It is shown theoretically that Bose condensation of spin-degenerated exciton polaritons results in spontaneous buildup of the linear polarization in emission spectra of semiconductor microcavities and therefore that linear polarization is a good order parameter for the polariton Bose condensation under unpolarized pumping. If spin degeneracy is lifted, an elliptically polarized light is emitted by the polariton condensate. The main axis of the ellipse rotates in time due to self-induced Larmor precession of the polariton condensate pseudospin. The polarization decay time is governed by the dephasing induced by the polariton-polariton interaction and is strongly dependent on the statistics of the condensed state. If the elliptical polarization preexists in the system as a result of pumping, the lifetime of the linear part of the polarization is also extremely sensitive to the degree of circular polarization induced in the system by pumping. This decay time can be used to measure the coherence degree of the condensate as a function of the polarization of the emitted light, as opposed to more conventional but harder particle counting experiments of the Hanbury Brown-Twiss type
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