Separate causal response from statistical fluctuations

For a Hermitian bosonic observable use the module's $G_R(t)=-i\Theta(t)\langle[A(t),A(0)]\rangle$. The advanced function instead has support at negative time with the opposite prefactor. For a stationary Hermitian autocorrelation, the advanced Fourier value is the complex conjugate of the retarded one on the real axis. Retarded poles lie below that axis for a stable damped model. These support properties distinguish the functions from a time-ordered correlation, whose ordering contains both positive- and negative-time information.

With a Hamiltonian perturbation minus f times A, the physical linear susceptibility equals minus our chosen retarded Green function. This explicit sign follows from first-order evolution and avoids silently importing another author's response convention. A static susceptibility can therefore be positive while our static retarded function is negative. A contact term may also enter particular conserved-current or stress responses. The dissipative part should be interpreted only after the source coupling, operator definition, and Fourier sign have been stated together.

Adding and subtracting the greater and lesser functions gives the statistical function and commutator spectrum. Their KMS ratio then implies $F(\omega)=\tfrac12\coth(\beta\omega/2)\rho(\omega)$. At positive frequency much smaller than temperature, the factor approaches $T/\omega$, the classical fluctuation–dissipation regime. At frequencies large compared with temperature it approaches one half, retaining vacuum fluctuations. A high-temperature approximation must therefore refer to the measured frequency window, not merely to a large numerical temperature in unrelated units. These relations concern nonzero-frequency connected fluctuations. An isolated static delta weight from a conserved or diagonal component requires separate treatment; the zero-frequency values used below are regular limits.

For fermions, the raw anticommutator spectrum and the statistical difference instead produce a hyperbolic tangent relation with consistent definitions. They cannot be substituted into this bosonic symmetrized formula by changing only a distribution symbol. Outside equilibrium, even a nearly stationary state may have independent occupation and response functions. A frequency-dependent effective temperature can sometimes summarize a restricted measurement, but assigning it does not establish a common Gibbs state or the validity of every equilibrium transport relation.

A conserved density offers a useful diagnostic. In a diffusive regime, a possible retarded form is $G_R^{nn}=-\chi Dk^2/(Dk^2-i\omega)$, where chi is the positive static susceptibility to a chemical-potential source and D is positive diffusivity. At nonzero k, taking frequency to zero gives minus chi. Taking spatial momentum to zero first at nonzero frequency instead gives zero: a spatially uniform conserved total density cannot oscillate in response at finite frequency in this ideal description. The two limits answer different physical questions.

This hydrodynamic form is a low-frequency, long-wavelength approximation, not a complete microscopic spectral function at arbitrarily large frequency. It needs matching to short-time behavior and any contact contributions when used outside that window. The pole at negative imaginary frequency expresses diffusion, and its spectral sign is consistent with the retarded convention. Stating the order of limits is essential when extracting a susceptibility, a transport coefficient, or a proposed violation of fluctuation–dissipation from a numerical calculation.