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

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.