Bose symmetry and CPT symmetry are two very fundamental symmetries of Nature. However, the validity of these symmetries in diverse phenomena must be verified by experiments. We propose new techniques to probe these two fundamental symmetries in the realm of mesons by using the Dalitz plot of a few three-body meson decays. Since these symmetries are very fundamental in nature, their violations, if any, are expected to be extremely small. Hence, observing their violations requires study of a huge data sample. In this context we introduce a new three-dimensional plot which we refer to as the Dalitz 'prism'. This provides an innovative means for acquiring the huge statistics required for such studies. Using the Dalitz plots and the Dalitz prisms we chart out the way to probe the violations of Bose and CPT symmetries in a significant manner. Since mesons are unstable and composite particles, testing the validity of Bose symmetry and the CPT symmetry in these cases are of paramount importance for fundamental physics. The statement that a state made up of two identical bosons does not alter under exchange of the two bosons is the dictum of Bose symmetry [1]. This along with the Fermi statistics [2] forms one of the cornerstones of modern physics, the famous spin-statistics theorem. Within the conventional Lorentz invariant and local quantum field theory, even a small violation of Bose symmetry is impossible. There have been therefore a lot of interest in experiments looking for Bose symmetry violation as a means of testing the present theoretical framework. Theoretical ideas and experimental investigations for Bose symmetry violations have looked at the spin-0 nucleus of oxygen 16 O [3, 4], molecules such as 16 O 2 and CO 2 [5-8], photons [9-14], pions [15] and Bose symmetry violating transitions [16-22]. Theoretically a scenario where Bose symmetry is not exact swings open doors to a plethora of avenues for new physics [23][24][25][26][27]. Like the Bose symmetry, the very nature of Lorentz invariant local quantum field theory encompasses another fundamental symmetry of Nature, namely the CPT symmetry. This symmetry combines the operations of charge conjugation (C), parity (P) and time reversal (T ). In the conventional settings of quantum field theory, the CPT symmetry is very closely related to both spin-statistics theorem and Lorentz invariance [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. However, CPT invariance and the spin-statistics theorem need not be connected [47][48][49], and there are examples of quantum field theories in the literature [50-52] that explicitly violate the CPT invariance. Under CPT transformation, a particle becomes its antiparticle and vice versa with the same three-momentum but with its helicity reversed. The CPT invariance also implies that a particle and its antiparticle must have the same mass, decay width and lifetime. It is important to note that if CPT invariance holds good but CP is violated, then partial rate asymmetries for a particle and its an...
