Extract transport from a low-frequency response

In an isotropic equilibrium medium in three spatial dimensions, choose the off-diagonal stress $T^{xy}$. Define its zero-momentum retarded correlator with minus i times the causal commutator, integrating over space and Fourier transforming time with $e^{i\omega t}$. Using the module's commutator spectral density, shear viscosity is half the positive small-frequency spectral slope. Spatial momentum is set to zero in this definition before taking the dissipative frequency limit. Real equilibrium contact terms require consistent treatment but do not supply the smooth imaginary slope.

The stress operator has energy dimension four in a relativistic four-dimensional theory. Integrating a two-stress product over space and time gives a frequency-space retarded function of energy dimension four. Dividing its spectral density by frequency therefore gives viscosity of dimension energy cubed. Entropy density has the same dimension, making their ratio dimensionless in natural units. These dimensional statements depend on the specified spatial integration and cannot be imported from the earlier single-oscillator coordinate correlator.

A simple illustrative transport spectrum is $\rho(\omega)=2A\Gamma\omega/(\omega^2+\Gamma^2)$ with positive A and relaxation rate Gamma. Its viscosity is $A/\Gamma$. At a fixed nonzero frequency, this spectrum can shrink as Gamma tends to zero, while its zero-frequency slope diverges. The collisionless and hydrodynamic operations consequently do not commute. A narrow peak must be resolved before taking the transport slope; looking at one finite-frequency point can suggest the opposite trend from the actual viscosity.

In weakly interacting field theories, nearly on-shell retarded and advanced factors can enhance repeated scattering contributions. Finite widths and ladder diagrams must be treated consistently, often leading to a kinetic integral equation. In gauge plasmas, screened soft exchange and collinear processes add further power-counting requirements. A damping time of one quasiparticle need not equal the momentum-relaxation time controlling viscosity. Replacing the full collision operator by one arbitrary lifetime can give the wrong coefficient or even the wrong coupling dependence.

A finite positive viscosity supplies dissipative stress in a gradient expansion near local equilibrium. It is not a complete constitutive law for arbitrarily rapid or large deformations. The signal frequency and wave number must be small compared with the relevant relaxation scales; additional relaxation-time terms can matter before a microscopic description becomes necessary. Similarly, a Euclidean stress correlator can constrain an integrated spectrum without tightly determining a very narrow transport slope. Numerical extraction needs honest resolution and model-dependence checks.

A controlled transport computation should identify the coupling order, screening assumptions, collision channels, and scale separation. Conservation laws constrain the collision operator and remove inappropriate relaxing modes. Gauge identities and spectral normalization supply complementary checks. The final comparison must use the same Kubo normalization as the reported quantity: changing from one off-diagonal component to a sum over a tensor projector changes numerical prefactors. A factor-of-two discrepancy can arise from defining the spectral function as a full commutator rather than half of it, so carry that definition through the calculation.