I would like to continue discussing canonical LQG based on chapter 2 from

http://arxiv.org/abs/1009.4475

**Critical Overview of Loops and Foams**

Authors: Sergei Alexandrov, Philippe Roche

(Submitted on 22 Sep 2010)

*Abstract: This is a review of the present status of loop and spin foam approaches to quantization of four-dimensional general relativity. It aims at raising various issues which seem to challenge some of the methods and the results often taken as granted in these domains. A particular emphasis is given to the issue of diffeomorphism and local Lorentz symmetries at the quantum level and to the discussion of new spin foam models. We also describe modifications of these two approaches which may overcome their problems and speculate on other promising research directions.*

As a summary, Alexandrov and Roche present (and discuss in detail) indications that canonical LQG in its present form suffers from quantization ambiguities (ordering ambiguities, unitarily in-equivalent quantizations, Immirzi-parameter), insufficient treatment of secondary second class constraints leading to anomalies, missing off-shell closure of constraint algebra, missing consistent definition of the Hamiltonian constraint and possibly lack of full 4-dim diffeomorphism invariance. They question some of the major achievements like black hole entropy, discrete area spectrum, uniquely defined kinematical Hilbert space (spin networks).

In chapter 3 Alexandrov and Roche discuss spin foam models which may suffer from related issues showing up in different form but being traced back to a common origin (secondary second class constraints, missing Dirac’s quantization scheme, …). I would like to discuss this chapter 3 in a separate thread. For the rest of this post I will try to present the most important sections from chapter 2, i.e. canonical LQG.

*Although we have raised the points already known by the experts in the field, they are rarely spelled out explicitly. At the same time, their understanding is crucial for the viability of these theories. ...*

*Thus, we will pay a particular attention to the imposition of constraints in both loop and SF quantizations. Since LQG is supposed to be a canonical quantization of general relativity, in principle, it should be straightforward to verify the constraint algebra at the quantum level. However, in practice, due to peculiarities of the loop quantization this cannot be achieved at the present state of knowledge. Therefore, we use indirect results to make conclusions about this issue. ...*

*Although LQG can perfectly incorporate the full local Lorentz symmetry, we find some evidences that LQG might have problems to maintaining space-time diffeomorphism symmetry at the quantum level. Thus, we argue that it is an anomalous quantization of general relativity which is not physically acceptable. ...*

*Since the action (2.1) possesses several gauge symmetries, 4 diffeomorphism symmetries and 6 local Lorentz invariances in the tangent space, there will be 10 corresponding first class constraints in the canonical formulation. Unfortunately, as will become clear in section 2.2, there are additional constraints of second class which cannot be solved explicitly in a Lorentz covariant way. To avoid these complications, one usually follows an alternative strategy. It involves the following three steps:*

... the boost part of the local Lorentz gauge symmetry is fixed from the very beginning by

choosing the so called time gauge, which imposes a certain condition on the tetrad field

...the three first class constraints generating the boosts are solved explicitly w.r.t. space components of the spin-connection; ...

... the boost part of the local Lorentz gauge symmetry is fixed from the very beginning by

choosing the so called time gauge, which imposes a certain condition on the tetrad field

...the three first class constraints generating the boosts are solved explicitly w.r.t. space components of the spin-connection; ...

*An important observation is that the spectrum (2.14) is proportional to the Immirzi parameter. This proportionality arises due to the difference between ... having canonical commutation relations with the connection. It signifies that this parameter, which did not play any role in classical physics, becomes a new fundamental physical constant in quantum theory. ...*

*A usual explanation is that it is similar to the theta-angle in QCD [35]. However, in contrast to the situation in QCD, the formalism of LQG does not even exist for the most natural value corresponding to the usual Hilbert–Palatini action. Moreover, the Immirzi parameter enters the spectra of geometric operators in LQG as an overall scale, which is a quite strange effect. Even stranger is that the canonical transformation (2.6) turns out to be implemented non-unitarily, so that the area operator is sensitive to the choice of canonical variables. To our knowledge, there is no example of such a phenomenon in quantum mechanics. ...*

*Below we will argue that the dependence on the Immirzi parameter is due to a quantum anomaly in the diffeomorphism symmetry, which in turn is related to a particular choice of the connection used to define quantum holonomy operators (2.7). ...*

Alexandrov and Roche do not address all weak points related to Thiemann's construction of the Hamiltonian but refer to

14] H. Nicolai, K. Peeters, and M. Zamaklar, “Loop quantum gravity: An outside view,” Class. Quant. Grav. 22 (2005) R193, arXiv:hep-th/0501114.

