The Illusion of Quantum Gravity
Lecture 11: The Failure of Loop Quantum Gravity. The Bottom Line is Red.
It’s overly complicated and leads to nothing much it appears. It cannot even derive Einstein’s classical relativity as a suitable limiting case. It’s a big buck for very little bang. This is what happens when pure mathematicians try to do theoretical physics. Angels dancing on the heads of pins? What Feynman called “philofawzy” with “rigor mortis”? ;-)
In contrast I claim to have solved the quantum gravity decisively by showing it is a pseudo-problem asking the wrong question from Nature.
“The Question is: What is The Question?” John A. Wheeler
Q. First what is the wrong question posed by “quantum gravity”?
A. “What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature. This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.”
“Whereas there is a rather direct link between (perturbative) string theory and classical space-time concepts, and string theory can therefore rely on familiar notions and concepts, such as the notion of a particle and the S-matrix, LQG must face up right away to the question of what an observable quantity is in the absence of a proper semiclassical space-time with fixed asymptotics.”
“Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.”
“Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections. … A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’.”
Also, Loop Quantum Gravity does not really deliver on its promise of 4D covariance and it is apparently not able to properly handle virtual zero point energy i.e. “off-mass-shell” processes:
“Quantum space-time covariance?
Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.”
“In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance. By weakening the requirements on the constraint algebra and by no longer insisting on off-shell clo•sure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modi•cations on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.”
“Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily.”
“Obviously, a brief introductory text such as this cannot do justice to the numerous recent developments in a very active field of current research. For this reason, we would like to conclude this introduction by referring readers to several ‘inside’ reviews for recent advances and alternative points of view, namely [9, 10, 11] for the canonical formulation, [12, 13, 14] for spin foams, and [15] for both. Readers are also invited to have a look at [16] for an update on the very latest developments in the subject.
2. The kinematical Hilbert space of LQG
There is a general expectation (not only in the LQG community) that at the very shortest distances, the smooth geometry of Einstein’s theory will be replaced by some quantum space or spacetime, and hence the continuum will be replaced by some ‘discretuum’. Canonical LQG does not do away with conventional spacetime concepts entirely, in that it still relies on a spatial continuum •as its ‘substrate’, on which holonomies and spin networks live (or ‘float’) — of course, with the idea of eventually ‘forgetting about it’ by considering ‘abstract spin networks’ and only the combinatorial relations between them. On this substrate, it takes as the classical phase space variables the holonomies of the Ashtekar connection,
…
Here, •a are the standard generators of SU(2) (Pauli matrices), but one can also replace the basic representation by a representation of arbitrary spin, de•noted … The Ashtekar connection A is thus a particular linear combination of the spin
Figure 1: LQG employs holonomies and fluxes as elementary conjugate variables.
…
Instead of building a Hilbert space as the space of functions over configurations of the Ashtekar connection, i.e. instead of constructing wave-functionals … LQG uses a Hilbert space of wave functionals which “probe” the geometry only on one-dimensional submanifolds, so-called spin networks. The latter are (not necessarily connected) graphs • … consisting of •finitely many edges (links). The wave functionals are functionals over the space of holonomies. In order to make them C-valued, the SU(2) indices of the holonomies have to be contracted using invariant tensors (i.e.vClebsch-Gordan coeffcients). … The spin network wave functions are thus labeled by (the spin network graph), by the spins {j} attached to the edges, and the intertwiners associated to the vertices. At this point, we have merely defined a space of wave functions in terms of rather unusual variables, and it now remains to define a proper Hilbert space structure on them. The discrete kinematical structure which LQG imposes does, accordingly, not come from the description in terms of holonomies and fluxes. After all, this very language can also be used to describe ordinary Yang-Mills theory. The discrete structure which LQG imposes is also entirely different from the discreteness of a lattice or naive discretisation of space (i.e. of a finite or countable set). Namely, it arises by ‘polymerising’ the continuum via an unusual scalar product. For any two spin network states, one defines this scalar product to be … where the integrals … are to be performed with the SU(2) Haar measure. The spin network wave functions depend on the Ashtekar connection only through the holonomies.
