We analyse the connections between the Wheeler DeWitt approach for two dimensional quantum gravity and holography, focusing mainly in the case of Liouville theory coupled to $c=1$ matter. Our motivation is to understand whether some form of averaging is essential for the boundary theory, if we wish to describe the bulk quantum gravity path integral of this two dimensional example. The analysis hence, is in a spirit similar to the recent studies of Jackiw-Teitelboim (JT)-gravity. Macroscopic loop operators define the asymptotic region on which the holographic boundary dual resides. Matrix quantum mechanics (MQM) and the associated double scaled fermionic field theory on the contrary, is providing an explicit "unitary in superspace" description of the complete dynamics of such two dimensional universes with matter, including the effects of topology change. If we try to associate a Hilbert space to a single boundary dual, it seems that it cannot contain all the information present in the non-perturbative bulk quantum gravity path integral and MQM.

中文翻译：

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Liouville 理论和矩阵模型：Wheeler DeWitt 的观点

我们分析了用于二维量子引力和全息的惠勒德威特方法之间的联系，主要关注与 $c=1$ 物质耦合的刘维尔理论。如果我们想描述这个二维例子的整体量子引力路径积分，我们的动机是了解某种形式的平均对于边界理论是否必不可少。因此，分析的精神类似于最近对 Jackiw-Teitelboim (JT)-重力的研究。宏观循环算子定义了全息边界对偶所在的渐近区域。相反，矩阵量子力学 (MQM) 和相关的双标度费米子场论提供了对此类二维宇宙与物质的完整动力学的明确“超空间幺正”描述，包括拓扑变化的影响。如果我们尝试将希尔伯特空间与单边界对偶联系起来，它似乎无法包含非微扰体量子引力路径积分和 MQM 中存在的所有信息。