Critical Overview of Loops and Foams

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.

Full text

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arXiv:1009.4475v1 [gr-qc] 22 Sep 2010
Critical Overview of Loops and Foams
Sergei Alexandrov and Philippe Roche
Laboratoire de Physique Th´ eorique & Astroparticules, CNRS UMR 5207,
Universit´ e Montpellier II, 34095 Montpellier Cedex 05, France
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.
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Contents
1Introduction3
2 Canonical approach
2.1SU(2) Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1Ashtekar–Barbero canonical formulation . . . . . . . . . . . . . . . . .
2.1.2Loop quantization: kinematical Hilbert space
2.1.3Geometric operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4Loop quantization: diffeomorphism and Hamiltonian constraints . . . .
2.2Lorentz covariant approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1Covariant canonical formulation . . . . . . . . . . . . . . . . . . . . . .
2.2.2Loop quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3Two quantizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 LQG in a covariant form . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 CLQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Immirzi parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Area spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3Black hole entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Diffeomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3Path integral approach
3.1 Spin foam models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 The strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Barrett–Crane model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3New vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1EPRL model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2FK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 The new models and LQG . . . . . . . . . . . . . . . . . . . . . . . . .
3.4Imposition of constraints revisited . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2A new look at simplicity and SF strategy . . . . . . . . . . . . . . . . .
3.4.3Secondary constraints and vertex amplitude . . . . . . . . . . . . . . .
3.4.4Closure constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5Comparison with the canonical approach . . . . . . . . . . . . . . . . . . . . .
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4 Roads to Quantum Gravity
4.1Canonical way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Path integral way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 GFT way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4Gravity as an effective theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1Introduction
Whereas string theory remains the most developed and active approach to quantum gravity,
during last years Loop Quantum Gravity (LQG) and Spin Foam (SF) models become more
and more popular. Following this growth of interest, there have appeared many reviews on
this subject [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Most of these reviews give a quite optimistic picture of
the developments in this domain, so that one could think that one has already at our disposal
a theory (or at least a model) of quantum gravity, which is derived in a rigorous way following
the standard well established quantization rules, internally self-consistent and able to produce
physical predictions. Besides, there have been several reviews comparing string theory and
LQG/SF approaches written mostly from the point of view of the latter [11, 12, 13]. On the
other hand, a critical overview of the recent advances almost does not exist in the literature. A
notable exception is the review [14] (see also its answer and criticism [15]) which concentrates
mostly on the Loop Quantum Gravity side.
The present review aims to fulfill this gap. It gives a picture of the present situation in
the domain of LQG and SF seen from the perspective which is based on the results obtained
by the authors during last years. Thus, it represents a personal viewpoint which might not
coincide with the viewpoint spread in the community. 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. Unfortunately, it is
extremely difficult to obtain exact results which may resolve the raised questions in one or
another way. Besides, even the quantization procedure which must be followed in the case
of gravity is not completely well determined and still subject to questioning. Therefore, our
analysis cannot be claimed to be a proof of any kind, but it is supported by various arguments
that we find compelling.
The central question to be answered below is whether one has a model at Planck scale
which is mathematically self-consistent, has a potential to reproduce general relativity in the
low-energy limit, and incorporates consistently its fundamental gauge symmetries. The last
condition is the main point of concern in this review. We accept the viewpoint that the
fundamental symmetries of general relativity, such as space-time diffeomorphism invariance
and local Lorentz invariance in the tangent space, appearing as we are working in the first
order formalism, must not be anomalous in the corresponding quantum theory.
Of course, as soon as the notion of manifold may disappear at the quantum level as it
seems to happen in LQG, or be replaced by a simplicial complex as in SF models, one should
precise what is meant by diffeomorphism invariance. For example, in the discrete setting this
issue is very non-trivial and has been discussed in [16, 17]. In this review we do not enter
the discussion of this extremely important problem, but subscribe to the commonly accepted
idea that the diffeomorphism symmetry in SF models is recovered as a result of summing
over all admissible triangulations. In general, we take the viewpoint that a symmetry remains
unbroken if the quantum theory properly implements the corresponding constraints of the
classical phase space. All our conclusions are based on this assumption and therefore should
be taken with care.
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. In particular, we use a Lorentz covariant approach [18], which allows to put
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LQG in a broader context and thereby to get insights on the fate of classical symmetries after
quantization.
In contrast, the SF approach is based on a covariant path integral. Nevertheless, con-
straints play an important role in this case too since the main idea of this approach is to get
quantum gravity by imposing certain constraints on the topological BF theory. During the
three previous years there was a great activity in this area. It was initiated by reconsider-
ing the old methods of imposing constraints and resulted in new techniques [19] and new SF
