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 his permission. Lubos Motl also has put up Smolin’s text on his weblog 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.
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. 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.
Thanks very much for all the time and work you put into your review.
While I disagree with a number of your assertions, both in point of detail
and of attitude, what is certainly very much appreciated is your evident
willingness to “get your hands dirty,” learn the technicalities and attack
key problems. It is very good that you do this, as indeed too few of us
loop people have taken the time to try to learn the details and attack
problems in string theory.
Some points you raise have been underappreciated. The issue of what
happens to the chiral anomaly, and whether there is fermion doubling in
LQG is one I have suggested to many graduate students and postdocs over
the years, but so far no one takes it up. It would be good to know if LQG
forces us to believe in a vector model of weak interactions.
At the same time, the major difficulties you raise were underestood to be
there more than ten years ago. This is especially true with respect to
issues concerning the hamiltonian constraint such as the algebra and
What is missing from your “review” is an appreciation of how the work done
over the last ten years addresses these difficulties.
Indeed the fact that much work in the field has been on spin foam models
is exactly because the problems you worry about do not arise in spin foam
models. I will explain this below. Other work, such as Thiemann’s master
constraint approach, also is motivated by a possible resolution of these
As you will appreciate, like any active community of 100+ people there is
a range of opinions about the key unsolved problems. I have the sense that
you are aware of only one out of several influencial points of view.
The view your concerns reflect is what one might call the “orthodox
hamiltonian” point of view towards LQG. According to this, the aim of
work in lqg is not so much to find the quantum theory of gravity as to
work through the excercise of quantizing a particular classical theory,
which is Einstein’s. From this point of view, the program would fail if
it turned out that there was not a consistent canonical quantization of
the Einstein’s equations.
While I will refer to my own views so as not to implicate anyone else, you
should beware that this is not necessarily the dominant view in the field.
It is a respectable view, and I have the greatest respect for my friends
who hold it. But, were it to fail, many of us would still believe that
loop quantum gravity is the most promising approach to quantum gravity.
This is not avoidence of hard problems, there are good physical reasons
for this assertion, which I’d like to explain.
What I and others have taken as most important about Ashtekar’s great
advance is the discovery that GR can be writen as a diffeomorphism
invariant gauge theory, where the configuration space is that of a
connection on a manifold Sigma, mod gauge transformations and Diff(Sigma).
This turns out to be true not only of Einstein’s theory in 4d but of all
the classical gravity theory we know, in all dimentions, including
supergravity, up to d=11, and coupled to a variety of matter fields.
This is a kinematical observation and it leads to a hypothesis at the
kinematical level, which is that the quantum theory of gravity, whatever
it is, is to be written in terms of states which come from the
quantization of this configuration space. This as you know, leads
directly to the diffeo classes of spin net states. Furthermore, given the
recent uniqueness theorems, that hilbert space is unique for spacetime
dimension 3 or greater. Thus, o long as the object is to construct a
theory based on diffeomorphism invariant states, it cannot be avoided.
The main physical hypothesis of LQG is not that the quantum Einstein
equations describe nature. It is that the hilbert space of diffeo classes
of spin nets, extended as needed for matter, p-form fields, supersymmetry
etc, is the correct arena for quantum gravitational physics. Given that
the theorems show that this hilbert space exists rigorously, this is a
well defined hypothesis about physics. It may hold whether or not the
Einstein equations quantized give the correct dynamics.
A lot already follows from this hypothesis. It gives us states,
discreteness of some geometric diffeo invariant observers, a physical
interpretation in terms of discrete quantum geometry etc.
But there is also a lot of freedom. We are free to pick the dimension,
topology, and algebra whose reps and intertwiners label the spinnetworks.
This then gives us a large class of diffeo invariant quantum gauge
theories, of which the choices that come from GR in d=4 are only one
example. These are possible kinematics for consistent background
independent quantum field theories.
Now let us come to dynamics. I believe the most important observation for
an understanding of quantum dyannics in this class of theories is that all
gravitational theories we know, in all dimensions, super or not, are
constrained topological field theories. (See my latest review,
hep-th/0408048, for details and references for all assertions here.) This
means they are related to BF theories by non-derivative constraints,
quadratic in the B fields.
A lot follows from this very general observation. It allows a direct
construction of spin foam models, by imposing the quadratic constraints in
the measure of the path integral for BF theory. This was the path
pioneered by Barrett and Crane. The construction of the Barrett Crane and
other spin foam models does not depend on the existence of a well defined
hamiltonian constraint. The properties that have been proven for it, such
as certain convergence results, also do not depend on any dynamical
results from the hamiltonian theory.
The relation to topological field theory is also sufficient to determine
the basic form of fields and states on boundaries. In 4d these give the
role of Chern-Simons theory in horizon and other boundary states. Thus,
it gives the basic quantum geometry of horizons.
Once we have the basic form of spin foam models, which follow from the
general relation to BF theories, we can consider the problem of dyanmics
in the following light. Given the choices made above, the spin foam
amplitudes are chosen from the invariants of the algebra which labels the
spin networks. There is then a large class of theories, differing by the
choice of the spin foam amplitudes. Each is a well defined spin foam
model, which gives amplitudes to propgate the spin network states based on
the chosen dimension and algebra.
The lack of uniqueness is unaviodable, because there is a general class of
theories, just like there is a general class of lattice gauge theories.
These theories exist, and the general program of LQG as some of us
understand it, is to study them.
