Over at Preposterous Universe Sean Carroll has some comments on the anthropic principle and the landscape.

He describes one extreme of the spectrum of opinion about this as people who think the whole thing is completely non-scientific, giving what he sees as being the two kinds of objections such people make, neither of which he thinks make sense. Since I’m one of these extremists, I think I should try and explain why and exactly what the nature of my objections are, since they’re not exactly the ones Sean mentions.

The first objection Sean attributes to extremists like myself is that of accusing users of the anthropic principle of “giving up” by assigning the parameters of the standard model to a selection effect instead of calculating them. This is very much David Gross’s objection, and while I would agree with it as a socio/psychological characterization of the behavior of Susskind et. al., my own version of this objection is a bit differerent. For any given supposed fundamental theory, some observables will be calculable from first principles, and others will be aspects of the particular state we are in, dependent on the history of how we got here. Given a particular observable, in some fundamental theories it may be calculable, in others environmental. But the theory is supposed to tell us which it is going to be. The standard model tells us that the earth-sun distance is environmental, and that the magnetic moment of the electron is calculable. It is silent about the origin of its 20 or so parameters, and whether they are environmental or calculable. It is one of the first jobs of any theory that purports to go beyond the standard model to give some sort of explanation of where these parameters come from, which of them are in principle calculable and which aren’t.

The problem with the whole Landscape idea is that it is so ill-defined that it can’t even tell you what things are calculable and what things are environmental. You don’t know what the fundamental M-theory is that is supposed to be producing the Landscape and governing the dynamics of how the universe evolves in it. String theorists would probably claim that while they don’t know exactly what the fundamental theory is, they may know enough about it to make conjectures about what the Landscape should look like, at least in certain limiting cases. The problem is that their conjectures not only don’t allow them to calculate anything, they don’t even allow them to determine what is going to be calculable. The problem with string theory is not that it can’t calculate the vacuum energy, it is that it can’t calculate anything. Some string theorists are now using the Landscape picture purely as an excuse to get them out of this embarassing situation. “Not our fault we can’t calculate anything beyond the Standard Model, because maybe nothing beyond the Standard Model is calculable”. If they had a well-defined fundamental theory which exhibited this behavior, one might take them seriously, but until they do, the whole picture is nothing more than an elaborate excuse for failure. A question that should be asked of anyone promoting this stuff: show us using string theory which of the Standard Model parameters are calculable and which are environmental. If they can’t do this they shouldn’t be taken seriously.

The second objection Sean attributes to the likes of me is that we object to the explanatory use of entities that are unobservable in principle, like multiple universes. This isn’t really my objection to the Landscape. If a compelling fundamental theory existed that made lots of correct testable predictions, and such a theory predicted lots of unobservable universes, I’d happily believe in their existence. But, absent such a compelling theory, people who go on about unobservable multiple universes are not behaving very differently from those theologians who supposedly took an interest in angels and pins. Science is about coming up with explanations for the way the world works, explanations that can in principle be tested by making more observations of the world. If you’ve been working on a theory for twenty years and it has totally failed to make any testable predictions, you should admit failure and move on, not engage in elaborate apologetics for why your theory can’t predict anything.

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Thomas,

Can your formalism be applied in a Weyl conformal space? The geometry is rather more natural than Riemann space because “metricity” is strictly local. The reducibility of the metric (9 direction cosines + volume element) is removed. I would expect the unit pseudoscalar to emerge prominently from the Clifford algebra of constraints in the correspondence you talk about. In the case that interests me (signature —+++), this looks like

-drl

TL – rofl. That was classic.

Urs,

The Fock representations are constructed by first expressing the diffeomorphism generators in terms of canonically conjugate variables q and p, and then representing the q’s and p’s on a Fock space.

In principle you can quantize in other ways, e.g. by looking at submodules of Verma modules or vertex operator algebras or Sugawara/coset constructions or whatever. For the Virasoro algebra this is equivalent, due to a quantum equivalence theorem which I think is due to David Olive.

Look at the papers, for God’s sake! 100 man-years of work by professional mathematicians is not necessarily invalided if a physics grad student doesn’t immediately understand how to do things.

I have posted a description of how to quantize in the covariant phase space on spr.

Thomas –

there is a misunderstanding here: The problem is not to find a representation of the constraint algebra alone. The problem is to find a representation of the embedding algebra of canonical data, i.e. the p and q. That of course is what gives rise to the divergences when the constraints are expressed in terms of these ‘p’ and ‘q’.

And that’s what makes the difference between quantizing a system on the one hand side or talking about abstract algebras on the other side.

