Green-function spectral weight and sum rules

Use energy E measured relative to the chemical potential, rather than angular frequency. For one canonical spin-orbital define the retarded Green function as a function of complex energy in the upper half-plane. Its spectral representation is an integral over a nonnegative spectral function. The spectral function has inverse-energy units and the Green function has the same units. Choosing energy avoids silently mixing a frequency integration measure with an omitted reduced Planck constant.

The spectral peaks come from matrix elements connecting many-body states that differ by one electron. Electron addition and removal both contribute to the retarded anticommutator spectrum. At zero temperature the chemical potential separates occupied removal weight from unoccupied addition weight, apart from states exactly at zero energy requiring a limiting prescription. These weights are probabilities of overlaps for a specified operator; they are not a count of independent little electrons moving on each plotted spectral branch.

Insert complete many-body energy eigenstates into the two terms of the anticommutator. Integrating over energy removes their delta functions and leaves the expectation of the equal-time anticommutator. For a normalized canonical orbital that operator is the identity, so the total spectral area equals one for each spin-orbital. Summing both spin spectra gives two, not one. Interactions redistribute the area but cannot change this algebraic normalization of the retained canonical operator.

Thermal occupation weights distinguish removal from addition. Multiplying the spectrum by the Fermi function and integrating gives the orbital occupation; the complementary weight is one minus that occupation. These are independent checks: a spectrum can have total area one yet put too much weight below the chemical potential. Momentum-resolved spectra obey corresponding rules for each momentum and spin. Projected noncanonical low-energy operators can have different anticommutators, so their reduced sum rules must be derived rather than copied unchanged.

An interacting atomic orbital provides two separated charge transitions, while hopping and lifetime effects can turn them into bands or broad peaks. A broadening procedure should conserve their integrated areas. Numerical smoothing that moves weight outside a finite plotting window can make the visible area too small without violating the exact full-axis sum rule. Renormalizing only the displayed window to unit area can then exaggerate the low-energy spectral residue and corrupt the inferred occupation.

A peak becoming narrow need not mean its area grows; height, width and integrated weight answer different questions. Experimental photoemission also multiplies occupied spectral weight by matrix elements and the Fermi function, then convolves with resolution and adds background. Its raw intensity is therefore not automatically a unit-normalized spectral function. The probability lab illustrates exact normalized weights in a two-outcome reference distribution, but it contains no Green function, energy axis, many-body transition matrix elements or photoemission response.