{"id":145,"date":"2005-01-29T12:17:49","date_gmt":"2005-01-29T16:17:49","guid":{"rendered":"http:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=145"},"modified":"2005-01-29T12:17:49","modified_gmt":"2005-01-29T16:17:49","slug":"loop-quantum-gravity-debate","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=145","title":{"rendered":"Loop Quantum Gravity Debate"},"content":{"rendered":"

A couple weeks ago, three string theorists, (Nicolai, Peeters and Zamaklar) posted on the arXiv a critical assessment<\/A> of loop quantum gravity. Today I received from Lee Smolin something he wrote responding to them, and I’m posting it here with his permission. Lubos Motl also has put up Smolin’s text on his weblog<\/A> this morning, but I thought it would be a good idea to provide a version that doesn’t include Lubos’s interspersed rantings. Smolin has some very interesting things to say, and his comments are well-worth reading by anyone who wants to understand what is going on in this field.<\/p>\n

Somewhat off-topic, I’d also like to mention a paper by Freidel and Starodubtsev from earlier this week called Quantum gravity in terms of topological observables<\/A>. The idea of trying to use topological quantum field theory to understand quantum gravity is one that I’ve always found appealing, and this paper is an interesting attempt to make this idea work. I don’t think I find it completely convincing, for one thing they seem to be breaking the topological invariance by hand. For another, TQFTs are very subtle QFTs, and the kind that might be relevant to gravity is still very far from well-understood.
\n
\nDear Friends,<\/p>\n

Thanks very much for all the time and work you put into your review.
\nWhile I disagree with a number of your assertions, both in point of detail
\nand of attitude, what is certainly very much appreciated is your evident
\nwillingness to “get your hands dirty,” learn the technicalities and attack
\nkey problems. It is very good that you do this, as indeed too few of us
\nloop people have taken the time to try to learn the details and attack
\nproblems in string theory.<\/p>\n

Some points you raise have been underappreciated. The issue of what
\nhappens to the chiral anomaly, and whether there is fermion doubling in
\nLQG is one I have suggested to many graduate students and postdocs over
\nthe years, but so far no one takes it up. It would be good to know if LQG
\nforces us to believe in a vector model of weak interactions.<\/p>\n

At the same time, the major difficulties you raise were underestood to be
\nthere more than ten years ago. This is especially true with respect to
\nissues concerning the hamiltonian constraint such as the algebra and
\nultralocality.<\/p>\n

What is missing from your “review” is an appreciation of how the work done
\nover the last ten years addresses these difficulties.
\nIndeed the fact that much work in the field has been on spin foam models
\nis exactly because the problems you worry about do not arise in spin foam
\nmodels. I will explain this below. Other work, such as Thiemann’s master
\nconstraint approach, also is motivated by a possible resolution of these
\nproblems.<\/p>\n

As you will appreciate, like any active community of 100+ people there is
\na range of opinions about the key unsolved problems. I have the sense that
\nyou are aware of only one out of several influencial points of view.<\/p>\n

The view your concerns reflect is what one might call the “orthodox
\nhamiltonian” point of view towards LQG. According to this, the aim of
\nwork in lqg is not so much to find the quantum theory of gravity as to
\nwork through the excercise of quantizing a particular classical theory,
\nwhich is Einstein’s. From this point of view, the program would fail if
\nit turned out that there was not a consistent canonical quantization of
\nthe Einstein’s equations.<\/p>\n

While I will refer to my own views so as not to implicate anyone else, you
\nshould beware that this is not necessarily the dominant view in the field.
\nIt is a respectable view, and I have the greatest respect for my friends
\nwho hold it. But, were it to fail, many of us would still believe that
\nloop quantum gravity is the most promising approach to quantum gravity.<\/p>\n

This is not avoidence of hard problems, there are good physical reasons
\nfor this assertion, which I’d like to explain.<\/p>\n

What I and others have taken as most important about Ashtekar’s great
\nadvance is the discovery that GR can be writen as a diffeomorphism
\ninvariant gauge theory, where the configuration space is that of a
\nconnection on a manifold Sigma, mod gauge transformations and Diff(Sigma).
\nThis turns out to be true not only of Einstein’s theory in 4d but of all
\nthe classical gravity theory we know, in all dimentions, including
\nsupergravity, up to d=11, and coupled to a variety of matter fields.<\/p>\n

This is a kinematical observation and it leads to a hypothesis at the
\nkinematical level, which is that the quantum theory of gravity, whatever
\nit is, is to be written in terms of states which come from the
\nquantization of this configuration space. This as you know, leads
\ndirectly to the diffeo classes of spin net states. Furthermore, given the
\nrecent uniqueness theorems, that hilbert space is unique for spacetime
\ndimension 3 or greater. Thus, o long as the object is to construct a
\ntheory based on diffeomorphism invariant states, it cannot be avoided.<\/p>\n