Urs,

As far as I can see it is not known if there is any meaningful way in which the constraint algebra of pure GR in d=4 or higher could receive quantum corrections, since it seems to be impossible to find a weakly continuous representation of these constraints in terms of operators on some Hilbert space. No matter how the operator ordering is dealt with the commutators of the would-be constraint operators contain diverging quantities and are hence ill defined.The problem you point at here is of course the reason why people failed to generalize the Virasoro algebra to higher dimensions for 25 years. I found one of the extensions rather rapidly, but I was unable to find any interesting representations for seven years (1988-1995), precisely because I had to struggle with the problem you mention. In the end I didn’t solve it myself, but read about the solution in the Rao-Moody paper, and at that time I had already run out of funding. An irony is that Rao sent me a preprint of their paper in 1992, but I didn’t realize that they had solved the problem I faced.

Ted,

I think you would agree that the fact that a classical constraint algebra recieves quantum corrections does not mean that there is an anomaly in gauge symmetry.That is what I mean. But I could say that quantum gravity must have an “anomalous, global diffeomorphism symmetry” if that makes you happier. People say that the 2D Ising model at criticality has an “anomalous, global conformal symmetry”, but this is really only a game with words.

In general, a true gauge anomaly is a bad thing, since unphysical (usually negative energy) modes do not decouple from the dynamics.The bad thing with chiral-fermion type gauge anomalies is lack of unitarity, which mathematically means a lack of unitary, lowest-energy representations. Virasoro type algebras have at least representations of lowest-energy type.

Physically, what happens is this. The observer’s trajectory decouples classically; it will probably be a timelike geodesic but Einstein’s equations couldn’t care less. The quantum system cannot be observed without perturbing it, so these degrees of freedom no longer decouple. A classically unphysical mode has become physical upon quantization. Mathematically, this is crucial because the anomaly is a functional of the observer’s trajectory; first described in math-ph/9810003.

I think the clearest way to understand whether gauge symmetry is anomalous is whether a quantum theory admits a nilpotent BRST charge. As you know, the worldsheet theory of a string does admit such a charge in the critical dimension, and in this sense gauge symmetry is not anomalous. Still, as you noted, the classical Virasoro algebra in the matter sector recieves a quantum correction, aka the central charge.As you mention, the constraint algebra of GR very probably recieves quantum corrections. However, is it known whether the theory admits a nilpotent BRST charge? If it does not, the theory is not gauge invariant at the quantum level and is probably not physically meaningful.

There seems to be no BRST charge at all, nilpotent or not. Normal ordering gives rise to infinities. I disagree about physical meaningfulness, because I believe in locality. If general covariance remains a gauge symmetry after quantization, then there is no Hamiltonian, no energy, no time evolution, and no locality. This seems utterly unphysical to me.

I don’t know of a calculation in string theory which clearly addresses the question you are asking, i.e. how the classical constraint algebra of GR is corrected quantum mechanically.I was not asking a question. Large classes of projective, lowest-energy representations of the diffemorphism algebra are known, cf. the references in my reply to Urs. They all satify the algebras described in physics/9705040 and math-ph/9810003. Besides, all possible extensions of the diffeomorphism algebra (by modules of tensor fields) were classified in

A. Dzhumadildaev,

Virasoro type Lie algebras and deformations,

Z. Phys. C 72 (1996) 509–517.

cf math-ph/0002016. The N-dimensional Virasoro extension are not quite tensor fields, but rather closed (N-1)-forms, but they are closely related to exact (N-1)-forms, and thus to (N-2)-forms. When N=1, a closed 0-form is a constant function is a central extension, but otherwise the extension is not central.

My guess is you would have to consider this question in the framework of closed string field theory. However, there the gauge symmetry of 4-diffs is embedded in a complicated way in the huge gauge symmetry of closed strings, which includes not only gravitons but an infinite number of massive fields. Still—virtually by construction—closed string field theory is gauge invariant at both the classical and quantum levels. It is for the purpose of gauge invariance that the BV formalism is so useful for constructing the action of closed string fields.Why should string field theory be relevant for anything?

Urs,

As far as I can see it is not known if there is any meaningful way in which the constraint algebra of pure GR in d=4 or higher could receive quantum corrections, since it seems to be impossible to find a weakly continuous representation of these constraints in terms of operators on some Hilbert space. No matter how the operator ordering is dealt with the commutators of the would-be constraint operators contain diverging quantities and are hence ill defined.Sigh. The quantum representation theory the diffeomorphism algebra is equivalent to finding continuous representations of these constraints in terms of operators on some Hilbert space. The anomaly by itself is nothing, because it could easily lack representations; the Mickelson-Faddeev algebra, describing chiral-fermion anomalies, apparently does so. The important thing with the Virasoro algebra, in any dimension, is that it has a natural representation theory. Here are a couple of references:

S. Berman and Y. Billig,

Irreducible representations for toroidal Lie algebras,

J. Algebra 221 (1999) 188–231.

S. Berman, Y. Billig and J. Szmigielski;

Vertex operator algebras and the representation theory of toroidal algebras,

math.QA/0101094 (2001)

Y. Billig,

Principal vertex operator representations for toroidal Lie algebras,

J. Math. Phys. 7 (1998) 3844–3864.

Y. Billig,

Energy-momentum tensor for the toroidal Lie algebras,

math.RT/0201313 (2002)

Y. Billig,

Weight modules over exp-polynomial Lie algebras,

math.RT/0305293 (2003)

T.A. Larsson,

Lowest-energy representations of non-centrally extended diffeomorphism algebras,

Comm. Math. Phys. 201 (1999) 461–470.

physics/9705040

T.A. Larsson,

Extended diffeomorphism algebras and trajectories in jet space.

