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

Kubo formulae for the shear and bulk viscosity relaxation times and the scalar field theory shear $\tau_\pi$ calculation