Figure 2: A simple spin network, embedded in the spatial hypersurface. The hypersurface is only present in order to provide coordinates which labe lthe positions ofthe vertices and edges. Spin network wave functions only probe the geometry along the one-dimensional edges and are insensitive to the geometry …
The kinematical Hilbert space … is then defined as the completion of the space of spin network wave functions w.r.t. this scalar product (5). The topology induced by the latter is similar to the discrete topology (‘pulverisation’) of the real line with countable unions of points as the open sets. Because the only notion of ‘closeness’ between two points in this topology is whether or not they are coincident, whence any function is continuous in this topology, this raises the question as to how one can recover conventional notions of continuity in this scheme.
The very special choice of the scalar product (5) leads to representations of operators which need not be weakly continuous: this means that expectation values of operators depending on some parameter do not vary continuously as these parameters are varied. Consequently, the Hilbert space does not admit a countable basis, hence is non-separable, because the set of all spin network graphs in is uncountable, and non-coincident spin networks are orthogonal w.r.t. (5). Therefore, any operation (such as a diffeomorphism) which moves around graphs continuously corresponds to an uncountable sequence of mutually orthogonal states in Hkin. That is, no matter how ‘small’ the deformation of the graph in •, the associated elements of Hkin always remain a finite distance apart, and consequently, the continuous motion in ‘real space’ gets mapped to a highly discontinuous one in Hkin. Although unusual, and perhaps counter-intuitive, as they are, these properties constitute a cornerstone for the hopes that LQG can overcome the seemingly unsurmountable problems of conventional geometrodynamics: if the representations used in LQG were equivalent to the ones of geometrodynamics, there would be no reason to expect LQG not to end up in the same quandary.
Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections.
Figure 3: The computation of the spectrum of the area operator involves the division of the surface into cells, such that at most one edge of the spin network intersects each given cell.
A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’. Because the measurement of lengths and distances ultimately requires an operational definition in terms of appropriate matter fields and states obeying the physical state constraints, ‘dynamical’ discreteness is expected to manifest itself in the spectra of the relevant physical observables. Therefore, let us now turn to a discussion of the spectra of three important operators and to the discussion of physical states.
3. Area, volume and the Hamiltonian
In the current setup of LQG, an important role is played by two relatively simple operators: the ‘area operator’ measuring the area of a two-dimensional surface S •, and the ‘volume operator’ measuringthe volume of a three-dimensional subset V ••. The latter enters the definition of the Hamiltonian constraint in an essential way. Nevertheless, it must be emphasized that the area and volume operators are not observables in the Dirac sense, as they do not commute with the Hamiltonian. To construct physical operators corresponding to area and volume is more difficult and would require the inclusion of matter (in the form of ‘measuring rod fields’). … These spin network states are thus eigenstates of the area operator. The situation becomes considerably more complicated for wave functions, which contain a spin network vertex which lies in the surface S; in this case the area operator does not necessarily act diagonally anymore (see figure 4). Expression (9) lies at the core of the statement tha tareas are quantised in LQG.
The construction of the volume operator follows similar logic, although it is substantially more involved. … [There is also an] operator which replaces the Hamiltonian evolution operator of ordinary quantum mechanics, and encodes all the important dynamical information of the theory (whereas the Gauss and diffeomorphism constraints are merely ‘kinematical’). More specically, together with the kinematical constraints, it defines the physical states of the theory, and thereby the physical Hilbert space Hphys (which may be separable, even if Hkin is not). To motivate the form of the quantum Hamiltonian one starts with the classical expression, written in loop variables. To this aim one re-writes the Hamiltonian in terms of Ashtekar variables, with the result … The key problem in canonical gravity is the definition and implementation of the Hamiltonian (scalar) constraint operator, and the verification that this operator possesses all the requisite properties. The latter include (quantum) space-time covariance as well as the existence of a proper semi-classical limit, in which the classical Einstein equations are supposed 6 For the special values … the last term drops out, and the Hamiltonian simplifies considerably. This was indeed the value originally proposed by Ashtekar, and it would also appear to be the natural one required by local Lorentz invariance (as the Ashtekar variable is, in this case, just the pullback of the four-dimensional spin connection). However, imaginary … obviously implies that the phase space of general relativity in terms of these variables would have to be complexified, such that the original phase space could be recovered only after imposing a reality constraint. In order to avoid the difficulties related to quantising this reality constraint, … is now usually taken to be real. With this choice, it becomes much more involved to rewrite (12) in terms of loop and flux variables.
