量子四面体的全纯分解
Holomorphic Factorization for a Quantum Tetrahedron
摘要：
本文给出了Hilbert空间H（j_1，. j_n），并建立了H（j_1，.，.）中恒等式分解的全纯因式分解公式，j_n）。有趣的是，出现在分解公式中的积分核原来是弦论体/边界对偶的n点函数。我们的结果为这个量提供了一个新的解释，在大的共形维数的限制下，它是辛流形的Kahler势的指数，其量子化给出H（j_1，...，j_n）。对于n=4的情形，所讨论的辛流形具有固定面面积的几何四面体的“形状”空间的解释，我们的结果提供了量子四面体的全纯相干态的描述.我们描述了如何全纯缠绕有关通常的真正的计算重叠。在大自旋的情况下，这些重叠系数的半经典分析，使我们能够获得一个明确的关系之间的真实和全纯描述的空间的形状的四面体。我们的结果与圈量子引力和自旋泡沫的主题直接相关，但也为体/边界对应的故事增加了一个有趣的新转折。
英文摘要：
We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H(j_1,..,j_n). Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives H(j_1,..,j_n). For the case n=4, the symplectic manifold in question has the interpretation of the space of "shapes" of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.
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