A.4 Complex structures and Bargmann-Fock quantization

We define complex coordinates on phase space by

z_{j} = \frac{1}{2} (q_{j} - i p_{j}), \overline{z}_{j} = \frac{1}{2} (q_{j} + i p_{j})

The standard choice of complex structure on phase space $M$ is given by

J_{0} \frac{\partial}{\partial q _{j}} = - \frac{\partial}{\partial p _{j}}, J_{0} \frac{\partial}{\partial p _{j}} = \frac{\partial}{\partial q _{j}}

and on coordinate basis vectors $q_{j}, p_{j}$ of the dual space $M$ by

J_{0} q_{j} = p_{j}, J_{0} p_{j} = - q_{j}

The complex coordinates satisfy

J_{0} z_{j} = i z_{j}, J_{0} \overline{z}_{j} = - i \overline{z}_{j}

so the $z_{j}$ are a basis of $M_{J_{0}}^{+}$ , the $\overline{z}_{j}$ of $M_{J_{0}}^{-}$ . They have Poisson brackets

{z_{j}, \overline{z}_{k}} = i δ_{j k}

In the Bargmann-Fock quantization, the state space is taken to be polynomials in the $z_{j}$ , with annihilation and creation operators

a_{j} = \frac{\partial}{\partial z _{j}}, a_{j}^{†} = z_{j}

来源与版本

正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 535、536

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

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