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 a 1/2 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.Mathematics Subject Classification. 81S99, 81T20, 81T55, 81U99, 34B08, 35J25, 11M36, 47B25, 47A10.Keywords. Quantum Theory, Quantum field theory on curved space backgrounds, Casimir effect, Scattering theory, Parameter dependent boundary value problems, Boundary value problems for second-order elliptic equations, Zeta and L-functions: analytic theory, Symmetric and selfadjoint operators, General theory of linear operators.

Casimir pistons with general boundary conditions

“…According to the formalism developed in [2,3,39], the boundary conditions that characterize a given selfadjoint extension of the differential operator in (2.3) are expressed in the following form…”

“…(2.5) Since the set of selfadjoint extensions of the differential operator in (2.3) defined over I is in one-to-one correspondence with the elements of the group U (4) [2], the matrix U in (2.5) must be an element of the unitary group U (4). This means that for any given U ∈ U (4) we obtain a corresponding selfadjoint extension of the second derivative operator in (2.3) defined on the domain [39] D…”

“…The large-z asymptotic expansion of the quantity in (4.18) can be obtained by following the argument presented in [39]. where (cf.…”

“…In order to perform such general analysis we exploit the results obtained in [2] which enable one to characterize all selfadjoint extensions of the Laplacian. By following the techniques employed in [39], we will utilize spectral zeta function regularization methods in order to derive explicit expressions for the desired Casimir energy and the corresponding force on the piston. We perform the analysis of the spectral zeta function of the piston configuration by relying primarily on methods from scattering theory.…”

Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity

“…In other physical contexts like scalar QFT on a line, point potentials serve to model impurities and provide external singular backgrounds where the bosons move [21]. The spectra of Hamiltonians with δ and δ ′ point interactions provide one-particle states in scalar (1 + 1)-dimensional QFT systems [22][23][24]. In particular, configurations of two pure delta potentials added to the free Schrödinger Hamiltonian have been used to describe scalar field fluctuations on external backgrounds [25], in terms of the corresponding scattering waves.…”