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Loop and Spin Foam Quantum Gravity: a Brief Guide for Beginners

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Loop and Spin Foam Quantum Gravity: a Brief Guide for Beginners AEI-2006-004 hep-th/0601129 January 18th, 2006 Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners Hermann Nicolai and Kasper Peeters Max-Planck-Institut f¨ur Gravitationsphysik Albert-Einstein-Institut Am M¨uhlenberg 1 14476 Golm, GERMANY hermann.nicolai, [email protected] Abstract: We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists. arXiv:hep-th/0601129 v2 16 Feb 2006 Contributed article to “An assessment of current paradigms in the physics of fundamental interactions”, ed. I. Stamatescu, Springer Verlag. 1. Quantum Einstein gravity ics approach [7], addresses several aspects of the problem that are currently outside the main focus The assumption that Einstein’s classical theory of of string theory, in particular the question of back- gravity can be quantised non-perturbatively is at the ground independence and the quantisation of geom- root of a wide variety of approaches to quantum etry. Whereas there is a rather direct link between gravity. The assumption constitutes the basis of sev- (perturbative) string theory and classical space-time eral discrete methods [1], such as dynamical tri- concepts, and string theory can therefore rely on fa- angulations and Regge calculus, but it also implic- miliar notions and concepts, such as the notion of a itly underlies the older Euclidean path integral ap- particle and the S-matrix, the task is harder for LQG, proach [2, 3] and the somewhat more indirect ar- as it must face up right away to the question of what guments which suggest that there may exist a non- an observable quantity is in the absence of a proper trivial fixed point of the renormalisation group [4–6]. semi-classical space-time with fixed asymptotics. Finally, it is the key assumption which underlies loop The present text, which is based in part on the and spin foam quantum gravity. Although the as- companion review [8], is intended as a brief intro- sumption is certainly far-reaching, there is to date ductory and critical survey of loop and spin foam no proof that Einstein gravity cannot be quantised quantum gravity 2, with special attention to some non-perturbatively, either along the lines of one of of the questions that are frequently asked by non- the programs listed above or perhaps in an entirely experts, but not always adequately emphasised (for different way. our taste, at least) in the pertinent literature. For In contrast to string theory, which posits that the the canonical formulation of LQG, these concern Einstein-Hilbert action is only an effective low en- in particular the definition and implementation of ergy approximation to some other, more fundamen- the Hamiltonian (scalar) constraint and its lack of tal, underlying theory, loop and spin foam gravity uniqueness. An important question (which we will take Einstein’s theory in four spacetime dimensions not even touch on here) concerns the consistent in- as the basic starting point, either with the conven- corporation of matter couplings, and especially the tional or with a (constrained) ‘BF-type’ formulation.1 question as to whether the consistent quantisation These approaches are background independent in of gravity imposes any kind of restrictions on them. the sense that they do not presuppose the existence Establishing the existence of a semi-classical limit, of a given background metric. In comparison to the in which classical space-time and the Einstein field older geometrodynamics approach (which is also for- equations are supposed to emerge, is widely re- mally background independent) they make use of garded as the main open problem of the LQG ap- many new conceptual and technical ingredients. A proach. This is also a prerequisite for understanding key role is played by the reformulation of gravity in the ultimate fate of the non-renormalisable UV di- terms of connections and holonomies. A related fea- vergences that arise in the conventional perturbative ture is the use of spin networks in three (for canon- treatment. Finally, in any canonical approach there ical formulations) and four (for spin foams) dimen- is the question whether one has succeeded in achiev- sions. These, in turn, require other mathematical in- ing (a quantum version of) full space-time covari- gredients, such as non-separable (‘polymer’) Hilbert ance, rather than merely covariance under diffeo- spaces and representations of operators which are morphisms of the three-dimensional slices. In [8] we not weakly continuous. Undoubtedly, novel concepts have argued (against a widely held view in the LQG and ingredients such as these will be necessary in community) that for this, it is not enough to check order to circumvent the problems of perturbatively the closure of two Hamiltonian constraints on