August 2, 2005

Background independence

Posted by Robert H.

Last week, Lee Smolin has put a paper on the arxive which on nearly fifty pages does not contain any formulas. Still, he claims it reveals the reason for the failures of string-theory: Background dependence. Lubos has already discussed it in his reference frame. Even so my boss has advised me that this is a waste of time I have written up some comments on that paper. I would love to read you comments, suggestions and criticism. I don’t think this is a waste of time (as long at it does not take too much of my time) as I assume (given for example the comments over at the Cosmic Variance) Smolin’s arguments would appeal especially to outsiders of the field. We should prevent too many people from getting the idea that “string theory is obviously wrong as it ignores the basic notion of background independence”. So I collected some thoughts that might help you arguing the next time you are envolved in this debate. Posted at August 2, 2005 3:20 PM UTC

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Re: Background independence

I have only just returned home and haven’t had the time to catch up with all the discussion that I have missed. But I feel like making the following comment, not particularly replying to anything you wrote:

A ‘background’ is really nothing but a saddle point of a path integral which we are trying to compute perturbatively. Hence perturbation theory and backgrounds are inseperable and denying the usefulness of backgrounds in quantum gravity amounts to denying the usefulness of perturbation theory. What one wants is background independence, in that the choice of saddle point must not affect the result, but background freedom seems to be calling for theories without a classical limit.

Posted by: Urs on August 3, 2005 8:52 PM | Permalink | Reply to this

Re: Background independence

The issue with background independence is far more subtle than that. The problem is that you introduce a reference structure, in this case a Minkowski metric, which is not inherently present in the theory. So you don’t just quantize one theory of gravity, but rather a family of theories parametrized by the choice of reference metric. Classically, it is easy to prove that no results depend on the reference, but QMly there might be subtleties.

There is a close analogy in gauge theories. Either you fix a reference gauge before quantization, or you quantize in a reference-free manner a la BRST. Leaving issues of beauty and practicality aside, both approaches should lead to equivalent results in principle. However, we know that there are subtleties (Gribov ambiguities, anomalies). To resolve these issues, it is essential to have a reference-free formalism like BRST.

Having said that, I should add that loop gravity is not background free in my sense. It is true that LQG does not involve any reference metric, but like any Hamiltonian approach it does involve a choice of foliation. So LQG does not lead to one single theory of QG, but rather to a family of theories parametrized by a reference foliation. To me, this is just as bad as a reference metric.

My concern here is about causality. It is a central axiom of QFT in flat space space that operators at spacelike separation commute. This becomes a major conceptual problem in QG, where the notion of spacelikeness is dynamical. If you introduce a reference metric or foliation, you quantize relative to this background causal structure, rather than relative to the physical causal structure defined by the dynamical metric.

Here one should point at a difference between path integrals and canonical quantization, recently emphasized by Hartle in gr-qc/0508001, around p 6. Both methods give the same results in flat space (for simple problems, at least), but in QG they cannot be equivalent. Causality is automatic in canonical quantization, but in path integrals there are histories which move backwards in time. Only in flat space is the path integral restricted to histories which do not double back on themselves in time, and this property makes path integrals and canonical quantization equivalent.

Hartle takes this as evidence that canonical quantization should be abandoned; I disagree. The major advantage of the path integral formalism is manifest covariance. However, the canonical formalism is not intrinsically non-canonical; it is just that the standard coordinatization of phase space singles out a privileged time direction. One can quantize canonically in a manifestly covariant way, provided that we explicitly introduce an observer. This is necessary to single out a privileged time direction in a covariant way; time is parallel to the observer’s trajectory. In fact, all the difficult conceptual issues with QG seem to evaporate once we take this step.

You might object that the observer is yet another reference object, but this is not so. I do not construct a family of theories parametrized by observer trajectories, but rather a single theory where the trajectory is dynamical; is obeys the geodesic equation and is quantized along with the fields.

Re: Background independence

the canonical formalism is not intrinsically non-covariant.

Re: Background independence

Not that anybody cares, but let me explicitly pinpoint my concerns about a background metric anyway.

When we expand around the Minkowski metric, we treat gravity as a flat space QFT of a spin-2 field. Any flat space QFT has the property that the correlation functions vanish outside the Minkowski lightcone, but not inside. However, the Minkowski lightcone is a reference object without physical meaning. In the true theory of gravity, it should be the physical lightcone that matters, whatever that means when the metric is quantized. Thus, a background metric seems to introduce a physical dependence on an unphysical object, the Minkowski lightcone, which clearly must be wrong.

Causality

When we expand around the Minkowski metric, we treat gravity as a flat space QFT of a spin-2 field. Any flat space QFT has the property that the correlation functions vanish outside the Minkowski lightcone, but not inside.

Correlation functions do not vanish outside the lightcone. Operator commutators vanish for spacelike separations.

Those are related, but not the same.

Secondly, even the above is false if you compute commutators of gauge-variant operators (in a gauge theory in flat space).

So you’d better formulate a gauge- (ie diffeomorphism- ) invariant question if you want to find a contradiction here.

At least in perturbation theory, about a solution to the field equations (eg, flat space), there is no problem in formulating the computation to be done. You mainly need to be careful gauge-fixing. DeWitt, Feynman and others, who formulated perturbation theory for quantum gravity (treated as a conventional gauge field theory), were definitely aware of the issues.

In String Theory, finding a well-posed version of this statement is difficult. But what is true is that, in field theory, causality implies certain analyticity properties of the S-matrix. These analyticity properties do hold true for the S-matrix of perturbative String Theory.

Posted by: Jacques Distler on August 8, 2005 6:04 AM | Permalink | PGP Sig | Reply to this

Re: Causality

100% ACK. In Maxwell theory, the gauge potential propagates instantaneously in ${A}_{0}=0$ gauge.

I remember Steve Carlip claiming in sci.physics.research that for gauge invariant observables in perturbative gravity causality indeed holds relative to the physical (and not background) metric up to higher order terms in the perturbation.

Posted by: Robert on August 8, 2005 8:06 AM | Permalink | Reply to this

Re: Causality

Correlation functions do not vanish outside the lightcone. Operator commutators vanish for spacelike separations.

What is the difference, apart from the trivial observation that operators can have non-zero expectation values? The point is that there can be no non-trivial correlations between events at spacelike separation, because no signal is fast enough to transmit the correlation.

In String Theory, finding a well-posed version of this statement is difficult. But what is true is that, in field theory, causality implies certain analyticity properties of the S-matrix. These analyticity properties do hold true for the S-matrix of perturbative String Theory.

Here you refer to Minkowski causality, not causality wrt the true metric, right? This makes sense if spacetime has asymptocally flat regions between which you can define the S-matrix, but this seems to me as a very strong assumption. Especially if spacetime is asymptotically de Sitter and no S-matrix exists, as observations indicate.

I fail to see what gauge dofs in electromagnism has to do with this. 4D gravity has physical dofs. The physical polarizations of the graviton field commute for separations which are Minkowski spacelike, even though the separation may be timelike wrt the physical metric. Surely there is no analog of this in EM.

A more relevant toy model would be 2D gravity. An extra field becomes physical because of the anomaly, so a background metric would amount to fixing a reference value for this Liouville field, say zero. So in perturbation theory, one should expand the Liouville field around the background value and treat the difference as a small perturbation. But I have only seen non-perturbative treatments using CFT. If background independence is so non-subtle, why don’t people just treat the Liouville field perturbatively?

Re: Causality

What is the difference, apart from the trivial observation that operators can have non-zero expectation values?

