The various relations linking the spectral functions and the Green’s functionse.g.,JAB(ε)=[ImGABR (ε)]/[1+γe−βε], are usually established via Lehmann’s approach. We demonstrate here how the same can be deduced elegantly through their respective superoperator representations. There are contexts in which the said relationships may not be strictly true and have to be modified through acδ(ε) term (c, a constant) e.g.,JAB(ε)=[ImGABR (ε)]/[1+γe−βε]+cδ(ε). The origin and unambiguous analysis of such situations are presented. A proper bilinear form for the causal Green’s function is derived.
The ‘displaced harmonic oscillator’ is analysed in detail, to illustrate the issues raised above.
The superoperator representation of the imaginary time Green functions is also given, in passing.
Volume 134, 2022
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