Lectures on the Geometry of QuantizationAmerican Mathematical Soc., 1997 - 137 páginas These notes are based on a course entitled ``Symplectic Geometry and Geometric Quantization'' taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading. |
Índice
Introduction The Harmonic Oscillator | 3 |
The WKB Method | 6 |
Symplectic Manifolds | 16 |
Quantization in Cotangent Bundles | 35 |
The Symplectic Category | 64 |
Fourier Integral Operators | 83 |
Geometric Quantization | 93 |
Algebraic Quantization | 111 |
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Términos y frases comunes
approximate solution associated asymptotic bundle Q C/C¹ canonical relation canonically isomorphic closed 1-form cohomology coisotropic submanifold composition conormal bundle constant corresponding cotangent bundle critical point defined definition denote diffeomorphism equivalence Example fiber foliation follows geometric groupoid half-density Hamilton-Jacobi hamiltonian vector field harmonic oscillator Hilbert space immersed lagrangian submanifold implies induces integral intersection kernel lagrangian embedding lagrangian immersion lagrangian subbundles lagrangian submanifold lagrangian subspace Lemma line bundle linear map Liouville class Maslov class metaplectic structure Morse family nondegenerate parallel section phase function Poisson algebra Poisson manifold polarization F prequantizable prequantization prequantum line bundle principal projection Proof pull-back quantization quantum mechanics satisfies Schrödinger equation Schwartz transform smooth manifold Sp(V subbundle symbol symplectic form symplectic groupoid symplectic manifold symplectic structure symplectic vector bundle symplectic vector space symplectomorphism tangent Theorem Tr bundle transverse vanishes vector bundle zero section
Pasajes populares
Página 134 - J.-L. Brylinski: Loop spaces, characteristic classes and geometric quantization. Progress in Mathematics vol. 107. Birkhauser, Boston, Mass.
Página 135 - Invariant deformations of the Poisson Lie algebra of a symplectic manifold and star-products.
Página 136 - ... Satake, The theorems of de Rham and the duality theorem of Poincare, (Seminar Notes, Tokyo) (in Japanese), 1957. [12] EH Spanier, Cohomology theory for general spaces, Ann. of Math., 4O (1948), 407427. [13] JP Serre, Faisceaux algebriques cohe"rents, Ann. of Math., 61 (1955), 197-278. [14] A. Weil, Sur les theoremes de de Rham, Comm. Math. Helv., 26 (1952), 119-145. [15] E. Luft, Eine Verallgemeinerung der Cechschen Homologietheorie, Bonn. Math. Schr. no. 8 (1959). (Math. Rev. Vol. 21, No. 6...
Página 136 - Quantification et analyse pseudo-differentielle, Ann. Sci. Ec. Norm. Sup. 21 (1988), 133-158.
Página 135 - Libermann, P., and Marie, C.-M., Symplectic Geometry and Analytical Mechanics, Reidel, Dordrecht, 1987.
Página 2 - As each skylark must display its comb, so every branch of mathematics must finally display symplectisation.
Página 136 - Symplectic groupoids, geometric quantization, and irrational rotation algebras, in Symplectic Geometry, Groupoids, and Integrable...