Loop quantum gravity, or LQG for short, is one of the most popular approaches to quantum gravity. It is in the second place right behind string theory. It is a theory that try to quantize gravity using just plain quantum mechanics as we already know it, without incorporating any other new principle. Compared to strings, a very humble theory.

The trick to make this approach to work is to describe general relativity using a set of new variables, called Ashtekar variables, and impose the quantum commutation relations for these variables instead of for position and momentum as is usual. These commutation relations are mathematical relations that in non-relativistic quantum mechanics are responsible for the uncertainty principle, that says that if a particle has a well-defined position at an instant it has no defined momentum and vice-versa. When you add relativity things become a little more complicated, but the spirit remains the same. Indeed, this method of quantizing a theory by imposing quantum commutation relations IS what we call to quantize a theory. Quantum mechanics is not a very understood theory. It works, but we don't know exactly what happens. What we do know is that if we expand the solution of classical equations in Fourier series and impose the quantum commutation relations for some variables, it works. That is how QED was quantized, and it worked with an astounding precision.

The approach is called LOOP quantum gravity because the variables to be quantized are variables known as Wilson loops. Formally, a Wilson loop is the trace (i.e., the sum of the diagonal components of a matrix) of the holonomy of a vector transported along some closed path (a loop) in spacetime. Holonomy is an operator that gives the resulting vector after the transport has been made. If you do this transport in a flat spacetime (with no gravitational fields), the resulting vector is the same as the initial vector. But if the spacetime is curved (like the surface of a sphere), the result is not the same vector.

Using this variables, physicists were able to find solutions for the resulting quantum equations for general relativity (GR). Some additional results have been found, for example, they found quanta of area and volume for the spacetime (something that physicists liked, because we think that spacetime is fundamentally not continuous, but discrete) and, like string theorists, the correct entropy formula for black holes (in special cases). LQG is far more simple than string theory, although this simplicity is relative once that the geometry involved in LQG is very sophisticated, and to date achieved as much successes as strings (theoretically, because experimentally, they're in the same level: no confirmation at all). There are some physicists that even believe that strings and LQG can be combined in a single theory, because both have interesting physics insights and results and some similarities. As the popularity of strings id coming down due to the lack of testable predictions, that of LQG is coming up. The only thing that rests for us physicists is keep working on the problem to see what Nature could reveal to us in the future.

Picture from the article: "Quantum gravity: The quantum of area?" - John Baez - Nature 421, 702-703 (13 February 2003). Original caption:

The trick to make this approach to work is to describe general relativity using a set of new variables, called Ashtekar variables, and impose the quantum commutation relations for these variables instead of for position and momentum as is usual. These commutation relations are mathematical relations that in non-relativistic quantum mechanics are responsible for the uncertainty principle, that says that if a particle has a well-defined position at an instant it has no defined momentum and vice-versa. When you add relativity things become a little more complicated, but the spirit remains the same. Indeed, this method of quantizing a theory by imposing quantum commutation relations IS what we call to quantize a theory. Quantum mechanics is not a very understood theory. It works, but we don't know exactly what happens. What we do know is that if we expand the solution of classical equations in Fourier series and impose the quantum commutation relations for some variables, it works. That is how QED was quantized, and it worked with an astounding precision.

The approach is called LOOP quantum gravity because the variables to be quantized are variables known as Wilson loops. Formally, a Wilson loop is the trace (i.e., the sum of the diagonal components of a matrix) of the holonomy of a vector transported along some closed path (a loop) in spacetime. Holonomy is an operator that gives the resulting vector after the transport has been made. If you do this transport in a flat spacetime (with no gravitational fields), the resulting vector is the same as the initial vector. But if the spacetime is curved (like the surface of a sphere), the result is not the same vector.

Using this variables, physicists were able to find solutions for the resulting quantum equations for general relativity (GR). Some additional results have been found, for example, they found quanta of area and volume for the spacetime (something that physicists liked, because we think that spacetime is fundamentally not continuous, but discrete) and, like string theorists, the correct entropy formula for black holes (in special cases). LQG is far more simple than string theory, although this simplicity is relative once that the geometry involved in LQG is very sophisticated, and to date achieved as much successes as strings (theoretically, because experimentally, they're in the same level: no confirmation at all). There are some physicists that even believe that strings and LQG can be combined in a single theory, because both have interesting physics insights and results and some similarities. As the popularity of strings id coming down due to the lack of testable predictions, that of LQG is coming up. The only thing that rests for us physicists is keep working on the problem to see what Nature could reveal to us in the future.

Picture from the article: "Quantum gravity: The quantum of area?" - John Baez - Nature 421, 702-703 (13 February 2003). Original caption:

*In loop quantum gravity, space is envisaged as a fabric of woven threads. Where these threads puncture a surface, such as the event horizon of a black hole, they define its area.*
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