The relation between effective action and vacuum energy in a kappa-deformed theory

TL;DR

This paper explores how the effective action and vacuum energy of a scalar field are affected by kappa-deformation of spacetime symmetry, revealing real and imaginary components linked to vacuum energy and particle creation.

Contribution

It demonstrates that using the kappa-deformed Casimir invariant as the proper-time Hamiltonian results in a complex effective action, connecting deformation to vacuum energy and excitation creation.

Findings

Real part of effective action equals half the sum of kappa-deformed zero mode frequencies.

Imaginary part indicates a creation rate of field excitations.

In the undeformed limit, results reduce to standard scalar field vacuum energy.

Abstract

In a quantum field with spacetime invariance governed by the Poincare algebra the one-loop effective action is equal to the sum of zero modes frequencies, which is the vacuum energy of the field. The first Casimir invariant of the Poincare algebra provides the proper time Hamiltonian in Schwinger's proper time representation of the effective action. We consider here a massive neutral scalar field with spacetime invariance governed by the so called kappa-deformed Poincare algebra. We show here that if in the kappa-deformed theory the first Casimir invariant of the algebra is also used as the proper-time hamiltonian the effective action appears with a real and an imaginary part. The real part is equal to half the sum of kappa-deformed zero mode frequencies, which gives the vacuum energy of the kappa-deformed field. In the limit in which the deformation disappears this real part reduces to half of the sum of zero mode frequencies of the usual scalar field. The imaginary part is proportional to the sum of the squares of the kappa-deformed zero mode frequencies. This part is a creation rate of field excitations in the situations in which it gives rise to a finite physically meaningful quantity. This is the case when the field is submitted to boundary conditions and properly renormalized, as we show in a related paper.