4. Implementation of the constraints
In canonical gravity, the simplest constraint is the Gauss constraint. In the setting of LQG, it simply requires that the SU(2)representation indices entering a given vertex of a spin network enter in an SU(2)invariant manner. More complicated are the diffeomorphism and Hamiltonian constraint. In LQG these are implemented in two entirely different ways. Moreover, the implementation of the Hamiltonian constraint is not completely independent, as its very definition relies on the existence of a subspace of diffeomorphism invariant states.
Let us start with the diffeomorphism constraint. Unlike in geometrodynamics, one cannot immediately write down formal states, which are manifestly diffeomorphism invariant, because the spin network functions are not supported on all of •, but only on one-dimensional links, which ‘move around’ under the action of a diffeomorphism. A formally diffeomorphism invariant state is obtained by ‘averaging’ over the diffeomorphism group, and more speci•cally by considering the formal sum … Although this is a continuous sum which might seem to be ill-defined, it can be given a mathematically precise meaning because the unusual scalar product (5) ensures that the inner product between a state and a … On the space of diffeomorphism averaged spin network states (regarded as a subspace of a distribution space) one can now again introduce a Hilbert space structure ‘by dividing out’ spatial diffeomorphisms, … As we said above, however, it is the Hamiltonian constraint which plays the key role in canonical gravity, as it this operator which encodes the 7 dynamics. Implementing this constraint on Hdi• or some other space is fraught with numerous choices and ambiguities, inherent in the construction of the quantum Hamiltonian as well as the extraordinary complexity of the resulting expression for the constraint operator [26]. The number of ambiguities can be reduced by invoking independence of the spatial background [10], and indeed, without making such choices, one would not even obtain sensible expressions, as we shall see very explicitly. In other words, the formalism is partly ‘on-shell’ in that the very existence of the (unregulated) Hamiltonian constraint operator depends very delicately on its ‘diffeomorphism covariance’, and the choice of a proper ‘habitat’, on which it is supposed to act in a well defined manner. A further source of ambiguities, which, for all we know, has not been considered in the literature so far, consists in possible dependent ‘higher order’ modifications of the Hamiltonian, which might still be compatible with all consistency requirements of LQG.
In order to write the constraint in terms of only holonomies and fluxes, one has to eliminate the inverse square root E-1/2 in (12) as well as the extrinsic curvature factors. This can be done through a number of tricks found by Thiemann [27… The attitude often expressed with regard to the ambiguities in the construction of the Hamiltonian 3 is that they correspond to different physics, and therefore the choice of the correct Hamiltonian is ultimately a matter of physics (experiment?), and not mathematics. However, it appears unlikely to us that Nature will allow such a great degree of arbitrariness at its most fundamental level: in fact, our main point here is that the infinitely many ambiguities which killed perturbative quantum gravity, are also a problem that other (to wit, non-perturbative) approaches must address and solve. …
3 The abundance of ‘consistent’ Hamiltonians and spin foam models (see below) is sometimes compared to the vacuum degeneracy problem of string theory, but the latter concerns different solutions of the same theory, as there is no dispute as to what (perturbative) string theory is. However, the concomitant lack of predictivity is obviously a problem for both approaches.
…
5. Quantum space-time covariance?
Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.
While there is general agreement as to what one means when one speaks of ‘closure of the constraint algebra’ in classical gravity (or any other classical constrained system [36]), this notion is more subtle in the quantized theory. 4 Letus therefore clarify first the various notions of closure that can arise: we see at least three different possibilities. The strongest notion is ‘off-shell closure’(or ‘strong closure’), where one seeks to calculate the commutator of two Hamiltonians … Although on-shell closure may perhaps look like a sufficient condition on the quantum Hamiltonian constraint, it is easy to see, at the level of simple examples, that this is not true. … In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance.
