Non-classical Energy Conservation in Multi-wave Systems: "Extra Energy", "Negative Energy" and "Annihilation of Energy"

The energy conservation is a general law of nature. In the classical physics, the energy W_{AB} of a conservative system {AB} that contains the objects A and B is equal to a sum of the positive energies W_A and W_B of the isolated objects A and B, W_{AB} = W_A+W_B. We show that the energy conservation does not exhibit the "classic law" if the physical objects are waves or they do have a wave nature of microscopic (quantum) objects. The "extra energy", "negative energy" and "annihilation of energy" are predicted for multi-wave (multi-beam) systems. The paradoxical phenomenon is demonstrated in context of the extraordinary transmission of light and matter through subwavelength aperture arrays assisted by surface waves [T.W. Ebbesen et al., Nature (London) 391, 667(1998)) and E. Moreno et al., Phys. Rev. Lett. 95, 170406 (2005)].