The calculated optical conductivity breaks the sum rule that the spectral weight should be conserved through the transition.

Thus, the sum rule might be written as:

Now take the sum rule in differentiation:

The sum rule in integration follows from it.

The sum rule tells us that for two functions u and v:

Therefore, the sum rule can be extended so it "accepts" addition and subtraction as follows:

This, combined with the sum rule for derivatives, shows that differentiation is linear.

The sum rule obtained in this way allows to calculate observable characteristics of the hadronic ground state.

By the sum rule in differentiation, this is:

This is called the addition law of probability, or the sum rule.