论文标题
探测普朗克量表：由于时空的量子结构而导致的时间演化运算符的修改
Probing the Planck scale: The modification of the time evolution operator due to the quantum structure of spacetime
论文作者
论文摘要
在NRQM中演变的传播器可以表示为时间演变操作员的矩阵元素：I.E $ g _ {\ rm nr}（x）= \ langle {\ langle {\ mathbf {x} _2} _2} _2} nr}（t）} | {\ MathBf {x} _1} \ rangle $在正常的eigenkets $ | {\ mathbf {x}} \ rangle $的位置操作员的$。在QFT中，不可能定义一个概念上有用的单粒子位置操作员或其eigenkets。也不可能将相对论（Feynman）繁殖者$ g_r（x）$解释为发展任何形式的单粒子波函数。尽管如此，但确实可以在qft中以矩阵元素$ \ langle {\ mathbf {x} _2} | {u _ {\ rm rm r}（t）（t）（t）} | {\ mathbf {x} _1} _1} _1} _1} _1} _1} _1} _1} _1} _1} _1} _1} _1} _1 _1} _1} _1} _1} _1} _1} _1 _1} _1 _1} _1} _1} _1} _1 _1} _1} _1} _1} _1} _1} _1 _1} _1} _1} _1 _1} _1 （非正常现象）kets $ | {\ mathbf {x}} \ rangle $由空间坐标标记。在接近但不太接近普朗克量表的介质尺度上，可以通过引入零点长度来将量子重力校正到传播器。事实证明，即使是该QG校正的传播器也可以表示为矩阵元素$ \ langle {\ MathBf {x} _2} | {u _ {\ rm qg}（\ rm qg}（t）} | {\ Mathbf {x Mathbf {x}} _1} _1} \ rangle $。我描述了这些结果并探讨了几个后果。事实证明，进化运算符$ u _ {\ rm qg}（t）$对于子planckian时间间隔而言是非自动的，而在时间间隔则保持统一的时间大于Planck时间。结果可以推广到任何超级弯曲的时空。
The propagator which evolves the wave-function in NRQM, can be expressed as a matrix element of a time evolution operator: i.e $ G_{\rm NR}(x)= \langle{\mathbf{x}_2}|{U_{\rm NR}(t)}|{\mathbf{x}_1}\rangle$ in terms of the orthonormal eigenkets $|{\mathbf{x}}\rangle$ of the position operator. In QFT, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator $G_R(x)$ as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in QFT, as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm R}(t)}|{\mathbf{x}_1}\rangle$ for a suitably defined time evolution operator and (non-orthonormal) kets $|{\mathbf{x}}\rangle$ labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this QG corrected propagator can be expressed as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm QG}(t)}|{\mathbf{x}_1}\rangle$. I describe these results and explore several consequences. It turns out that the evolution operator $U_{\rm QG}(t)$ becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalised to any ultrastatic curved spacetime.
