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The free quantum states of topologically massive electrodynamics and gravity in 2+1 dimensions are found explicitly. It is shown that in both theories the states are described by infrared-regular polarization tensors containing a regularization phase which depends on the spin. This is done by explicitly realizing the quantum algebra on a functional Hilbert space and by finding the Wightman function to define the scalar product on such a Hilbert space. The physical properties of the states are analysed defining creation and annihilation operators. For both theories, a canonical and covariant quantization procedure is developed. The higher-order derivatives in the gravitational Lagrangian are treated by means of a preliminary Dirac procedure. The closure of the Poincare algebra is guaranteed by the infrared-finiteness of the states which is related to the spin of the excitations through the regularization phase. Such a phase may have interesting physical consequences.