, At the same time, the corresponding antiparticle is associated with a negative energy IR where the energy has the spectrum in the range (??, ?mass]or later we will arrive at the point where energy=-mass. Therefore in FQT a particle and its antiparticle automatically belong to the same IR and have the same masses because the ring R p is finite and has the property of strong cyclicity. The fact that in FQT a particle and its antiparticle belong to the same IR makes it possible to conclude that, in full analogy with the case of standard dS theory (see the preceding section), there are no neutral particles in the theory, the very notion of a particle and its antiparticle is only approximate and the electric charge and the baryon and lepton quantum numbers can be only approximately conserved. As shown in Sec. 8.7, if one tries to replace nonphysical annihilation and creation operators (a, a * ) by physical operators (b, b * ) related to antiparticles then the symmetry on quantum level is inevitably broken, My original motivation for investigating FQT was as follows. Let us take standard QED in dS or AdS space, write the Hamiltonian and other operators in angular momentum basis and replace standard IRs for the electron, positron and photon by corresponding modular IRs. One might treat this motivation as an attempt to substantiate standard momentum regularizations

, This poses a problem whether there are physical reasons for such a choice of the vacuum. As explained in Sec. 8.9, the spin-statistics theorem can be treated as a requirement that standard quantum theory should be based on complex numbers. This requirement also excludes the existence of neutral elementary particles. Since FQT can be treated as the modular version of both, dS and AdS standard theories, supersymmetry in FQT is not prohibited, contrast to standard theory, the vacuum can be chosen such that the vacuum energy is not infinite but zero

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