# Unitarity (physics)

In quantum physics, unitarity is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event is always 1.

More precisely, the operator which describes the progress of a physical system in time must be a unitary operator. When the Hamiltonian is time-independent the unitary operator is $e^{-i \hat{H} t}$.

Similarly, the S-matrix that describes how the physical system changes in a scattering process must be a unitary operator as well; this implies the optical theorem.

In quantum field theory one usually uses a mathematical description which includes unphysical fundamental particles, such as longitudinal photons. These particles must not appear as the end-states of a scattering process. Unitarity of the S-matrix and the optical theorem in particular implies that such unphysical particles must not appear as virtual particles in intermediate states. The mathematical machinery which is used to ensure this includes gauge symmetry and sometimes also Faddeev–Popov ghosts.

Since unitarity of a theory is necessary for its consistency, the term is sometimes also used as a synonym for consistency, and is sometimes used for other necessary conditions for consistency, in particular the condition that the Hamiltonian is bounded from below. This means that there is a state of minimal energy (called the ground state or vacuum state). This is needed for the second law of thermodynamics to hold.

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 a 2-body 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

$|M|^2 \leq \mbox{Im}(M)$

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 cannot increase too much with energy or they must decrease as quickly as a certain formula dictates.