[15] T. Thiemann, “Loop quantum gravity: An inside view,” Lect. Notes Phys. 721 (2007) 185–263, arXiv:hep-th/0608210.

In the following Alexandrov and Roche present alternative quantization approaches developped by Alexandrov et al.; this does not really solve these issues but it clarifies the main problems, i.e. quantization ambiguities, inequivalent quantization schemes, secondary second class constraints missed by using the Poisson structure instaed of the Durac brackets.

*The reduction of the gauge group originates from the first two steps in the procedure leading to the AB canonical formulation on page 6. Therefore, it is natural to construct a canonical formulation and to quantize it avoiding any partial gauge fixing and keeping all constraints generating Lorentz transformations in the game. The third step in that list (solution of the second class constraints) can still be done and the corresponding canonical formulation can be found in [51]. However, this necessarily breaks the Lorentz covariance. On the other hand, it is natural to keep it since the covariance usually facilitates analysis both at classical and quantum level. Thus, we are interested in a canonical formulation of general relativity with the Immirzi parameter, which preserves the full Lorentz gauge symmetry and treats it in a covariant way. ...*

*The presence of the second class constraints is the main complication of the covariant canonical formulation. They change the symplectic structure on the phase space which must be determined by the corresponding Dirac bracket. ...*

*They identify a two-parameter family of inequivalent (!) connections; their main results are*

...First of all, the connection lives now in the Lorentz Lie algebra so that its holonomy operators belong to a non-compact group. This is a striking distinction from LQG where the compactness of the structure group SU(2) is crucial for the discreteness of geometric operators and the validity of the whole construction.

...The symplectic structure is not anymore provided by the canonical commutation relations of the type (2.3) but is given by the Dirac brackets.

...In addition to the first class constraints generating gauge symmetries, the phase space to be quantized carries second class constraints. Although they are already taken into account in the symplectic structure by means of the Dirac brackets, they lead to a degeneracy in the Hilbert space constructed ignoring their presence ...

...First of all, the connection lives now in the Lorentz Lie algebra so that its holonomy operators belong to a non-compact group. This is a striking distinction from LQG where the compactness of the structure group SU(2) is crucial for the discreteness of geometric operators and the validity of the whole construction.

...The symplectic structure is not anymore provided by the canonical commutation relations of the type (2.3) but is given by the Dirac brackets.

...In addition to the first class constraints generating gauge symmetries, the phase space to be quantized carries second class constraints. Although they are already taken into account in the symplectic structure by means of the Dirac brackets, they lead to a degeneracy in the Hilbert space constructed ignoring their presence ...

(Remarkably some of these issues identified by Alexandrov and Roche in the generalized canonical approach will show up in the "new SF" models which have become popular over the last years)

*Thus, we see that the naive generalization of the SU(2) spin networks to their Lorentz analogues is not the correct way to proceed. A more elaborated structure is required. The origin of this novelty can be traced back to the presence of the second class constraints which modified the symplectic structure and invoked a connection different from the usual spinconnection. ...*

*The spectrum (2.40) depends explicitly on the parameters a, b entering the definition of the connection. This implies that the quantizations based on different connections of the two-parameter family are all inequivalent. ...*

*Finally, we notice that the projected spin networks are obtained by quantizing the phase space of the covariant canonical formulation ignoring the second class constraints. Therefore they form what can be called enlarged Hilbert space and as we mentioned above this space contains many states which are physically indistinguishable. To remove this degeneracy one has to somehow implement the second class constraints at the level of the Hilbert space. ...*

(The authors claim that this is what is missed in all SF models)

Alexandrov and Roche present two special choices for the connection (on physical grounds) in order to demonstrate how one could proceed

1) LQG in a covariant form from which the standard time-gauge LQG framework can be recovered

2) CLQG which leads to physically different results!

