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**Obtaining general relativity in the appropriate limit**

**Obtaining general relativity in the appropriate limit**In order to understand the current status, let us remark that there are two limits which can be taken in order to obtain a spacetime of large scale in loop quantum gravity: large quantum numbers (spins), or many quantum numbers, i.e. very fine spin networks. Much is known about the limit of large spins, where the number of quanta is fixed. Here, one finds strong evidence that the theory reproduces general relativity on large scales both in the canonical and in the covariant approach. The resulting semiclassical picture however corresponds to Regge-gravity on a given lattice which is specified by the underlying graph on which the (coherent) quantum state labelled by the large spins lives. This limit is usually referred to as the “semiclassical” limit in the literature and should not be confused with the following:

On the other hand, one can leave the quantum numbers arbitrary, in particular maximally small, and only increase the number of quanta. This corresponds to a continuum limit and it should be a priori preferred over the large spin limit in my opinion. In particular, the large spin and continuum limit do not need to commute and may in principle lead to different physics, even on macroscopic scales. The problem with this approach is that we know only very little about the dynamics in this sector of the theory, apart from several concrete proposals for its implementation. The dynamics on large scales is then expected to emerge via a coarse graining procedure. More discussion on this point of view can be found here. See also this brand new paper for a dynamical emergence of such a sector consistent with FRW cosmology.

As an additional subtlety, we can consider arbitrary superpositions of spin networks, in other words we can have quantum superpositions of “lattices”. The impact on the dynamics of this feature is so far unclear and may strongly depend on the regularisation details of the Hamiltonian, e.g. whether it is graph preserving or not, and thus superselecting.

Concerning a limit to obtain quantum field theory on curved spacetimes, we point out these pioneering works.

To conclude, the situation of whether general relativity emerges in the continuum limit is so far unclear, whereas there is strong evidence for a Regge-truncation thereof emerging in the large spin limit. Whether one is satisfied with one or the other limit, or a combination of both, also depends on the following problem.

To conclude, the situation of whether general relativity emerges in the continuum limit is so far unclear, whereas there is strong evidence for a Regge-truncation thereof emerging in the large spin limit. Whether one is satisfied with one or the other limit, or a combination of both, also depends on the following problem.

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**Local Lorentz invariance**

**Local Lorentz invariance**It is sometimes suggested that loop quantum gravity is not locally Lorentz invariant in the sense that modified dispersion relations might arise which could be in conflict with observation. Unfortunately, our current understanding of loop quantum gravity does not allow us to answer whether there are Lorentz violations, and how severe they might be. In order to make a meaningful statement, one would essentially have to identify a quantum state corresponding to Minkowski space, which should be thought of as a (possibly infinite) superposition of lattices, put matter fields thereon, and track their dynamics, including back-reaction, on a coarse grained scale where geometry can already be considered smooth. This is a formidable task, and currently out of reach not because of lacking proposals for how to define the problem, but mainly due to computational complexity.

In order to judge certain statements that one might find in the literature or on the internet, one should keep the following in mind to avoid confusion:

**Discrete eigenvalues of geometric operators don't imply Lorentz violations**

**One might naively think that discrete eigenvalues of geometric operators violate special relativity: if an observer at rest measures a certain discrete eigenvalue of, say, an area, what does another observer measure who is not at rest? The short answer is that he might in principle measure any value for the area, as long as it is in the (discrete) spectrum of the area operator. However, the expectation value can still transform properly according to special relativity. A well-known example of this is the theory of angular momentum: while the eigenvalues of one of the components of the angular momentum are always (half)-integers, expectation values transform properly according to the continuous rotation symmetry. This point has been made for example here, with further discussion here. Similar conclusions are also drawn in other contexts.**

**Internal gauge groups do not determine isometries of the spacetime**

**Different formulations of loop quantum gravity, canonically or covariant, use different**

*internal*gauge groups. The analogue of these groups in QCD is thus SU(3), and not the Lorentz group, which is a global symmetry of Minkowski space.

While the covariant path integral formulations of the Lorentzian theory use either SL(2, C) or SU(2) in a gauge-fixed version, the Lorentzian canonical theory in 3+1 dimensions can be formulated using either SU(2) [45], SO(1,3) (see here, here, here, and here), or SO(4). This is because one is coding the spatial metric and its momentum in a connection, whereas the signature of spacetime in the canonical formalism is determined by the Hamiltonian constraint, more precisely a relative sign between two terms. In fact, the structure of spacetime, coded in the hypersurface deformation algebra, is already set at the level of metric variables, and completely insensitive of the additional gauge redundancy that one introduces by passing to connection variables. Also, it does not matter for this whether the connection that one uses can be interpreted as the pullback of some manifestly covariant spacetime connection. While it is a possibility that only a certain choice of variables or internal gauge group leads to a consistent quantum theory in agreement with current bounds on Lorentz violations, such a conclusion cannot be drawn given our current understanding of the theory.

