Spectral functions and Green’s functions: A critique
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The various relations linking the spectral functions and the Green’s functionse.g.,J AB (ε)=[ImG AB R (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.,J AB (ε)=[ImG AB R (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.
KeywordsSpectral functions Green’s functions Lehmann’s approach superoperator representation displaced harmonic oscillator
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