*Although the commutativity of the connection is a nice property, there is another possibility of imposing an additional condition to resolve the quantization ambiguity, which has a clear physical origin. Notice that the Lorentz transformations and spatial diffeomorphisms, which appear in the list of conditions (2.34), do not exhaust all gauge transformations. What is missing is the requirement of correct transformations under time diffeomorphisms generated by the full Hamiltonian. Only the quantity transforming as the spin-connection under all local symmetries of the theory can be considered as a true spacetime connection. ...*

*In particular, it involves a Casimir of the Lorentz group and hence this spectrum is continuous. But the most striking and wonderful result is that the spectrum does not depend on the Immirzi parameter! Moreover, one can show that this parameter drops out completely from the symplectic structure ...*

*Thus, it [the Immirzi parameter] remains unphysical as it was in the classical theory, at least at this kinematical level. ...*

*From our point of view, this is a very important fact which indicates that LQG may have troubles with the diffeomorphism invariance at the quantum level. ...*

*But as we saw in the previous subsection, there is an alternative choice of connection, suitable for the loop quantization, which respects all gauge symmetries. Besides, the latter approach, which we called CLQG, leads to results which seem to us much more natural. For example, since it predicts the area spectrum independent on the Immirzi parameter, there is nothing special to be explained and there is no need to introduce an additional fundamental constant. Moreover, the spectrum appears to be continuous which is very natural given the non-compactness of the Lorentz group and results from 2+1 dimensions (see below). Although these last results should be taken with great care as they are purely kinematical and obtained ignoring the connection non-commutativity, in our opinion, the comparison of the two possibilities to resolve the quantization ambiguity points in favor of the second choice. ...*

The authors have serious doubts that the discrete area spectrum should be taken as a physical result:

*In fact, there are two other more general issues which show that the LQG area spectrum is far from being engraved into marble. First, the area operator is a quantization of the classical area function and, as any quantization, is supplied with ordering ambiguities ...*

*Second, the computation of the area spectrum has been done only at the kinematical level. The problem is that the area operator is not a Dirac observable. It is only gauge invariant, whereas it is not invariant under spatial diffeomorphisms and does not commute with the Hamiltonian constraint. This fact raises questions and suspicions about the physical relevance of its spectrum and in particular about the meaning of its discreteness, even among experts in the field [77, 78]. ...*

In the following Alexandrov and Roche show that there are to different interpretations on quantization of gravity, namely the Dirac scheme and the relational interpretation

*The difference between the two interpretations and the importance of this issue has been clarified in [77, 81]. Namely, the authors of [77] proposed several examples of low dimensional quantum mechanical constrained systems where the spectrum of the physical observable associated to a partial observable is drastically changed. This is in contradiction with the expectation of LQG that the spectrum should not change. Then in [81] it was argued that one should not stick to the Dirac quantization scheme but to the relational scheme. Accepting this viewpoint allows to keep the kinematical spectra unchanged. Thus, the choice of interpretation for physical observables directly affects predictions of quantum theory and clearly deserves a precise scrutiny. ...*

*Whereas the relational viewpoint seems to be viable, the work [77] shows that if we adhere only to the first interpretation, which is the most commonly accepted one, then it is of upmost importance to study the spectrum of complete observables. Unfortunately, up to now there are no results on the computation of the spectrum of any complete Dirac observable in full LQG. ...*

*In our opinion, all these findings and the above mentioned issues clearly make the discreteness found in LQG untrustable and suggest that the CLQG spectrum (2.52) is a reasonable alternative. ...*

I don't think that this means that LQG is wrong and CLQG is right. But it definitly means that even canonical LQG is far from being unique!

