Quantization triggered by breaking commutativity of symmetries. Discrete spectrum and tunnelling transfer controlled by distortion

By an important physical example of electron moving in a planar Penning trap we explain the general role of resonances making the symmetry algebra of a system to be noncommutative. After tuning at resonance the classical system can obtain quantum behavior, for instance, well distinguishable energy levels and opportunities for tunnelling transfer. Quantum properties are controlled by distortions in geometry of the system and by anharmonic parts of potentials. These controlling instruments determine the effective Hamiltonian over the noncommutative, non-Lie symmetry algebra which is the very object subjected to quantization. Tuning at resonance creates a (gyroscopic) symplectic structure as a mechanism for appearing additional forces, calls on a new (gyron) Hamiltonian and hence generates a specific mechanics in the space of symmetries. This mechanics can become quantum although the particle itself (core) remains to be classical. The point is that the dynamics in the gyroscopic structure is decelerated thus making the de Broglie wavelength much bigger in directions transverse to the classical trajectory and critically increasing the probability of quantum translocations of the particle.