Topic summary
Linear representation
Extracted from the Wikipedia article Representation theory.
Unitary representations
A unitary representation of a group G is a linear representation φ of G on a real or (usually) complex Hilbert spaceV such that φ(g) is a unitary operator for every g ∈ G. Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl, and this has inspired the development of the theory, most notably through the analysis of representations of the Poincaré group by Eugene Wigner. One of the pioneers in constructing a general theory of unitary representations (for any group G rather than just for particular groups useful in applications) was George Mackey, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.