  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.
