Quantum dots for demonstration of additional dimensions of spacetime

By the example of electron mesoscopic systems, we show the impossibility of constraints of the quantum principle of superposition imposed by the superselection rule. This rule was introduced by Wick, Wightman, and Wigner in order to avoid the violation of Lorentz invariance due to the absence of physical invariance under rotations by an angle of 2π in states which are a coherent superposition of states with an even and odd number of fermions. We describe a mesoscopic system (a semiconductor double quantum dot at low temperatures) where such superpositions are realized; this is confirmed by experiments. We suggest a new experiment which explicitly demonstrates the absence of physical invariance under rotations by an angle of 2π. We note that an alternative to the superselection rule is the existence (along with x, y, z, and t) of additional spinor (Grassmann) dimensions of spacetime introduced in quantum field theory for realization of supersymmetry. It is proved that additional dimensions are real; their physical meaning is clarified for nonrelativistic systems of fermions.