Loop Quantum Gravity

Loop Quantum Gravity is a theory that contends against String Theory to combine gravity and quantum mechanics to create a theory of everything.
 

Let's review some background information. In Quantum mechanics, the background is fixed, and particles move on the stage. It is background dependent. General relativity focuses on how the stage changes, making it background independent. What Loop quantum Gravity tries to do is quantize general relativity while also preserving its background independence.

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It starts by recognizing the difference between both theories. The Wheeler De-Witt equation is an attempt of describing the quantum evolution of space-time and it is based on the idea of ADM Formalism.

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ADM starts by defining the abstract space of spaces. These are 3-dimensional spacial slices of 4-dimensional space-time. As you move through this space of spaces, which can be treated as a coordinate system, the geometry of space-time changes. Think of it as sort of like a stop-motion animation. The Wheeler De-Witt quantifies the analogies for properties such as position and momentum, and as a result, gives an equation of motion.

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However, the equation turned out to be unsolvable. However, we can introduce a new idea that may fix this called connections. Connections are mathematical functions that can tell us information about how something like a vector can change over time. They can be used to represent information about space-time by representing space-time as a system of connections rather than regular coordinates. 

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In the 80s, Abey Ashkatar tried a different type of connection that instead of vectors, uses spinors, which are like vectors except they represent spin as well. He was able to rewrite General Relativity in terms of these spinor-connections (now called Ashkatar Variables). It turns out that this new space of spaces looks pretty similar to a space of fields in QFT, showing a definite connection.

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Here is where we get to the loop part in loop quantum gravity. Using Ashkatar variables, Lee Smolin and Carlo Rovelli solved the previously unsolvable Wheeler De-Witt Equation using the Ashkatar Variables. They evaluated the connections over closed loops because the loops connected toward themselves. Using these loops, they could define the geometry of 3d space, each loop acting like a closed circuit of gravitational field.

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This new space of loops can be quantized neatly and is background independent. Think of it as General Relativity made of blocks. At large scales, it would look normal, but at the Planck scale, it would look pixelated. That is Loop Quantum Gravity.
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Loop Quantum Gravity has had some successes, in predictions of hawking radiation and black hole entropy. However, there are still problems with how this new background independence extends to time. 

Another problem that Quantum mechanics and General relativity has is how it treats time. While Quantum mechanics treats time completely separately, General Relativity treats it like it's just another dimension.