unitärt

Unitärt is a term used mainly as the adjective form of unitary in several languages, most often in mathematical and scientific contexts. The English equivalent is unitary. In many German-language sources the adjective appears as unitär, and transliterations or simplified spellings may render it as unitärt in older or non-standard texts. The core idea across usages is a relation to unity, preservation, or a single cohesive structure, depending on the domain.

In mathematics, unitary refers to structures that preserve a Hermitian inner product. A linear operator U on a complex inner product space is unitary if U†U = UU† = I, where U† is the adjoint (conjugate transpose) of U. Equivalently, U is invertible with U−1 = U†. A matrix representation of a unitary operator is a unitary matrix, satisfying U†U = I. The set of all n×n unitary matrices forms the unitary group U(n); those with determinant 1 constitute the special unitary group SU(n).

In physics, unitarity expresses the conservation of probability. Quantum time evolution is described by unitary operators, ensuring that the total probability remains 1. The S-matrix in scattering theory is required to be unitary, reflecting probability conservation in transitions between states. In quantum computing, quantum gates are modeled as unitary operations, enabling reversible transformations of quantum states; a finite universal gate set can approximate any unitary operation.

Unitary representations and harmonic analysis study how groups act by unitary operators on Hilbert spaces, preserving the inner product structure. The term can also appear in non-mathematical contexts, such as references to unitary governance (a unitary state), though this sense differs from the mathematical one.