models [20, 21]. These models have been shown to possess some attractive features and in
particular it was claimed that they are consistent with LQG at least at the kinematical level.
Thus, it might look like one has a beautiful coherent picture where two different quantization
approaches lead to equivalent consistent quantum gravity models.
However, we suggest to revisit the constraint imposition once more, now for the new SF
models. For this purpose it is extremely useful to take into account the canonical structure of
the theory so that the preceding covariant analysis of LQG will be very helpful. Such approach
immediately reveals various fallacies of the new models and weak points in the interpretation
of their results.
Our main conclusions regarding the status of the two quantization approaches are the
following:
• 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.
• There is an alternative quantization following the same loop ideas, the so called Covari-
ant LQG (CLQG), which has a potential to resolve the drawbacks of LQG. However, it
is supplied with some serious technical obstacles (consisting mainly in finding a represen-
tation of the algebra of connections) preventing yet the realization of this quantization
program.
• The claim [21] that the recently introduced spin foam models [20, 21] have the same
boundary states as the kinematical states of LQG cannot be formulated as such because
they have completely different representations as functionals of connection.
• The new spin foam models in the presence of a finite Immirzi parameter represent quan-
tizations which do not respect the standard Dirac rules and we argue that they are
incompatible with a self-consistent canonical quantization. Moreover, any SF model
derived by the usual strategy “first quantize, then constrain” (see section 3.1.2), in-
cluding the models without the Immirzi parameter, does not implement consistently
all constraints of general relativity and therefore cannot properly describe its quantum
dynamics.
• A spin foam quantization consistent with the canonical one can be achieved by modifying
the association of geometric bi-vectors to generators of the gauge algebra and by relaxing
the closure constraint. The vertex amplitude should also be modified and in general is
given by the integral formula (3.73) with a non-trivial measure which however remains
still unknown.
Given these statements, we have to conclude that neither the canonical loop approach nor
its spin foam cousin were able to provide so far a model which can be claimed to be free
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from inconsistencies and anomalies. This does not mean however that the ideas behind these
approaches are not reasonable. They might well be relevant and even indispensable for a
theory of quantum gravity. For example, as argued in [22], there is a remarkable convergence
of various modern approaches to quantum gravity, which all lead to an effective spacetime
dimension 2 at the Planck scale. This feature is exhibited very explicitly in the loop and spin
foam quantizations. However, in our opinion, their present realization is not satisfactory and
requires a serious reconsideration. In fact, in this review we consider some of the modifications,
which have been already suggested, and point out their advantages and loopholes.
The review is not technical in the sense that the details of proofs and derivations, if they
are not subject of a critical discussion, are omitted. Instead, we concentrate mostly on the
ideas behind the loop and SF quantizations, their results, properties and interpretation. On
the other hand, the review does not address such important issues as the problem of time,
prediction power on the low energy limit, inclusion of matter, etc. Besides, we do not consider
some branches of LQG and SF such as, for example, Loop Quantum Cosmology (LQC) [23]
and evaluation of the graviton propagator [24]. Since these branches are based on results
and ideas of the two main approaches, they seem to have even less firm ground than those
approaches themselves. Therefore, for example, if LQG in its present form fails to provide a
consistent quantization of general relativity, it is highly unlikely that LQC can do better.
Let us briefly describe the content of the review. Chapter 2 is devoted to the loop approach
to quantum gravity which is a type of canonical quantization. In the first section of this chapter
we recall the basics of LQG following the standard presentation of this domain. Then in section
2.2 we briefly review the Lorentz Covariant approach, which allows to look at LQG from a
different angle and suggests an alternative loop quantization, as discussed in section 2.3. The
implications of these results are further analyzed in section 2.4, where we also discuss various
controversial issues and constructions of LQG. The closing section of this chapter presents a
summary of our main conclusions regarding the status of the loop quantization.
Chapter 3 deals with the SF quantization which can be seen as a discretized path integral
for general relativity. First, we give a brief introduction to general ideas of the SF approach and
present the strategy followed in most of the SF models in 4 dimensions. Then we introduce the
main SF models existing in the literature and realizing the strategy mentioned above: section
3.2 reviews the Barrett–Crane model and section 3.3 presents the new models. Since we want
to revise their derivation, we discuss its main steps in detail and critically analyze their relation
to LQG. Then in section 3.4 we reconsider the imposition of constraints in the new models
and observe a few sources of potential mistakes. The main issue, in our opinion, is that the
SF models quantize the symplectic structure not of general relativity, but of BF theory. This
results in various inconsistencies demonstrated on a simple example in subsection 3.4.1. This
does not however exhaust all problems which become apparent under the thorough analysis of
the constraint imposition in the following subsections. At the same time this analysis reveals
some connections to the structures appearing in the covariant approach to the canonical loop
quantization and suggests a way to cure the problems. So we finish this chapter by section
3.5 comparing the situations we arrived at in the spin foam and loop approaches to quantum
gravity.
Finally, in chapter 4 we speculate on possibilities to overcome the problems exposed in
this review. We consequently discuss the canonical (loop) approach, path integral (spin foam)
quantization and group field theory reformulation of SF models. The last section 4.4 is devoted
to a brief discussion of holography and the perspective on gravity as an emergent theory put
in the context of LQG and spin foams.
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