*However, since the spectrum (2.52) is independent of the Immirzi parameter, the challenge now is to find such counting which gives the exact coefficient 1/4, and not just the proportionality to the horizon area. In fact, the last point is the weakest place of the LQG derivation comparing to all other derivations existing in the literature. ...*

*Besides, there are two other points which make the LQG derivation suspicious. First, it is not generalizable to any other dimension. If one draws direct analogy with the 4-dimensional case, one finds a picture which is meaningless in 3 dimensions and does not allow to formulate any suitable boundary condition in higher dimensions. ...*

(This aspect is discussed in a series of papers by Thiemann et al., but I have to admit that I did not check for results relevant in this context)

Last but not least Alexandrov and Roche focus on diffeomorphism invariance, non-separability of LQG Hilbert space and non-uniqueness of the Hamiltonian constraint.

*In fact, there still remain some continuous moduli depending on the relative angles of edges meeting at a vertex of sufficiently high valence [96]. Due to this the Hilbert space HGDiff is not separable and if one does not want that the physics of quantum gravity is affected by these moduli, one is led to modify this picture. To remove this moduli dependence, one can extend Diff(M) to a subgroup of homeomorphisms of M consisting of homeomorphisms which are smooth except at a finite number of points [97] (the so called “generalized diffeomorphisms”). If these points coincide with the vertices of the spin networks, the supposed invariance under this huge group will identify spin networks with different moduli and solve the problem. However, this procedure has different drawbacks. First, the generalized diffeomorphisms are not symmetries of classical general relativity. Moreover, they transform covariantly the volume operator of Rovelli–Smolin but not the one of Ashtekar–Lewandowski which is favored by the triad test [39]. This analysis indicates that these generalized diffeomorphisms should not be implemented as symmetries at quantum level and, as a result, we remain with the unsolved problem of continuous moduli. ...*

*In the Dirac formalism the constraints Hi only generate diffeomorphisms which are connected to the identity. Therefore, there is a priori no need for defining HGDiff to be invariant under large diffeomorphisms. On the other hand, in LQG these transformations, forming the mapping class group, are supposed to act trivially. This is justified in [2] (section I.3.3.2) to be the most practical option given that the mapping class group is huge and not very well understood. ...*

*Thus, the simplest option taken by LQG might be an oversimplification missing important features of the right quantization. ...*

*That means that LQG may simply miss the full diffeomorphism symmetry, especially all structures related to large diffeomorphisms may be lost. ... Moreover, a huge arbitrariness is hidden in the step suggesting to replace the classical Poisson brackets, as for example (2.22), by quantum commutators. In general, this is true only up to corrections in h and on general ground one could expect that the Hamiltonian constructed by Thiemann may be modified by such corrections. This is a bit disappointing situation for a would be fundamental quantum gravity theory. In principle, all this arbitrariness should be fixed by the requirement that the quantum constraints reproduce the closed Dirac constraint algebra. However, the commutators of quantum constraint operators are not under control ...*

Here's the summary of chapter 2

*Trying to incorporate the full Lorentz gauge symmetry into the standard LQG framework based on the SU(2) group, we discovered that*

**LQG is only one possible quantization of a twoparameter family of inequivalent quantizations**. All these quantizations differ by the choice of connection to be used in the definition of holonomy operators — the basic building blocks of the loop approach. LQG is indeed distinguished by the fact that the corresponding connection is commutative. Nevertheless, a more physically/geometrically motivated requirement selects another connection, which gives rise to the quantization called CLQG. Although the latter quantization has not been properly formulated yet, it predicts the area spectrum which is continuous and independent on the Immirzi parameter, whereas LQG gives a discrete spectrum dependent on the IP. We argued that**these facts lead to suspect that LQG might be an anomalous quantization of general relativity**:**in our opinion they indicate that it does not respect the 4d diffeomorphism algebra at quantum level**. If this conclusion turns out indeed to be true, LQG cannot be physically accepted. At the same time, CLQG is potentially free from these problems. But due to serious complications, it is far from being accomplished and therefore the status of the results obtained so far, such as the area spectrum, is not clear.**We also pointed out that some of the main LQG results are incompatible either with other approaches to the same problem or with attempts to generalize them to other dimensions**.**We consider these facts as supporting the above conclusion that LQG is not, in its present state, a proper quantization of general relativity.**So much for today!