Effective action for a free scalar field in the presence of spacetime foam

We model spacetime foam by a gas of virtual wormholes. For a free scalar field we derive the effective Lagrangian which accounts for the interaction with spacetime foam and contains two additional non-local terms. One term describes the scattering of scalar particles on virtual wormholes and explicitly reproduces the Pauli-Villars regularization procedure. The second term describes the back reaction of particles on the number density of wormholes and introduces a self-interaction between particles.