Consider two (bosonic) local operators $O\left(x\right)$, $O\prime \left(y\right)$, both of which have vanishing expectation value:

$⟨O\left(x\right)⟩=⟨O\prime \left(y\right)⟩=0$

Now consider three objects:

1. The 2-point Green’s Function (aka the the time-ordered product)$⟨T\left(O\left(x\right)O\prime \left(y\right)\right)⟩$
2. The Wightman function$⟨O\left(x\right)O\prime \left(y\right)⟩$
3. The commutator$⟨\left[O\left(x\right),O\prime \left(y\right)\right]⟩$

Which, if any of these, vanishes when $x,y$ are spacelike separated?

1. The Green’s function doesn’t vanish.
2. The Wightman function doesn’t vanish.
3. The commutator vanishes, but only if the operators are gauge-invariant. The commutators of gauge-variant operators, in general, do not vanish for spacelike separations.

Does that clarify things for you?

Especially if spacetime is asymptotically de Sitter and no S-matrix exists, as observations indicate.

Nobody has a clue what the gauge-invariant observables of quantum gravity in de Sitter space are.

As to the rest of your comment, understanding the situation in field theory is a prerequisite to tackling the situation in gravity. Understanding the situation in asymptotically-flat backgrounds is a prerequisite to tackling other backgrounds.

And, finally, understanding the situation in De Witt-Feynman perturbative quantum gravity is a prerequisite to tackling other, more sophisticated formalisms (like String Theory).

Posted by: Jacques Distler on August 9, 2005 2:46 PM | Permalink | PGP Sig | Reply to this

Re: Causality

By non-trivial correlation function I of course mean

<A(x)B(y)> - <A(x)> <B(y)>.

But this has no bearing on the argument, of course. The problem remains that two operators separated by a Minkowski spacelike distance commute, even if their separation is physically timelike. Of course, the last statement is murky when the metric is quantized.

Nevertheless, classically it is ok to expand around a reference metric, modulo questions of convergence. But when you posit that the metric asymptotically is flat, rather than just being expanded around flat metric, you make an assumption about physics which needs not be correct even on the classical level. And which, indeed, seems to be incorrect in our universe.

But surely somebody must have tested whether expansion around a Minkowski metric is ok in 2D. Because there we have complete control over the situation, with an exact CFT solution available. In 2D, the Liouville field is the metric, and expanding the Liouville field around zero is equivalent to expanding the metric around Minkowski. Conversely, if you cannot even trust the expansion around the Minkowski metric even in 2D, how can you attach any significance to it in 4D?

Re: Causality

The problem remains that two operators separated by a Minkowski spacelike distance commute, even if their separation is physically timelike. Of course, the last statement is murky when the metric is quantized.

I think you are fooling yourself by computing the commutator of non-gauge-invariant operators. Operators, which are not gauge-invariant, do not commute for spacelike separations, even in field theory.

Or maybe you’re not. Exactly what commutator are you calculating?

Posted by: Jacques Distler on August 9, 2005 7:21 PM | Permalink | PGP Sig | Reply to this

Re: Causality

OK, I have not calculated. But what you need is a local operator which is not gauge. In 2D gravity coupled to a single scalar, the energy-momentum tensor should do. At least within the context of lightcone quantization, this is not a gauge variable because the central charge, but the theory is still unitary.

Actually, this is probably not a very good example, because I don’t think that the lightcone will be moved. To find a local, non-gauge operator which moves the lightcone I have to resort to handwaving, but I think that all we need to do is to give the scalar field a mass. Then the Weyl symmetry disappears, and the anomaly moves to the diffeomorphism algebra. At least that is what happens in the massless case if we choose to gaugefix the Weyl symmetry rather than diffeomorphisms.

But let me return to the main topic of this entry, namely the claim that background independence is not very important in quantum gravity. If this were true, then it should be straightforward to recover the correct results for 2D gravity, including the intricacies of the Liouville mode, by expanding the metric around the flat one. Has this been done?

It seems to me that there will be severe problems. Only a discrete set of values for the central charge c < 1 leads to acceptable theories. I doubt that you can find such a discrete spectrum by expanding around a background metric. Unless you can reproduce the correct theory of 2D gravity in a background-dependent approach, how can you trust your results in 4D?

Discrete Series

Only a discrete set of values for the central charge $c<1$ leads to acceptable theories. I doubt that you can find such a discrete spectrum by expanding around a background metric.

I am not sure what you are talking about.

There is a discrete series of unitary conformal field theories with $c<1$. That statement is utterly independent of whether you study those theories on a fixed background geometry (the context in which it was first discovered), or attempt to couple them to dynamical worldsheet gravity.

Posted by: Jacques Distler on August 10, 2005 3:14 PM | Permalink | PGP Sig | Reply to this

Independent of what, again?

But let me return to the main topic of this entry, namely the claim that background independence is not very important in quantum gravity.

Nobody has claimed this!

When you perturbatively expand about a given background $B$ (= classical solution including, possibly, a choice of gauge) you want the result to be the same as that obtained by perturbing about a nearby background $B\prime$ so that, while the perturbative expansion makes use of a chosen background, it is indepenent of that choice, at least in the connected component of such choices.

There are detailed arguments (as rigorous as can be expected in this context, as far as I am aware), that this is indeed a property enjoyed by string (field) theory, open as well as closed. Though probably the open case is better understood. I can point you to a wealth of literature on this, if desired.

Maybe you want to argue that it would be desireable to have a method which is not just independent, but (as Jacques has put it here) manifestly independent of the choice of $B$ or $B\prime$, so that you don’t ever have to mention any choice of $B$ (as I have put it here).

In that case we would all quite agree!

Or are you arguing that perturbative expansion about a given background in quantum gravity itself is an insufficient tool for computing (graviton, say) scattering amplitudes in that background?

(I note that nobody seems to like my attempts to make this discussion more substantial by first making it more precise, but right now I see evidence to feel affirmed.)

If this were true, then it should be straightforward to recover the correct results for 2D gravity, including the intricacies of the Liouville mode, by expanding the metric around the flat one. Has this been done?

It seems to me that there will be severe problems.

So you are looking for a toy example computation in perturbative quantum gravity for the case of 1+1dimensional gravity coupled to some ‘matter’. I can’t help out with a reference right now, but I would like to point out that it is unreasonable to expect such a perturbative quantization to see all nonperturbative effects.

Posted by: Urs on August 11, 2005 10:17 AM | Permalink | Reply to this

Re: Independent of what, again?

In 2D gravity, just about everything of interest is non-perturbative. If this is any guideline to 4D gravity, failure to see such effects seems rather drastic.

Graviton scattering amplitudes is not the first thing that I would expect quantum gravity to answer. Rather, I would expect answers to the conceptual issues, like the problem of time. Actually, I think I have a very good, albeit non-original, suggestion for the nature of time: the ticks of a clock. The standard objection to this definition is that it only works on the clock’s worldline. However, this objection might be less serious than one could think. Formally at least, all fields can be approximated by data living on the clock’s worldline - do a Taylor expansion.

My concerns about causality applies perhaps even more to canonical quantization. They were formulated better than I am able to do in gr-qc/0412059. The author Savvidou might be one of Isham’s former students.

Re: Independent of what, again?

In 2D gravity, just about everything of interest is non-perturbative. If this is any guideline to 4D gravity, failure to see such effects seems rather drastic.

Ok, so the concern expressed here is:

Perturbative quantization sees only perturbative effects.

I think we all agree on that.

Probably the implication you have in mind is:

So how do you know that non-perturbative effects don’t ruin the uselfulness of your perturbative quantization of 4D gravity in string theory?

