标题： 七球量化 显示英文标题

标题： Seven Sphere Quantization

摘要： 共向接触流形通常描述经典动力系统。 通过适当将薛定谔方程与接触流形中的每条路径相关联来实现量子化。 我们通过将标准接触七球视为四元数酉群的齐次空间来对其量子化，以构建对称辛旋量丛上费多索夫形式连接的接触类似物。 我们证明，要求形式连接的收敛性自然地过滤了对称辛旋量丛，并在每个相应的子丛上产生一个精确的平坦连接。 关键要素是对四元数酉群的霍尔斯坦-普里马科夫机制的一般化。 从形式量子化到真实量子化的过渡确定了四元数酉群的酉不可约表示，其维度随着形式变形参数趋近于其经典极限而趋于无穷大。 出现有限维表示并不令人意外，因为接触七球是闭合的，并且物理上描述了广义的位置、动量和时间变量。 显示英文摘要

摘要： Co-oriented contact manifolds quite generally describe classical dynamical systems. Quantization is achieved by suitably associating a Schr\"odinger equation to every path in the contact manifold. We quantize the standard contact seven sphere by treating it as a homogeneous space of the quaternionic unitary group in order to construct a contact analog of Fedosov's formal connection on symplectic spinor bundles. We show that requiring convergence of the formal connection naturally filters the symplectic spinor bundle and yields an exact flat connection on each corresponding subbundle. A key ingredient is a generalization of the Holstein--Primakoff mechanism to the quaternionic unitary group. The passage from formal to bona fide quantization determines unitary irreducible representations of the quaternionic unitary group, whose dimensions tend to infinity as the formal deformation parameter approaches its classical limit. This appearance of finite-dimensional representations is not surprising since the contact seven sphere is closed and physically describes generalized positions, momenta and time variables.