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FURTHER STUDY
Quantum gravity, mathematical explanation:    • The Trouble with Gravity: Why Can't Q...  
String Theory vs Loop Quantum Gravity:    • String theory vs Loop quantum gravity...  
How QM ruins GR:    • Quantum Gravity: How quantum mechanic...  
General Relativity explained:    • General Relativity Explained simply &...  

CHAPTERS
0:00 How gravity models evolved
2:22 Is Quantum Gravity even necessary?
6:23 3D Bronstein Cube
7:56 Why can't we quantize gravity?
11:19 Ways that we could quantize gravity
12:59 Why don't we fit the other forces into General Relativity?
14:26 String theory and Loop quantum gravity
16:52 Why should we care about quantum gravity?

SUMMARY
The universe seems to be quantum not classical. But General Relativity is classical. When we try to use Einstein's theory to make a quantum model of gravity, we get nonsense results. Why is quantum gravity the most difficult problem in physics? Is a quantum model of gravity even necessary? why can’t we fit the other three forces into the framework of General Relativity instead?

Reasons to quantize gravity: All the other fields in nature are quantized. Why would nature make an exception for only the gravitational fiield? Also, General Relativity breaks down at the singularity at the Big Bang and inside black holes. A zero size singularity seems absurd, and likely unphysical. It probably means there’s a breakdown in the theory.

We can see a problem just by looking at the equation for General Relativity. The left side describes curvature of spacetime, which is classical. But the right-hand side is matter, which is quantum. So, we have two incompatible types of mathematics.

A 3D Bronstein cube can illustrate what a quantum gravity theory would look like. We need to either take General Relativity and quantize it, or take quantum field theory and incorporate gravity into it.

All the quantum theories of the standard model take classical theories and make it quantum by taking certain variables such as momentum and turn them into operators. This procedure however doesn't seem to work with gravity, because we get all kinds of infinities, that cannot be corrected or renormalized.

What makes General Relativity unique is that it is a theory of space-time itself, not stuff happening in spacetime, which is the case with the other three fundamental forces. Gravity results from the geometry and curvature of space-time. The other forces describe events happening within this background geometry, not the background itself.

Another factor that makes quantizing gravity difficult is because it is very weak, making it nearly impossible to do experiments. So inventing a quantum theory of gravity becomes mostly a thought experiment.

We can attempt to treat gravity as a field just like other fields. An excitation in this field would be the graviton. The exchange of gravitons between two particles would result in an attraction. This is what string theory attempts to do because a graviton emerges in the math of string theory.

The second way is to quantize spacetime itself. This is what Loop quantum gravity attempts to do. But this would mean spacetime could exist in a superposition of various different geometries. This is problematic because spacetime would be dynamic in quantum gravity. So we can’t ask for example what is the probability of finding an electron at a certain location, because there is no objective way to specify what location is that we are talking about. This is because spacetime itself would be in a superposition.

Why don’t we attempt to fit everything into the framework of general relativity instead of trying to fit general relativity into quantum mechanics? This has been attempted. In 1919, Theodore Kaluza, came up with general relativity 5 dimensions instead of 4. The found the laws of classical electromagnetism in his 5 dimensional equations. But it was obviously wrong, because we live in a 4 dimensional universe, 3 spatial dimensions and 1 of time, not 5.

Swedish physicist, Oskar Klein suggested that perhaps the newly hypothesized 5th dimension was very small, so small that it could not be detected. This idea was put on the back burner because at the same time he published his paper, quantum field theory took off.
#quantumgravity
To of the most popular approached today to quantizing gravity is String Theory and Loop Quantum Gravity. This video discusses the difference between these two theories, and their and pros and cons.