The answer is: Because we know how small they are. Since this has just recently been discussed in the blogosphere, let me just quote Jacques Distler from the recent discussion over at CosmicVariance:

But given that you don’t know what the non-perturbative theory is, can you be sure that non-perturbative contributions are under control?

Because, just as in field theory, the strength of the leading nonperturbative effects can be estimated from the growth of large-orders in perturbation theory.

In weakly-coupled field theory the $h$-loop amplitudes grow like ${g}^{2h}h!$, leading to the familiar (instantonic) ${e}^{-1/{g}^{2}}$ effects.

In string theory, the $h$-loop amplitudes grow like ${g}^{2h}\left(2h\right)!$, leading to stronger nonperturbative effects, of order ${e}^{-1/g}$. This is stronger than in field theory, but still controllably-weak at weak coupling.

We even have a good (but not great) understanding of what those leading nonperturbative effects are in may weakly-coupled string theory backgrounds.

There is, by now, a rather large literature on the subject…

Next you wrote:

Rather, I would expect answers to the conceptual issues, like the problem of time. Actually, I think I have a very good, albeit non-original, suggestion for the nature of time: the ticks of a clock.

This makes it almost sound as if the ‘problem of time’ were about identifying the ‘nature of time’. Certainly you have something else in mind, but it is not quite clear to me.

I am not sure in which respect you hope a choice of timelike curve is less severe than a choice of folitation. While not equivalent, both are closely related.

On the other hand, I am also not sure about the objections against a choice of foliation in canonical gravity. It is precisely the Hamiltonian constraint which ensures that no observable in the end depends on that choice.

The only restriction it gives is that it forces a spacetime product topology $\sim \Sigma ×ℝ$ upon us. Otherwise it seems fine - if you can make sense of the Hamiltonian constraint, that is.

So for instance in 1+1D gravity an ADM split works nicely and produces the ordinary results.

Posted by: Urs on August 11, 2005 5:38 PM | Permalink | Reply to this

Re: Independent of what, again?

Actually, I believe that string perturbation theory makes a quite strong assumption about physics: that the metric is asymptotically flat, because otherwise you don’t have an S-matrix. I.e., not only is the asymptotic metric expanded around a flat metric, but it actually is flat. This would not really be background independent even in your weak sense, would it?

The most striking thing in 2D gravity is the anomaly. This can be formulated in different ways; usually it is a Weyl anomaly, but if you start by gauge-fixing Weyl symmetry you find a diff anomaly instead. You can cancel the anomaly by adding extra fields, but no matter what you do, the quantum theory will have more degrees of freedom than its classical counterpart. This is an invariant statement which is true in both canonical and path-integral quantization.

Now let us assume that we can learn something about 4D gravity by looking at 2D gravity. As was argued by Roman Jackiw in gr-qc/9511048, p 20, the existence of diff anomalies will generalize from 2D to 4D. However, if you only quantize the fields, there can be no diff anomalies in 4D; this is because the relevant cocycles (there are four of them) are functionals of the clock’s worldline.

Hence the importance of introducing a physical clock (or observer, as I usually call it). Both a timelike curve and a foliation define a clock, but the former is local and material, whereas the latter is global and background. As a bonus, one obtains a way to quantize some general-covariant theories in a way which indeed is manifestly background free. The theories are not gravity, but regularizations thereof, obtained by replacing all fields by their p-jets. Background independence comes about because the space of p-jets is invariant under diffeomorphisms.

I think that you would react strongly if somebody claimed to have quantized 2D gravity without finding anomalies. Why should 4D gravity be different?

Re: Independent of what, again?

Actually, I believe that string perturbation theory makes a quite strong assumption about physics: that the metric is asymptotically flat, because otherwise you don’t have an S-matrix.

I wouldn’t say this is an assumption. It is rather a fact that it is hard to define the quantum gravity observables when there is no S-matrix available.

You can still perturb about non-asymptotically flat backgrounds, just that you don’t know how to compute observables in such a case.

I think that you would react strongly if somebody claimed to have quantized 2D gravity without finding anomalies. Why should 4D gravity be different?

Yes, we agreed on that about a year ago. I still, admittely, don’t know what it should mean for string perturbation theory.

But there might be a toy laboratory example for this question, too. One can write down $N=\left(2,1\right)$ string theories that describe strings which propagate in 1+1-dimensional target spaces, namely on the worldsheets of the ‘ordinary’ type I,II, Het strings, where they are interpreted as the quanta of the worldsheet supergravity fields.

(1)$\left(2,1\right)\to \mathrm{I},\mathrm{II},\mathrm{Het}\phantom{\rule{thinmathspace}{0ex}}.$

The question you are asking is:

What does the existence of the worldsheet anomaly in I,II,Het worldsheet supergravity theories mean for its stringy perturbation theory in terms of $\left(2,1\right)$ strings?

I don’t now. But surely somebody does.

Posted by: Urs on August 12, 2005 10:05 AM | Permalink | Reply to this

4D diff anomalies

Let us forget about string perturbation theory and look back at the two past posts:

TL> I think that you would react strongly if somebody claimed to have quantized 2D gravity without finding anomalies. Why should 4D gravity be different?

US> Yes, we agreed on that about a year ago.

Compare this with Jacques Distler’s statement that can be found here:
Well, since gravitational anomalies don’t arise in D=4, it’s not surprising that you “don’t see any problem”.

So something odd is going on here. You and I, and probably Jackiw and Nicolai as well, agree that 4D diff anomalies must arise in analogy with the 2D case. Distler, backed up by standard wisdom and Weinberg’s book (ch. 22), asserts that no such thing exists. Who is right?

Actually, both are right. The no-go theorem is based on certain axioms, mainly that the only thing we quantize are the fields. It is circumvented by explicitly introducing, and quantizing, the clock’s worldline. It leads to rather drastic conseqences, e.g. the Hamiltonian ceases to be a constraint, so we don’t have to sacrifice locality.

But why is nobody, not even the people who state that 4D diff anomalies must exist, interested in learning about the relevant cocycles and anomalous representations?

Re: 4D diff anomalies

I must admit that I am really still feeling out of my depth concerning the details of this issue. For instance I am not sure if the anomaly we see on the string worldsheet is the same as what would be called a ‘gravitational anomaly’ for the worldsheet gravity theory. Is it?

Posted by: Urs on August 12, 2005 11:40 AM | Permalink | Reply to this

Re: 4D diff anomalies

Well, you can consider the following references:

In section 5.1 of hep-th/0501114, Nicolai - Peeters - Zamaklar consider the free string as an example how quantization should work, and compare it to LQG where it does not work this way. So the free string should be 2D gravity.

The point that Weyl anomalies can be traded for diff anomalies was made by Jackiw in hep-th/9501016.

The point that one can learn something about 4D gravity from 2D gravity was emphasized by Jackiw in gr-qc/9511048. In particular, on p 21 he writes

“What does any of this teach us about the physical four-dimensional model? We believe that an extension in the constraint algebra will arise for all physical, propagating degrees of freedom: for matter fields, as is seen already in two dimensions, and also for gravity fields, which in four dimensions (unlike in two) carry physical energy. How to overcome this obstruction to quantization is unclear to us, but we expect that the resulting quantum theory will be far different from its classical couterpart.”

This is precisely my philosophy. However, nobody is perfect, and just because Nicolai and Jackiw write these things does not mean that they are true. But I think they are. And if diff anomalies do arise in 4D, you will need non-trivial reps of the diff algebra. Constructing such reps is not trivial, except in 1D. Jackiw continues:

“Especially problematic is the fact that flat-space calculations of anomalous Schwinger terms in four dimensions yield infinite results, essentially for dimensional reasons.”