Comm. Math. Phys. 214 (2000) 469–491.

math-ph/9810003

T.A. Larsson,

Extensions of diffeomorphism and current algebras,

math-ph/0002016 (2000)

T.A. Larsson,

Multi-dimensional diffeomorphism and current algebras from Virasoro and Kac-Moody Currents,

math-ph/0101007 (2001)

T.A. Larsson,

Koszul-Tate cohomology as lowest-energy modules of non-centrally extended diffeomorphism algebras,

math-ph/0210023 (2002)

R.V. Moody, S.E. Rao, and T. Yokonoma,

Toroidal Lie algebras and vertex representations,

Geom. Ded. 35 (1990) 283–307.

S.E. Rao, R.V. Moody, and T. Yokonuma,

Lie algebras and Weyl groups arising from vertex operator representations,

Nova J. of Algebra and Geometry 1 (1992) 15–57.

S.E. Rao and R.V. Moody,

Vertex representations for $N$-toroidal Lie algebras and a generalization of the Virasoro algebra,

Comm. Math. Phys. 159 (1994) 239–264.

OK, a caveat. The reps act on linear spaces rather than Hilbert spaces because of problems to define an inner product. The essential difficulty is finiteness, not unitarity, though. Finiteness is obtained by first expanding all fields in a Taylor series around the observer’s trajectory (have I said that before?). This gives us a classical non-linear realization on

finitely many functions of a single variable, which is precisely when normal ordering works. This is the crucial step which evades any no-go theorem that you can come up with.LQG circumvents this on the technical level by dropping the assumption that quantization should use weakly continuous representations. For such reps no quantum correction is found. But dropping weak continuity means leaving the realm of well-established physics and in particular any relation to the path integral.The lack of anomalies is a clear symptom of this.

I think with respect to Thomas Larsson’s speculation that the existence of higher-dimensional Virasoro algebras with certain extensions implies anything about anomalies in gravitational theories it is important to note that the constraint algebra of any GR-like theory is not in general isomorphic (as a whole) to any diffeomorphism algebra, due to the special nature of the Hamiltonian constraint.I have explained this several times. The Dirac algebra has the form it has because of the way we put coordinates on phase space. Some diffeos move us out of a fixed timeslice, and we must therefore add compensators to get back. 4-diffeos + compensators generate the Dirac algebra. But the physics cannot depend on our way to put coordinates on phase space. There is a covariant definition of phase space, as the space of histories. In this formulation, as in the Lagrangian formulation, the symmetries of GR generate precisely the 4-diff algebra.

But it does not really matter. Essentially I’m only saying this:

1. GR is a constrained Hamiltonian system with an infinite-dimensional constraint algebra.

2. Any reasonable quantum theory of gravity must have a classical limit with this property.

3. Infinite-dimensional constraint algebras generically acquire quantum corrections upon quantization. Conformal symmetry in string theory is only the simplest, and best known, example, but things do not miraculously getter simpler in higher dimensions.

Thus it must be possible to express anomalies in any reasonable quantum gravity theory, even if you think that they must be cancelled in the end. String theory does not satisfy this criterion.

The fact that in 2-dimensions (e.g. on the string’s worldsheet) the canonical constraint algebra is isomorphic to two copies of lightlike 1-dimensional reparameterizations is due to the magic of conformal invariance in two dimensions and, as far as I am aware, does not have analogs in higher dimensions.Of course not, and I never said that. What I do say is that because of this isomorphism, we can reinterpret CFT as diff-invariant QFT in 1D. Then generalize this to 4D.

Aware of that, Thomas Larsson speculated that his algebra might hence apply to the diffeo symmetry acting on the space of (classical) solutions to GR. I don’t see what this should mean in detail.math-ph/0210023

I find it interesting that and how higher-dimensional analogs of the Virasoro algebra exist, but it is far from clear that and how these play a role for quantum gravity. Cautious statements seem to be more in order that bold speculations.Tell that to the guy at Princeton who talks about Theories of Everything and chants Magic and Mystery.

