[Paper Review] Functional determinants in the presence of zero modes

Klaus Kirsten, Alan J. McKane|ArXiv.org|Jul 1, 2005

Spectral Theory in Mathematical Physics3 references3 citations

TL;DR

This paper presents a contour integration method to compute functional determinants for second-order differential operators, particularly addressing the challenge of zero modes. By reformulating the zeta function via contour integrals and analytically continuing the resulting expressions, the authors derive a closed-form formula for the functional determinant in the presence of zero modes, with the key result being $-\ln\det L = -\ln(2f_0)$, where $f_0$ is derived from the zero mode's derivative at the boundary.

ABSTRACT

We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible we illustrate the general ideas using the Laplacian with Dirichlet boundary conditions on the interval. Afterwards, we indicate how more general operators as well as general boundary conditions can be covered.

Motivation & Objective

• To develop a systematic method for computing functional determinants in quantum field theory and mathematical physics when zero modes are present.
• To extend contour integration techniques—previously applied to non-degenerate cases—to operators with zero modes, where standard zeta function regularization fails.
• To derive a closed-form expression for the functional determinant that isolates the contribution from the zero mode using boundary data of the zero mode.
• To generalize the method to Sturm-Liouville operators with arbitrary boundary conditions and potentials, ensuring applicability beyond symmetric or simple cases.

Proposed method

• Use contour integration in the complex $k$-plane to represent the zeta function of the operator, starting from the implicit eigenvalue equation $\sin k = 0$ for the Laplacian.
• Modify the zeta function representation by replacing $\ln \sin k$ with $\ln(\sin k / k)$ to remove the singularity at $k=0$, enabling contour deformation.
• Shift the integration contour to the imaginary axis, using the identity $\sinh k / k$ for $k$ imaginary, and express the zeta function as an integral involving $\ln(\sinh k / k)$.
• Apply analytic continuation via subtraction of asymptotic expansions at $k \to \infty$ and $k \to 0$, enabling evaluation at $s=0$.
• For zero modes, replace the boundary value $u_k(1)$ with a regularized function $f_k$, defined via $u_k(1) = -k^2 f_k$, to handle the singularity at $k=0$.
• Derive the final determinant formula by evaluating the logarithmic derivative of $f_k$ at $k=0$, leading to $\zeta_L'(0) = -\ln(2f_0)$.

Experimental results

Research questions

• RQ1 How can functional determinants be computed when the operator possesses a zero mode, which invalidates standard zeta function regularization?
• RQ2 What modification to contour integration techniques is required to handle the singularity introduced by the zero mode in the eigenvalue equation?
• RQ3 Can a closed-form expression for the determinant be derived that explicitly depends on the zero mode’s derivative at the boundary?
• RQ4 How can the method be generalized to Sturm-Liouville operators with arbitrary boundary conditions and potentials?
• RQ5 What is the role of the function $f_k$ in regularizing the determinant computation when $u_k(1)$ vanishes at $k=0$?

Key findings

• The functional determinant for the Laplacian on $[0,1]$ with Dirichlet boundary conditions is $\det L = 2$, consistent with known results.
• For operators with a zero mode, the determinant is given by $\zeta_L'(0) = -\ln(2f_0)$, where $f_0 = -u_0'(1)/2$ and $u_0$ is the zero mode.
• The method successfully handles the singularity at $k=0$ by replacing $u_k(1)$ with $f_k$, which remains finite and non-zero at $k=0$.
• The asymptotic behavior at $k \to \infty$ is preserved in the regularization, ensuring the same contribution to the determinant as in the non-zero mode case.
• The approach generalizes to Sturm-Liouville operators with arbitrary boundary conditions through a first-order system formulation and matrix boundary conditions.
• The method remains valid even when eigenvalues are negative, provided the branch cut for the complex root is chosen away from the positive real axis.