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The microscopic formulas for the shear viscosity $\eta$, the bulk viscosity$\zeta$, and the corresponding relaxation times $\tau_\pi$ and $\tau_\Pi$ ofcausal dissipative relativistic fluid-dynamics are obtained at finitetemperature and chemical potential by using the projection operator method. Thenon-triviality of the finite chemical potential calculation is attributed tothe arbitrariness of the operator definition for the bulk viscous pressure.Weshow that, when the operator definition for the bulk viscous pressure $\Pi$ isappropriately chosen, the leading-order result of the ratio, $\zeta$ over$\tau_\Pi$, coincides with the same ratio obtained at vanishing chemicalpotential. We further discuss the physical meaning of the time-convolutionless(TCL) approximation to the memory function, which is adopted to derive the mainformulas. We show that the TCL approximation violates the time reversalsymmetry appropriately and leads results consistent with the quantum masterequation obtained by van Hove. Furthermore, this approximation can reproduce anexact relation for transport coefficients obtained by using the f-sum rulederived by Kadanoff and Martin. Our approach can reproduce also the result inBaier et al.(2008) Ref. \cite{con} by taking into account the next-ordercorrection to the TCL approximation, although this correction causes severalproblems.

Shear viscosity, Bulk viscosity and Relaxation Times of Causal Dissipative Relativistic Fluid-Dynamics at Finite Temperature and Chemical Potential