The main physical hypothesis of LQG is not that the quantum Einstein
\nequations describe nature. It is that the hilbert space of diffeo classes
\nof spin nets, extended as needed for matter, p-form fields, supersymmetry
\netc, is the correct arena for quantum gravitational physics. Given that
\nthe theorems show that this hilbert space exists rigorously, this is a
\nwell defined hypothesis about physics. It may hold whether or not the
\nEinstein equations quantized give the correct dynamics.<\/p>\n

A lot already follows from this hypothesis. It gives us states,
\ndiscreteness of some geometric diffeo invariant observers, a physical
\ninterpretation in terms of discrete quantum geometry etc.<\/p>\n

But there is also a lot of freedom. We are free to pick the dimension,
\ntopology, and algebra whose reps and intertwiners label the spinnetworks.
\nThis then gives us a large class of diffeo invariant quantum gauge
\ntheories, of which the choices that come from GR in d=4 are only one
\nexample. These are possible kinematics for consistent background
\nindependent quantum field theories.<\/p>\n

Now let us come to dynamics. I believe the most important observation for
\nan understanding of quantum dyannics in this class of theories is that all
\ngravitational theories we know, in all dimensions, super or not, are
\nconstrained topological field theories. (See my latest review,
\nhep-th\/0408048, for details and references for all assertions here.) This
\nmeans they are related to BF theories by non-derivative constraints,
\nquadratic in the B fields.<\/p>\n

A lot follows from this very general observation. It allows a direct
\nconstruction of spin foam models, by imposing the quadratic constraints in
\nthe measure of the path integral for BF theory. This was the path
\npioneered by Barrett and Crane. The construction of the Barrett Crane and
\nother spin foam models does not depend on the existence of a well defined
\nhamiltonian constraint. The properties that have been proven for it, such
\nas certain convergence results, also do not depend on any dynamical
\nresults from the hamiltonian theory.<\/p>\n

The relation to topological field theory is also sufficient to determine
\nthe basic form of fields and states on boundaries. In 4d these give the
\nrole of Chern-Simons theory in horizon and other boundary states. Thus,
\nit gives the basic quantum geometry of horizons.<\/p>\n

Once we have the basic form of spin foam models, which follow from the
\ngeneral relation to BF theories, we can consider the problem of dyanmics
\nin the following light. Given the choices made above, the spin foam
\namplitudes are chosen from the invariants of the algebra which labels the
\nspin networks. There is then a large class of theories, differing by the
\nchoice of the spin foam amplitudes. Each is a well defined spin foam
\nmodel, which gives amplitudes to propgate the spin network states based on
\nthe chosen dimension and algebra.<\/p>\n

The lack of uniqueness is unaviodable, because there is a general class of
\ntheories, just like there is a general class of lattice gauge theories.
\nThese theories exist, and the general program of LQG as some of us
\nunderstand it, is to study them.<\/p>\n

>From a modern, renormalization group point of view, the first phsyical
\nquestion to be answered is which of these theories lead to evolution that
\nis sensible, i.e. which spin foam ampltidues are convergent in some
\napprorpiate sense. The second physical question is to classify the
\nuniversality classes of the spin foam models and, having done this, learn
\nwhich classes of theories have a good low energy behaivor that reproduces
\nclassical GR and QFT.<\/p>\n

It is of course of interest to ask whether some of these theories follow
\nfrom quantizing classical theories like GR and supergravity, by various
\nmethods. But no one should mind if the most successful spin foam model, in
\nterms of both matheamtical elegance and physical results, was not the
\nquantization of a classical theory, but only reproduced the classical
\ntheory in the low energy limit. How could one object from a physics point
\nof view, were this true?<\/p>\n

This is the point of view from which many of us view the problems with the
\nhamiltonian constraint you describe.
\nThe next thing to be emphasized is that there is no evidence that a
\nsuccessful spin foam model must have a corresponding quantum hamiltonain
\nconstraint. There are even arguments that it should not. These have not
\npursuaded everyone in the community, and this is proper, for the
\nhealthiest situation is to have differing views about open problems. But
\nit has persuaded many of us, which is why many people in the field turned
\nto the study of spin foam models after the difficulties you describe were
\nunderstood, more than ten years ago.<\/p>\n

For example, Fotini Markopoulou argued that, as the generators of
\ninfinitesimal spatial diffeos do not exist in the kinematnical hilbert
\nspace, while generators of finte spatial diffeos do exist, the same should
\nbe true for time evolution. This implies that there should only be
\namplitudes for finite evolutions, from which she proposed one could
\nconstruct causal spin foam models.<\/p>\n

This was partly motivated by the issue ultralocality. (Btw, you dont
\nemphasize the paper that first raised this worry, which was my
\ngr-qc\/9609034). The worry arises because moves such as 2 to 2 moves
\nnecessary for propagation do not occur in the forms of the hamiltonian
\nconstraint constructed by Thiemann, Rovelli and myself, or Borissov.
\nThis is because they involve two nodes connected by a finite edge.<\/p>\n

