in analogy to the relations above with s o ( 3 ; 1 ) replaced by the full Poincaré algebra. By just cancelling the i 's, the result THT = − H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with time-reversal invariance, there would be a state Π(T)Ψ with negative energy − E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT = H, and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, Π is not a representation.