In quantum physics, unitarity is (or a unitary process has) 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.
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.
[2] In quantum mechanics, every state is described as a vector in Hilbert space.
When a measurement is performed, it is convenient to describe this space using a vector basis in which every basis vector has a defined result of the measurement – e.g., a vector basis of defined momentum in case momentum is measured.
For a physical state that is measured after it has evolved in time, the probability amplitude can be described either by the inner product of the physical state after time evolution with the relevant basis vectors, or equivalently by the inner product of the physical state with the basis vectors that are evolved backwards in time.
Since by the Born rule the norm determines the probability to get a particular result in a measurement, unitarity together with the Born rule guarantees the sum of probabilities is always one.
Furthermore, unitarity together with the Born rule implies that the measurement operators in Heisenberg picture indeed describe how the measurement results are expected to evolve in time.
That the time evolution operator is unitary, is equivalent to the Hamiltonian being Hermitian.
Equivalently, this means that the possible measured energies, which are the eigenvalues of the Hamiltonian, are always real numbers.
The S-matrix is used to describe how the physical system changes in a scattering process.
It is in fact equal to the time evolution operator over a very long time (approaching infinity) acting on momentum states of particles (or bound complex of particles) at infinity.
Thus it must be a unitary operator as well; a calculation yielding a non-unitary S-matrix often implies a bound state has been overlooked.
Unitarity of the S-matrix implies, among other things, the optical theorem.
just implies the S-matrix is 1, no interaction occur and all states remain unchanged.
In order to see what the right-hand side is, let us look at any specific element of this matrix, e.g. between some initial state
The mathematical machinery which is used to ensure this includes gauge symmetry and sometimes also Faddeev–Popov ghosts.
According to the optical theorem, the probability amplitude M (= iT) for any scattering process must obey 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[which?]