论文标题
量子空间，量子时间和相对论量子力学
Quantum Space, Quantum Time, and Relativistic Quantum Mechanics
论文作者
论文摘要
我们将空间和时间视为希尔伯特空间中平等基础上的真正的量子自由度。受量子重力的考虑，我们专注于处理线性，一阶哈密顿量和动量约束的范式，这导致了时间和空间翻译的新兴特征。与常规处理不同，我们表明，可以通过将相对论分散关系应用于特征值，而不是将它们视为操作员值得值的方程式，可以在我们的范式中统一相对论量子力学中的klein-gordon和dirac方程。随着时间和空间在希尔伯特（Hilbert）空间的平等基础上进行处理，我们显示出通过希尔伯特（Hilbert）空间的单一基础变化来实现的对称转换，从而使它们具有更强的量子机械基础。诸如洛伦兹（Lorentz）变换之类的全球对称性修改了希尔伯特（Hilbert）空间的分解；和当地的对称性，例如$ u（1）$仪表对称性在坐标的基础上是对角线，并且不会改变希尔伯特空间的分解。我们简要讨论了该范式到量子场理论和量子重力的扩展。
We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being treated on an equal footing in Hilbert space, we show symmetry transformations to be implemented by unitary basis changes in Hilbert space, giving them a stronger quantum mechanical footing. Global symmetries, such as Lorentz transformations, modify the decomposition of Hilbert space; and local symmetries, such as $U(1)$ gauge symmetry are diagonal in coordinate basis and do not alter the decomposition of Hilbert space. We briefly discuss extensions of this paradigm to quantum field theory and quantum gravity.