However, the missing moves are there in spin foam models. This concretely
\nconfirms Fotini’s argument. In fact, as Reisenberger and Rovelli argued,
\ninvariance under boosts generated by spacetime diffeo requires that they
\nbe there. For one can turn a 1-3 move into a 2->2 (0r 1->4 into 2-> 3)
\nmove by slicing the spin foam differently into a sequance of spinnetworks
\nevolving in time.<\/p>\n

So we have two arguments that suggest 1) that the problem of ultralocaity
\ncomes from requiring infinitesimal timelike diffeos to exist in a theory
\nwhere infinitesimal spacelike diffeos do not exist and 2) the problem
\nis not present in a path integral approach where there are only
\namplitudes for finite timelike diffeos.<\/p>\n

One can further argue that if there were a regularization of the
\nhamiltonian constraint that produced the amplitudes necesary for
\npropagation and agreed with the spin foam ampltidues, it would have to be
\nderived from a point splitting in time as well as space. This suggests
\nthat there is a physical inadquancy of defining dynamics through the
\nhamiltonian constraint, in a formalism where one can regulate only in
\nspace and not in time.<\/p>\n

Let me also add that there is good reason to think that the other issues
\nsuch as the algebra of constraints arise because of the issue of
\nultralocality. Thiemann’s constraints have the right algebra for an
\nultralocal theory.<\/p>\n

It was for these and other reasons that some of us decided ten years ago
\nto put the problems of the hamiltonian constraint to one side and
\nconcentrate on spin foam models. That is, we take the canonical methods
\nas having been good enough to give us a kinematical frameowrk for a large
\nclass of diffeo invariant gauge theories, but unnecessary and perhaps
\ninsufficient for studying dynamics.<\/p>\n

At the very least, making a point splitting regularization in both space
\nand time seems a much more difficult problem and hence is less attractive
\nthan spin foam methods where one can much more easily get to the physics.
\nGiven that the relation to BF theory gives us an independent way to define
\nthe dynamics, and path integral methods are more directly connected to
\nmany physical questions we want to investigate, there seemed no reason to
\nhold back progress on the chance that the problems of the hamiltonian
\nconstraint can be cleanly resolved.<\/p>\n

Nothing I’ve said here means that I am not highly supportive of Thomas’s
\nand others efforts to resolve the problems of the hamiltonian dynamics-I
\nam. But it must be said that a “review” of LQG that focues on this issue
\nmisses the significance of much of the work done the last ten years.<\/p>\n

Let me make an analogy. No one has proved perturbative finiteneess of
\nsuperstring theory past genus two. I could, and have even been tempted to,
\nwrite a review of the problem, highlighting the heroic work of a few
\npeople like d’Hoker and Phong to resolve it. I think it would be useful
\nif someone did that, as their work is underappreciated. But it would be
\nvery unfair of me to call this a review of, or introduction to, the state
\nof string theory. Were I to do so, I would rightly be criticized as
\nfocusing on a very hard problem that most people in the field have for
\nmany years felt was not crucial for the development of the theory. This is
\nnot a perfect analogy to what you have done in your “review”, but it is
\npretty close.<\/p>\n

There are other mis-statments in your review. For example, there are
\ncertainly results at the semiclassical level. Otherwise there could not be
\na lively literature and debate about predictions stemming from LQG for
\nreal experiments. See my recent hep-th\/0501091 for an introduction and
\nreferences. Of course semiclassical states do not necessarily fit into a
\nrigorous framework-after all, WKB states are typically not normalizable.
\nBut I would suggest that it may be too much to require that results in QFT
\nthat make experimental predictions be first discovered through rigorous
\nmethods. At the standards of particle physics levels of rigor, there are
\nsemiclassical results, and these do lead to nontrivial predictions for
\nnear term experiments. It is possible that a more rigorouos treatment
\nwill in time lead to a rigorous understanding of how classical dynamics
\nemerges-and that is a very important problem. But given that AUGER and
\nGLAST may report within two years, may I suggest that it is reasonable to
\ndo what we can do now to draw predictions from the theory.
\nIn closing let me emphasize again that your efforts are very well
\nappreciated. I hope this is the beginning of a dialogue, and that you will
\nbe interested to explore other aspects of LQG not covered by or addressed
\nin your review.<\/p>\n

Sincerely yours,<\/p>\n

Lee Smolin<\/p>\n","protected":false},"excerpt":{"rendered":"

A couple weeks ago, three string theorists, (Nicolai, Peeters and Zamaklar) posted on the arXiv a critical assessment of loop quantum gravity. Today I received from Lee Smolin something he wrote responding to them, and I’m posting it here with … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-145","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=145"}],"version-history":[{"count":0,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/145\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}