The theory of relativity directly relates the four dimensions of our universe (time, motion, space, and the speed of light). Quantum mechanics is the study of measurements in extremely small quantities. I understand that if you distort one dimension of the universe, another must be distorted as well. But how does this relate to Quantum mechanics and how does it branch off from the M-Theory?

Basically, Quantum Mechanics and General relativity clash with each other. They show two different pictures of the universe. M-theory reconciles these two views. If you are interested in M-theory you should watch this program –> http://www.pbs.org/wgbh/nova/elegant/program_d.html

*Beware it’s three hours long*

That’s really the BIG Question. Unfortunately, nobody has a good answer. (And I DO keep up with the latest developments in physics) The crux of the problem seems to be that, at the quantum level, time really isn’t relevant. The whole issue requires a pretty endomorphic math background to even begin to wrap one’s mind around. I have that, but I’ve been wrestling with it for years and haven’t come up with a good answer yet. I’ll keep trying, though.

The four dimensions are usually understood as the three space and the one time. relativity’s consequences explain a good deal of the macro universe, while quantum explains pretty much everything else, as far as I understand it

Quantum Mechanics has been reconciled with *Special* Relativity via Quantum Field Theory. Relativistic electrons, for example, are described by the Dirac Equation. No one has figured out how to quantize the wave function in the much more complex space-time metric of General Relativity, however. There’s a Nobel Prize waiting for you if you can do it.

At the root of quantum mechanics is the Heisenberg uncertainty principle, which states that it is impossible to know the simultaneous magnitudes of certain pairs of dynamical variables (called “canonical pairs”) with arbitrary precision. An example of which are the position and momentum of a particle. Now, the reason why QM and GR are incompatible is that according to GR the gravitational field is a property of the geometry of space-time, which predicts that this field is propagated with a finite velocity (that of light), as opposed to infinitely fast, as Newton had believed. A moment’s thought will make you realize that this property requires that a moving (i.e. accelerating) mass necessarily transfers energy and momentum to the gravitational field, just as is the case of a charge in relation to the electromagnetic field, both result in the emission of waves (gravitational in the case of the former, electromagnetic in the case of the latter). Since a quantum theory of matter is required to fit the observed phenomena, it is also necessary to have a quantum theory of gravitational fields, for the same reason that we have to have a quantum theory of electromagnetic radiation-because of the interaction between mass (or charge) and the corresponding field. But since the gravitational waves are carried as ripples in the space-time framework itself, this would require a quantization of the coordinates of space and time themselves. This is a very difficult task that hasn’t been successfully performed to-date.

Hope this is clear.

I added later the following additional explanation why it is so difficult to quantize space-time: The coordinates of space and time are not canonical in the usual sense, in that we do not have canonical momentum coordinates, which would permit quantization by the normal procedure (postulating the usual commutation relations).