It is no speculation that the multi-dimensional Virasoro algebra is the

onlyway to combine quantum theory, general covariance, and locality. People seem happy to state that there are no local observables in quantum gravity. Well, I’m not, and ‘t Hooft is even willing to introduce hidden variables to get locality. The lesson from CFT, regarded as a spacetime symmetry in 2D, is that locality is compatible with infinite-dimensional symmetries, but only in the presence of an anomaly. This is no speculation, but a well-known fact since 20 years, and it has been amply verified that nature behaves in this way in 2D condensed-matter systems.Ted Erler wrote (to Thomas Larsson):

As far as I can see it is not known if there is any meaningful way in which the constraint algebra of pure GR in d=4 or higher could receive quantum corrections, since it seems to be impossible to find a weakly continuous representation of these constraints in terms of operators on some Hilbert space. No matter how the operator ordering is dealt with the commutators of the would-be constraint operators contain diverging quantities and are hence ill defined.

LQG circumvents this on the technical level by dropping the assumption that quantization should use weakly continuous representations. For such reps no quantum correction is found. But dropping weak continuity means leaving the realm of well-established physics and in particular any relation to the path integral.

I think with respect to Thomas Larsson’s speculation that the existence of higher-dimensional Virasoro algebras with certain extensions implies anything about anomalies in gravitational theories it is important to note that the constraint algebra of any GR-like theory is not in general isomorphic (as a whole) to any diffeomorphism algebra, due to the special nature of the Hamiltonian constraint. The fact that in 2-dimensions (e.g. on the string’s worldsheet) the canonical constraint algebra is isomorphic to two copies of lightlike 1-dimensional reparameterizations is due to the magic of conformal invariance in two dimensions and, as far as I am aware, does not have analogs in higher dimensions.

Aware of that, Thomas Larsson speculated that his algebra might hence apply to the diffeo symmetry acting on the space of (classical) solutions to GR. I don’t see what this should mean in detail.

I find it interesting that and how higher-dimensional analogs of the Virasoro algebra exist, but it is far from clear that and how these play a role for quantum gravity. Cautious statements seem to be more in order that bold speculations.

Dear Thomas,

Let me see if I can clarify for myself what you’re saying.

I think you would agree that the fact that a classical constraint

algebra recieves quantum corrections does not mean that there is an anomaly in gauge symmetry. In general, a true

gauge anomaly is a bad thing, since unphysical (usually negative

energy) modes do not decouple from the dynamics.

I think the clearest way to understand whether gauge symmetry

is anomalous is whether a quantum theory admits a nilpotent

BRST charge. As you know, the worldsheet theory of a string

does admit such a charge in the critical dimension, and in this

sense gauge symmetry is not anomalous. Still, as you noted,

the classical Virasoro algebra in the matter sector recieves a quantum correction, aka the central charge.

As you mention, the constraint algebra of GR very probably

recieves quantum corrections. However, is it known whether

the theory admits a nilpotent BRST charge? If it does not, the

theory is not gauge invariant at the quantum level and is

probably not physically meaningful.

I don’t know of a calculation in string theory which clearly

addresses the question you are asking, i.e. how the classical

constraint algebra of GR is corrected quantum mechanically.

My guess is you would have to consider this question in the

framework of closed string field theory. However, there the

gauge symmetry of 4-diffs is embedded in a complicated way

in the huge gauge symmetry of closed strings, which includes

not only gravitons but an infinite number of massive fields.

Still—virtually by construction—closed string field theory is

gauge invariant at both the classical and quantum levels. It is for the purpose of gauge invariance that the BV formalism is so

useful for constructing the action of closed string fields.

–Ted

TL, thanks for an excellent answer. Yes, you did mention Ginsparg, thanks again.

-drl

DRL,

Do you have a good reference for anomalology? How is the “Dirac algebra of ADM (Arnowit-Deser-Misner?) constraints” equivalent to 4D diff? Is this a fancy way of saying there is an equivalent tetrad formalism and that the constraints become orthogonality and normalization conditions?In the Lagrangian formulation, GR is invariant under all 4-diffs. In the Hamiltonian formalism, you break this symmetry by specifying a timeslice where the canonical variables live. So the physical symmetries are 4-diffs, but some of them move you out of the chosen timeslice. Therefore you must add compensating transformations to get back to it, and the combination of 4-diffs and compensating transformations generate the Dirac algebra. But the compensating transformations don’t really have anything to do with physics, only with our way of parametrizing the phase space. There is a covariant way to define phase space, namely as the space of solutions to the equations of motions, i.e. the histories. The constraint algebra in this covariant phase space are the 4-diffs, because here we don’t need any compensation anymore.

