Unitarity bound

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In theoretical physics, a unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that probabilities are numbers between 0 and 1 whose sum is conserved. Unitarity implies, among other things, the optical theorem. According to the optical theorem, the imaginary part of a probability amplitude Im(M) of the forward scattering is related to the total cross section, up to some numerical factors. Because | M | ² for the forward scattering process is one of the terms that contributes to the total cross section, it cannot exceed the total cross section i.e. Im(M). The inequality

implies that the complex number M must belong to a certain disk in the complex plane. Similar unitarity bounds imply that the amplitudes and cross section can't increase too much with energy or they must decrease as quickly as a certain formula dictates.