>From a modern, renormalization group point of view, the first phsyical
question to be answered is which of these theories lead to evolution that
is sensible, i.e. which spin foam ampltidues are convergent in some
approrpiate sense. The second physical question is to classify the
universality classes of the spin foam models and, having done this, learn
which classes of theories have a good low energy behaivor that reproduces
classical GR and QFT.
It is of course of interest to ask whether some of these theories follow
from quantizing classical theories like GR and supergravity, by various
methods. But no one should mind if the most successful spin foam model, in
terms of both matheamtical elegance and physical results, was not the
quantization of a classical theory, but only reproduced the classical
theory in the low energy limit. How could one object from a physics point
of view, were this true?
This is the point of view from which many of us view the problems with the
hamiltonian constraint you describe.
The next thing to be emphasized is that there is no evidence that a
successful spin foam model must have a corresponding quantum hamiltonain
constraint. There are even arguments that it should not. These have not
pursuaded everyone in the community, and this is proper, for the
healthiest situation is to have differing views about open problems. But
it has persuaded many of us, which is why many people in the field turned
to the study of spin foam models after the difficulties you describe were
understood, more than ten years ago.
For example, Fotini Markopoulou argued that, as the generators of
infinitesimal spatial diffeos do not exist in the kinematnical hilbert
space, while generators of finte spatial diffeos do exist, the same should
be true for time evolution. This implies that there should only be
amplitudes for finite evolutions, from which she proposed one could
construct causal spin foam models.
This was partly motivated by the issue ultralocality. (Btw, you dont
emphasize the paper that first raised this worry, which was my
gr-qc/9609034). The worry arises because moves such as 2 to 2 moves
necessary for propagation do not occur in the forms of the hamiltonian
constraint constructed by Thiemann, Rovelli and myself, or Borissov.
This is because they involve two nodes connected by a finite edge.
However, the missing moves are there in spin foam models. This concretely
confirms Fotini’s argument. In fact, as Reisenberger and Rovelli argued,
invariance under boosts generated by spacetime diffeo requires that they
be there. For one can turn a 1-3 move into a 2->2 (0r 1->4 into 2-> 3)
move by slicing the spin foam differently into a sequance of spinnetworks
evolving in time.
So we have two arguments that suggest 1) that the problem of ultralocaity
comes from requiring infinitesimal timelike diffeos to exist in a theory
where infinitesimal spacelike diffeos do not exist and 2) the problem
is not present in a path integral approach where there are only
amplitudes for finite timelike diffeos.
One can further argue that if there were a regularization of the
hamiltonian constraint that produced the amplitudes necesary for
propagation and agreed with the spin foam ampltidues, it would have to be
derived from a point splitting in time as well as space. This suggests
that there is a physical inadquancy of defining dynamics through the
hamiltonian constraint, in a formalism where one can regulate only in
space and not in time.
Let me also add that there is good reason to think that the other issues
such as the algebra of constraints arise because of the issue of
ultralocality. Thiemann’s constraints have the right algebra for an
It was for these and other reasons that some of us decided ten years ago
to put the problems of the hamiltonian constraint to one side and
concentrate on spin foam models. That is, we take the canonical methods
as having been good enough to give us a kinematical frameowrk for a large
class of diffeo invariant gauge theories, but unnecessary and perhaps
insufficient for studying dynamics.
At the very least, making a point splitting regularization in both space
and time seems a much more difficult problem and hence is less attractive
than spin foam methods where one can much more easily get to the physics.
Given that the relation to BF theory gives us an independent way to define
the dynamics, and path integral methods are more directly connected to
many physical questions we want to investigate, there seemed no reason to
hold back progress on the chance that the problems of the hamiltonian
constraint can be cleanly resolved.
Nothing I’ve said here means that I am not highly supportive of Thomas’s
and others efforts to resolve the problems of the hamiltonian dynamics-I
am. But it must be said that a “review” of LQG that focues on this issue
misses the significance of much of the work done the last ten years.
Let me make an analogy. No one has proved perturbative finiteneess of
superstring theory past genus two. I could, and have even been tempted to,
write a review of the problem, highlighting the heroic work of a few
people like d’Hoker and Phong to resolve it. I think it would be useful
if someone did that, as their work is underappreciated. But it would be
very unfair of me to call this a review of, or introduction to, the state
of string theory. Were I to do so, I would rightly be criticized as
focusing on a very hard problem that most people in the field have for
many years felt was not crucial for the development of the theory. This is
not a perfect analogy to what you have done in your “review”, but it is
There are other mis-statments in your review. For example, there are
certainly results at the semiclassical level. Otherwise there could not be
a lively literature and debate about predictions stemming from LQG for
real experiments. See my recent hep-th/0501091 for an introduction and
references. Of course semiclassical states do not necessarily fit into a
rigorous framework-after all, WKB states are typically not normalizable.
But I would suggest that it may be too much to require that results in QFT
that make experimental predictions be first discovered through rigorous
methods. At the standards of particle physics levels of rigor, there are
semiclassical results, and these do lead to nontrivial predictions for
near term experiments. It is possible that a more rigorouos treatment
will in time lead to a rigorous understanding of how classical dynamics
emerges-and that is a very important problem. But given that AUGER and
GLAST may report within two years, may I suggest that it is reasonable to
do what we can do now to draw predictions from the theory.
In closing let me emphasize again that your efforts are very well
appreciated. I hope this is the beginning of a dialogue, and that you will
be interested to explore other aspects of LQG not covered by or addressed
in your review.
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