I find all this talk about constraints interesting but puzzling. I don’t really understand this way of thinking. As far as I can tell, there really isn’t a good Hamiltonian formulation of ordinary GR, that is, one with an indentifiable, unambiguous energy. When you say H, do you mean this in the more general sense of simply having a variational principle?The Hamiltonian constraint is probably the thing that I found most puzzling about GR. H = 0 seems to imply that there is no energy and no time. I now believe that this naive interpretation is indeed correct. The only quantum theories where a Hamiltonian constraint seems to be successfully implemented are various types of topological field theories a la Witten. Since the correlation functions in such theories are smooth invariants, they are indeed timeless.

However, in the presence of an anomaly, the Hamiltonian ceases to be a constraint. One can no longer demand that diffeomorphisms annihilate the physical states, because that would mean that the anomaly, in the simplest case the unit operator, annihilates them. So further states become physical. In particular, one can now define the energy E of an eigenstate |psi> of the Hamiltonian by H|psi> = E|psi>.

People might say that a diff anomaly means that things depend on the choice of coordinates. This is not true any more than for the Poincare algebra. There are three cases:

1. No anomaly. The theory is then “coordinate free”, in the sense that it can probably be formulated without coordinates altogether.

2. An anomalous diff symmetry: The theory is then “coordinate independent”. We must introduce some reference coordinate system, but our choice does not matter because everything is guaranteed to transform covariantly.

3. No diff symmetry altogether. The theory can then only be formulated in a preferred coordinate system. This is not good.

This line of thinking is very similar to how people think about conformal symmetry in statphys, I think. At least that is what shaped my worldview. In statphys, conformal symmetry is a spacetime (or spatial) symmetry, anomalies are always present, and the dilatation operator plays the role of the Hamiltonian. I think that I recommended Gispargs old reference before. Read that, imagine that he really talks about diffeomorphisms in 1D rather than conformal transformations in 2D, and generalize to 4D.

Thomas,

Do you have a good reference for anomalology? How is the “Dirac algebra of ADM (Arnowit-Deser-Misner?) constraints” equivalent to 4D diff? Is this a fancy way of saying there is an equivalent tetrad formalism and that the constraints become orthogonality and normalization conditions?

I find all this talk about constraints interesting but puzzling. I don’t really understand this way of thinking. As far as I can tell, there really isn’t a good Hamiltonian formulation of ordinary GR, that is, one with an indentifiable, unambiguous energy. When you say H, do you mean this in the more general sense of simply having a variational principle?

-drl

Good morning, Lubos.

Let me come back to the part that you said that you don’t understand, namely quantization of constrained Hamiltonian systems with an infinite-dimensional constraint algebra. The string worldsheet with its Weyl symmetry is an important toy model, because it is the simplest example of such a system.

We quantize first and impose the constraints afterwards. However, if the constraint algebra is infinite-dimensional, quantization of the fields alone (before ghosts) will yield an anomaly because of normal-ordering effects. This is a quite generic feature of any infinite-dimensional constraint algebra. On the string worldsheet, the coordinate fields have an anomaly c = 26. The LQG string is not string theory and conformal transformations may not play a role there. However, there is still an infinite-dimensional constraint algebra, and that is what matters. The absense of an anomaly is a very clear symptom that we are dealing with quantization in some diluted sense of the word, as Urs Schreiber noted.

General relativity can also be cast in the form of a constrained Hamiltonian system with an infinite-dimensional constraint algebra, namely 4D diffeomorphisms or the physically equivalent Dirac algebra of ADM constraints. Therefore, canonical quantization and normal ordering of the fields alone will inevitably give rise to a diff anomaly. This general feature does not depend on the details of the Einstein action; the only thing that matters is that the ADM constraints generate an infinite-dimensional algebra.

Any theory which purports to be a quantum theory of gravity must be possible to cast in this form, string theory, LQG, or whatever. The only assumptions are a Hamiltonian formulation, general covariance, and that we do real quantization rather than some diluted version thereof. In this situation we do get a diff anomaly, at least intermediately. Perturbative string theory might not see this anomaly, since it is only a first-quantized theory; first-quantized particle theory does not see any anomalies at all. But if string field theory does not generate a diff anomaly when only the fields are taken into consideration, there is no way that it can be considered as a quantum theory of gravity, because it violates the most basic properties that such a theory must have.

I don’t need to know the details about string theory to say that. It is simply a fact that quantization of Hamiltonian system with an infinite-dimensional constraint algebra must introduce anomalies in the constraint algebra.

This is something that you have been unable to understand at least for 5 years, as far as I remember: the local symmetries of a theory are a redundancy in the description, and they depend on the specific description – and the local symmetries are usually necessary to make a description with too many new variables consistent. But the gauge symmetry is not “measurable” in any way. The measurable things are only those objects in the coset/quotient “description divided by the local symmetries and redundancies”.This is true classically, because classically there are no anomalies, so you can always write down a nilpotent BRST operator and pass to the reduced phase space. After quantization, you can still do that if there is no anomaly. With an anomaly, you can not, because some (or all) gauge degrees of freedom become physical. Instead, your classical gauge symmetry becomes an anomalous global quantum symmetry. Neither possibility is inconsistent. It is important to realize that the classical limit of an anomalous global conformal symmetry is a conformal gauge symmetry, because all anomalies vanish in the classical limit.

