球面上 GJMS 算子的乘性反常与 Casimir 能量相匹配
Multiplicative Anomaly matches Casimir Energy for GJMS Operators on Spheres
AI总结：
本文推广 Shintani-Mizuno 公式计算 ζ 正则化乘积的乘性反常，并证明有限温度 Einstein 宇宙中 GJMS 算子的 Casimir 能量等价于构成移位 Laplace 算子的累积乘性反常。
摘要：
利用 Feynman 参数化，推导了一个用于计算线性因子的 ζ 正则化乘积的乘性反常或缺陷的显式公式，该公式推广了 Shintani-Mizuno 公式。首先，将其应用于 n 维球面，重现了文献中的已知结果。随后，将该框架应用于有限温度下的闭合 Einstein 宇宙，即 S^1_β×S^{n-1}。在此过程中表明，GJMS 算子的标准 Casimir 能量与构成它们的移位 Laplace 算子的累积乘性反常相一致。据我们所知，Casimir 能量与乘性反常之间的这一等价关系此前尚未被注意到，它为乘性反常的物理意义带来了新的转折，如今使二者处于同等地位。文中还讨论了一种考虑到构成性 Laplace 算子之间乘性反常的、涌现出的改进型 Casimir 能量。
英文摘要：
An explicit formula to compute the multiplicative anomaly or defect of $ζ$-regularized products of linear factors is derived, by using a Feynman parametrization, generalizing Shintani-Mizuno formulas. Firstly, this is applied on $n$-spheres, reproducing known results in the literature. Then, this framework is applied to a closed Einstein universe at finite temperature, namely $S^1_β\times S^{n-1}$. In doing so, it is shown that the standard Casimir energy for GJMS operators coincides with the accumulated multiplicative anomaly for the shifted Laplacians that build them up. This equivalence between Casimir energy and multiplicative anomaly, unnoticed so far to our knowledge, brings about a new turn regarding the physical significance of the multiplicative anomaly, putting both now on equal footing. An emergent improved Casimir energy, that takes into account the multiplicative anomaly among the building Laplacians, is also discussed.
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