[Paper Review] Casimir energy, the cosmological constant and massive gravitons

Remo Garattini|ArXiv.org|Oct 13, 2005

Quantum Electrodynamics and Casimir Effect1 references3 citations

TL;DR

This paper investigates the cosmological constant problem by treating it as an eigenvalue in the Wheeler-DeWitt equation using a variational approach with Gaussian wave functionals. It computes zero-point energy (ZPE) of gravitons on a Schwarzschild background via zeta function regularization and renormalization, showing that massive gravitons lead to a finite, renormalized cosmological constant consistent with observation when quantum corrections are properly handled.

ABSTRACT

The cosmological constant appearing in the Wheeler-De Witt equation is considered as an eigenvalue of the associated Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. The case of massive gravitons is discussed.

Motivation & Objective

• To address the cosmological constant problem by treating Λ as an eigenvalue in the Wheeler-DeWitt equation.
• To compute the zero-point energy (ZPE) of massless and massive gravitons propagating on a Schwarzschild background.
• To apply zeta function regularization and renormalization to handle ultraviolet divergences arising in the ZPE computation.
• To derive a renormalization group equation for the cosmological constant in the context of quantum gravity.
• To explore whether massive gravitons can lead to a finite, physically viable cosmological constant.

Proposed method

• Uses a variational approach with Gaussian trial wave functionals to approximate the Wheeler-DeWitt equation to one loop in a Schwarzschild background.
• Treats the cosmological constant as an eigenvalue of a Sturm-Liouville problem derived from the WDW equation.
• Applies zeta function regularization to handle divergences in the ZPE of transverse-traceless (TT) graviton modes.
• Introduces a renormalization procedure to subtract infinities, with a derived renormalization group equation for the cosmological constant.
• Extends the regularization and renormalization framework to massive gravitons by modifying the mode spectrum and mass-dependent dispersion relations.
• Employs functional integration and supermetric formalism to express the WDW equation in a form amenable to variational and regularization techniques.

Experimental results

Research questions

• RQ1 Can the cosmological constant be treated as an eigenvalue in the Wheeler-DeWitt equation using a variational method?
• RQ2 What is the contribution of quantum fluctuations of gravitons (massless and massive) to the zero-point energy in a Schwarzschild spacetime?
• RQ3 How does zeta function regularization handle the ultraviolet divergences in the ZPE computation for TT graviton modes?
• RQ4 What is the role of renormalization and the renormalization group equation in stabilizing the cosmological constant to observed values?
• RQ5 Can massive gravitons lead to a finite, renormalized cosmological constant consistent with observations?

Key findings

• The zeta function regularization yields a finite expression for the zero-point energy of TT graviton modes, with a divergent term proportional to $1/ u$ and logarithmic dependence on the renormalization scale.
• The divergent part of the ZPE is removed via a renormalization procedure, leading to a finite effective cosmological constant.
• The renormalization group equation derived from the regularization procedure governs the scale dependence of the cosmological constant.
• For massive gravitons, the same regularization and renormalization framework leads to a finite ZPE contribution, with the mass modifying the logarithmic and divergent structure of the energy density.
• The final renormalized cosmological constant is finite and potentially consistent with the observed value of $\sim 10^{-47}\,\mathrm{GeV}^4$, depending on the mass scale and renormalization parameters.
• The method provides a consistent quantum gravity framework where the cosmological constant emerges as a physical observable from quantum fluctuations of gravitons in curved spacetime.