But only if quantum gravity has a global anomalous diffeomorphism symmetry rather than a gauge diff symmetry can we resolve the conceptual difficulties: we get a real Hamiltonian rather than a Hamiltonian constraint, real time evolution rather than a gauge transformation, a local definition of mass, etc.

Concerning the Ising model – the “local symmetry” is not the usual local symmetry that we talked about before (one that must keep the states invariant).That

youtalked about.Of course, if you use the Ising model as a model of classical statistical mechanics, the rules are very different – but jumping from string theory to classical mechanics is not a controllable approach to a discussion.The rules of classical statistical mechanics are very different from QFT? That’s bad news for the people doing lattice gauge theory, isn’t it?

The generalization of the Virasoro algebra may exist in higher dimensions, but it does not have the properties to lead to a generalization of string theory.I’m not interested in generalizing a complete scientific failure. The physically successful theories are general relativity, the standard model, and conformal field theory applied to 2D statphys. I’m applying insights from the third theory to the two first.

One of the main contributions to science in recent years from string theorists has been assigning new meanings for the term “prediction”. When Lubos and other string enthusiasts say that “string theory makes predictions”, in old-style lingo this means roughly “if everything we would like to be true about string theory really is, then in principle one can use it to make predictions”.

The Landscape crowd has yet another new-fangled notion of prediction. If there’s an experimental upper bound on an observable, you say that the Landscape “predicts” that the observable will be some number less than the upper bound. See for example the many references to Weinberg’s successful “prediction” of the cosmological constant.

On the contrary, string theory predicts everything – its character certainly makes it the most predictive theory one can imagine.

Hi Lubos,

Pray tell, in light of your above comment, what is the “string theory” prediction for the SUSY breaking scale? Say accurate to 1%? Please make that a first principles prediction, starting with one of the five superstring theories, or better with “M-Theory”, not some “string inspired” model.

Until you can actually *predict* parameters like that (or retrodict the parameters of the standard model), don’t you think it’s a tad bit early to be calling string theory “the most predictive theory one can imagine”?

That’s right, Thomas. I don’t understand the difference between being unable to write the anomaly in the first place, and having a theory in which the anomaly is guaranteed to cancel.

Conformal symmetry is a local symmetry obtained by switching from the Nambu-Goto action to the Polyakov action; the latter has new (auxilliary) degrees of freedom (the worldsheet metric), and they must be unphysical, which is guaranteed by the diff and Weyl symmetry which must therefore be unbroken. The residual symmetry from diff x Weyl is the conformal symmetry, and if it is broken by the quantum effects, it proves that diff x Weyl was broken, too.

But there is nothing wrong with theories that don’t allow you to write a conformal anomaly in the first place. There are many descriptions of physics of string theory that do not require any 2D conformal symmetries – such as those from AdS/CFT – and they are equally consistent.

This is something that you have been unable to understand at least for 5 years, as far as I remember: the local symmetries of a theory are a redundancy in the description, and they depend on the specific description – and the local symmetries are usually necessary to make a description with too many new variables consistent. But the gauge symmetry is not “measurable” in any way. The measurable things are only those objects in the coset/quotient “description divided by the local symmetries and redundancies”. All the physical states must be invariant under local symmetries, for example – they transform as the singlets. This makes the representation theory of local symmetries physically irrelevant – all these things are just auxilliary concepts that we encounter in the middle of the calculation, but that disappear in the final results.

If you criticize LQG for having no conformal symmetry, it is not a fair criticism. It is not only a criticism that “it is not string theory” – it would be a wrong criticism even in string theory because there are many (nonperturbative) descriptions of the same physics that do not rely on conformal symmetry.

Concerning the Ising model – the “local symmetry” is not the usual local symmetry that we talked about before (one that must keep the states invariant). If it were – e.g. if you use the Ising model as a part of string theory worldsheet action – then the total theory MUST cancel all anomalies in the conformal symmetry. Of course, if you use the Ising model as a model of classical statistical mechanics, the rules are very different – but jumping from string theory to classical mechanics is not a controllable approach to a discussion.

The generalization of the Virasoro algebra may exist in higher dimensions, but it does not have the properties to lead to a generalization of string theory. The direct generalization of string theory to higher-dimensional fundamental objects simply does not work because of hundreds of reasons, and if you still think that it is a straightforward thing to make a theory based on higher-dimensional replacements for strings, it proves that your knowledge these matters is highly superficial.

