  The Jagiellonian Positron Emission Tomograph (J-PET) was constructed as a
prototype of the cost-effective scanner for the simultaneous metabolic imaging
of the whole human body. Being optimized for the detection of photons from the
electron-positron annihilation with high time- and high angular-resolution, it
constitutes a multi-purpose detector providing new opportunities for studying
the decays of positronium atoms. Positronium is the lightest purely leptonic
object decaying into photons. As an atom bound by a central potential it is a
parity eigenstate, and as an atom built out of an electron and an anti-electron
it is an eigenstate of the charge conjugation operator. Therefore, the
positronium is a unique laboratory to study discrete symmetries whose precision
is limited in principle by the effects due to the weak interactions expected at
the level of (~10$^{-14}$) and photon-photon interactions expected at the level
of (~10$^{-9}$). The J-PET detector enables to perform tests of discrete
symmetries in the leptonic sector via the determination of the expectation
values of the discrete-symmetries-odd operators, which may be constructed from
the spin of ortho-positronium atom and the momenta and polarization vectors of
photons originating from its annihilation. In this article we present the
potential of the J-PET detector to test the C, CP, T and CPT symmetries in the
decays of positronium atoms.