I think that it is quite important that the difficulty Jackiw refers to has been eliminated; the first non-trivial reps were constructed by Rao and Moody in 1994. It is very frustrating to know that the key obstruction to quantum gravity has been identified and eliminated, and that nobody cares. I realize that I may sometimes have appeared quite aggressive, and that I have not always expressed my point very clearly; with a background in statphys and algebra it is not so easy to find the right formulations for particle theorists. But it is extremely frustrating when you have something important to say and nobody cares for a very long time; I published the first paper on the multi-dimensional Virasoro algebra in 1989. That ‘t Hooft, Smolin and perhaps even Witten contemplate that QM itself needs a revision does not make me think higher about the alternatives.

Not only string theorists can make progress in sharing their feelings.

Re: 4D diff anomalies

What I meant is that I don’t fully understand the relation between the ‘diff-anomalies’ that you are talking about and that what is called a ‘gravitational anomaly in the presence of chiral fermions’. Might be an embarrassing ignorance, but that’s how it is.

Another point to be aware of is the following: you quote Jackiw (it seems) as saying

We believe that an extension in the constraint algebra will arise for all physical, propagating degrees of freedom
[also in 4 dimensions]

Fine, but that requires any constraint algebra at all in the first place. But ordinary EH gravity in 4D does not admit any weakly continuous rep of their ADM constraint algebra, centrally extended or not, it seems.

You have that algebra, fine, but as long as this algebra does not arise from a quantization procedure of something its just an abstract algebra and tells us nothing about anything. (I know you have proposed something along these lines, but I am sceptical, to be frank.)

Posted by: Urs on August 12, 2005 2:38 PM | Permalink | Reply to this

Re: 4D diff anomalies

There is no relation between the chiral fermion-type anomaly and the new anomalies that I talk about. The former arise from coupling chiral fermions to gauge fields, the latter from normal-ordering effects when you express the fields in terms of Taylor coefficients. The latter can not be seen in field theory proper, because the cocycles are functionals of the observer’s trajectory. Unless you introduce this trajectory, you cannot formulate this type of anomalies.

The difference is clearer in Yang-Mills theory. Chiral-fermion anomalies are proportional to the third Casimir operator, whereas the observer-dependent anomaly is proportional to the second Casimir; it is also known as the central extension, although it does not commute with diffeomorphisms. The third Casimir vanishes for the standard model, the second Casimir does not.

I am not saying that all anomalies lead to consistent theories - some do and some don’t. An necessary condition for consistency is that the anomalous algebra possesses unitary representations. It appears that the Mickelsson-Faddeev algebra, which pertains to third-Casimir anomalies, has no good unitary reps at all, which would be a simple algebraic argument why such anomalies must be inconsistent. I gave a simple plausibility argument for that in math-ph/0501023.

About the constraint algebra: the difference between the Dirac algebra and the 4-diffeo algebra is an artefact of introducing a foliation. This is an unphysical choice. It must still be possible to express 4-diffeo modules, like tensor fields and connections, in the language of the Dirac algebra. In any covariant formalism, the relevant constraint algebra is the 4-diffeo algebra. The canonical formalism is not inherently non-covariant - canonical quantization means that Poisson brackets are replaced by commutators, and convariance that space and time are treated on an equal footing. It is quite well known that phase space can be given a covariant definition.

However, it would be nice if someone transcribed the 4-diffeo cocycles and representations to the Dirac algebra. I don’t think that I will try that - the project seems very cumbersome and not very rewarding. However, if you want to give it a try, your are most welcome.

I will be out of town over the weekend. Back Tuesday.

Credo: lowest-energy representations

The fundamental lesson from the free string is not really the anomalies, but that the constraint algebra has a lowest-energy representation on the kinematical Hilbert space. This observation is independent of the value of the central charge. What makes the string special is that we consider time-dependent gauge transformations.

In general, consider conventional canonical quantization of some gauge theory, say Yang-Mills. The spatial gauge generators J^a(x) generate an anomaly-free algebra, which thus can be modded out. However, the time-dependent gauge transformations

J^a(t,x) = exp(iHt) J^a(x) exp(-itH)

are also realized on the kinematical Hilbert space, and we must not neglect those. The Fourier transformed generators J^a(m), where 4-momentum m = (m_0, m_i), carry energy m_0 since

[H, J^a(m)] = m_0 J^a(m).

By definition, the vacuum state has minimal energy. Hence every operator which carries negative energy, including the gauge generators, must annihilate the vacuum, and the kinematical Hilbert space must carry a lowest-energy representation of the algebra of time-dependent gauge transformations.

This is the general argument why lowest-energy representations are important to quantization of gauge theories. Without such representations, we cannot quantize a gauge theory along the lines of the free string. Of course, nobody can deny the pragmatic succcess of QED and the renormalization program, but it is much less satisfactory than the free string from a purely mathematical point of view.

Thus, all I am saying is that one should not ignore time-dependent gauge transformations, and that energy in the kinematical Hilbert space is bounded from below. This inevitably lead to lowest-energy representations and thus to anomalies. I frankly see how these assumptions can be wrong.

Re: 4D diff anomalies

BTW: Today I found an envelope in my letterbox here at the institute, coming from Seswara Rao of Tata Institute, Mumbai. It contained a copy of

S. Rao Partial classification of modules for Lie-algebra of diffeomorphisms of $d$-dimensional torus Journ. Math. Phys. 45 8 (2004), pp. 3322

The abstract says:

We consider the Lie-algebra of the group of diffeomorphisms of a $d$-dimensional torus which is also known to be the algebra of derivations on a Laurent-polynomial ring $A$ in $d$ commuting variables denoted by $\mathrm{Der}\left(A\right)$. […] Earlier Larsson constructed a large class of modules, the so-called tensor fields, based on $g{\ell }_{d}$ modules which are also $A$ modules. We prove that they exhaust all $\left(A,\mathrm{Der}A\right)$ irreducible modules.

There was no further notice in the envelope, so I can only guess that this is meant as a comment on discussions like the one here.

Posted by: Urs on August 22, 2005 1:27 PM | Permalink | Reply to this

Re: 4D diff anomalies

First of all, we should understand that that the thing Thomas insists on calling “2D gravity” is something peculiar to the Polyakov string: a theory in which Weyl transformations are treated as a gauge symmetry. This is a very peculiar theory, unlike anything you would try to quantize in higher dimensions (well, unless you are interested in conformal supergravity, which you probably aren’t).

The Weyl symmetry is, of course, anomalous at the quantum level. But, unlike other anomalies, it receives corrections at all loop-orders (if you evaluate it in perturbation theory).

Thomas says that it can be traded for a gravitational anomaly. That is half-true. The real gravitational anomaly is measured by ${c}_{L}-{c}_{R}$, which, you will notice, only receives contributions at 1-loop (as you would expect for “real” gravitational anomalies). The Weyl anomaly is measured by ${c}_{L}+{c}_{R}$, which receives corrections at all loop orders.

Thomas is much impressed by the fact that, when the Weyl anomaly is uncancelled (as in string theory outside the critical dimension), you can sometimes successfully understand the resulting theory as one in which the Weyl mode is a dynamical degree of freedom in the theory.

(The worldsheet theory of) noncritical string theory, however, is not much like gravity in higher dimensions. A better model, which has been much studied, is dilaton-gravity in 1+1 dimensions.

Thomas is also impressed by the fact that one can quantize an “anomalous” $U\left(1\right)$ gauge theory in 1+1 dimensions. The resulting theory has an additional degree of freedom, a chiral scalar. There are many variations on the story, but let me explain the simplest.

