Yang-Mills analogues of the ambiguities in defining LQG geometric operators

Changing the connection variables underlying loop quantum gravity
also changes the geometric operators. They measure geometry with respect
to the metric encoded in the Yang-Mills electric field $E^a_i$.   

tl;dr: In some cases there are such analogues, but they are rather awkward.

A comment to a recent post on the current status of the issue of the spectra of geometric operators in loop quantum gravity raised the question of whether such ambiguities can also be found in Yang-Mills theory. The question if of course very interesting, however I am not aware of any reference commenting on it. So let me try.

To begin with, let us recall what is the issue in the context of loop quantum gravity. There, since one is in the context of general relativity, observables (= gauge invariant quantities) have to be constructed. One way to do this is to construct an observable algebra at the classical level and then to quantise it. Such a choice of observable algebra then effectively leads to a specific choice of quantisation variables, which as a result may change the properties of geometric operators, e.g. their spectra.

An instructive example is provided by considering general relativity coupled to a scalar field. Here, the canonical variables are the spatial metric $q_{ab}$, its conjugate momentum $P^{ab}$, the scalar field $\phi$, as well as its momentum $\pi$. One can now construct connection variables in the gravitational sector as follows. We first extend our theory to incorporate a local SO(3) gauge invariance by going over to the variables $e_{ai}$ and $K_{ai}$, where $e_{ai}$ is simply a tetrad for the metric, $q_{ab} = e_{ai} e_{b}^i$, and $K_{ai} e_{b}^i = K_{ab}$ defines the extrinsic curvature (entering $P^{ab}$). Instead of $e_{ai}$, we will actually use the densitised co-tetrad $E^{a}_j$, which is conjugate to $K_a^i$. In order to erase the additional degrees of freedom, we have to introduce the Gauss law $K_{a[i} E^{b}_{j]} = 0$.

The key step in order to arrive at the connection variables underlying loop quantum gravity is now the following. From the tetrad, we construct the spin connection $\Gamma_a^i$, which annihilates the tetrad in a suitably defined covariant derivative. We can now construct the connection $A_{a}^i = \Gamma_a^i + \beta K_{a}^i$, which turns out to have canonical brackets with $E^{a}_i / \beta$. Here, $\beta$ is a priori a non-zero real number that we are free to choose. We can now proceed to the quantum theory, where the configuration space is the space of all holonomies obtained from $A_a^i$.

If we now consider the area operator, then its eigenvalues will scale with $\beta$. This follows since fluxes in the quantum theory are constructed from $E^{a}_i / \beta$, while the physical geometry is measured only by $E^{a}_i$. However, it is the (squared) fluxes rescaled with $\beta$ which act via multiplication with $\sqrt{j(j+1)}$ on spin network edges labelled by the spin $j$. Therefore, the real number $\beta$ (known as the Barbero-Immirzi parameter) already introduces an ambiguity.

Let’s come back now to the scalar field. Instead of just taking $\beta$ to be a real number, we could also take it to be a function of the scalar field. In order to ensure canonicity of the variables, some changes in the scalar field momentum $\pi$ are then necessary, however this introduces no problems. Then, the area operator is changed in such a way that it does not measure area any more, but $\beta(\phi) \cdot \text{area}$. An operator measuring area could then be recovered by inserting an appropriate operator for the scalar field.

While this last construction may seem a bit awkward, it can in fact be well motivated: The choice $\beta = \phi^2$ follows if one wants an observable algebra which is defined in relation to a constant mean curvature gauge fixing in the context of a conformally coupled scalar field, as discussed here in detail.

Let us now come back to the issue of a similar phenomenon in Yang-Mills theory. As we have seen above, the appearance of such ambiguities in LQG is related to the fact that the connection used there is assembled from an affine piece, the spin connection, and a tensorial piece, the extrinsic curvature. The tensorial piece can be modified by an arbitrary scalar function.

Without such a split however, a similar ambiguity does not occur. Consider for example the Gauss law $D_a E^a_i = 0$ of Yang-Mills theory. It acts as $A_a^i \rightarrow A_a^i - D_a \lambda^i$. If we would simply rescale $A_a^i$ by a constant, we would immediately find that the transformation properties of $A_a^i$ under the Gauss law have been spoiled, since the affine piece would now be rescaled. Instead, if we rescale the metric as above and then construct the affine piece of the connection from it, the constant would drop and the affine part of the connection would remain intact. In the case of a non-trivial scalar function, the spin connection would change, since it would then be the spin connection with respect to the rescaled metric. But then the new Gauss law would again be intact.

So can we make such splits in general? To the best of my knowledge not. In the case of SU(2) Yang-Mills theory in 3 dimensions, we could just repeat the construction outlined above. We could solve the Gauss law and introduce a metric and its conjugate momentum as SU(2) - invariant degrees of freedom. For this, we would need to demand that $E^{ai} E^{b}_i$ is a positive definite 3-metric, something that one naturally demands in GR, but no necessarily in Yang-Mills theory. Then however, the above construction can be repeated and a new connection, based on the rescaled “metric” and its momentum could be performed.

The more radical changes in the geometric operators which resulted from reducing the components of the metric via gauge fixing don’t seem to have an analogue in Yang-Mills theory, as there simply is nothing left to gauge fix once one would be at the “metric” level.

What about other gauge groups? It seems that similar constructions are not possible in general, at least when reducing to a metric and its conjugate. The only other working example in 3 dimensions that I know is to use the groups SO(4) or SO(1,3). Then however, the corresponding Yang-Mills electric fields $E^{a}_{IJ}$ with $I,J = 0,1,2,3$ have to be subject to an additional constraint, known as the simplicity constraint $E^{a}_{[IJ} E^{b}_{KL]} = 0$. This is natural in the context of general relativity, but again probably not in the case of Yang-Mills theory. In $D+1$ dimensions, it turns out that SO(D+1) and SO(1,D) work, again subject to the simplicity constraints.

To conclude, the reason for the ambiguity in the geometric operators in loop quantum gravity is the freedom in choosing a metric on which the connection variables are based. An example for a good reason to change this metric is the construction of observables in order to access the physical Hilbert space. In Yang-Mills theory on the other hand, one always starts with connection variables, and it is highly unnatural to temporarily pass to similar metric variables in order to construct a different connection.