By weakening the requirements on the constraint algebra and by no longer insisting on off-shell closure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modifications on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.
6. Canonical gravity and spin foams
Attempts to overcome the difficulties with the Hamiltonian constraint have led to another development, spin foam models [37, 38, 39]. These were originally proposed as space-time versions of spin networks, to wit, evolutions of spin networks in ‘time’, but have since developed into a class of models of their own, disconnected from the canonical formalism. Mathematically, spin foam models represent a generalisation of spin networks, in the sense that group theoretical objects (holonomies, representations, intertwiners, etc.) are attached not only to vertices and edges (links), but also to higher dimensional faces in a simplicial decomposition of space-time.
The relation between spin foam models and the canonical formalism is based on a few general features of the action of the Hamiltonian constraint operator on a spin network (for a review on the connection, see [40]). As we have discussed above, the Hamiltonian constraint acts, schematically, by adding a small plaquette close to an existing vertex of the spin network (as in figure 5). In terms of a space-time picture, we see that the edges of the spin network sweep out surfaces, and the Hamiltonian constraint generates new surfaces, as in figure 7; but note that this graphical representation does not capture the details of how the action of the Hamiltonian affects the intertwiners at the vertices. Instead of associating spin labels to the edges of the spin network, one now associates the spin labels to the surfaces, in such a way that the label of the surface is determined by the label of the edge which lies in either the initial or final surface.
In analogy with proper-time transition amplitudes for a relativistic particle, it is tempting to define the transition amplitude between an initial spin network state and a final one … There are many questions one could ask about the physical meaning of this expression, but one important property is that (just as with the relativistic particle), the transition amplitude will project onto physical states (formally, this projection is effected in the original path integral by integrating over the lapse function multiplying the Hamiltonian density). One might thus consider (25) as a way of defining a physical inner product. In order to make contact with statistical partition sums, and because path integrals with oscillatory measures are difficult to handle, one next applies a formal Wick rotation to (25), replacing the Feynman weight with a Boltzmann weight, as is usually done in Euclidean quantum field theory. However, in making these steps one should always remember that there is no Osterwalder-Schrader type reconstruction theorem in quantum gravity, and therefore the derivation remains formal. Alternatively, one can adopt Hawking’s point of view that the world really is Euclidean, and simply take the Euclidean analog of (25) as the basic definition of the theory.
…
The simplest context in which to study these ideas is (2+1)gravity, because it is a topological (‘BF-type’) theory, that is, without local degrees of freedom, which can be solved exactly (see e.g. [41, 42, 43] and [44] for a more recent analysis of the model within the spin foam picture). … When one tries to formulate spin foam models in four dimensions, the relation to the canonical quantisation approach becomes less clear … Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily. For space-time covariance, one expects 22 and 04 moves (and their time-reversed partners) as well, see figure 9. It has been argued [38] that these missing moves can be obtained from the Hamiltonian formalism by a suitable choice of operator ordering. In section 4 we have used an ordering, symbolically denoted by FEE, in which the Hamiltonian first opens up a spin network and subsequently glues in a plaquette. If one chooses the ordering to be EEF , then the inverse densitised vielbeine can open the plaquette, thereby potentially inducing a 22 or 04 move. However, ref. [27] has argued strongly against this operator ordering, claiming that in such a form the Hamiltonian operator cannot even be densely defined. In addition, the derivation sketched here is rather symbolic and hampered by the complexity of the Hamiltonian constraint [51]. Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.
7. Spin foams and discrete gravity
In view of the discussion above, it is thus perhaps best to view spin foam models as models in their own right, and, in fact, as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable. To clarify the relation between spin foam models and earlier attempts to define a discretised path integral in quantum gravity, we recall that the latter can be roughly divided into two classes, namely:
•
Quantum Regge Calculus (see e.g. [52]), where one approximates space-time by a triangulation consisting of a fixed number of simplices, and integrates over all edge lengths, keeping the ‘shape’ of the triangulation fixed;
•
Dynamical Triangulations (see e.g. [53, 54, 55]), where the simplices are assigned •fixed edge lengths, and one sums instead over different triangulations, but keeping the number of simplices fixed (thus changing only the ‘shape’, but not the ‘volume’ of the triangulation).
