量子力学需要（至少）复结构
The Requirement of (at least) Complex Structure for Quantum Mechanics
浏览论文内容
中文总结 AI 辅助
本文论证许多量子力学的实值构造实为复结构的表示，并证明在态归一化和严格能量守恒下，复结构是时间演化的最低要求，但不排除超复数表述。
中文摘要 AI 辅助
本文论证，许多量子力学的实值构造仅是“名义上”为实的，因为这些构造中的算符和态被限制为在希尔伯特空间上强加一个复结构。也就是说，在这些表述中，表示复数的一对元素之间的复代数运算得以保留。因此，将这些构造视为“实的”是错误的，因为它们实际上是复线性代数的一种表示。随后，本文证明，在态归一化和严格能量守恒的假设下，这种复结构（至少）是允许量子力学中时间演化所必需的。这只是一个最低要求，因为我们的论证并不排除用量子力学中的超复数实体（如四元数）来表述量子力学的可能性。
英文摘要
It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.
