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Quantization in riemannian space

P. V. Elyutin, V. D. Krivchenkov

Theoretical and Mathematical PhysicsSep 1, 19732 citations

Flux quantization and quantum mechanics on Riemann surfaces in an external magnetic field

Jens Bolte, Frank Steiner

The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.

Journal of Physics A Mathematical and GeneralAug 21, 199112 citations

Holomorphic Quantization

WORLD SCIENTIFIC eBooksSep 17, 20150 citations

Berry Curvature and Riemann Curvature in Kinematic Space with Spherical Entangling Surface

Xing Huang, Chen‐Te Ma

Abstract We discover the connection between the Berry curvature and the Riemann curvature tensor in any kinematic space of minimal surfaces anchored on spherical entangling surfaces. This new holographic principle establishes the Riemann geometry in kinematic space of arbitrary dimensions from the holonomy of modular Hamiltonian, which in the higher dimensions is specified by a pair of time‐like separated points as in CFT1 and CFT2. The Berry curvature that we constructed also shares the same property of the Riemann curvature for all geometry: internal symmetry; skew symmetry; first Bianchi identity. We derive the algebra of the modular Hamiltonian and its deformation, the latter of which can provide the maximal modular chaos to the modular scrambling modes. The algebra also dictates the parallel transport, which leads to the Berry curvature exactly matching to the Riemann curvature tensor. Finally, we compare CFT1 to higher dimensional CFTs and show the difference from the OPE block.Abstract We discover the connection between the Berry curvature and the Riemann curvature tensor in any kinematic space of minimal surfaces anchored on spherical entangling surfaces. This new holographic principle establishes the Riemann geometry in kinematic space of arbitrary dimensions from the holonomy of modular Hamiltonian, which in the higher dimensions is specified by a pair of time‐like separated points as in CFT1 and CFT2. The Berry curvature that we constructed also shares the same property of the Riemann curvature for all geometry: internal symmetry; skew symmetry; first Bianchi identity. We derive the algebra of the modular Hamiltonian and its deformation, the latter of which can provide the maximal modular chaos to the modular scrambling modes. The algebra also dictates the parallel transport, which leads to the Berry curvature exactly matching to the Riemann curvature tensor. Finally, we compare CFT1 to higher dimensional CFTs and show the difference from the OPE block.

arXiv:2003.12252Fortschritte der PhysikDec 16, 20206 citations

Bosonization of Chiral Fermions on Higher Genus Riemann Surfaces

Shyuichi Ojima

We give the method of bosonization of chiral fermions on arbitrary genus Riemann surface using KN basis. We calculate the bosonized correlation functions between chiral fermions and those including U (1) current with the help of KN operator formalism.

Progress of Theoretical PhysicsJun 1, 19900 citations

HOLOMORPHIC QUANTIZATION

WORLD SCIENTIFIC eBooksJun 23, 20130 citations

Riemann surface and quantization

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