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When we say that light behaves as a particle and as a wave, we say that light is an electromagnetic wave, and it's a photon. Both EM waves and photons have physical definition. On the contrary an electron wave is just the probability amplitude, in other words the definition for the so called wave nature of electron is not physical as it is for a photon.

My question is how can we physically define the photon? Is a photon just a lump in the electromagnetic wave, or is it a point on the wave?

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When we say that light behaves as a particle and as a wave, we say that light is an electromagnetic wave, and it's a photon.

The discussion about light or more broad about electromagnetic radiation needs a focus about what are EM radiation, EM waves and photons. In a nutshell

  • light is not a particle, it consists of photons
  • the photon has an oscillating magnetic and an oscillating electric field component, both perpendicular to each other and to the direction of propagation
  • the synchronous acceleration of the - emitting this time - electrons give us a technical solution for electromagnetic waves (radio waves)
  • last not least the radiation from a thermic source is not a wave.$^1$

Does this imply that there is a fundamental difference in wave properties of fermions and bosons?

That is correct, but suppressed by the double use of the word spin.

For fermions it is clearly something to do with the magnetic interaction between the fermions (remember there is a magnetic quantum number) and with the deflection of fermions at the edges of obstacles (giving a periodic distribution of them behind the obstacle).Without investigating anything about the interactions between the edge of the obstacle (with surface electrons) and the fermions, Young's interpretation about interference from different points is the accepted interpretation. Although it later became clear that nothing interferes (photons at low energies do not interact), interference is still part of the formation.

For bosons there is a helicity of the arrangement of the magnetic and the electric field components of the photon, expressed by the left and right hand rule. It is a convention, what for a direction are showing the free fingers. But once fixed we get the result, that for photons from electrons and from protons we get rules for different hands.$^2$

$^1$ Different from a radio waves such radiation hasn‘t any periodically changing amplitude (be this a geometrical property like in water waves are a probability amplitude).

$^2$ In addition to helicity, a photon can clearly exhibit rotation, which is given by the rotating subatomic particle or when passing through a dielectric material. This time, the magnetic and electric field components rotate along the axis of propagation (if you move your thumb in the direction of propagation, you should rotate the other two fingers involved clockwise or anticlockwise).

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