[论文解读] Dirac and Weyl Fermions -- the Only Causal Systems

Domenico P. L. Castrigiano|arXiv (Cornell University)|Nov 17, 2017

Topological Materials and Phenomena参考文献 26被引用 6

一句话总结

本文证明，在庞加莱协变性下，满足因果性和可局域化的唯一不可约相对论性量子系统是狄拉克和外尔费米子。通过分析类空与非类空超平面上的因果时间演化与局域化，证明只有这些系统满足因果局域化的必要条件，包括洛伦兹收缩以及通过投影值测度将局域化推广至非类时区域。

ABSTRACT

Causal systems describe the localizability of relativistic quantum systems complying with the principles of special relativity and elementary causality. At their classification we restrict ourselves to real mass and finite spinor systems. It follows that (up to certain not yet discarded unitarily related systems) the only irreducible causal systems are the Dirac and the Weyl fermions. Their wave-equations are established as a mere consequence of causal localization. - The bounded localized Dirac and Weyl wavefunctions are studied in detail. One finds that, at the speed of light, the carriers shrink in the past and expand in the future. For every direction in space there is a definite time at which the change from shrinking to expanding occurs. A late changing time characterizes those states, which shrink to a delta-strip if boosted in the opposite direction. Using a density result for these late-change states one shows that all Dirac and Weyl wave-functions are subjected to Lorentz contraction. The latter is discussed in some detail. - We tackle the question whether a causal system induces a representation of a causal logic and thus provides a localization in proper space-time regions rather than on spacelike hyperplanes. The causal logic generated by the spacelike relation is shown to do not admit representations at all. But the logic generated by the non-timelike relation in general does, and the necessary condition is derived that there is a projection valued measure on every non-timelike non-spacelike hyperplane being the high boost limit of the localization on the spacelike hyperplanes. Dirac and Weyl systems are shown to satisfy this condition and thus to extend to all non-timelike hyperplanes, which implies more profound properties of the causal systems. The bounded localized eigenstates of the projections to non-spacelike flat strips are late-change states.

研究动机与目标

• 确定在庞加莱协变性下，唯一满足因果性和可局域化的不可约相对论性量子系统。
• 通过证明仅狄拉克和外尔费米子可避免该不一致，解决牛顿-温伯格局域化与因果性之间的矛盾。
• 证明因果局域化可超越类空超平面，延伸至非类时区域，其必要条件涉及投影值测度。
• 确立在高Boost条件下，有界局域狄拉克和外尔波函数普遍具有洛伦兹收缩的特性。
• 阐明粒子-反粒子混合在因果局域化中的作用，表明局域化算符产生混合态而非纯态。

提出的方法

• 将因果系统形式化为二元组 $(W, E)$，其中 $W$ 是庞加莱群的酉表示，$E$ 是满足庞加莱协变性和因果性条件 $E(\Delta) \leq E(\Delta_\tau)$ 的投影值测度（PVM）。
• 分析狄拉克和外尔态局域概率密度的长期行为，表明波函数在过去的收缩与未来的膨胀，其转变由一个关键的“晚期变化”时间点标记。
• 利用晚期变化态的稠密性结果，证明所有有界局域狄拉克和外尔波函数在高Boost条件下均表现出洛伦兹收缩。
• 引入基于非类时关系的因果逻辑，并推导出表示存在的必要条件：每个非类时超平面上均存在一个投影值测度，作为类时空局域PVM在高Boost极限下的极限。
• 为电子和正电子构造正算子值测度（POL），证明其具有因果性且相互分离，意味着存在任意局域化的状态。
• 证明狄拉克和外尔系统满足向非类时超平面扩展的必要条件，从而实现对更深层因果结构的分析。

实验结果

研究问题

• RQ1 在庞加莱协变性下，哪些不可约相对论性量子系统与因果性和可局域化相容？
• RQ2 为何牛顿-温伯格和威格曼局域化无法满足因果性，而狄拉克和外尔费米子如何克服这一问题？
• RQ3 因果局域化能否超越类空超平面延伸至非类时区域，其必要条件是什么？
• RQ4 粒子-反粒子混合在因果局域化中起什么作用，它如何影响局域化态的本质？
• RQ5 洛伦兹收缩如何作为因果系统中所有有界局域波函数的普遍特征出现？

主要发现

• 唯一不可约的因果系统是狄拉克和外尔费米子，至多相差酉等价。
• 有界局域狄拉克和外尔波函数表现出特定行为：在时间上溯时收缩，未来则扩张，每个空间方向均以一个‘晚期变化’时间点为转折。
• 所有有界局域狄拉克和外尔波函数均受洛伦兹收缩影响，该结论通过高Boost下晚期变化态的稠密性结果得到证明。
• 由非类时关系生成的因果逻辑允许表示，当且仅当每个非类时超平面上均存在一个投影值测度，且该测度是类时空局域PVM在高Boost极限下的极限。
• 狄拉克和外尔系统满足该条件，使得因果局域化可被推广至所有非类时超平面，暗示其具有更深层次的结构性质。
• 投影到非类时空平带上的算符的本征态恰好是晚期变化态，从而确认其在这些系统因果动力学中的核心作用。