Is taking the “square root” of the densitized inverse triad irregular in loop quantum gravity?

Question

In loop quantum gravity, the canonical (Ashtekar) variables are chosen to be the densitized inverse triad $\mathbf{E}$ and some rotation connection field $\mathbf{A}$. To get the ordinary triad from $\mathbf{E}$, we need to take its "square root". But we know as long as the volume factor, i.e. the square root of the determinant of $\mathbf{E}$ is nonzero, we always have two solutions for the triad. They are related by a branch cut around $|\mathbf{E}| = 0$. We also know that if this determinant is negative, the corresponding triad will be imaginary! OK, you might now say there's no problem as long as we stick to solutions where $|\mathbf{E}|$ is always positive everywhere.

But here is where another problem comes in. The state about which loop quantum gravity expansions are made is the state where $\mathbf{E}=0$ everywhere. Excitations over this state by Wilson loops contain wavefunctional components with negative determinants as frequently as we have components with positive determinants. Does that mean loop quantum gravity predicts imaginary distances, negative areas and imaginary volumes? And besides, aren't we expanding about a highly irregular solution?

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Another related question is: People sometimes multiply the Hamiltonian constraint by the factor $\sqrt{| \mathbf{E} |}$, but this goes to zero whenever $| \mathbf{E} | = 0$, which actually weakens the constraints by introducing additional solutions which were previously forbidden. Is such a procedure really justified?

Answer 1

Yes, absolutely, the "proper distances vanish" is a singular point of the configuration space of general relativity and it is not permissible to "expand around it".

And you are also totally right about the "imaginary roots" problem. The map was chosen to be bilinear exactly because the loop quantum gravity physicists wanted the areas to be quantized in a simple way. So the triad $E$ really measures the proper areas and the proper distances must be taken as the square root of it.

There is nothing physically legitimate about this operation - it is just a trick to pretend that the metric tensor is equivalent to a bulk gauge field. While the counting of components of the SU(2) gauge field could work, the physics doesn't work and the field redefinition is not one-to-one. The appearance of the imaginary roots is just one manifestation of the problem.

There are other manifestations of the illegitimacy of the field redefinition from gravity to the "new variables". In particular, the quantization of the areas themselves proves that the map is only valid locally on the configuration space, but not globally. The holonomies or Wilson loops constructed out of the gauge field are periodic variables; that's essentially why their canonical momenta are quantized - and the canonical momenta are the areas.

However, in the proper gravity, with the continuous metric tensor, there is no quantization of the areas. In fact, such a quantization violates the local Lorentz symmetry because the "spin network" always picks a preferred reference frame, much like the luminiferous aether. This brutally breaks the Lorentz symmetry and destroys inertia: the entropy density is pretty much Planckian and an object moving through this material will instantly dissipate its energy to the "spin network degrees of freedom" and stop.

So the field redefinition is only a game that is valid locally at the configuration space - because of the right counting of the degrees of freedom - but is incompatible globally, because it is not one-to-one, and it incompatible with the required dynamics for the degrees of freedom. In the new variables, one can never get a theory that behaves as general relativity.

That's why the word "gravity" in "loop quantum gravity" is a misnomer. There can't be any gravity.

Answer 2

No, LQG does not predict imaginary distances, negative areas and imaginary volumes. It is sufficient to remember the geometrical interpretation of the triad. A triad with a negative determinant is simply a left handed rather than a right handed triad. Therefore it does not describes funny spaces with imaginary volumes, but just the same good old space with where the basic triad is left handed rather than right handed. This simply guides you in giving the proper definitions of area and volume, putting the right sign and the right absolute values where it belongs. There may be papers where those absolute values have been placed wrong, but when things are done properly, no imaginary volumes ! carlo rovelli