QFT Over the Finite Line. Heat Kernel Coefficients, Spectral Zeta Functions and Selfadjoint Extensions

Abstract

Following the seminal works of Asorey–Ibort–Marmo and Muñoz–Castañeda–Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint extension of the Laplace operator over the finite line [0, L]. The derivative of the corresponding spectral zeta function at s = 0 (partition function of the corresponding quantum field theory) is obtained. To compute the correct expression for the a1/2 heat kernel coefficient, it is necessary to know in detail which non-negative selfadjoint extensions have zero modes and how many of them they have. The answer to this question leads us to analyze zeta function properties for the Von Neumann–Krein extension, the only extension with two zero modes.