马约拉纳场算符 (Majorana field operator)

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Woit · 2025-10-20

已登记出处中的最早位置：§47.2 · zh-CN。

位置	语言	登记角色
§47.2	zh-CN · 英文上下文	定义入口
§47.2	en · 中文上下文	定理入口

来源短摘录与定位

§47.2 · zh-CN · machine_draft · machine-draft-not-reviewed

定义（马约拉纳场算符）

马约拉纳场算符由下式给出

$$\hat{\Psi }(\mathbf{x},t)=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}\sum _{s=\pm }(a_s(\mathbf{p})u_s(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a_s^{†}(\mathbf{p})\overline{u_s(\mathbf{p})}{e}^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

§47.2 · en · machine_draft · machine-draft-not-reviewed

Definition (Majorana field operator)

The Majorana field operator is given by

$$\hat{\Psi }(\mathbf{x},t)=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}\sum _{s=\pm }(a_s(\mathbf{p})u_s(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a_s^{†}(\mathbf{p})\overline{u_s(\mathbf{p})}{e}^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

If one uses commutation instead of anticommutation relations, the Hamiltonian operator will have eigenstates with arbitrarily negative energy, and there will be problems with causality due to observable operators at space-like separated points not commuting. These two problems are resolved by the use of anticommutation instead of commutation relations. The multi-particle state space for the theory has occupation numbers 0 or 1 for each value of p and for each value of $s=\pm$ . Like the case of the real scalar field, the particles described by these states are their own antiparticles. Unlike the case of the real scalar field, each particle state has a ${\mathbf{C}}^2$ degree of freedom corresponding to its spin $\begin{array}{l}\frac{1}{2}\end{array}$ nature.

来源覆盖、定位与记录

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