arXiv 1910.033002019-10gr-qc

通过标度不变圈引力实现时间和大爆炸的量子化

Quantization of time and the big bang via scale-invariant loop gravity

Charles H. -T. Wang, Marcin Stankiewicz

备注：: 7 pages, 4 figures, version to appear in PLB
发表信息：: Physics Letters B 800 (2020) 135106
DOI：: 10.1016/j.physletb.2019.135106

摘要：

我们认为背景独立量子化的一般尺度不变的引力理论与物质，这支持了保守的外尔电流最近建议作为一个自然的时间流。对于标量度量系统，构造了一个共形的Ashtekar-Barbero连接变量类，它可以用自旋网络量子化。最重要的是，量子态在爱因斯坦坐标系中可以分离为尺度变换生成元的本征态和自旋网络态。本征值由额外的量子数组成，包括一种新类型的基频$ω$和能量$E = \hbar ω$相对于一个新的本地时间$τ$携带的每个自旋网络顶点。离散分布的$τ$值作为"时间量子"对应于与经典理论中的积分外尔电流相关的函数时间。圈量子几何的伊米尔齐模糊性被尺度对称性消除了。为了探索早期宇宙的量子行为，新的形式主义适用于一个尺度不变的均匀和各向同性的宇宙模型耦合到一对标量与说明性的数值模拟。爱因斯坦框架体积是量化的最近改进和正则化的聚合物表示与任意的Immirzi参数。由此产生的膨胀宇宙的量子态的幺正演化具有正的能谱。Immirzi参数的重新缩放等效于在不改变动力学的情况下的时间平移。当乔丹标架体积的期望值趋于零时，大爆炸可以在过去的时限内被识别。值得注意的是，量子化的大爆炸并没有被大反弹所取代-这是目前圈量子宇宙学中普遍存在的情况。

英文摘要：

We consider the background-independent quantization of a general scale-invariant theory of gravity with matter, which supports a conserved Weyl current recently suggested as a natural flow of time. For scalar-metric systems, a conformal class of Ashtekar-Barbero connection variables is constructed, which can be quantized using spin networks. Crucially, the quantum states become separable into the eigen states of the generator of the scale transformation and spin-network states in the Einstein frame. The eigen values consist of additional quantum numbers including a new type of fundamental frequency $ω$ and energy $E = \hbar ω$ with respect to a new local time $τ$ carried by every spin-network vertex. The discretely distributed $τ$ values as the "quanta of time" correspond to a functional time related to the integrated Weyl current in the classical theory. The Immirzi ambiguity of loop quantum geometry is removed by scale symmetry. To probe the quantum behaviour of the early Universe, the new formalism is applied to a scale-invariant homogenous and isotropic cosmological model coupled to a doublet of scalars with illustrative numerical simulations. The Einstein-frame volume is quantized in recently improved and regularized polymer representations with an arbitrary Immirzi parameter. The resulting unitary evolution of the quantum state of an expanding universe has a positive energy spectrum. A rescaling of the Immirzi parameter is equivalent to a translation in time without changing dynamics. The big bang can be identified in the past time limit when the expectation values of the Jordan-frame volume tend to zero. Remarkably, the quantized big bang is not replaced by a big bounce---a prevalent scenario in present loop quantum cosmology.

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