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32
Index and dynamics of quantized contact transformations. Annales de l’institut Fourier
, 1997
"... Abstract. Quantized contact transformations are Toeplitz operators over a contact manifold (X, α) of the form Uχ = ΠAχΠ, where Π: H 2 (X)→L 2 (X) is a Szego projector, where χ is a contact transformation and where A is a pseudodifferential operator over X. They provide a flexible alternative to the ..."
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Cited by 31 (4 self)
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Abstract. Quantized contact transformations are Toeplitz operators over a contact manifold (X, α) of the form Uχ = ΠAχΠ, where Π: H 2 (X)→L 2 (X) is a Szego projector, where χ is a contact transformation and where A is a pseudodifferential operator over X. They provide a flexible alternative to the Kahler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine ind(Uχ) when the principal symbol is unitary, or equivalently to determine whether A can be chosen so that Uχ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms g—by showing that Ug duplicates the classical transformation laws on theta functions. Using the CauchySzego kernel on the Heisenberg group, we calculate the traces on theta functions of each degree N. We also study the quantum dynamics generated by a general q.c.t. Uχ, i.e. the quasiclassical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on χ. Our principal results are proofs of equidistribution of eigenfunctions φNj and weak mixing properties of matrix elements (BφNi, φNj) for quantizations of mixing symplectic maps. The problem of quantizing symplectic maps and of analyzing the dynamics of the quantum system is
Quantum computation and the localization of modular functors
"... Kevin Walker, and Zhenghan Wang. Their work has been the inspiration for this lecture. The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum ..."
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Cited by 27 (6 self)
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Kevin Walker, and Zhenghan Wang. Their work has been the inspiration for this lecture. The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus = 0 surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation. 1 The Picture Principle Reality has the habit of intruding on the prodigies of purest thought and encumbering them with unpleasant embellishments. So it is astonishing when the chthonian hammer of the engineer resonates precisely to the gossamer fluttering of theory. Such a moment may soon be at hand in the practice and theory of quantum computation. The most compelling theoretical question, “localization, ” is yielding an answer which points the way to a solution of Based on lectures prepared for the joint Microsoft/University of Washington celebration
A magnetic model with a possible ChernSimons phase
 Commun. Math. Phys
"... A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a co ..."
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Cited by 27 (3 self)
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A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a complete mathematical description: the quantum double of the SO(3)−ChernSimons modular functor at q = e 2πi/ℓ+2 which we call DEℓ. The Hamiltonian H◦,ℓ defines a quantum loop gas. We argue that for ℓ = 1 and 2, G◦,ℓ is unstable and the collapse to Gǫ,ℓ ∼ = DEℓ can occur truly by perturbation. For ℓ ≥ 3 G◦,ℓ is stable and in this case finding Gǫ,ℓ ∼ = DEℓ must require either ǫ> ǫℓ> 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state G◦,ℓ of H◦,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gǫ,ℓ described by a quotient algebra. By classification, this implies Gǫ,ℓ ∼ = DEℓ. The fundamental point is that nonlinear structures
Universality and scaling of zeros on symplectic manifolds
"... Abstract. This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebrogeometric generalization studied in our prior work involves random holomorphic sections H 0 (M, L N) of the powers of any positive line bundle L → ..."
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Cited by 18 (12 self)
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Abstract. This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebrogeometric generalization studied in our prior work involves random holomorphic sections H 0 (M, L N) of the powers of any positive line bundle L → M over any complex manifold. Our main interest is in the statistics of zeros of k independent sections (generalized polynomials) of degree N as N → ∞. We fix a point P and focus on the ball of radius 1 / √ N about P. Under a microscope magnifying the ball by the factor √ N, the statistics of the configurations of simultaneous zeros of random ktuples of sections tends to a universal limit independent of P, M, L. We review this result and generalize it further to the case of prequantum line bundles over almostcomplex symplectic manifolds (M, J, ω). Following [SZ2], we replace H 0 (M, L N) in the complex case with the ‘asymptotically holomorphic ’ sections defined by Boutet de MonvelGuillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an mdimensional setting of the KacRice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random ktuples of asymptotically holomorphic sections belong to the same universality class as in the complex case. 1.
Szegö kernels and a theorem of Tian
 Int. Math. Res. Notices
, 1998
"... A variety of results in complex geometry and mathematical physics depend upon the analysis of holomorphic sections of high powers L ⊗N of holomorphic line bundles L → M over compact Kähler manifolds ([A][Bis][Bis.V] [Bou.1][Bou.2][B.G][D][Don] [G][G.S][K][Ji] [T] [W]). The principal tools have been ..."
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Cited by 17 (3 self)
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A variety of results in complex geometry and mathematical physics depend upon the analysis of holomorphic sections of high powers L ⊗N of holomorphic line bundles L → M over compact Kähler manifolds ([A][Bis][Bis.V] [Bou.1][Bou.2][B.G][D][Don] [G][G.S][K][Ji] [T] [W]). The principal tools have been Hörmander’s L 2estimate on
Topological quantum field theory with corners based on the Kauffman bracket
 COMMENTARII MATHEMATICI HELVETICI
, 1997
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On the WittenReshetikhinTuraev representations of the mapping class groups
 Proc. Amer. Math. Soc
, 1999
"... We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representa ..."
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Cited by 14 (5 self)
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We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image. Recall that an object in a cobordism category of dimension 2+1 is a closed oriented surfaces Σ perhaps with some specified further structure. A morphism M from Σ to Σ ′ is (loosely speaking) a compact oriented 3manifold perhaps with some specified further structure, called a cobordism, whose boundary is the disjoint union of −Σ and Σ ′. A morphism M ′ from Σ ′ to Σ ′ ′ is composed with a morphism from Σ to Σ ′ by gluing along Σ ′ , inducing any required extra structure from the structures on M and M ′. Also the extra structure on a 3manifold must induce the extra structure on the boundary. A TQFT in dimension 2+1 is then a functor from
Discrete Quantum Causal Dynamics
 International Journal of Theoretical Physics
, 2003
"... We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolut ..."
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Cited by 10 (5 self)
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We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences  such as covariance  follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1
Direct sum decompositions and indecomposable TQFTs
 J. Math. Phys
, 1995
"... Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and i ..."
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Cited by 9 (0 self)
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Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and indecomposable twodimensional theories are classified. 1.
A Uribe; The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)connections on the torus are the same
 Commun. Math. Phys
"... Abstract. We prove that, for the moduli space of flat SU(2)connections on the 2dimensional torus, the Weyl quantization and the quantization using the quantum group of SL(2, C) are the same. This is done by comparing the matrices of the operators associated by the two quantizations to cosine funct ..."
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Cited by 8 (6 self)
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Abstract. We prove that, for the moduli space of flat SU(2)connections on the 2dimensional torus, the Weyl quantization and the quantization using the quantum group of SL(2, C) are the same. This is done by comparing the matrices of the operators associated by the two quantizations to cosine functions. We also discuss the ∗product of the Weyl quantization and show that it satisfies the producttosum formula for noncommutative cosines on the noncommutative torus. Contents