Consider a single, right-moving, Weyl fermion, of charge-1, coupled to an abelian gauge field in 1+1 dimensions. The theory is anomalous. When you quantize it, à la Jackiw, you find that there is an extra left-moving chiral scalar, whose couplings to the gauge field are just such as to cancel the anomaly. If you “fermionize” this chiral scalar, you obtain a charge-1 left-moving Weyl fermion, and the resulting theory is plainly recognizable as a garden-variety non-anomalous theory, the (massless) Schwinger model.

There are higher dimensional analogues of Jackiw’s mechanism. They go under the general rubric of the “Green-Schwarz Anomaly-Cancellation Mechanism”. They occur whenever the anomaly polynomial factorizes, and there’s an appropriate $p$-form field in the theory.

The difference is that the higher-dimensional versions don’t have the stupid interpretation of making the gauge-mode (the thing that became the chiral scalar in Jackiw’s treatment) dynamical.

Posted by: Jacques Distler on August 12, 2005 4:09 PM | Permalink | PGP Sig | Reply to this

P.S. on Jackiw

Just a further note.

Even in 1+1 dimensions, if you consider both gauge and gravitational anomalies, the anomaly polynomial (except in the simplest example, cited above) typically doesn’t factorize, so you can’t cancel both the gauge and gravitational anomalies with a single chiral scalar.

(By gravitational anomaly, I mean the real thing, not the Weyl anomaly.)

Jackiw’s mechanism is also only tailored to cancel the local part of the gauge anomaly. Again, except for the simplest example, the global gauge anomaly is uncancelled by this mechanism.

Posted by: Jacques Distler on August 12, 2005 4:41 PM | Permalink | PGP Sig | Reply to this

Re: 4D diff anomalies

Nicolai et al argued that loop gravitists could learn something from the free string regarded as 2D gravity. Since LQGists are interested in 4D gravity, the lesson should have something to do with that, shouldn’t it? However, Lee Smolin argues that 2D gravity is very special and has no bearing on 4D. It seems like you agree with Smolin.

It is quite conceivable that you can cancel anomalies by adding extra fields - the Liouville mode does this, right? The important thing is that the quantum theory will have more degrees of freedom than what you naively expect from the classical theory. There are good reasons to expect this to happen in gravity. Without anomalies, there can be no local observables in quantum gravity. However, the world certainly looks local. Correlation functions appear to exist, and to diverge when the points approach is other. In particular, anomalous dimensions do not depend on the metric.

The key insight is that the explicit introduction of the observer’s trajectory (or clock’s worldline) changes the rules of the game. Without it, you cannot have any interesting general-covariant quantum theory. In particular, you cannot have any correlation functions which diverge when point coalesce, despite the fact that the notion of closeness is independent of the metric structure.

Curved backgrounds?

Urs, you wrote:

When you perturbatively expand about a given background B (= classical solution including, possibly, a choice of gauge) you want the result to be the same as that obtained by perturbing about a nearby background B’ so that, while the perturbative expansion makes use of a chosen background, it is indepenent of that choice, at least in the connected component of such choices.
There are detailed arguments (as rigorous as can be expected in this context, as far as I am aware), that this is indeed a property enjoyed by string (field) theory, open as well as closed.

Are you claiming that you can do perturbation theory around curved backgrounds? If you can compare the results for two backgrounds B and B’, it seems to me that at least one of them must be curved, however little. I thought that QFT over a curved space was a tricky subject.

OTOH, if both B and B’ are flat, they are the same up to a gauge, right?

Re: Curved backgrounds?

Are you claiming that you can do perturbation theory around curved backgrounds? If you can compare the results for two backgrounds $B$ and $B\prime$, it seems to me that at least one of them must be curved, however little. I thought that QFT over a curved space was a tricky subject.

Yes, the background can be curved.

For generic curvature the computation will be very hard. Not so for curved backgrounds on which the worldsheet CFT is exactly solvable. This includes notably WZW backgrounds where the background has a group metric and a Kalb-Ramond field is present which provides the parallelizing torsion of that group.

This is true for the holographically interesting ${\mathrm{AdS}}_{3}×{S}^{3}$ background, for instance.

(${\mathrm{AdS}}_{5}×{S}^{5}$ is again harder because here instead of Kalb-Ramond flux one has RR-flux, which makes things messy, and which makes all the recent progress on semiclassical strings in this background so interesting.)

Lots of string perturbation theory has been done on WZW backgrounds.

(But, yes, WZW backgrounds happen not to be phenomenologically realistic.)

In general, every background that satisfies the string background equations of motion can be used to expand about by computing CFT string diagrams. This is equivalent to saying that every 2D susy CFT of central charge $15$ defines a background about which you may perturb. These are all generically curved (if they have a geometric interpretation)!

Posted by: Urs on August 12, 2005 9:36 AM | Permalink | Reply to this

Re: Curved backgrounds?

I was under the impression that QFT on fixed curved backgrounds (at least in the hyperbolic case) is understood by now at least in principle. And concretely, you can caluculate anything you want like the first curvature correction to the Lamb shift or whatever.

For PW: I don’t want to say that there are no QFT problems remaining but I think that introducing a curved background does not make matters less understood.

It’s technically more complicated than the flat space case and of course you lack a unique vacuum. But relative to your preferred vacuum things go thru as usual.

Posted by: Robert on August 12, 2005 9:50 AM | Permalink | Reply to this

Re: Background independence

It might be helpful to point out that many (but not all) of the issues related to the “background/quantum split”, that you are worried about, occur in the following simpler system.

Consider a gauge theory on a manifold, $M$, in which the gauge bundle is nontrivial. Since the bundle is nontrivial, you cannot use $A=0$ as a “reference connection”, about which to do your perturbation theory. Instead, you need to expand about some nontrivial connection,

$A={A}_{0}+\delta A$

treating $\delta A$ as your quantum field. You then might worry whether physics depends on the arbitrary choice of reference connection, ${A}_{0}$.

Posted by: Jacques Distler on August 8, 2005 7:56 AM | Permalink | PGP Sig | Reply to this

Classical or Quantum

Y’know, it occurs to me, on rereading your question, that perhaps you are not even really asking about the quantum theory at all.

You could ask a question about the signal propagation in the classical theory – whether they propagate along the light-cones determined by the “background” metric, ${g}_{0}$, or the along the light-cones determined by the full metric, $g={g}_{0}+h$.

This question is addressed by the `sci.physics.research` posting by Steve Carlip, that Robert mentioned earlier.

Posted by: Jacques Distler on August 9, 2005 3:49 PM | Permalink | PGP Sig | Reply to this

Re: Classical or Quantum

Come on! In the classical theory, you don’t have to worry about different notions of causality. You solve the equations of motion first, and use the solution metric to define the lightcone. In quantum theory, you need the notion of spacelikeness from the outset, because operators on a spacelike surface commute by definition.

Re: Classical or Quantum

In the classical theory, you don’t have to worry about different notions of causality. You solve the equations of motion first, and use the solution metric to define the lightcone.

Yes, but the point was that even if you compute the lightcones perturbatively at the classical level, the lightcones come out those of the perturbed metric $g+h$.

Same for the quantum theory: You compute the propagation of some ‘probe’, but in the presence of some perturbative gravitational field $\eta +h$, so that there are $h$-vertices in your diagrams, and these will ensure that the propagation is causal with respect to $\eta +h$.

(Unfortunatly I don’t know which post by Steve Carlip precisely Robert was referring to. Maybe he could provide the link.)

