# Unitarity (physics)

In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics.[1] A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space.

## Hamiltonian evolution and scattering matrix

Time evolution described by a time-independent Hamiltonian is represented by a one-parameter family of unitary operators, for which the Hamiltonian is a generator: ${\displaystyle U(t)=e^{-i{\hat {H}}t/\hbar }}$. The expectation value of the Hamiltonian is conserved under the time evolution that the Hamiltonian generates.[2] If the Hamiltonian itself has an intrinsic time dependence, as occurs when interaction strengths or other parameters vary over time, then computing the family of unitary operators becomes more complicated (see Dyson series). In the Schrödinger picture, the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture, the time dependence is incorporated into the observables instead.[3]

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.

## Optical theorem

Unitarity of the S-matrix implies, among other things, the optical theorem. The optical theorem in particular implies that 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.

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 ${\displaystyle |M|^{2}}$ 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

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

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.