In this paper, Dirac Quantization of 3D gravity in the first-order formalism is attempted where instead of quantizing the connection and triad fields, the connection and the triad 1-forms themselves are quantized. The exterior derivative operator on the space of differential forms is treated as the ‘time’ derivative to compute the momenta conjugate to these 1-forms. This manner of quantization allows one to compute the transition amplitude in 3D gravity which has a close, but not exact, match with the transition amplitude computed via LQG techniques. This inconsistency is interpreted as being due to the non-quantizable nature of differential geometry.

中文翻译：

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量子引力可量化几何的必要性


在本文中，狄拉克 3 的量子化D尝试一阶形式主义中的引力，其中不是量化连接和三元组场，而是量化连接和三元组 1 形式本身。微分形式空间上的外导数算子被视为“时间”导数，以计算这些 1-形式的动量共轭。这种量化方式允许我们计算 3 中的跃迁幅度D重力与通过 LQG 技术计算的转变幅度有接近但不精确的匹配。这种不一致被解释为是由于微分几何的不可量化性质造成的。