Microscopic formula for transport coefficients of causal hydrodynamics

The Green-Kubo-Nakano formula should be modified in relativistic hydrodynamics because of the problem of acausality and the breaking of sum rules. In this Rapid Communication, we propose a formula to calculate the transport coefficients of causal hydrodynamics based on the projection operator method. As concrete examples, we derive the expressions for the diffusion coefficient, the shear viscosity coefficient, and corresponding relaxation times.