Quantum mechanics without quanta

In this paper, I argue that light is a continuous classical electromagnetic wave, while the observed so-called quantum nature of the interaction of light with matter is connected to the discrete (atomic) structure of matter and to the specific nature of the light-atom interaction. From this point of view, the Born rule for light is derived, and the double-slit experiment is analysed in detail. I show that the double-slit experiment can be explained without using the concept of a "photon", solely on the basis of classical electrodynamics. I show that within this framework, the Heisenberg uncertainty principle for a "photon" has a simple physical meaning not related to the fundamental limitations in accuracy of the simultaneous measurement of position and momentum or time and energy. I argue also that we can avoid the paradoxes connected with the wave-particle duality of the electron if we consider some classical wave field - an "electron wave" - instead of electrons as the particles and consider the wave equations (Dirac, Klein-Gordon, Pauli and Schrodinger) as the field equations similar to Maxwell equations for the electromagnetic field. It is shown that such an electron field must have an electric charge, an intrinsic angular momentum and an intrinsic magnetic moment continuously distributed in the space. It is shown that from this perspective, the double-slit experiment for "electrons", the Born rule, the Heisenberg uncertainty principle and the Compton effect all have a simple explanation within classical field theory. The proposed perspective allows consideration of quantum mechanics not as a theory of particles but as a classical field theory similar to Maxwell electrodynamics.