In this paper we present prospects for using the J-PET detector to search for discrete symmetries violations in a purely leptonic system of the positronium atom. We discuss tests of CP and CPT symmetries by means of ortho-positronium decays into three photons. No zero expectation values for chosen correlations between ortho-positronium spin and momentum vectors of photons would imply the existence of physics phenomena beyond the Standard Model. Previous measurements resulted in violation amplitude parameters for CP and CPT symmetries consistent with zero, with an uncertainty of about 10-3. The J-PET detector allows to determine those values with better precision thanks to a unique time and angular esolution combined with a high geometrical acceptance. Achieving the aforementioned is possible due to application of polymer scintillators instead of crystals as detectors of annihilation quanta.
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