I've always thought that the reason we had trouble unifying Quantum mechanics and General relativity is:
Quantum mechanics is defined "on" space time and time doesn't vary because of energy. Time is considered a constant (or more like an independent variable.) Whereas GR states that energy bends spacetime and that time is not a constant (or it varies).
The second thing is that energy is not Globally conserved. While in Quantum mechanics everything is based on energy conservation. And hence, both are not completely compatible.
There might be some mathematical incompatibilities like re-normalization which doesn't work for GR. I don't know why though.
Then how does string theory solve these problems? Isn't the theory just an Idea that string exist and their vibrational state determines the type of particle. Then how does it relate to time being considered as a variable, energy conservation problems and space time curvatures?
How does it try to unite GR with QM?
Answer
What you are referring to is called Background Independence.
Among theoreticians, different attitudes towards background independence dominate. Some consider it extremely important (like the founding fathers of Loop Quantum Gravity), others think of it as merely a peculiar feature of the low-energy theory.
The truth is, ofcourse, that any physical theory has to be judged on the basis of predictions that it makes, not on the basis of which approach you find more appealing. This is exactly why quantum gravity research has gone astray a long time ago: the experimental vacuum forces scientists to speculate.
I will wrap it up with a brief overview of how theories like LQG and string theory treat background independence.
LQG tries to capture the insight of Einstein's GR (whcich is exactly that theories of gravity have to be background-independent). It therefore presents a quantization procedure which doesn't make any reference to any specified background.
String theory is originally formulated as a theory of some physical entity (a string) living on the fixed background. The fluctuations of the string are conjectured to behave like the fluctuations of the background in which the string lives. As an indirect proof of this claim: one of the modes in the spectrum of the strings corresponds precisely to the graviton (a perturbation of the background spacetime); RG-flow equations for the worldsheet conformal invariance turn out to imply Einstein's equations for the background spacetime.
String theory is definitely not manifestly background-independent. But this doesn't mean that it isn't background-independent! The question of whether it is background-independent or not is, to my knowledge, still unsettled.
There have been claims made by respected superstring theorists that superstrings might be a first-quantized perturbative version of some background-independent theory. This could be M-theory (though most of the searches for the fundamental formulation of M-theory were carried out in the background-dependent setting, lol); or this could be formulated on the boundary through AdS/CFT.
To conclude: background independence is a beautiful physical insight of General Relativity, there's no doubt about that. But we accept physical theories based on how well they can predict results of experiments, not by how appealing their fundamental principles seem to us. Both superstrings and LQG have yet to give a single numerical prediction verified by experiments (don't take me the wrong way, they give plenty of mutually contradicting predictions, but none of them are experimentally accessible now or in not-so-far future).
Speculations are allowed to satisfy whatever fundamental principles we want them to, really.
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