In condensed matter physics, sum rules are used widely by both experimentalists and theorists. One can even go as far to say that sum rules provide us with a framework within which theories must exist, i.e. theories cannot violate the constraints put forth by these sum rules. In this sense, they are of vast importance, and any theory of, for example, the dielectric function should be checked against these constraints.
Even though these sum rules are used often, their physical meaning is not always apparent because they can be written in many forms. Let me use the Thomas-Reiche-Kuhn sum rule (a.k.a the f-sum rule) to illustrate some of these points. This sum rule can be formulated as so:
where is the Fourier-transformed number density operator. In this formulation, one can see the physical principles behind the sum rule most clearly:
If one adds up the energies of the transitions made from the ground state to higher energy states (in this case by perturbing the density), this should be equal to the total energy put into the system.
The TRK sum rule can be understood quite simply, therefore, as an energy conservation law for a many-body system. This is why these sum rules are so important — they are many-body manifestations of conservation laws.
The Thomas-Reiche-Kuhn sum rule is often written in the following way as well:
where is the dynamic structure factor.
Furthermore, TRK can be formulated in terms of the inverse longitudinal dielectric function as so:
where is the plasma frequency. Also, it can be written in a form more familiar to optical spectroscopists, who often plot the optical conductivity:
So while there are many sum rules (and many formulations of each sum rule as seen above for the TRK), one should always keep in mind that they derive from rather general physical principles, which are unfortunately sometimes hidden in the way they are written.