[Paper Review] On two complementary approaches aiming at the definition of the determinant of an elliptic partial differential operator

E. Elizalde|ArXiv.org|Aug 30, 1995

Advanced Mathematical Modeling in Engineering4 references3 citations

TL;DR

This paper unifies two complementary approaches—mathematical zeta function regularization and physical renormalization techniques—for defining the determinant of an elliptic partial differential operator with complex spectrum. It establishes a precise bridge between them, showing how physical intuition from simple cases leads to clear mathematical conditions for zeta function and determinant definition, yielding an explicit formula connecting both frameworks.

ABSTRACT

We bring together two apparently disconnected lines of research (of mathematical and of physical nature, respectively) which aim at the definition, through the corresponding zeta function, of the determinant of a differential operator possessing, in general, a complex spectrum. It is shown explicitly how the two lines have in fact converged to a meeting point at which the precise mathematical conditions for the definition of the zeta function and the associated determinant are easy to understand from the considerations coming up from the physical approach, which proceeds by stepwise generalization starting from the most simple cases of physical interest. An explicit formula that establishes the bridge between the two approaches is obtained.

Motivation & Objective

• To reconcile two distinct research lines: mathematical zeta function regularization and physical renormalization methods for defining determinants of elliptic differential operators.
• To clarify the mathematical conditions required for the convergence and analytic continuation of the zeta function associated with such operators.
• To demonstrate how physical reasoning—starting from simple, physically relevant cases—can guide and validate the mathematical construction of the determinant.
• To derive an explicit formula that explicitly connects the physical and mathematical approaches to the determinant definition.
• To provide a unified framework where physical intuition and rigorous mathematical analysis converge for operators with complex spectra.

Proposed method

• Utilizes zeta function regularization as the core mathematical tool to define the determinant of an elliptic differential operator.
• Applies stepwise generalization from simple physical cases (e.g., scalar Laplacians on compact manifolds) to build physical intuition.
• Analyzes the spectral properties of elliptic operators with complex eigenvalues to determine conditions for zeta function convergence.
• Derives an explicit bridge formula that relates the zeta-regularized determinant from the mathematical approach to the physical renormalization procedure.
• Employs analytic continuation techniques on the zeta function ζ(s) = Tr(A^{-s}) to define det(A) = exp(-ζ'(0)).
• Validates the consistency of the physical approach by showing it satisfies the same mathematical conditions as the rigorous zeta function method.

Experimental results

Research questions

• RQ1 How can the zeta function regularization of an elliptic differential operator with complex spectrum be rigorously defined?
• RQ2 In what way do physical renormalization procedures for determinants align with mathematical zeta function methods?
• RQ3 What are the precise mathematical conditions under which the zeta function of an elliptic operator is well-defined and analytically continuable?
• RQ4 Can a unified formula be derived that explicitly connects the physical and mathematical approaches to the determinant?
• RQ5 How does stepwise generalization from simple physical cases inform the construction of the determinant in the general case?

Key findings

• The zeta function of an elliptic differential operator with complex spectrum admits a well-defined analytic continuation under standard spectral conditions, enabling the definition of the determinant.
• The physical approach—starting from simple cases like the scalar Laplacian—provides a clear and intuitive path to the mathematical conditions required for zeta function convergence.
• An explicit bridge formula is derived that equates the zeta-regularized determinant from the mathematical approach with the renormalized determinant from physical methods.
• The convergence and analyticity of the zeta function are shown to be naturally understood through physical reasoning, particularly in the context of quantum field theory applications.
• The two approaches are not only complementary but converge at a precise mathematical point, where physical intuition and rigorous analysis coalesce.
• The paper establishes that the determinant definition via zeta regularization is both mathematically sound and physically meaningful, especially in the context of path integrals and effective actions.