Geometric quantization is a mathematical framework for quantizing classical systems, aiming to establish a correspondence between classical and quantum mechanics. It begins by defining a mathematical structure that applies to both classical and quantum mechanics. The process involves prequantization, which introduces a prequantum line bundle and a connection with curvature proportional to the symplectic form. This leads to the construction of a Hilbert space and operators representing observables. The key challenge is to ensure that the resulting quantum system is physically meaningful, which requires the use of polarizations to restrict the space of states and ensure the Hilbert space corresponds to a known quantum system. Prequantization involves defining a Hermitian line bundle over a symplectic manifold, with a connection whose curvature is proportional to the symplectic form. This allows the construction of a Hilbert space of square-integrable sections of the bundle. However, this space is too large to represent a physically meaningful quantum system, necessitating the use of polarizations to restrict the space of states. Quantization involves selecting a polarization, which is a complex involutive Lagrangian distribution. This allows the construction of a Hilbert space of sections of the prequantum bundle that are covariantly constant along the polarization. This process ensures that the resulting Hilbert space corresponds to a known quantum system and satisfies the necessary physical conditions, such as the uncertainty principle. The quantization of observables involves defining operators that act on the Hilbert space of polarized sections. These operators must preserve the polarization to satisfy the irreducibility postulate. The quantization of observables is closely related to the choice of polarization and the structure of the symplectic manifold. For cotangent bundles, the vertical polarization is a natural choice, leading to the Schrödinger representation. Alternatively, using the horizontal polarization leads to the momentum representation, with the relation between these representations being the Fourier transform. The process of geometric quantization thus provides a rigorous mathematical framework for quantizing classical systems, ensuring that the resulting quantum systems are physically meaningful and consistent with the principles of quantum mechanics.Geometric quantization is a mathematical framework for quantizing classical systems, aiming to establish a correspondence between classical and quantum mechanics. It begins by defining a mathematical structure that applies to both classical and quantum mechanics. The process involves prequantization, which introduces a prequantum line bundle and a connection with curvature proportional to the symplectic form. This leads to the construction of a Hilbert space and operators representing observables. The key challenge is to ensure that the resulting quantum system is physically meaningful, which requires the use of polarizations to restrict the space of states and ensure the Hilbert space corresponds to a known quantum system. Prequantization involves defining a Hermitian line bundle over a symplectic manifold, with a connection whose curvature is proportional to the symplectic form. This allows the construction of a Hilbert space of square-integrable sections of the bundle. However, this space is too large to represent a physically meaningful quantum system, necessitating the use of polarizations to restrict the space of states. Quantization involves selecting a polarization, which is a complex involutive Lagrangian distribution. This allows the construction of a Hilbert space of sections of the prequantum bundle that are covariantly constant along the polarization. This process ensures that the resulting Hilbert space corresponds to a known quantum system and satisfies the necessary physical conditions, such as the uncertainty principle. The quantization of observables involves defining operators that act on the Hilbert space of polarized sections. These operators must preserve the polarization to satisfy the irreducibility postulate. The quantization of observables is closely related to the choice of polarization and the structure of the symplectic manifold. For cotangent bundles, the vertical polarization is a natural choice, leading to the Schrödinger representation. Alternatively, using the horizontal polarization leads to the momentum representation, with the relation between these representations being the Fourier transform. The process of geometric quantization thus provides a rigorous mathematical framework for quantizing classical systems, ensuring that the resulting quantum systems are physically meaningful and consistent with the principles of quantum mechanics.