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arXiv:1608.00031 · Journal of Physics A Mathematical and Theoretical

The geometry of real reducible polarizations in quantum mechanics

Carlos Prieto, Raffaele F. Vitolo

Published Jan 11, 2017• 2 citations
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Abstract

Abstract The formulation of geometric quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to quantum mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the physics literature.

DOI
10.1088/1751-8121/aa58c7

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Abstract The formulation of geometric quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to quantum mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the physics literature.
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations we find a compatibility condition restricting the set of admissible connections. In the special case when phase space is a cotangent bundle this compatibility condition has many solutions, and the resulting quantum theory not only reproduces the well-known geometric quantization scheme, but also allows to quantize all interesting observables. For Kähler manifolds there is no compatibility condition, but a canonical choice for the parameters. The explicit form of the observables however remains undetermined.
Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold MMM is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of MMM with the quotient space G/HG/HG/H, where HHH is the isotropy group of an arbitrary fixed point of MMM, we show that quantum mechanics on G/HG/HG/H possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle G(G/H,H)G(G/H, H)G(G/H,H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.
Abstract. The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework. 1.