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arXiv · 1211.4117

The multiplicative anomaly of three or more commuting elliptic operators

Victor Castillo-Garate·Eduardo Friedman·Marius Mantoiu
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Abstract

Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\frac{\det_ζ\Big(\prod_{i=1}^n A_i\Big)}{\prod_{i=1}^n \det_ζ(A_i)} attached to n (suitable) operators A_1,...,A_n. We show that if the A_i are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely M_n(A_1,...,A_n)^{m_1+...+m_n} =\prod_{1\le i<j\le n}M_2(A_i,A_j)^{m_i+m_j}, where m_j is the order of A_j. The proof relies on Wodzicki's 1987 formula for the pairwise multiplicative anomaly M_2(A,B) of two commuting elliptic operators.

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Victor Castillo-Garate, Eduardo Friedman, Marius Mantoiu. 2012-11-17. The multiplicative anomaly of three or more commuting elliptic operators. https://arxiv.org/abs/1211.4117

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