Posted by: Urs on August 11, 2005 10:26 AM | Permalink | Reply to this

Re: Classical or Quantum

Urs,

have you actually done this calculation or do you have a reference or do you (like me) just believe it’s true?

In the classical setting, the precise formulation of the problem is I think clear: You formulate it as an initial value problem and then work out the domain of dependence of the solution on a change of initial conditions in a (small) bounded region. Then the news about the change of the initial conditions should travel along the physical lightlike geodesics.

Regarding Carlip’s posting, maybe my memory failed me. Sorry. What I can find is this and this .

Posted by: Robert on August 11, 2005 10:54 AM | Permalink | Reply to this

Re: Classical or Quantum

or do you (like me) just believe it’s true?

Yes. I would however be hard-pressed figuring out what there is to be checked, its’s sort of obvious, isn’t it? But that’s why I wanted to see what Carlip really was talking about.

Then the news about the change of the initial conditions should travel along the physical lightlike geodesics.

So first do this exactly and then drop all terms except those of first order. That’s your ‘perturbative result’, classically.

Doesn’t the entire agrument reduce to noting that within the domain of convergence we can Taylor expand any function around any point?

Same holds for the path integral, and its independence from the chosen background we perturb about (up to the fact that most path integrals are fictitious, of ocurse, ;-)

Posted by: Urs on August 11, 2005 11:34 AM | Permalink | Reply to this

Re: Classical or Quantum

Doesn’t the entire agrument reduce to noting that within the domain of convergence we can Taylor expand any function around any point?

I don’t quite understand what you mean by this. What I had in mind was you write $g=\eta +\kappa h$ and plug this into the Lagrangian. Then you obtain your equations of motion (at some point you have to gauge fix of course). These will roughly look like

(1)${\partial }^{\mu }{\partial }_{\mu }h=\mathrm{interactions}$

where you keep all terms up to some power of $\kappa$ in the interactions. This PDE should be hyperbolic (in nice gauges) and you can give $h$ and the first fundamental form on a Cauchy surface.

Now, naively you would think that if you change your Cauchy data in some region $K$, this spreads out along the characteristics of ${\partial }^{\mu }{\partial }_{\mu }$ that is, the solution only changes in regions time- or lightlike to $K$ (measured with $\eta$).

If things work out correctly however, everything conspires and the solution changes according to the regions that are causal to $K$ with respect to $g$.

This, I think, is non-trivial and not just a statement about stuff being analytic.

Posted by: Robert on August 11, 2005 1:00 PM | Permalink | Reply to this

Re: Classical or Quantum

Robert,

maybe I am overlooking something, or not expressing myself well, but I cannot quite see how it is nontrivial (in the classical case).

Take an exact solution, and split it as $g+h$. Expand everything in sight in powers of $h$. By dropping higher powers you get the perturbation series that you are talking about. Lightcones and everything is now determined by your solution to the given order of $h$.

I believe this is the same idea that Steve Carlip expresses in the first of the two postings that you cited, where he says:

note that one way to get the equations for $j$ is to write down the full Einstein equations for ${g}_{0}+h+j$, find the linearized equation for $j$, and then expand the result in powers of $h$, keeping only terms with fewer than n powers of $h$.

He states his conclusion more carefully than I did, by saying that he can’t see how this could possibly give light cones coming from 0th order in $h$. But in fact, the light cones will surely be determined precisely to that order above which we threw away all terms. How else?

Posted by: Urs on August 11, 2005 1:59 PM | Permalink | Reply to this

Re: Classical or Quantum

Maybe it’s easier to see using some path integral reasoning:

Say we have a theory of gravity $g$ coupled to matter $\varphi$ with action $L\left(g,\varphi \right)$. The path integral is

(1)$Z=\int \left[\mathrm{Dg}\right]\left[D\varphi \right]\phantom{\rule{thickmathspace}{0ex}}\mathrm{exp}\left(\mathrm{iL}\left(g,\varphi \right)\right)$

and you are worrying about the fact that the metric is not fixed but ‘fluctuates’.

However, decomposing the path integral as

(2)$Z=\int \left[\mathrm{Dg}\right]\int \left[D\varphi \right]\phantom{\rule{mediummathspace}{0ex}}\mathrm{exp}\left(i{L}_{g}\left(\varphi \right)\right)\phantom{\rule{thinmathspace}{0ex}},$

where

(3)${L}_{g}\left(\cdot \right):=L\left(g,\cdot \right)$

is the Lagrangian for the matter $\varphi$ on the fixed background geometry $g$, shows that in principle the path integral involves doing ordinary quantum field theory

(4)${Z}_{g}:=\int \left[D\varphi \right]\phantom{\rule{thickmathspace}{0ex}}\mathrm{exp}\left({\mathrm{iL}}_{g}\left(\varphi \right)\right)$

in curved space and summing the result over all geometries

(5)$Z=\int \left[\mathrm{Dg}\right]\phantom{\rule{thickmathspace}{0ex}}{Z}_{g}\phantom{\rule{thinmathspace}{0ex}}.$

A simple argument of this sort is also used in that Maldacena paper which Hawking has based his recent argument on. There the correlation functions of some matter theory on a fixed geometry $g$ are computed, and it is noted that a certain consistency condition coming from holography is satisfied only if we don’t forget to sum up all these correlations for different background geometries $g$.

Posted by: Urs on August 11, 2005 10:51 AM | Permalink | Reply to this

Re: Classical or Quantum

Well, except that this argument is quite formal and that the effective action $S=\mathrm{log}{Z}_{g}$ will be horribly nonlocal.

Posted by: Robert on August 11, 2005 10:59 AM | Permalink | Reply to this

Re: Classical or Quantum

except that this argument is quite formal

Right, it’s of course very ‘formal’, but less so than the idea which was expressed here (and elsewhere) before, that perturbative quantum gravity has some intrinsic problem because of ‘fluctuating causality’. In as far as we believe to be able to make sense of path integrals (which is a big assumption, of course), this argument holds, doesn’t it?

Posted by: Urs on August 11, 2005 11:44 AM | Permalink | Reply to this

Re: Classical or Quantum

Im pretty sure causality is more or less guarenteed to be ok, so long as one is satisfied that G = g0 + h is a good approximation. (Alternatively in the quantum version, one might worry that there is no good prequantization bundle definition)

There might be some fuzzy issues if the background metric has no killing vectors, but c’est la vie, even in the classical case thats hard to deal with.

But then it begs the question, why are we so sure we can do that in the first place, particularly very close to the Planck scale and in regions of strong curvature? Just b/c the instanton configurations are under control, doesn’t necessarily guarentee that the rest of the nonperturbative spectrum is ok. Or am I missing something?

Posted by: Haelfix on August 11, 2005 11:40 PM | Permalink | Reply to this

Re: Classical or Quantum

doesn’t necessarily guarentee that the rest of the nonperturbative spectrum is ok.

Not sure. Which rest?

Posted by: Urs on August 12, 2005 1:58 PM | Permalink | Reply to this

Re: Background independence

“We should prevent too many people from getting the idea that string theory is obviously wrong as it ignores the basic notion of background independence”

Finally back home again and settled in. I haven’t read through the numerous comments on this thread, but here’s two cents worth anyway:

I think I appreciate that String theory does not ignore background independence in the sense that many people seem to think it does because of target spaces etc. However (and this is a very big however) it still fails to be background independent in the deep sense that Lee is trying to refer to, although he fails by advocating standard LQG ideas, which (as every String theorist knows) also fail to be truly background independent.

