  In this paper we provide a quantum field theoretical study on the shear and
bulk relaxation times. First, we find Kubo formulas for the shear and the bulk
relaxation times, respectively. They are found by examining response functions
of the stress-energy tensor. We use general properties of correlation functions
and the gravitational Ward identity to parametrize analytical structures of the
Green functions describing both sound and diffusion mode. We find that the
hydrodynamic limits of the real parts of the respective energy-momentum tensor
correlation functions provide us with the method of computing both the shear
and bulk viscosity relaxation times. Next, we calculate the shear viscosity
relaxation time using the diagrammatic approach in the Keldysh basis for the
massless $\lambda\phi^4$ theory. We derive a respective integral equation which
enables us to compute $\eta\tau_\pi$ and then we extract the shear relaxation
time. The relaxation time is shown to be inversely related to the thermal width
as it should be.
