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A projective representation is a group homomorphism into the projective linear group of a vector space. Choosing invertible linear representatives makes multiplication hold up to nonzero scalar factors. A projective unitary representation chooses unitary operators as representatives, whose scalar factors have modulus one.

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy

A projective unitary representation is a projective representation whose chosen representatives are unitary operators. The phase factors disappear on rays.