Both approaches are usually based on a positive signature (Euclidean) metric, where the Boltzmann factor is derived from, or at least motivated by, some discrete approximation to the Einstein-Hilbert action, possibly with a cosmological constant [but see [56, 57] for some recent progress with a (Wick•rotated) ‘Lorentzian’ dynamical triangulation approach which introduces and exploits a notion of causality on the space-time lattice]. In both approaches, the ultimate aim is then to recover continuum space-time via a refinement limit in which the number of simplices is sent to infinity. Establishing the existence of such a limit is a notoriously difficult problem that has not been solved for four-dimensional gravity. In fact, for quantum Regge models in two dimensions such a continuum limit does not seem to agree with known continuum results [58, 59, 60, 61] (see however [62]).
From the point of view of the above classifica•tion, spin foam models belong to the first, ‘quantum Regge’, type, as one sums over all spins for a given spin foam, but does not add, remove or replace edges, faces or vertices, at least not in a first step. Indeed, for the spin foams discussed in the foregoing section, we have so far focused on the partition sum for a single given spin foam. An obvious question then concerns the next step, or more specically the question how spin foam models can recover (or even only define) a continuum limit. The canonical setup, where one sums over all spin network states in expressions like (25), would suggest that one should sum over all foams … where Z• denotes the partition function for a given spin foam, and where we have allowed for the possibility of a non-trivial weight w• depending only on the topological structure (‘shape’) of the foam. The reason for this sum would be to achieve formal independence of the triangulations. In a certain sense this would mimic the dynamical triangulation approach (except that one now would also sum over foams with a different number of simplices and different edge lengths), and thus turn the model into a hybrid version of the above approaches. However, this prescription is far from universally accepted, and several other ideas on how to extract classical, continuum physics from the partition sum Z•have been proposed. … The key issue is then to ensure that the final result does not depend on the way in which the triangulations are performed and refined. The refinement limit is motivated by the fact that it does appear to work in three space-time dimensions: more specifically, for large spins, the 6j symbol which appears in the Ponzano-Regge model approximates the Feynman weight for Regge gravity [63, 64]. … At present, there is little evidence that triangulation independence can be realised in non-topological theories, or that the problems related to the continuum limit will not reappear in a different disguise.
8. Predictive (finite) quantum gravity?
What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature.
This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.
At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences, but the problem reappears through the lack of uniqueness of the canonical Hamiltonian. For spin foams (or, more generally, discrete quantum gravity) the problem is no less virulent. The known finiteness proofs all deal with the behaviour of a single foam, but, as we argued, these proofs concern the infrared rather than the ultraviolet. Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far, but, as we saw, there is an embarras de richesse of physically distinct models, again reflecting the non-uniqueness that manifests itself in the infinite number of couplings associated with the perturbative counterterms. Indeed, fixing the ambiguities of the non-perturbative models by ad hoc, albeit well-motivated, assumptions is not much different from defining the perturbatively quantised theory by •fixing infinitely many coupling constants ‘by hand’.
Finally, let us remark that in lattice gauge theories, the classical limit and the UV limit can be considered and treated as separate issues. As for quantum gravity, this also appears to be the prevailing view in the LQG community.
However, the continuing failure to construct viable physical semi-classical states, solving the constraints even in only an approximate fashion, seems to suggest (at least to us) that in gravity the two problems cannot be solved separately, but are inextricably linked —also in view of the fact that the question as to the precise fate of the two-loop divergence (37) can then no longer be avoided.
…
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Lecture 10
Comment 19: Is Smolin’s paper the end of the story?
“Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach.”
Comment 19: In other words the Loop Quantum Gravity theorists are not really able to derive Einstein’s 1915 equation
Guv + kTuv = 0
As a limiting case.
I am able to do this in my emergent gravity theory from
B ~ (dTheta)(Phi) – (Theta)(dPhi)
Where Theta and Phi are the two Goldstone phases of the vacuum Planck Higgs field whose bare vacuum manifold is S2 excluding Calabi-Yau parameters. B is the curved piece of the Einstein-Cartan invariant 1-form tetrad field. Einstein’s metric field is obtained trivially from the mathematical form the equivalence principle.