diffeo- quantised gravity (that novel ingredients are neces- morphism invariant states, but that it is rather the off- sary is, in any case, not just the point of view of LQG shell closure of the constraint algebra that should be but also of most other approaches to quantum grav- made the crucial requirement in establishing quan- ity). Nevertheless, it is important not to lose track of tum space-time covariance. the physical questions that one is trying to answer. Many of these questions have counterparts in the Evidently, in view of our continuing ignorance spin foam approach, which can be viewed as a ‘space- about the ‘true theory’ of quantum gravity, the time covariant version’ of LQG, and at the same time best strategy is surely to explore all possible av- as a modern variant of earlier attempts to define a enues. LQG, just like the older geometrodynam- discretised path integral in quantum gravity. For in- stance, the existence of a semi-classical limit is re- 1In the remainder, we will often follow established (though perhaps misleading) custom and summarily refer to this frame- 2Whereas [8] is focused on the ‘orthodox’ approach to loop work of ideas simply as “Loop Quantum Gravity”, or LQG for short. quantum gravity, to wit the Hamiltonian framework. 2 lated to the question whether the Einstein-Hilbert ac- in [16, 17]4. Readers are also invited to have a look tion can be shown to emerge in the infrared (long at [18] for an update on the very latest developments distance) limit, as is the case in (2+1) gravity in the in the subject. Ponzano-Regge formulation, cf. eq. (38). Regarding the non-renormalisable UV divergences of perturba- 2. The kinematical Hilbert space of LQG tive quantum gravity, many spin foam practitioners seem to hold the view that there is no need to worry There is a general expectation (not only in the LQG about short distance singularities and the like be- community) that at the very shortest distances, the cause the divergences are simply ‘not there’ in spin smooth geometry of Einstein’s theory will be re- foam models, due to the existence of an intrinsic cut- placed by some quantum space or spacetime, and off at the Planck scale. However, the same statement hence the continuum will be replaced by some ‘dis- applies to any regulated quantum field theory (such cretuum’. Canonical LQG does not do away with con- as lattice gauge theory) before the regulator is re- ventional spacetime concepts entirely, in that it still moved, and on the basis of this more traditional un- relies on a spatial continuum Σ as its ‘substrate’, on derstanding, one would therefore expect the ‘correct’ which holonomies and spin networks live (or ‘float’) theory to require some kind of refinement (contin- — of course, with the idea of eventually ‘forgetting 3 uum) limit , or a sum ‘over all spin foams’ (corre- about it’ by considering abstract spin networks and sponding to the ‘sum over all metrics’ in a formal only the combinatorial relations between them. On path integral). If one accepts this point of view, a this substrate, it takes as the classical phase space key question is whether it is possible to obtain re- variables the holonomies of the Ashtekar connection, sults which do not depend on the specific way in which the discretisation and the continuum limit are a m performed (this is also a main question in other dis- he[A]= exp Amτadx , P e crete approaches which work with reparametrisation Z invariant quantities, such as in Regge calculus). On with Aa := 1 ǫabcω + γKa . (1) m − 2 mbc m the other hand, the very need to take such a limit is often called into question by LQG proponents, who Here, τa are the standard generators of SU(2) (Pauli claim that the discrete (regulated) model correctly matrices), but one can also replace the basic repre- describes physics at the Planck scale. However, it is sentation by a representation of arbitrary spin, de- then difficult to see (and, for gravity in (3+1) di- noted by ρj (he[A]). The Ashtekar connection A is mensions has not been demonstrated all the way in thus a particular linear combination of the spin con- a a single example) how a classical theory with all the nection ωmbc and the extrinsic curvature Km which requisite properties, and in particular full space-time appear in a standard (3+1) decomposition. The covariance, can emerge at large distances. Further- parameter γ is the so-called Barbero-Immirzi pa- more, without considering such limits, and in the ab- rameter. The variable conjugate to the Ashtekar sence of some other unifying principle, one may well connection turns out to be the inverse densitised ˜ m m remain stuck with a multitude of possible models, dreibein Ea := eea . Using this conjugate vari- whose lack of uniqueness simply mirrors the lack of able, one can find the objects which are conjugate uniqueness that comes with the need to fix infinitely to the holonomies.
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