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.[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.
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:
U(t)=e-it/\hbar}
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. The measurement operator is diagonal in this basis.[3]
The probability to get a particular measured result depends on the probability amplitude, given by the inner product of the physical state
|\psi\rangle
\{|\phii\rangle\}
e-i\hat{Ht/\hbar}
\left\langle\phii\left|e-i\hat{Ht/\hbar}\psi\right.\right\rangle=\left\langle\left.e-i\hat{H(-t)/\hbar}\phii\right|\psi\right\rangle
But by definition of Hermitian conjugation, this is also:
\left\langle\phii\left|e-i\hat{Ht/\hbar}\psi\right.\right\rangle=\left\langle\left.\phii\left(e-i\hat{Ht/\hbar}\right)\dagger\right|\psi\right\rangle= \left\langle\left.\phiie-i\hat{H\dagger(-t)/\hbar}\right|\psi\right\rangle
Since these equalities are true for every two vectors, we get
\hat{H}\dagger=\hat{H}
e-i\hat{Ht/\hbar}
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.
See main article: Optical theorem.
Unitarity of the S-matrix implies, among other things, the optical theorem. This can be seen as follows:[5]
The S-matrix can be written as:
S=1+iT
T
T=0
Unitarity of the S-matrix:
S\daggerS=1
is then equivalent to:
-i\left(T-T\dagger\right)=T\daggerT
The left-hand side is twice the imaginary part of the S-matrix. 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
|I\rangle
\langleF|
\left\langleF\left|T\daggerT\right|I\right\rangle=\sumi\left\langleF|T\dagger|Ai\right\rangle\left\langleAi|T|I\right\rangle
where is the set of possible on-shell states - i.e. momentum states of particles (or bound complex of particles) at infinity.
Thus, twice the imaginary part of the S-matrix, is equal to a sum representing products of contributions from all the scatterings of the initial state of the S-matrix to any other physical state at infinity, with the scatterings of the latter to the final state of the S-matrix. Since the imaginary part of the S-matrix can be calculated by virtual particles appearing in intermediate states of the Feynman diagrams, it follows that these virtual particles must only consist of real particles that may also appear as final 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 probability amplitude M (= iT) for any scattering process must obey
|M|2=2\operatorname{Im}(M)
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. For example, Froissart bound says that the total cross section of two particles scattering is bounded by
cln2s
c
s