nLab geometric quantization - contents

geometric quantization higher geometric quantization

geometry of physics: Lagrangians and Action functionals + Geometric Quantization

Prerequisites

• symplectic geometry, higher symplectic geometry

  • n-plectic geometry, multisymplectic geometry

  • symplectic infinity-groupoid

Prequantum field theory

• prequantum field theory

• prequantum circle n-bundle = extended Lagrangian

  • prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology

  • prequantum 0-bundle = action functional

• quantomorphism group, quantomorphism n-group

  • symplectomorphism, contactomorphism

  • Poisson algebra, Hamiltonian vector field

Geometric quantization

• polarization

  • metaplectic correction

  • Bohr-Sommerfeld leaf

  • orientation in generalized cohomology

• quantization

  • geometric quantization

  • geometric quantization by push-forward

  • geometric quantization of symplectic groupoids

  • geometric quantization of non-integral forms

  • motivic quantization

• space of states

  • coherent state

• quantum operator

Applications

• Borel-Weil theorem, Borel-Weil-Bott theorem

  • orbit method

  • Schubert calculus

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