论文标题
由于普遍的不确定性关系而引起的粒子路径的分形特性
Fractal properties of particle paths due to generalised uncertainty relations
论文作者
论文摘要
我们在最近提出的量子几何形状的“涂片空间”模型中确定了粒子路径的Hausdorff尺寸，$ d _ {\ rm h} $。该模型引入了额外的自由度来描述背景的量子状态，并引起普遍的不确定性原理（GUP）和扩展的不确定性原理（EUP），而无需引入修改后的换向关系。我们将我们的结果与基于修改的换向因子的GUP模型中的Hausdorff维度进行了比较，并表明最小长度以不同的方式进入相关公式。然后，我们确定涂抹动量空间中粒子路径的Hausdorff尺寸，$ \ tilde {d} _ {\ rm h} $，并证明最小动量是双重的至最小长度。对于足够粗的粒状路径，$ d _ {\ rm h} = \ tilde {d} _ {\ rm h} = 2 $，如规范量子力学。但是，随着决议接近最小量表，与规范理论中的对应物相比，每个表示中路径的尺寸都不同。 GUP诱导的更正增加了$ D _ {\ rm H} $，而EUP诱导的更正相对于其规范值，$ \ tilde {d} _ {d} _ {\ rm h} $，相对于其规范值，极端情况对应于$ d _ {\ rm h} = 3 $，$ h} = 3 $} 1 $。这些结果表明，GUP和EUP以根本不同但免费的方式影响了粒子路径的分形特性。
We determine the Hausdorff dimension of a particle path, $D_{\rm H}$, in the recently proposed `smeared space' model of quantum geometry. The model introduces additional degrees of freedom to describe the quantum state of the background and gives rise to both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP) without introducing modified commutation relations. We compare our results to previous studies of the Hausdorff dimension in GUP models based on modified commutators and show that the minimum length enters the relevant formulae in a different way. We then determine the Hausdorff dimension of the particle path in smeared momentum space, $\tilde{D}_{\rm H}$, and show that the minimum momentum is dual to the minimum length. For sufficiently coarse grained paths, $D_{\rm H} = \tilde{D}_{\rm H} = 2$, as in canonical quantum mechanics. However, as the resolutions approach the minimum scales, the dimensions of the paths in each representation differ, in contrast to their counterparts in the canonical theory. The GUP-induced corrections increase $D_{\rm H}$ whereas the EUP-induced corrections decrease $\tilde{D}_{\rm H}$, relative to their canonical values, and the extremal case corresponds to $D_{\rm H} = 3$, $\tilde{D}_{\rm H} = 1$. These results show that the GUP and the EUP affect the fractal properties of the particle path in fundamentally different, yet complimentary, ways.
