Are the spectra of the LQG geometric operators really discrete?

tl;dr: Sometimes yes, sometimes no, depending on how one passes to the physical Hilbert space.

The title for this post is shamelessly borrowed from this paper, where Bianca Dittrich and Thomas Thiemann have discussed this question some time ago, concluding that the answer could be negative at the level of the physical Hilbert space, i.e. after solving the Hamiltonian constraint. Carlo Rovelli then wrote a rebuttal, arguing that discreteness should remain also at the physical level. Since those papers, some time has passed and new light has been shed on this issue from various angles, to be reviewed in this post.

To begin with, let us briefly recall the area operator in loop quantum gravity. It is constructed roughly as $\text{A}(S) = \int_S \sqrt{ E^i E_i }$ and acts diagonally on spin networks, with an eigenvalue proportional to $l_p^2 \sum_i \sqrt{j_i(j_i+1)}$, where $j_i$ is the spin label of the edges intersecting the surface $S$. The operator thus has a discrete spectrum and a spectral gap.

This operator is a priori only defined on the kinematical Hilbert space, where the spatial diffeomorphism constraint and the Hamiltonian constraint still have to be implemented. In particular, it is not invariant w.r.t. their gauge flow. The question is thus whether solving these constraints in some suitable way might change the spectrum of the area operator.

Constraints can be solved classically (reduced phase space quantisation) as well as at the quantum level (Dirac quantisation). At the quantum level, little is known about the issue of physical spectra, since we do not have a sufficient understanding of the solution space of the Hamiltonian constraint operator. As for the diffeomorphism constraint, we know how to solve it (roughly by going over to diffeomorphism averages of spin networks), but we then have the problem that the area operator does not preserve the diffeomorphism-invariant states. However, we can simply go over to the volume operator, which when evaluated on the whole spatial slice (e.g. for compact slices) does preserve these states. Then, the spectral properties of the volume operator simply pass to the diffeomorphism-invariant Hilbert space, indicating that the physical spectra might coincide with the kinematical ones. However, a study in 2+1 dimensions has shown that that the spectra can become continuous at the level of the physical Hilbert space.

As soon as we allow ourselves to solve the constraints classically, many things can happened. This is connected with a freedom in the choice of quantisation variables, or more precisely, the choice of Dirac observables and gauge fixings. As a seminal example, Kristina Giesel and Thomas Thiemann have considered LQG coupled to Brown-Kuchar dust, where the dust fields provide us with a physical coordinate system and the Ashtekar-Barbero variables at given values of the physical dust coordinates are Dirac observables. Since the dust fields don’t interfere with the gravitational sector at the level of the symplectic structure, the LQG geometric operators simply pass to the level of the physical Hilbert space, thus giving and example of where their physical spectra indeed are discrete and furthermore coincide with the kinematical ones.

However, different choices of gauge fixings or Dirac observables are possible. Another example has been discussed in the context of a constant mean curvature clock for GR conformally coupled to a scalar field. Here, due to the non-minimal coupling, matter and geometry mix in the symplectic structure and the Dirac observables commuting with the constant mean curvature gauge condition correspond to a conformally rescaled metric $\tilde q_{ab} = \phi^2 q_{ab}$. From this metric, one can again pass to connection variables and construct a loop quantisation, resulting in a “twiddled” area operator which has the same spectral properties as the standard area operator, however a different interpretation. Since it is constructed from the twiddled metric, it measures a twiddled area, which is related to the usual area by a multiplication of $\phi^2$. Yet another example of canonical variables leading to a changed are spectrum due to non-minimal coupling has been given here.

So are the physical (untwiddled) areas discrete in this case? This obviously depends on the quantisation of $\phi$, since we have to extract the geometric area from the twiddled one. For scalar fields, different representations are available, e.g. following Thiemann’s original proposal, its upgrade to the Bohr compactification of the real line as a gauge group, or the representation discussed here. As an example, in the last case $\hat \phi(x)$ acts by multiplication of an arbitrary (sufficiently smooth) function $\phi(x)$, and thus the eigenvalues of the induced untwiddled area operator can take any value. However, this simple argument only makes sense if we forget about our intended gauge fixing and simply use the variables suggested by it. Otherwise, we should solve the Hamiltonian constraint for $\phi$, resulting in a much more complicated and generally non-local expression.

While an area operator could be reconstructed from the scalar field and the twiddled area operator, this might be too complicated for other choices of variables. An example is provided by Lovelock gravity, a higher-derivative generalisation of general relativity with the same phase space. In the LQG context, one finds that the appropriate area operator analogue now measures Wald entropy (on non-rotating and non-distorted isolated horizons) for a suitable choice of variables suggested by the canonical analysis of the Lovelock action. The area of the surface can not simply be recovered from this expression, and the fate its spectrum remains unclear.

While in the above examples the basic structure of the connection variables remained the same, e.g. having gauge group SU(2), this can also be tampered with. For example, choosing the diagonal metric gauge, one obtains Abelian connection variables with the Bohr compactification of the real line as a gauge group. Then, all real numbers are eigenvalues of the area operator, as opposed to the SU(2) case. Somewhat differently, one can choose the radial gauge and corresponding connection variables, where some of the area operates (the radial-angular ones) remain the same as in standard LQG, while others (the angular-angular) become volume operators of 2+1 gravity, which then have different spectral properties.

In conclusion, we saw that the spectrum of geometric operators may be strongly influenced by the choice of variables we make, which is in turn influenced by the choice of gauge fixings / Dirac observables, i.e. ultimately the passage to the physical sector of the theory. This shows that the final spectra of geometric operators do depend on this choice and may or may not have discrete eigenvalues. One may conjecture that similar effects might also appear when solving the constraints at the quantum level, but this remains to be shown in concrete examples.