Zero-Point Energy of a Scalar Field in $q$-Deformed Euclidean Space

TL;DR

The paper develops a $q$-deformed Euclidean framework to study the zero-point energy of a scalar field, introducing Jackson calculus and a $q$-delta formalism to regularize vacuum fluctuations. It demonstrates that, globally over the entire $q$-deformed space, the vacuum energy vanishes for a massless Klein–Gordon field, while locally (in finite regions) the vacuum energy density can be enormous, reflecting strong short-distance fluctuations. The analysis combines a mass-series expansion, $q$-deformed Laplacians, and the $q$-delta structure to show how global cancellations arise, contrasting with persistent local energy densities. These results suggest that $q$-deformed noncommutative geometry can reconcile large Planck-scale fluctuations with negligible large-scale vacuum energy, offering insights relevant to the cosmological constant problem and the role of space-time discreteness in QFT.

Abstract

We examine the energy of a scalar field in its ground state within $q$-deformed Euclidean space. Specifically, we compute the total vacuum energy of the entire $q$-deformed Euclidean space, originating from the scalar field's ground-state energy. Our results show that, for a massless scalar field, the total vacuum energy vanishes. In contrast, when evaluating the average ground-state energy over finite, localized regions of the $q$-deformed Euclidean space, we find that the vacuum energy density can assume significant values.

Figures (4)

• Figure 1: Eigenvalue equations of $q$-exponentials.
• Figure 2: Addition theorems for $q$-exponentials.
• Figure 3: Invertibility of $q$-exponentials.
• Figure 4: Eigenvalue equation of twisted $q$-exponential.