unitarity

Unitary refers to a property of linear operators on complex inner product spaces that preserve the inner product. An operator U is unitary if U* U = U U* = I, where U* is the adjoint (the conjugate transpose in finite dimensions). Equivalently, <Ux, Uy> = <x, y> for all vectors x and y, which also implies ||Ux|| = ||x|| and that U is invertible with U^{-1} = U*.

In finite dimensions, unitary operators correspond to unitary matrices. Their columns form an orthonormal basis, and eigenvalues have absolute value 1. The determinant of a unitary matrix has modulus 1. The product of unitary operators is unitary, and the inverse of a unitary operator is itself unitary.

Key examples include rotations in complex space and, more broadly, all unitary matrices. The discrete and continuous Fourier transforms are unitary on suitable function spaces when appropriately normalized.

Unitary operators play a central role in quantum mechanics, where evolution in time is described by a unitary operator to preserve total probability. The unitarity of the scattering matrix (S-matrix) expresses conservation of probability in particle interactions. In quantum field theory, unitarity constraints influence the behavior of amplitudes.

In mathematics, unitary representations preserve inner products, leading to the concept of unitary groups U(n) and unitary equivalence of representations. In numerical linear algebra and signal processing, unitary matrices preserve norms and are numerically stable under multiplication; they underpin algorithms such as the QR decomposition. The spectral theorem shows that unitary operators on a complex Hilbert space can be diagonalized by a unitary change of basis.