String theory backgrounds do not have any anomalies in local symmetries. They don’t have a conformal anomaly on the worldsheet. String theory is only called string theory if the perturbative portion of it is anomaly-free conformal field theory. Bosonic string theory in flat spacetime whose dimension differs from 26 is not a string theory.You don’t seem to understand the difference between writing down an anomaly and then cancelling it out, and being unable to write down the anomaly in the first place. LQG cannot write down the conformal anomaly in the first place. But neither LQG nor string theory can write down the diff anomalies in 4D, which exist mathematically (multi-dimensional Virasoro algebra), and must be present physically to ensure locality.

It’s not clear to me why you want to extrapolate 2D conformal anomaly to higher (four) dimensions. It seems to me that you are confusing the worldsheet and the spacetime.2D conformal transformations are isomorphic to (twice) 1D diffeomorphims. A general-covariant theory transforms covariantly under diffeomorphisms. I generalize this property from 1D to 4D.

The rest of your text is comparably weird. You seem to be looking for a theory that has an anomaly. Why do you want it? A theory with an anomaly in local symmetries is not really a theory, it is a garbage.A local symmetry can very well have an anomaly, provided that you don’t try to impose a physical state condition. It is then no longer a gauge symmetry on the quantum level. The prime example is the application of CFT to 2D phase transitions. E.g., the 2D Ising model has local conformal symmetry with an anomaly (c = 1/2), but it is perfectly consistent: it is unitary, and

realized in nature. Ultimately, that’s what physics is about. It is sad that you don’t understand that.But it is not a FLAW, as you seem to be saying, it is one of the critical virtues of string theory. Sorry, there may be someone else who will see something coherent in your text, but it does not make any sense to me.Some people seem to think that locality is worth many sacrifices – ‘t Hooft even seems to be willing to give up quantum mechanics to get it. However, hidden variables are not necessary. You can combine quantum theory, general covariance, and locality, but only with a diff anomaly.

At any rate, it is easy to verify that a generalization of the Virasoro algebra exists in any dimension. It is thus a diff anomaly, something that Weinberg claims does not exist in 4D. Missing an anomaly is a gross oversight. Your reaction is in fact very LQGish: “I don’t like this anomaly, hence it does not exist”. It is good that you reveal your LQG mentality.

Sorry, “Peter” should be “Thomas” in the previous post.

I was trying hard, but it’s still not comprehensible to me what is the “missing anomaly syndrome”.

Let me try to explain you a couple of elementary points, Peter.

String theory backgrounds do not have any anomalies in local symmetries. They don’t have a conformal anomaly on the worldsheet. String theory is only called string theory if the perturbative portion of it is anomaly-free conformal field theory. Bosonic string theory in flat spacetime whose dimension differs from 26 is not a string theory.

It’s not clear to me why you want to extrapolate 2D conformal anomaly to higher (four) dimensions. It seems to me that you are confusing the worldsheet and the spacetime. It’s the whole point of perturbative string theory that the objects must be one-dimensional (therefore strings), and therefore the worldvolume is 2-dimensional (worldsheet).

Virtually nothing in perturbative string theory can be generalized to higher dimensions of the object. String theory has higher-dimensional objects, but they are never as fundamental as the string. String is the only object that can generate a consistent spacetime theory, and all the magic of 2D conformal symmetry and complex calculus is essential.

The rest of your text is comparably weird. You seem to be looking for a theory that has an anomaly. Why do you want it? A theory with an anomaly in local symmetries is not really a theory, it is a garbage. It is the reason why many things that we can write down are nonsensical at quantum level. Do you really want string theory to reproduce this garbage? The purpose of string theory is certainly NOT to reproduce any idiotic inconsistent pseudo-theory that someone invents. Indeed, many problems and inconsistencies are solved AUTOMATICALLY by string theory’s basic rules. The framework just does not allow us to write down backgrounds that would have some kinds of problems that can appear in other theories.

But it is not a FLAW, as you seem to be saying, it is one of the critical virtues of string theory. Sorry, there may be someone else who will see something coherent in your text, but it does not make any sense to me.

Lubos,

String theory suffers from a missing anomaly syndrome, very similar to Loop quantum gravity. It follows from the following observations:

1. In quantization of the string, a conformal anomaly generically arises. In the special case of c = 26, it can be cancelled against ghosts, but generically it is there, also when c = 26 before cancellation. You know this.

2. Things don’t get simpler in higher dimensions. Canonical quantization gave rise to an anomaly already in 1+1D, so canonical quantization of a diff-invariant theory in 3+1D will also give rise to an anomaly. You realize this, because you are not foolish enough to believe that the anomaly will miraculously disappear in higher dimensions.

3. There are no diff anomalies in field or string theory in 4D. You know this (or if you don’t know it, check out Weinberg’s second book, chapter 22.

Taken together, this implies that canonical quantization in 4D must give rise to a diff anomaly which isn’t present in string theory. This can mean one of two things

1. No second-quantized Hamiltonian formulation of string theory exists.

2. A second-quantized Hamiltonian formulation of string theory exists and is anomalous, which is inconsistent with the path-integral formalism.

