To give a possible explanation of the 3-dimensionality of space, a matrix version of the Wheeler-DeWitt equation for quantum gravity is developed. In this equation, the number of spatial dimensions D is associated with a discrete variable. In this framework, a toy quantum-cosmological model for the birth of the universe from nothing is constructed. Here, nothing means a precosmic state with no classical space-time, but constituted by a collection of potential universes in different dimensions. The tunneling probability for the birth of a closed (D + 1)-dimensional Friedmann-Lemaitre-Robertson-Walker space-time is calculated, and two particular cases are distinguished: (1) In the interpretation of parallel universes, we find that there is no preferred number of spatial dimensions. Then, the 3-dimensionality of space is explained by the anthropic principle. (2) In the interpretation of a single universe, we find a solution showing that, under certain assumptions, a 3-dimensional space is preferred.

中文翻译：

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矩阵惠勒-德维特方程，量子宇宙隧道学和三维空间

为了对空间的3维性做出可能的解释，开发了量子引力的Wheeler-DeWitt方程的矩阵形式。在该等式中，空间尺寸D的数量与离散变量相关联。在此框架中，构建了一个从无到有的宇宙诞生的玩具量子宇宙学模型。在这里，没有什么表示没有古典时空的前状态，而是由不同维度的潜在宇宙的集合构成的。封闭物（D+ 1）计算弗里德曼-勒迈特-罗伯逊-沃尔克的时空，并区分两种特殊情况：（1）在解释平行宇宙时，我们发现没有优选的空间维数。然后，通过人类原理来解释空间的三维。（2）在单个宇宙的解释中，我们找到了一种解决方案，该解决方案表明，在某些假设下，首选3维空间。