On renormalisation of λϕ4field theory in curved space-time. I

Abstract

Renormalisation of lambda phi ⁴ theory in curved space-time is considered in the interaction picture. A generalisation of normal ordering to curved space-time is introduced, based on the construction of adiabatic particle states in Robertson-Walker space-time. Dimensional regularisation is used to define uniquely the divergent quantities which are removed by normal ordering. It is shown that this normal ordering is sufficient to make finite all physical processes including vacuum polarisation to first order in lambda . An alternative and equivalent procedure is given which requires renormalisation of the mass and of the constant which couples the field to the Ricci scalar. The stress tensor is found to be finite to first order in lambda and it is shown that if the free-field theory in a Robertson-Walker universe predicts that particles are created by the gravitational field with a black-body spectrum then this spectrum is maintained when first-order self-interactions are taken into account. Finally, some aspects of the renormalisation of second-order physical processes are discussed.