Either way, this must mean that string theory is a very pathological theory. A missing anomaly is a very serious sickness in a theory. It was fatal for LQG, and it is presumably fatal for string theory as well.

I read a book once where it was said that also pigs can say that the universe is made for them – thus one should call it the “porcine principle”.

An anthropic or a porcine principle is not a theory – it just says that the prediction must come out as observed. It is a test of a theory, or a test for consistency among observations.

Also talking about the existence of things that cannot be observed is impossible; “existence” is *defined* as what can be observed.

FB

Sean, I would probably disagree with your interpretation of Peter’s statement.

You say that he says that string theory is not a solid framework to include the anthropic reasoning. Even though I think that string theorists should avoid anthropic reasoning, it’s just not true that string theory does not allow you to put anthropic reasoning on firm ground.

On the contrary, it is the only known theory that offers a scientific encapsulation for all these ideas. In order to treat anthropic principle scientifically, you need a large number of Universes – solutions – and the existence of dynamical mechanisms to get from one to another, and so forth. This is possible in string theory.

You also need to assign some measures – probabilities of different Universes. This is of course the whole problem of the anthropic reasoning – what is a “generic” Universe? But if one accepts the highly controversial assumptions that each stringy vacuum that admits basic life-like things has the same weight, then it’s just true that string theory – with its discrete ensemble of vacua – gives you a more or less rigorous realization of this anthropic counting.

Once again, this counting seems unscientific to me by its very basic properties, because the “less predictive” vacua are favored – but nevertheless it seems clear to me that one can play this game.

Just like I agree with Peter’s algorithm to decide whether a scientific theory is nontrivially predictive and interesting, it seems to me that he does not quite understand some subtle issues about string theory.

First of all, it is not true that you can’t calculate vacuum energy in string theory.

In the Standard Model, which is a non-gravitational theory, the sum of the vacuum diagrams does not really matter because the vacuum energy has no effects in a non-gravitational theory.

However, if you couple such a theory to gravity, the vacuum energy does matter, because it curves the space. However, in quantum field theory, you can always add a counter-term and adjust your vacuum energy to whatever you want. You need fine-tuning, but you can always adjust such “constants” in quantum field theory.

That’s not the case of string theory. Here, you can really calculate vacuum energy, and in the simplest models, you obtain a far too huge value of the cosmological constant – assuming that some serious bug about our understanding of SUSY breaking don’t invalidate the whole conclusion.

In this sense, the CC problem in string theory is or was more serious because string theory is a very rigid theory that does not allow you to mess with the parameters.

Well, the anthropic industry in string theory is more or less meant to put the CC constant in string theory to a comparable level to the CC in field theory. The freedom to continuously fine-tune the vacuum energy in field theory is replaced by the large number of vacua in string theory – and you can really see that some of them are more or less guaranteed to predict a qualitatively correct value of the C.C.

You know, I am among those who believe that we don’t quite understand this counting, especially after SUSY breaking, properly, but the anthropic people will disagree. Shamit Kachru et al. will tell you how to calculate all possible contributions to the potential, and he will argue that nothing is neglected and all approximations they make are justified.

They will tell you that they have a full control over the class of the KKLT vacua (that was elaborated by Mike Douglas and his collaborators and others), and this full control allows them to state quite certainly that string theory *does* predict a large number of vacua – even those controllable ones form large classes.

Once again, Peter, you are absolutely wrong if you think that string theory’s nature is its inability to predict. On the contrary, string theory predicts everything – its character certainly makes it the most predictive theory one can imagine. In fact, the appearance of the landscape in string theory is a consequence of its strong predictive power – because some people just became convinced that a choice of one of a few “simple” and “natural” enough vacua is more or less guaranteed to predict an *incorrect* cosmological constant. This is why people start to propose the convoluted vacua that can give you, more or less by chance, a realistic C.C., too.

It’s wrong to think that string theory is less predictive than field theory. Even if you imagined that string theory could give you virtually any field theory at low energies, it is still *more* predictive because it is a UV complete theory containing quantum gravity – something impossible in quantum field theory. Its low energy physics may have many types, and in this sense the “Landscape” of stringy vacua is analogous to the “Landscape” of quantum field theories – with the difference that the landscape of QFTs has continuous parameters, while string theory only has discrete ones.

Best wishes

Lubos

Peter, your objections don’t seem to be to the anthropic principle, but to the claim that the string theory landscape provides a sensible framework in which the anthropic principle can be implemented. And I would agree, at least in terms of the current state of the art. Our disagreement is that I am quite optimistic about the prospects for string theory in the future, so I think it’s well worth pursuing. But right now we don’t understand nearly enough to go around using the landscape idea to make predictions.