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This is a 2013 thread I created on sciforums that discusses the pros and cons of Loop Quantum Gravity and a number of insights into it versus a response by a working theoretical Physicist (two actually), where they measure it against String Theory.
BY MARKM125:
"Considering this thread is about LQG, I'll type a quick post about the subject.
What is LQG?
Loop Quantum Gravity is an attempt to quantize General Relativity by introducing gauge fields (like any QFT) onto a curved background. In doing so, two basic symmetries are demanded of the theory - gauge invariance, and invariance under the diffeomorphism group.
Most readers are probably familiar with the first, if not, here's the basic idea: A local gauge transformation is a transformation performed on a Lagrangian dependent on the point in spacetime (i.e. it's local). For example, a transformation of the form AОјв†'AОј+ПµBОјAОјв†'AОј+ПµBОј (Where AОјAОј is the vector potential. You can verify this by simply plugging this in to the equations giving the E and B fields in terms of the vector potential. ) is a global gauge transformation - it will leave the classical Maxwell Equations invariant, but it doesn't depend on the coordinates. Instead, a local GT will have such a position dependence. The result is then this: the gauge transformations of a particular form (i.e. Multiplication by a unitary matrix) will form a Lie Group, which in tern has several generators. The generators then can be associated with vectors fields, which are then quantized into particles. These particles are then your gauge bosons, such as photons, W bosons, Z bosons, and gluons. For example: the group U(1)U(1) is a symmetry group of QED, and gives rise to the photon and its interactions.
Diffeomorphism invariance is a property of general relativity. It's the ultimate form of relativity - as long as we perform some continuous deformation of the background spacetime, we still have a consistent theory. Thus, any theory with the spirit of GR should, as is felt by proponents of cannonical approaches to quantum gravity, contain this symmetry. Essentially, DI demands that we construct a theory that has no dependence on the background spacetime. We can associate this mathematical idea with general covariance - choose any frame you wish, and physics is the same.
Now, one of the critical equations in the field of quantum gravity is the Wheeler-DeWitt equation, taking the form
Hв€ЈП€вџ©=0Hв€ЈП€вџ©=0
Where П€П€ is the "wavefunction of the universe", or simply, the quantum representation of one possible state of spacetime. In analogy to quantum mechanics, we should find the solutions to this equation, use them to construct a general Hilbert (or Fock? Not sure) space, and then express the quantum state of spacetime, and hence the gravitational field.
Following a mathematical reformulation of GR in terms of Ashtekar variables, it was determined that solutions could take the form of "spin netowrks" (see John Baez's page for a ton of info) that evolve in time as "spin foams". Then the dynamics of the gravitational field can be expressed as a superposition of these spin foams. Spin foams can be thought of as the repeated action of a scalar constraint в€ЈП€(t1)вџ©в€јeв€'iHОґtв€ЈП€(t0)вџ©в€ЈП€(t1)вџ©в€јeв€'iHОґtв€ЈП€(t0)вџ©.
That's the basic structure. Here are the consequences:
1. Firstly, the graviton propagator reduces to the inverse square law over long distances.
2. The symmetries of GR are retained.
3. The usual action remains, consisting of the Ricci Scalar with additional quantum corrections.
4. The use of "Wilson Loops" in the theory preserves gauge invariance and results in a minimum area of 10в€'6610в€'66 meters squared. The square root is roughly the Planck length.
If you're interested into delving into the theory, there are various lecture notes on the subject. I'd recommend the one by Pietro Dona and Simmone Spezial."
http://www.sciforums.com/threads/loop-quantum-gravity-lqg.135750/