**At which order do we expect possible Lorentz violations?**

In order to parametrise the violation of Lorentz invariance in a model-independent way, one usually constrains the free parameters $c_n$ in a modified dispersion relation

$$ E^2 = m^2 + p^2 + \sum_{n\geq 3} c_n \frac{p^n}{E_{\text{pl}}^n} $$

where $E_{\text{pl}} = \sqrt{\hbar c / G}$ is the Planck mass. It is worthwhile to formulate a general expectation at what order one expects the first quantum corrections to appear. The two reasonable alternatives here are n = 3 or n = 4, with the difference being a first quantum correction in $\sqrt{\hbar}$ or in $\hbar$. While an appropriate calculation as outlined above will finally have to decide this question, we can make two observations. First, in the case of loop quantum cosmology, where we know how to compute the quantum corrections to the classical theory, we obtain a correction of the order $\hbar$, i.e. the effective Friedmann equation

$$ H^2 \sim \rho \left(1-\frac{\rho}{\rho_{\text{crit}}} \right), ~~~ \rho_{\text{crit}} \sim \frac{1}{G^2 \hbar} $$

Second, the concept of area has a much more fundamental status in loop quantum gravity than that of length, and one would thus expect quantum corrections as an expansion in $l_{\text{pl}}^2 = \hbar G / c^3$, rather than $l_{\text{pl}}$. Both of these observations hint that one would expect that a first non-vanishing correction in the dispersion relation should appear at order n=4.

Experimentally, it turns out that there are very strong constraints on the n = 3 terms, whereas constraints on n = 4 are much weaker, yet still restrictive, and currently still plagued by some astrophysical uncertainties. Thus, even if one expects Lorentz- violating effects to arise from loop quantum gravity at order n = 4, this is not necessarily in conflict with current experiments. Still, the stringent bounds put by experiment might rule out loop quantum gravity once we will be able to properly derive Lorentz violations from it.

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**Testability and ambiguities**

**Testability and ambiguities**A general problem for theories of quantum gravity is to come up with testable predictions which are independent of free parameters in the theory that can be tuned in order to hide any observable effect below the measurement uncertainty. Furthermore, one would like to have a uniquely defined fundamental theory whose dynamics depend only on a finite number of free parameters.

In loop quantum cosmology, much progress towards predictions for observable effects have been made, however the choice of parameters in the models, such as the e-foldings during inflation, can so far hide any observable effects. Also, there are different approaches to the dynamics of cosmological perturbations.

Within full loop quantum gravity (and also loop quantum cosmology), the Barbero-Immirzi parameter $\beta$, a free real parameter entering, e.g., the spectra of geometric operators, constitutes a famous ambiguity. It enters the classical theory in a canonical transformation which is not implementable as a unitary transformation or even an algebra automorphism at the quantum level and thus constitutes a quantisation ambiguity. Since this parameter can enter physical observables, one would like to fix it by an experiment or derive its (only consistent) value by theoretical means. An example for how to fix it by theoretical means is to consider black hole entropy computations and match them to the expected Bekenstein-Hawking entropy. Within the original approach to black hole entropy from loop quantum gravity, this gives a certain value for $\beta$, however this computation neglects a possible running of the gravitational constant, i.e. it identifies the high and low energy Newton constants. More recently, it has been observed that the Bekenstein-Hawking entropy can also be reproduced by an analytic continuation of $\beta$ to the complex self-dual values $\pm i$, which interestingly also agrees with a computation of the effective action. In fact, the classical theory is easiest when expressed in self-dual variables, so that it might turn out that the value $\beta = \pm i$ could be favoured also in the quantum theory. The current problem is however that the quantum theory is ill-defined for complex $\beta$ and the above mentioned results were obtained via analytic continuation from real beta.

In addition to quantisation ambiguities resulting from the choice of variables, as above (see also this and this post), there are quantisation ambiguities in the regularisation of the Hamiltonian constraint, and thus the dynamics of the theory. These go somewhat beyond factor ordering, as the techniques used in regularising the Hamiltonian also involve classical Poisson bracket identities which are used to construct otherwise ill-defined operators. The requirement of anomaly freedom of the quantum constraint algebra already removed many of those ambiguities in Thiemann’s original construction. However, the precise notion of anomaly freedom has been criticised on the ground that it corresponds to a certain “on-shell” notion, which is however the one that makes sense in the context of Thiemann's construction. More recent work based on a slightly changed quantisation seems to make significant progress towards the goal of implementing a satisfactory “off-shell” version of the quantum constraint algebra, at least in simplified toy models. The possible regularisations then seem to be very constrained, although no unique prescription has emerged so far.

