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As Dr. Michio Kaku says, that in the current state of Physics, we have the 'Theory of the Big' that is black holes, the Big Bang, cosmology, astrophysics on one hand and the 'Theory of the Small' that is Quantum Mechanics, Atomic and Nuclear Physics on the other. He says that both of these fields are based on different mathematics, different principles and presently both these theories hate each other. Is String Theory the only leading theory that can unite both these theories?

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Short answer: No, there are other theories as well that unite them.

Long answer: While string theory is the most discussed “theory of everything,” other ones such as loop quantum gravity exist (see the Wikipedia page on quantum gravity). Each of these theories have downsides. String theory is often presented in a glorified light. In reality, the math is tedious to the point that setting up the equations to solve requires a computer. The mathematics is inconsistent as well (there are several types of string theory). No theories have yet provided experiments that could verify their correctness.