Excerpts & my commentaries on:
AEI-2006-004 hep-th/0601129 January 18, 2006
arXiv:hep-th/0601129 v1 18 Jan 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, kasper.peeters@aei.mpg.de
“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.
Contributed article to “An assessment of current paradigms in theoretical physics”.
1. Quantum Einstein gravity
The assumption that Einstein’s classical theory of gravity can be quantised non-perturbatively is at the root of a wide variety of approaches to quantum gravity. The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity. Although the assumption is certainly far-reaching, there is to date no proof that Einstein gravity cannot be quantized non-perturbatively, either along the lines of one of the programs listed above or perhaps in an entirely different way.
In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. 1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients. A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.
The present text, which is based in part on the companion review [7], is intended as a brief introductory and critical survey of loop and spin foam quantum gravity2, with special focus on some of the questions that are frequently asked by non-experts, but not always adequately emphasized (for our taste, at least) in the pertinent literature.
[1 In the remainder, we will follow established (though perhaps slightly misleading) custom and summarily refer to this frame•work of ideas simply as “Loop Quantum Gravity”, or LQG for short.
2 while our review [7] is focused on the ‘orthodox’ approach] to loop quantum gravity — to wit the Hamiltonian framework — the present text also addresses the more recent spin foam developments. Even though the connections between these approaches are not as strong as one might expect, they do share some historical background and philosophy.
These concern in particular the definition and implementation of the Hamiltonian (scalar) constraint and its lack of uniqueness. Another important question (which we will not even touch on here) concerns the consistent incorporation of matter couplings, and especially the question as to whether the consistent quantisation of gravity imposes any kind of restrictions on them. Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach. Explaining the emergence of classical space-times is also a prerequisite for understanding the ultimate fate of the non-renormalisable UV divergences that arise in the conventional perturbative treatment. The latter question also arises in the ‘covariant’ spin foam approach in the form of the existence (or nonexistence) of a proper ‘continuum limit’.
A further question in any canonical approach to quantum gravity is whether one has succeeded in achieving (a quantum version of) full space-time covariance, rather than merely covariance under the diffeomorphisms of the three-dimensional slices. For someone unfamiliar with the concepts of LQG, it is not easy to see whether and how this requirement is met. In the presently known canonical set-up of LQG, it is only possible to establish on-shell closure of the constraint algebra, which means that partial use of the (diffeomorphism) constraint must be made in checking the commutator of two Hamiltonian constraint operators. In [7] we have argued that this is not enough, and that it is rather the off-shell closure of the constraint algebra that should be made the crucial requirement in establishing quantum space-time covariance. Space-time covariance is also an issue in discrete approaches, and thus spin foam quantum gravity, although the problem appears in a different guise. Whereas in conventional lattice discretisation the main question was whether and in which sense it is possible to ‘approximate’ general coordinate transformations on discrete sets of points and links, the key question in modern approaches which work with reparametrisation invariant quantities (proper lengths, etc.), as in Regge calculus, is now whether it is possible to obtain results which do not depend on the way in which the discretisation and the continuum limit are performed.
In view of our continuing ignorance about the ‘true theory’ of quantum gravity, the best strategy is surely to explore all possible avenues, including non-string approaches to quantum gravity. LQG, just like the older geometrodynamics approach [8], addresses several aspects ofthe problem that are currently outside the main focus of string theory, in particular the question of background independence and the quantisation of geometry.”
Comment 20 In my alternative theory the curved space-time geometry should not be quantized top -> down because it is a “More is different” bottom -> up emergent structure from the “organizational order” macro-quantum vacuum ODLRO Goldstone phases of the first Higgs field that forms at 10^-33 inflation to the edge of the ledge on the cosmic landscape before the final plunge off the ledge to make the hot big bang. Quantizing geometry is a silly as quantizing temperature, or quantizing elasticity. The ODLRO is already non-perturbative and background independent. There are still spin 2 quanta relative to the dynamical curved ODLRO spacetime of course. This explains why no giant Schrodinger Cats, why macro-spacetime physics is local and what the correspondence principle really is.