To conclude, it is so far not possible to extract definite predictions from full loop quantum gravity due to the existence of ambiguities in its construction, whose influence on the dynamics is not well understood. However, there is promising recent work on the removal of such ambiguities, which could eventually allow for clear predictions which would allow to falsify the theory.

In order to better understand the theory called LQG, it is necessary to make the distinction between the spin-foam (covariant) and the canonical formulation. Although many people beleive that these two are the same theory, they are really two different theories. In the canonical LQG

ReplyDeletethe spacetime is topologically Sx[a,b] and there are colored graphs (i.e. spin networks) embedded in a smooth 3-manifold S. A physical state is a linear combination of the spin-network states such that it satisfies the hamiltonian constraint. The classical limit in this case is

defined by veryfying whether (hbar)Im(log Psi(A)) satisfies the Hamilton-Jacobi equation for the canonical GR in the limit hbar --> 0 and A is the Ashtekar-Barbero connection on S. This is a very difficult task, so that one uses the spin-foam formulation. In the spin-foam formulation, the theory is formulated on a triangulation of Sx[a,b], which is not the same as the smooth manifold Sx[a,b]. Hence obtaining the classical limit requires performing 2 limits, as you point out, i.e. the limit hbar --> 0, or equivalently j(triangle) --> infinity, and the smooth limit, i.e. N --> infinity, where N is the number of 4-simplices in T(Sx[a,b]). The easiest way to study these two limits is to use the effective action formalism. One then obtains that the first limit gives the area-Regge action. This is a problem, because area-Regge theory does not define always a metric geometry, so it is not equivalent to the usual Regge discretization of GR. Hence one needs to modify the usual spin-foam formulation in order to implement the constraints which reduce the area-Regge theory to the length-Regge theory.

As far as the Lorentz invariance is concerned, it is not problematic for the spin-foam formulation, because the unitary irreps of the Lorentz group are (j,p) where j is an SU(2) spin and p is a real number. Also one can assume that the area of a triangle in a spacetime triangulation is determined by the flat metric in a 4-simplex which contains that triangle. The Lorentz invariance is more problematic in the canonical case, because it is not manifest.

Thank you for pointing out these differences between canonical LQG and the spinfoam formulation. I didn’t include them in my post, because I thought that they would distract too much from the core issues.

DeleteAbout Lorentz invariance: I am not sure if we speak about the same thing. As emphasised in the post, the notion of “Lorentz invariance” appears in LQG both when choosing the internal gauge group (which can be the Lorentz group), and in the context of the question how the quantum geometry described by your physical state behaves (e.g. how matter fields propagate on it). While there are spinfoam models which are “Lorentz invariant” w.r.t. the the former notion, this is not clear for the second one, and thus for what people usually mean by Lorentz invariance.

In canonical LQG the local Lorentz invariance can be established by constructing the generators of local Lorentz rotations from holonomy and flux operators (if there is fermionic matter one has to add the matter contributions to the generators) and then verify that the algebra closes. This is a difficult task and anomalies may appear. In the spin-foam case, the analog of the local Lorentz transformations would be the Lorentz transformations that preserve the flat metric in each 4-simplex (provided that such a metric exists in every 4-simplex).

ReplyDeleteThank you for the clarification. It still seems to me that the notion of local Lorentz invariance that you refer to is not what one means when one asks whether LQG is locally Lorentz invariant from a phenomenological point of view. There, the question of how matter propagates, morally from 4-simplex to 4-simplex, is the relevant one.

ReplyDeleteMatter propagates in a spin foam according to the discretized form of the corresponding matter-field action. In the case of fermions there is a problem to formulate such an action because the fermions couple to the tetrads, and the tetrads are not well-defined in a spin-foam geometry. That was the reason to propose a spin-cube formulation, which is a generalization of the spin-foam formulation, based on the substitution of the spin-foam group by a 2-group (in the QG case the Lorentz group is replaced by the Poincare 2-group). The consequence is that the tetrads enter the formalism explicitely, through the edge lengths, so that a spin foam (a colored 2-complex) is replaced by a spin cube, which is a colored 3-complex. Hence in a spin cube the edges carry the 2-group irreps, the triangles carry the corresponding 1-intertwiners and the tetrahedrons carry the corresponding 2-intertwiners as oposed to a spin foam where the triangles carry the group irreps and the tetrahedrons carry the corresponding intertwiners.

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