This paper considers geometric quantization of T ∗ (K), first using the vertical polarization and then using a natural Kähler polarization obtained by identifying T ∗ (K) with the complexified group KC. The firs ..."
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Cited by 42 (11 self)
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Let K be a connected Lie group of compact type and let T ∗ (K) be its cotangent bundle. This paper considers geometric quantization of T ∗ (K), first using the vertical polarization and then using a natural Kähler polarization obtained by identifying T ∗ (K) with the complexified group KC. The first main result is that the Hilbert space obtained by using the Kähler polarization is naturally identifiable with the generalized Segal–Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal–Bargmann transform introduced by the author. This means that in this case the pairing map is a constant multiple of a unitary map. For both results it is essential that the halfform correction be included when using the Kähler polarization. These results should be understood in the context of results of K. Wren and of the author with B. Driver concerning the quantization of 1+1dimensional Yang–Mills theory. Together with those results the present paper may be seen as an instance of “quantization commuting with reduction.”
Matrix algebras converge to the sphere for quantum GromovHausdorff distance
 Mem. Amer. Math. Soc
"... Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how t ..."
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Cited by 35 (6 self)
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Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups. On perusing the theoretical physics literature which deals with string theory and related parts of quantum field theory, one finds in many scattered places assertions that the complex matrix algebras, Mn, converge to the twosphere, S 2, (or to related spaces) as n goes to infinity. Here S 2 is viewed as synonymous with the algebra C(S 2) of continuous complexvalued functions on S 2 (of which S 2 is the maximalideal space). Approximating the sphere by matrix algebras is attractive for the following reason. In trying to carry out quantum field theory on S 2 it is natural to try to proceed by approximating S 2 by finite spaces. But “lattice ” approximations coming from choosing a finite set of points in S 2 break the very important symmetry of the action of SU(2) on S 2 (via SO(3)). But SU(2) acts naturally on the matrix algebras, in a way coherent with its action on S 2, as we will recall below. So it is natural to use them to approximate C(S 2). In this setting the matrix algebras are often referred to as “fuzzy spheres”. (See [33], [34], [17], [22], [24] and references therein.) When using the approximation of S 2 by matrix algebras, the precise sense of convergence is usually not explicitly specified in the literature. Much of the literature is at a largely algebraic level, with indications that the notion of convergence which is intended involves how structure constants and important formulas change as n grows. See, for
BerezinToeplitz Quantization of Compact Kähler Manifolds
 in Quantization, Coherent States and Poisson Structures, Proceedings of the XIV Workshop on Geometric Methods in Physics, Bia̷lowie˙za
"... Abstract. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the BerezinToeplitz quantization of compact Kähler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follo ..."
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Cited by 35 (7 self)
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Abstract. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the BerezinToeplitz quantization of compact Kähler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed. Invited lecture at the XIV th workshop on geometric methods in physics, Bia̷lowie˙za,
On integrability of infinitesimal actions
 Amer. J. Math
"... We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semidirect product associated to an infinitesimal action of one integrable Lie algebroid on another is int ..."
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Cited by 33 (5 self)
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We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semidirect product associated to an infinitesimal action of one integrable Lie algebroid on another is integrable. This generalizes recent results of Dazord and Nistor.
YangMills theory and the SegalBargmann transform
 COMMUN. MATH. PHYS
, 1999
"... We use a variant of the SegalBargmann transform to study canonically quantized YangMills theory on a spacetime cylinder with a compact structure group K. The nonexistent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The SegalBargmann ..."
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Cited by 33 (23 self)
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We use a variant of the SegalBargmann transform to study canonically quantized YangMills theory on a spacetime cylinder with a compact structure group K. The nonexistent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The SegalBargmann transform is then a unitary map from the L² space over the space of connections to a holomorphic L² space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by et∆A/2 followed by analytic continuation. Here ∆A is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory. On the gaugetrivial subspace, consisting of functions of the holonomy around the spatial circle, the SegalBargmann transform becomes et∆K/2 followed by analytic continuation, where ∆K is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that ∆A reduces to ∆K on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized