Why can people learn about QED and QCD and not about:QGD(i.e.Quantum gravitodynamics)?What are then the difficulties (both physically and mathematically speaking)that prevented theoretical physicists to develop such a theory??
We currently lack the theory you describe, which is called 'the quantum theory of gravity" (or, if you wanted to name it in the form QxD, it would be QGD, quantum geometrodynamics).
Myriad problems face researchers in this field. Entirely new branches of mathematics and have been discovered in its pursuit, and many of them are still in their infancy. Developing a quantum theory of gravity is to present-day physics as building a sky-scraper is to cavemen. Quite a lot of "infrastructure" has to be developed before the skyscraper itself can be built.
Yes, the math itself is what gives us problems. We have a rough idea of how a quantum theory of gravity should look. We currently are nowhere even close to being able to design experiments able to operate at the requisite level of precision, so there is currently no experimental evidence -- it's all pure theory.
The two biggest competing theories are string theory and loop quantum gravity. Of the two, string theory is the most ambitious, since it is not only a theory of gravity, but a theory of all four fundamental interactions (five if you include the Higgs mechanism).
Loop quantum gravity, on the other hand, is purely a quantization of GR, and many people see it as sort of a contrivance.
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