"The wave is actually probability in the sense that it assigns probability to the space coordinates of detecting photon at a certain time. Now, the wave transports energy & momentum."
As you say that you quote words from a book, then, you have to know that there are many books and many authors, each one with his/her opinion. There are four basic opinions about the wave-function:
- Standard Quantum Mechanics - we don't know what is the wave-function, and we can only calculate results of experiments.
- Bohmian Mechanics - the wave-function is a field, not one that is known to us from the classical physics (e.g. not e.m., not gravitational), and this field guides particles like electrons, photons, which trajectories to follow. E.g. in interference experiments, these particles are guided to go with bigger probability to some places than to others.
- G.R.W. (Ghirardi-Rimini-Weber) - the wave-function is a wave with no structure of particles in it, but when meeting classical apparatuses, it produces an effect of discrete responses.
- MWI (many-worlds interpretation) which I am not going to discuss.
(There are also other interpretations but the most elaborated are the ones I quoted above.)
Now, to different details of your question, (my answers are in agreement to the most widespread views, not biased by one interpretation or another):
We don't know that the wave is probability. Probability is a mathematical thing that we write on the paper. In nature exists matter, under different forms: waves that carry by themselves their energy and momentum, waves whose energy and momentum are transmitted with the help of the particles of some medium (as in the case of water waves or the sound), definite objects big or small, and eventually other forms not yet known to us.
Now, if the waves were waves-of-probability as you are told, then how could they carry energy and momentum?
"Unlike light, electron-wave is not made by electric or magnetic fields."
Who says that a wave has to be electromagnetic? The gravitation waves aren't electromagnetic. The sound is not electromagnetic.
But, unfortunately, we don't know what is the wave of which you talk, i.e. the wave-function. It's more than a century that we stay with this problem. The electron-wave yes carries energy and momentum, however, we are unable to decide which one of the different interpretation is correct, and we also admit that there may exist another variant.
A. Peres, said once that it may be that we have to reconsider our concepts about space, time, and others.