It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors the dimensions of the spaces can be freely chosen, and the geometries-which are curved in general-can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensors to the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective action which is invariant under the spatial diffeomorphism due to the underlying orthogonal group symmetry of the randomly connected tensor network. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.
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机译:It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. 随着张量的变化,可以自由选择空间的尺寸,并且几何形状 - 这是一般的弯曲 - 可以改变。 我们给出了任意尺寸的紧急扁平扭矩的明确可解决的例子,以及从张量到一般弯曲病例的几何形状的对应关系。 由于随机连接的张量网络的底层正交组对称性,所示的突击空间中的扰动动力被描述为通过在空间扩散的底层对称性下不变。 还示出了空间之间存在各种相变,包括在连续变化的张量的连续变化之外的延伸和点状。
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