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Vacuum energy for generalised Dirac combs at $T = 0$

Michael Bordag, Jose M Muñoz-Castañeda, Lucía Santamaría-Sanz

TL;DR

The paper addresses the quantum vacuum energy of a (1+1)-D scalar field in a periodic background formed by a generalized Dirac comb of $\delta$ and $\delta'$ potentials. It combines a lattice reformulation as a one-parameter family of selfadjoint extensions (piston boundary conditions) with spectral zeta-function methods to derive a finite, per-unit-cell vacuum energy $E_{\rm comb}^{fin}$, computed from the single-potential scattering data and integrated over Bloch quasi-momenta. For the $\delta$-$\delta'$ comb, the authors obtain an explicit spectral function and a single convergent integral expression for the energy, showing that the vacuum force can be repulsive, attractive, or zero, and that non-perturbative features emerge near infrared regimes. The framework generalizes to other compactly-supported combs and clarifies the role of ultraviolet divergences (independent of lattice spacing), providing a basis for extensions to finite temperature and lattice-phonon interaction analyses.

Abstract

The quantum vacuum energy for a hybrid comb of Dirac $δ$-$δ'$ potentials is computed using the energy of the single $δ$-$δ'$ potential over the real line that makes up the comb. The zeta function of a comb periodic potential is the continuous sum of zeta functions over the dual primitive cell of Bloch quasi-momenta. The result obtained for the quantum vacuum energy is non-perturbative in the sense that the energy function is not analytical for small couplings

Vacuum energy for generalised Dirac combs at $T = 0$

TL;DR

The paper addresses the quantum vacuum energy of a (1+1)-D scalar field in a periodic background formed by a generalized Dirac comb of and potentials. It combines a lattice reformulation as a one-parameter family of selfadjoint extensions (piston boundary conditions) with spectral zeta-function methods to derive a finite, per-unit-cell vacuum energy , computed from the single-potential scattering data and integrated over Bloch quasi-momenta. For the - comb, the authors obtain an explicit spectral function and a single convergent integral expression for the energy, showing that the vacuum force can be repulsive, attractive, or zero, and that non-perturbative features emerge near infrared regimes. The framework generalizes to other compactly-supported combs and clarifies the role of ultraviolet divergences (independent of lattice spacing), providing a basis for extensions to finite temperature and lattice-phonon interaction analyses.

Abstract

The quantum vacuum energy for a hybrid comb of Dirac - potentials is computed using the energy of the single - potential over the real line that makes up the comb. The zeta function of a comb periodic potential is the continuous sum of zeta functions over the dual primitive cell of Bloch quasi-momenta. The result obtained for the quantum vacuum energy is non-perturbative in the sense that the energy function is not analytical for small couplings

Paper Structure

This paper contains 10 sections, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some basics formulas on spectral zeta functions
  3. The comb as a piston
  4. REMARK.
  5. Spectral zeta function for the crystal
  6. The finite quantum vacuum energy at zero temperature for generalised Dirac combs.
  7. General formulas
  8. Some comments on $E_{\rm comb}^{fin}$ and $E_0^{fin}(\theta)$.
  9. $E_{\rm comb}^{fin}$ for the $\delta$-$\delta^\prime$ comb.
  10. Conclusions and further comments

Figures (3)

  • Figure 1: Quantum vacuum energy as a function of the distance $a$ for different values of the $\delta \delta^{\prime}$ couplings.
  • Figure 2: Quantum vacuum energy for $a=0.5$ in the coupling space $\gamma$-$\Omega$.
  • Figure 3: Plot of $E_{\delta\delta'{\rm comb}}^{fin}(\Omega=-1,w_0\to 0)/( w_0\log{w_0})$ for $a=1$. The $w_0$ axis is in logarithmic scale