[Paper Review] Product Einstein Manifolds, Zeta-Function Regularization and the Multiplicative Anomaly

Andrei A. Bytsenko, Floyd L. Williams|ScholarWorks@UMassAmherst (University of Massachusetts Amherst)|Jun 16, 1997

Advanced Operator Algebra Research36 references8 citations

TL;DR

This paper investigates zeta-function regularization of Laplace-type operators on product Einstein manifolds, deriving explicit expressions for the zeta function and the multiplicative anomaly in irreducible rank 1 symmetric spaces. It establishes a precise relationship between the global zeta function on product spaces and the individual zeta functions of the factors, revealing the structure of the multiplicative anomaly in this geometric setting.

ABSTRACT

The global additive and multiplicative properties of Laplace type operators acting on irreducible rank 1 symmetric spaces are considered. The explicit form of the zeta function on product spaces and of the multiplicative anomaly is derived.

Motivation & Objective

• To analyze the global additive and multiplicative properties of Laplace-type operators on irreducible rank 1 symmetric spaces.
• To derive the explicit form of the zeta function on product manifolds constructed from such symmetric spaces.
• To compute and characterize the multiplicative anomaly in the context of zeta-function regularization on product Einstein manifolds.
• To clarify the interplay between spectral geometry and quantum field theory regularization in curved, symmetric backgrounds.

Proposed method

• The authors employ zeta-function regularization techniques to compute the spectral zeta function of Laplace-type operators on product manifolds.
• They decompose the zeta function on the product space into contributions from the individual factors using the properties of zeta functions on symmetric spaces.
• The multiplicative anomaly is derived from the difference between the zeta function of the product and the product of the individual zeta functions.
• The analysis is restricted to irreducible rank 1 symmetric spaces, which include spheres, projective spaces, and their hyperbolic counterparts.
• The paper uses known spectral data and functional equations for zeta functions on symmetric spaces to derive explicit expressions.
• The derivation relies on the structure of the spectrum and the functional form of the zeta function in Einstein manifolds.

Experimental results

Research questions

• RQ1 How does the zeta function of a Laplace-type operator on a product Einstein manifold decompose in terms of the zeta functions on the individual factors?
• RQ2 What is the explicit form of the multiplicative anomaly in zeta-function regularization on such product spaces?
• RQ3 How do the global spectral properties of the Laplacian on product manifolds differ from the product of the spectral properties on the components?
• RQ4 What role does the curvature and symmetry of rank 1 spaces play in determining the multiplicative anomaly?
• RQ5 Can the multiplicative anomaly be expressed in closed form for irreducible rank 1 symmetric spaces?

Key findings

• The zeta function on a product of irreducible rank 1 symmetric spaces is expressed as a sum of zeta functions of the individual factors, with corrections due to the multiplicative anomaly.
• The multiplicative anomaly is explicitly computed and shown to arise from the non-tensorial nature of the zeta function under product constructions.
• The anomaly is non-zero and depends on the curvature and dimension of the symmetric spaces involved.
• The result confirms that zeta-function regularization on product manifolds does not factorize simply, even in the case of Einstein manifolds.
• The paper provides a closed-form expression for the multiplicative anomaly in terms of the zeta functions of the individual symmetric spaces.
• The findings are consistent with known results in quantum field theory on curved spacetimes and extend them to symmetric product geometries.