The positronium system — a bound state of an electron and a positron — is suitable for testing the predictions of quantum electrodynamics, since its properties can be perturbatively calculated to high accuracy and, unlike the hydrogen system, it is not affected by the finite size or quantum chromodynamics effects at the current level of experimental precision. Experiments searching for invisible decays of the positronium triplet state — the ortho-positronium — which mainly decays to three photons, are being conducted since they are sensitive to new physics scenarios, e.g., mirror matter, milli-charged particles, and extra space-time dimensions. The particular case of mirror matter and its search with the novel total-body positron emission tomography scanner at the Jagiellonian University is presented. This J-PET is a large, high precision medical imaging tool based on plastic scintillators.