Background independence in GR, as far as I am aware, can best be understood in terms of twistor theoretic ideas. Etc etc. I’ve spoken about this often enough!

Posted by: Kea on August 11, 2005 10:57 PM | Permalink | Reply to this

Re: Background independence

The distinction between background independence and diffeomorphism invariance is merely a semantic one. Every reasonable theory can be formulated in a coordinate-free way, but this statement is devoid of physical content and completely uninteresting. Analogously, you can make every theory scale invariant by adding a compensating dilaton field. So you may say that a massive theory is scale invariant, whereas a massless theory is scale invariant without a background dilaton. Such a viewpoint may have some merits, but mainly it is a source of confusion.

In contrast, the existence of diff anomalies is a matter of physics. In hep-th/9501016 Jackiw discusses 2D gravity coupled to matter from a somewhat unusual viewpoint. Usually, you gauge-fix diffeomorphisms and find a Weyl anomaly. If you instead gauge-fix the Weyl symmetry, a diff anomaly pops up instead. This generalizes to massive scalars, which are not Weyl invariant. This is a matter of substance rather than just semantics.

Re: Background independence

The distinction between background independence and diffeomorphism invariance is merely a semantic one.

I don’t think I would agree with this statement. To me, both are really quite different concepts. Diffeo invariance is a sort of gauge invariance, while ‘background’ has (at least) two meanings which are not really related to gauge invariance. I have tried to discuss these two meanings in another comment.

Posted by: Urs on August 4, 2005 6:31 PM | Permalink | Reply to this

Re: Background independence

No matter what you define, people (there seems to be many in each camp) will keep using the term diff invariance in two different ways: background independence and a coordinate-free formalism. That the same term is used for two things is unfortunate and leads to confusion, but in the end it does not matter; it does not affect the outcome of any calculation. However, only background independence has a physical meaning, and it is used in analogy with scale invariance.

That diff symmetry is not a gauge symmetry on the quantum level in 2D gravity with gauge-fixed Weyl symmetry does matter, however.

Re: Background independence

No matter what you define, people (there seems to be many in each camp) will keep using the term diff invariance in two different ways: background independence and a coordinate-free formalism

Not sure if many people would really equate ‘diff inavriance’ with ‘background independence’.

(In any case, it is not true: e.g. I can fix some Riemann normal coordinates when writing down the EH action and the metric would still be a dynamical non-background field.)

Even if, my whole point here is that once such sloppy nomencalture is replaced by sufficiently well defined concepts, much of the argument that is going on will evaporate into tautology. (And we are talking about ‘philosophical’ arguments about what a QG theory should look like, not the outcome of any calculations.)

Posted by: Urs on August 5, 2005 5:34 PM | Permalink | Reply to this

Re: Background independence

I more or less agree with the paper. All theories of nature should have a background dependant formulation, or else you are guarenteed to have no classical solutions.

But let me try to play devils advocate.

There are some General Relativists who are still concerned that linearized gravity need not equate to full GR. So just b/c your theory outputs Einstein Hilbert locally, it could still fail or be non unique globally.

Also, there is the rather troubling point of view inherited from semiclassical gravity. A priori, there is a back reaction on the gravitationaly induced stress energy tensor from quantum effects that should change the dynamics. This has caused unending amounts of confusion in the ‘qft in curved spacetime’ field, pretty much b/c its hard to find divergence free covariantly conserved quantities to make sense of things. I personally often have difficulty reconciling QFT in curved spacetime with what String theory is trying to tell us.

For instance, in the former one tends to seperate the metric into two parts (the geometry and small perturbations thereof like what the graviton can induce), its tacitly assumed that you can then pull the graviton term over into the matter side of the equation and treat it like a fluid. Now b/c of technical renormalization and covariance issues arising from the quantization procedure, its hard to set the background metric = flat space, instead you have to keep the background field arbitrary. Otoh in string theory, flat space is more or less priviledged.

Finally to take your Yang Mills example. Assume we have picked a zero section, and a local trivialization the worry is that you have now obscured yourself from the global meaning of say the connection, along with all the nonuniqueness that can entail. eg the local lie algebra that we observe can have several nonequivalent structure groups. The worry then is when we start using ST for cosmology, its not quite clear what that even means anymore.

Posted by: haelfix on August 4, 2005 3:47 PM | Permalink | Reply to this

Local vs. Global

Finally to take your Yang Mills example.

[…]

The worry then is when we start using ST for cosmology, its not quite clear what that even means anymore.

I feel that at the point which you are referring to here Robert is being a little vague in that somehow a choice of background is compared to a choice of gauge. While there is something to this, shouldn’t we sharpen this argument a little?

But apart from that what you say raises a point which I find quite interesting. We have a pretty good idea what a string field is locally. What is actually known about the nature of string fields globally?

There must be something intersting going on here. Aspects of this question should be known seceretly in the guise of anomaly cancellation in the respective effective field theories. So for instance the condition that a string field for the spin aspects of heterotic strings be globally definable is equivalent to the $\mathrm{SO}\left(32\right)/{E}_{8}×{E}_{8}$ condition (as briefly recalled recently).

But it seems that it would help to understand more conceptually and more generally what a string field ‘really is’, globally (or ‘intrinsically’, if you like).

Posted by: Urs on August 4, 2005 5:05 PM | Permalink | Reply to this

Re: Background independence

Robert,

In case you care, I can tell you that I pretty much agree with what you write.

I should maybe add that I believe the comparison to Yang-Mills theory that you make could be sharpened, in my humble opinion. What should be compared is GR coupled to any fields with YM coupled to charged fields. The issue of gauge fixing in both cases is really a little different from the general issue of background dependence, I believe.

I feel however that the entire discussion is suffering a little from the fact that different people in different situations seem to mean different things when talking about ‘backgrounds’ in physics. My feeling is that once we adopt and all agree on a clear-cut definition of terms, half of the confusion (if any) and half of the apparent disagreement will dissolve.

I checked with Wikipedia and it seems that nobody has yet written an entry on ‘background independence in physics’ or anything like that. I would want to propose some definitions and have them discussed here. Maybe we end up with something that we could post to Wikipedia for future reference.

Here is my proposal, the main point being that there are at least two different meanings of all the relevant terms floating around:

Background (in theoretical physics)

1) A parameter ${p}_{i}$ of a Lagrangian $\left(\left\{{p}_{i}\right\},\left\{{\varphi }_{i}\right\}\right)↦L\left(\left\{{p}_{i}\right\},\left\{{\varphi }_{i}\right\}\right)$ that is not varied, as opposed to a dynamical field ${\varphi }_{i}$ :

(1)$\delta L\left(\left\{{p}_{i}\right\},\left\{{\varphi }_{i}\right\}\right):=\sum _{i}\frac{\delta }{\delta {\varphi }_{i}}L\left(\left\{{p}_{i}\right\},\left\{{\varphi }_{i}\right\}\right)\delta {\varphi }_{i}\phantom{\rule{thinmathspace}{0ex}}.$

2) A solution to classical equations of motion about which the quantum theory is computed perturbatively.

Background Independence

1) Given a theory $L\left(\left\{{p}_{i}\right\},\left\{{\varphi }_{i}\right\}\right)$ with background parameters $\left\{{p}_{i}\right\}$ and dynamical fields $\left\{{\varphi }_{i}\right\}$ the theory $\stackrel{˜}{L}\left(\left\{{p}_{i}{\right\}}_{i\ne {i}_{0}},\left\{{\varphi }_{i},{p}_{{i}_{0}}\right\}\right):=L\left(\left\{{p}_{i},{p}_{{i}_{0}}\right\},\left\{{\varphi }_{i}\right\}\right)\phantom{\rule{thinmathspace}{0ex}},$ which is given is given by the same Lagrangian functional but with the parameter ${p}_{0}$ regarded as a dynamical field instead of a background parameter, is said to be independent of the background ${p}_{{i}_{0}}$.

