Separate spectral support from thermal occupation

Retain the module's forward transform $\int dt\,e^{i\omega t}$ and define the bosonic spectral density as the Fourier transform of $C^>-C^<$, with no extra factor of i. Define the retarded Green function as minus i times the causal commutator. Its analytic continuation lies in the upper half frequency plane, and its boundary value has imaginary part equal to minus half the spectral density. This convention gives a negative static coordinate Green function for a stable oscillator; physical susceptibility to a Hamiltonian source minus fA is its negative.

For a Hermitian operator in equilibrium, the spectral density is real and odd in frequency, with nonnegative positive-frequency weight for an autocorrelation. The full retarded function follows from a dispersion integral, possibly with ultraviolet subtractions and contact terms for composite operators. Such real local terms can matter for static response even though they do not produce a smooth dissipative spectrum. Stating only an imaginary part without its analytic and subtraction information does not always specify a complete field-theory Green function.

Away from isolated zero-frequency weights, combining KMS with the commutator definition gives $C^>=(1+n_B)\rho$ and $C^<=n_B\rho$, with $n_B(\omega)=1/(e^{\beta\omega}-1)$. These formulas include negative frequencies algebraically; the negative Bose factor there combines with the odd spectral density to give the correct raw weights. For a fermionic annihilation operator, the corresponding positive spectral object is built from the anticommutator. Its raw greater and lesser factors are $1-n_F$ and $n_F$. Reusing the bosonic commutator subtraction would therefore give the wrong fermionic normalization.

For a unit-mass oscillator with $H=(p^2+\Omega^2q^2)/2$ and $[q,p]=i$, the spectrum has two delta peaks of opposite signs. Its positive-frequency weight is independent of temperature because the canonical commutator is a c-number. The equal-time coordinate variance, however, grows as $(n_B+1/2)/\Omega$. Interactions can broaden or shift the peaks and change the spectrum itself. The free result is a useful distinction between state occupation and spectral structure, not a claim that every thermal spectral function is temperature independent.

The positive Euclidean correlator used previously is $G_E(i\omega_n)=1/(\omega_n^2+\Omega^2)$. Its spectral denominator is $\nu-i\omega_n$, opposite to the retarded denominator. Consequently, under these explicit conventions the retarded boundary value is minus the continued Euclidean function. Substituting $i\omega_n\to\omega+i0$ into its analytic expression and retaining that overall minus sign yields the displayed retarded oscillator. Other authors absorb the sign into their Euclidean definition, so formulas must be compared at the level of definitions.

A finite-width ansatz must also respect retarded analyticity and the desired sum rules. Simply replacing delta peaks by arbitrary positive bumps can violate the odd-frequency structure or the high-frequency normalization. Conversely, a narrow peak may leave only a subtle signature in Euclidean data. Reconstructing it requires more than taking a numerical Fourier transform of a finite list. Use positivity, asymptotic behavior, resolution tests, and an honestly stated ansatz when extracting spectral information from uncertain thermal data.