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arXiv 2409.08932gr-qchep-thquant-ph
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$SU(\infty)$ 量子引力与宇宙学

$SU(\infty)$ Quantum Gravity and Cosmology

Houri Ziaeepour

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19 pages; 1 figure; published version
发表信息：
Symmetry 16 (2024) 1672
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AI总结：

本文提出 $SU(\infty)$-QGR 抽象方法，基于宇宙为孤立量子系统且具有无限可对易观测量的公理，预测量子引力存在自旋-1中介粒子，并证明经典时空作为涌现的洛伦兹几何出现。

摘要：

我们强调了量子宇宙学和引力的一种抽象方法的结构和性质，称为 $SU(\infty)$-QGR。从宇宙作为孤立量子系统的概念出发，其主要公理是存在无限多个相互对易的观测量。因此，宇宙的希尔伯特空间表示 $SU(\infty)$ 对称性。这个宇宙整体是静态且拓扑的。然而，量子涨落在其量子态中引起局域聚类，并将其划分为近似孤立的子系统，这些子系统表示 $G \times SU(\infty)$，其中 $G$ 是一般有限秩的内部对称性。由于全局的 $SU(\infty)$ 对称性，子系统与宇宙的其余部分纠缠。除了表征 $G$ 表示的参数外，它们的量子态还依赖于四个连续参数：其中两个表征 $SU(\infty)$ 的表示，一个维度参数源于不同子系统比较 $SU(\infty)$ 表示的可能性；第四个参数是一个可测量量，用作被选为时钟的任意子系统所记录的时间。它引入了子系统的相对动力学，由定义在连续参数的 (3+1) 维空间上的对称不变的等效拉格朗日量表述。在最低量子阶，该拉格朗日量是 $SU(\infty)$ 和内部对称性的杨-米尔斯场论。我们识别了共同的 $SU(\infty)$ 对称性及其与引力的相互作用。因此，$SU(\infty)$-QGR 预测了量子引力（QGR）的自旋-1 中介粒子。显然，这与经典引力相矛盾。然而，我们表明，一个无法检测引力量子性的观察者会将其效应感知为连续参数平均值空间的曲率。我们展示了这种涌现的经典时空的洛伦兹几何。

英文摘要：

We highlight the structure and properties of an abstract approach to quantum cosmology and gravity, dubbed $SU(\infty)$-QGR. Beginning from the concept of the Universe as an isolated quantum system, the main axiom of is the existence of an infinite number of mutually commuting observables. Consequently, the Hilbert space of the Universe represents $SU(\infty)$ symmetry. This Universe as a whole is static and topological. Nonetheless, quantum fluctuations induce local clustering in its quantum state and divide it into approximately isolated subsystems representing $G \times SU(\infty)$, where $G$ is a generic finite-rank internal symmetry. Due to the global $SU(\infty)$ subsystems are entangled to the rest of the Universe. In addition to parameters characterizing the representation of $G$, their quantum states depend on four continuous parameters: two of them characterize the representation of $SU(\infty)$, a dimensionful parameter arises from the possibility of comparing representations of $SU(\infty)$ by different subsystems; the fourth parameter is a measurable used as time registered by an arbitrary subsystem chosen as a clock. It introduces a relative dynamics for subsystems, formulated by a symmetry-invariant effective Lagrangian defined on the (3+1)D space of the continuous parameters. At lowest quantum order, the Lagrangian is a Yang--Mills field theory for both $SU(\infty)$ and internal symmetries. We identify the common $SU(\infty)$ symmetry and its interaction with gravity. Consequently, $SU(\infty)$-QGR predicts a spin-1 mediator for quantum gravity (QGR). Apparently, this is in contradiction with classical gravity. Nonetheless, we show that an observer who is unable to detect the quantumness of gravity perceives its effect as curvature of the space of average values of the continuous parameters. We demonstrate Lorentzian geometry of this emergent classical spacetime.

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