Aims. A new mechanism of particle acceleration, based on the resonant interaction of a classical electromagnetic wave (EM) with a quantum wave (associated with a relativistic particle), is explored. Methods. In a model illustrative calculation, we study the fate of a Klein Gordon wave subjected to the intense radio frequency waves generated in the vicinity of an active galactic nuclei (AGN). In the framework of the paper, we examine a quantum wave associated with a relativistic particle, and it

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... is shown that the group velocity of the wave approaches the speed of light, implying that the particles resonantly exchange energy with EM waves, eventually leading to acceleration of particles to very high energies. Results. For typical parameters of under accreting Eddington radio AGN, it is shown that the resonant energization could catapult particles to extreme energies ∼ 10 16−20 eV.