2) A theory whose perturbative quantization does not depend on a fixed but arbitrary choice of classical background is said to be background independent.

(Even though I stated these definitions in some pseudo-formal fashion it is clear that we could make these definitions much more precise if desired. But I think the main point should be clear.)

It should go without mention but I will mention it nevertheless that we all want the metric not to be a background in sense 1) but that there is nothing wrong with (and it is in fact desirable for) it being a background in sense 2).

Also, perturbative string theory is well-known to be background-indepent in the second sense (at least under infinitesimal shifts of classical solutions) and is trivially background indepent (of the metric) in the first sense (since the metric tensor is not fixed, otherwise there would be just a single classical solution for it, which there is not).

What perturbative string theory is not is (by definition) being in a form that would allow quantization without ever using any (fixed but arbitrary) background in sense 2), namely a classical solution to its equations of motion.

I strongly believe that those people who complain that perturbative string theory is not a background indepentent quantization of gravity really mean (and should really be saying) that it is not a non-perturbative quantization of gravity, i.e. one in which you never have to mention a background in sense 2).

Which is true, trivially, and debated by nobody.

Posted by: Urs on August 4, 2005 5:58 PM | Permalink | Reply to this

Re: Background independence

In your definition(s) of “background” you should be slightly more general as the background could be more general structure than numerical parameters ${p}_{i}$ as it could for example envolve things like topology etc.

Second, I tried to argue that your definition 2) is a special case of 1): By writing your fluctuation ansatz as $\mathrm{field}=\mathrm{background}+\mathrm{fluctuation}$ the background is exactly one of those parameters that is not varied as in 1).

Posted by: Robert on August 5, 2005 3:16 PM | Permalink | Reply to this

Re: Background independence

Hi Robert,

In your definition(s) of ‘background’ you should be slightly more general as the background could be more general structure than numerical parameters ${p}_{i}$ as it could for example envolve things like topology etc.

Agreed. I guess it would be fair to say that we could always encode everything we want in a list of numerical parameters, but, as I said,

we could make these definitions much more precise.

Next you wrote:

Second, I tried to argue that your definition 2) is a special case of 1): By writing your fluctuation ansatz as field=background+fluctuation the background is exactly one of those parameters that is not varied as in 1).

I see what you mean. While this would probably be compatible with how I stated the definitions, I am not sure if it is a helpful way to look at things. Maybe it is, but allow me to ponder this a little:

I feel that replacing a dynamical field ${g}_{\mu \nu }$ by a field ${h}_{\mu \nu }$ defined by

(1)${h}_{\mu \nu }={g}_{\mu \nu }-{g}_{\mu \nu }^{\left(0\right)}$

should not be an example of the step which I have indicated in part 1) of my definition of background independence above. Instead, it seems to be rather a redefinition of the dynamical fields as in

(2)${\stackrel{˜}{\varphi }}_{i}={\stackrel{˜}{\varphi }}_{i}\left(\left\{{\varphi }_{j}{\right\}}_{j}\right)\phantom{\rule{thinmathspace}{0ex}}.$

I would think the hallmark of taking a background dependent theory $S$ and freeing some of its parameters by declaring them to be dynamical fields, thus getting a less background dependent theory $\stackrel{˜}{S}$, is that varying the Lagrangian of $\stackrel{˜}{S}$ really involves a ‘higher dimensional’ space as for $S$ (in the vague but hopefully obvious sense). This is not the case when we switch from ${g}_{\mu \nu }$ to ${g}_{\mu \nu }-{g}_{\mu \nu }^{\left(0\right)}$.

But actually my main point in distinguishing the two definitions of background was that they are really disparate concepts used in different contexts.

There are background free theories in sense 1) which can be quantized by expanding about a background in sense 2), and quite some of the discussion that I have seen did not distinguish between the word ‘background’ in these two cases and ended in confusion because of that.

I am not saying that this applies to your dicussion, I know that you are perfectly aware of this rather trivial (though important) distinction, of course. I just felt it might help to state once and for all clearly that there is this difference.

In as far as one is interested in the endless argument about whether string or LQG or whatever follow the ‘right philosophy’ this is sort of crucial.

Are you interested in us writing up a brief discussion of ‘background independence’ that we would submit to Wikipedia? Of course I could just go ahead if I liked, but I would enjoy having this discussed first.

Posted by: Urs on August 5, 2005 5:25 PM | Permalink | Reply to this

Re: Background independence

Exactly.

In one loop truncated nonrenormalizable gravity, there is the ‘perturbative back reaction’ problem, b/c gravity gravitates and one is always forced by hand to run consistency checks to make sure the approximation remains valid in the dynamics. The problem is completely absent if you have the nonperturbative sector ironed out.

ST is of course even better in this regards because it is renormalizable.

Posted by: Haelfix on August 5, 2005 1:40 AM | Permalink | Reply to this

Re: Background independence

>ST is of course even better in this regards because it is renormalizable

Has this been proved (in the sense that e.g. QED is)? I’m not completely in touch, but I think I would have heard of it. And when you say ST, do you mean all five (I IIA IIB, SO and E8)? Or are you thinking M-th and weak coupling limit + compactification etc?

Also, is the non-pertubative aspect(s) of ST really all worked out?

As you may have guessed, I’m also devil-advocating. My views on the whole quantum geometry/string theory battle are pretty much neutral.

I gather the main complaint about ST is that one has to put in the “minimum” we are expanding around. Perhaps the aim of the others is to have a theory, find the minimum, then expand pertubatively if that is what one wants to do. I must say, however, that I am originally from a ST background (at least, I know more about it than Q-Geom - I never did any real work on it) so don’t anyone take the above paragraph as authoritative.

D

I notice our friend Kea is silent, but I imagine an argument converting noone is not what we want.

Posted by: David Roberts on August 5, 2005 2:26 AM | Permalink | Reply to this

UV Finite

ST is of course even better in this regards because it is renormalizable

String Theory is better, even, than that.

A renormalizable theory has UV divergences which can be removed by appropriate counterterms. String Theory is UV-finite. No renormalization is needed (or, truth be told, possible).

Really, String Theory behaves, in many ways, like a theory with a cutoff. But, unlike any field theory cutoff you’ve ever thought of, this one does not affect the global symmetries (Poincaré, …) of the theory. And, as long as you can verify BRST-invariance, the local symmetries (e.g., infinitesimal diffeomorphisms) are preserved as well.

Also, is the non-pertubative aspect(s) of ST really all worked out?

Oh, yes, I just finished them this evening. Alas, they are too lengthy to include in the margins of this weblog comment. ;-)

Seriously, if the subject was all worked out already, there wouldn’t be much future to working on it, would there?

I gather the main complaint about ST is that one has to put in the “minimum” we are expanding around. Perhaps the aim of the others is to have a theory, find the minimum, then expand pertubatively if that is what one wants to do.

I would say that the main complaint is the lack of a nonperturbative definition of the theory. Smolin argues that we’d have a better chance of figuring that out if we tried to recast what we know in a manifestly background-independent fashion.

I’m generally sympathetic (read: I don’t, currently, have a better proposal for how to attack this rather formidable challenge), but I don’t think any of the approach that Smolin and his colleagues have been pursuing hold out much promise.

Posted by: Jacques Distler on August 5, 2005 6:12 AM | Permalink | PGP Sig | Reply to this