Re: What's wrong with loop quantum gravity

From: Lubos Motl (motl_at_feynman.harvard.edu)
Date: 10/07/04 

Date: Thu, 7 Oct 2004 11:55:34 -0400

On Thu, 7 Oct 2004, Ralph Hartley wrote:

> > LM: It is critical that the number of parameters is finite, otherwise
> > the theory could not be defined.
>
> In the sense that a theory with one parameter can be defined, there are
> theories with infinitely many that can also be defined.

No, it cannot. A theory is only fully defined once the parameters are set,
and if there is an infinite number of them, it is impossible to determine
all of them and define the theory.

> A single real parameter is not a finite amount of information. It carries
> the same amount of information as an infinite number.

No, it does not. For any theory that depends on its parameters in a
continuous way - and virtually all meaningful theories in physics ever
proposed fall into this category - one can simply count how many
parameters it depends on, and two is always more than one. The
dimensionality of the parameter space is a well-defined quantity. An
infinite number of yet-to-be-determined, physically relevant real
parameters is unacceptable for a set of ideas to be called a "theory".

> But you can't determine that just by counting parameters. A theory with
> infinitely many can be more rigid than a theory with just one.

No, a theory with infinitely many parameters cannot be rigid at all, and
it certainly cannot be more rigid than a theory with one parameter.

> I could do it in a continuous way as well, but the formulation wouldn't be
> any less perverse :-)

Show us, I don't think that this would be possible, even in a perverse
way. Would it be some continuous, but highly unsmooth dependence? ;-)

> I'm pretty sure I could prove that there exist theories for which no
> physicist cold determine how many parameters there "really" are, or even if
> there are finitely many.

I'm pretty sure that such "theories" are irrelevant for natural science.
Maybe they are useful for religion?

> Perhaps, but it is your game as well.

I am not playing any games. I am talking about absolutely physical and
essential physical theories - of the type that was awarded by the Nobel
prize on Tuesday: pure QCD has no dimensionless parameters, for example,
and this is what makes it really powerful, predictive, rigid, and
convincing.

You are probably talking about something very different, and I think that
this very different thing does not belong to science.

> If you count alternate formulations as the same theory, I'm not even
> sure "number of parameters" is well defined, or is ever different from
> 1.

QCD has no dimensionless constants; QED has one; the Standard Model has
roughly 19+10 of them. You should learn how to count them before you make
nonsensical statements like "it is not possible to count". It's not that
hard.

> For example, consider a pertubative quantum gravity theory, which needs
> infinitely many parameters to completely define.

First of all, such a thing is not a "theory", and second of all, I really
don't intend to "consider" this kind of nonsense.

> You have repeated that over and over, but you haven't proven it. I don't
> see how you could, in light of the many predictions that *are* made by
> theories that *do* have infinitely many parameters.

Never. One can make an effective theory with a limited validity, and say
that there is a lot of new physics beyond it, but this physics is
irrelevant for some category of questions that we want to ask (physics at
low energies), at least if the questions should only be answered with a
limited accuracy. But such a theory must still have a finite number of
parameters that matter. If a set of ideas or rules has infinitely many
adjustable parameters, then it is *not* a theory and it should not be
considered by any scientist.

We are often using the word "theory" in different meanings. Above, I am
using the most frequent meaning - a set of rules and ideas that are able
to predict phenomena. The theory of special relativity is called a
"theory" but it is really a "metatheory" - a set of principles that every
"theory" (in the previous sense) must satisfy.

> A theory can make predictions without being defined *completely*. If it has
> undetermined parts, it can't predict them by definition, but it can predict
> other things.

An (incomplete) theory can only be used to predict a limited set of events
AB, without claiming to predict other things such as CD, if the truncation
to AB can be justified (or at least if it is true, at least in some
sense). If it is so, then the parameters affecting CD do *not* belong to
this theory. Only the parameters affecting AB are a part of this theory,
and the number of them must again be finite.

> Note that your argument applies *equally* well to theories with a single
> parameter, as those with infinitely many. Both carry an infinite amount of
> "physically relevant information".

No. Nineteen real parameters is always more than one real parameter and
everyone who is able to distinguish integers from each other must know
why. An infinite number of physically relevant real continuous adjustable
parameters makes a set of ideas useless and unacceptable as a scientific
theory.

> I suspect that the only reason you want to make the distinction is that
> otherwise you would have to reject theories with even one parameter, and
> too many good theories are in that class.

All these ridiculous statements are based on your assumption that one
equals infinity, which is not a correct assumption.

> > A completely non-predictive theory cannot be correct; in fact, it should
> > not even be called a theory.
>
> It would be useless, but there are lots of useless, but true, facts.

No, there are no *theories* like that. A single fact has no adjustable
parameters either, otherwise it is not a fact. ;-) For example, if you say
that the Bush approval rate is X where X is adjustable, then it is not a
fact but a vague combination of words without any information value.

> You say, "String theory is the most predictive theory of fundamental
> physics we have."
>
> I say, "Isn't that an open question?"

I say "No, according to what we know, it is not an open question."

> You seam to be trying to say, "Yes, but when we know the answer it won't be."

No, if it were an open question, I could not state it as a statement. It
is only you who is trying to make an open question out of it. I've
explained how predictivity is determined, and why the statement about
string theory is true.

> I say, "Answers first, inferences from the answers later. Open questions
> can go either way."

You're saying many things like that. Whatever.

> The number of "realistic string vacua." Basically the number of different
> "versions" of string theory

There are no "versions" of string theory. There is one theory only and it
has many states - even many superselection sectors (backgrounds) or
classical solutions.

> You can claim that string theory is unique. But it doesn't make predictions
> independent of the vacuum, so if you don't know which vacuum applies it's
> just a mathematical structure, not a physical theory.

You would have to explain what you mean by a "physical theory", because
otherwise your sentence makes no sense. In normal language, a quantum
physical theory is a set consisting of a Hilbert space, operators with
amplitudes that can be calculated from a finite amount of input and
interpreted as results of (any) experiments, and string theory undoubtedly
is a physical theory. The only (slightly) open question is whether it is a
correct theory of our Universe.

> No. I'm talking about string theory in *all* these cases.

Then you're incorrect in virtually *all* of these cases.

> This is the case where lots of vacua are stable, and agree with all the
> experiments we have done, or if you like with all that we could do. But
> there are so many that you can't get any new predictions out of it.

That's simply not true. You could have said this untrue statement about
any successful theory in history of science before the theory calculated
the answer for a given question - and it would be an equally wrong
statement in all these cases. You could say that Mendelejev's table was
useless because it had too many atoms and one cannot make anything with
such a complicated table. Your current statement is comparably
unjustified.

> This is the case where string theory is completely wrong, has nothing to do
> with our world, but doesn't make enough predictions for it to be proven wrong.

String theory certainly cannot be "completely" wrong. As far as the rough
scheme goes, string theory has already been proved correct.

> > I don't understand what it means. How can something in physics ever be
> > correct if it cannot predict?
>
> If you think you have been promised a completely predictable universe then
> I suppose it can't.

What can't? String theory certainly *can* predict, and it is the most
predictive theory we ever had. The question is whether the really relevant
detailed predictions can be extracted with the tools we have and we will
have, and whether they are correct.

> But there is certainly nothing incoherent in the possibility that the
> universe really *is* constructed from the vibration modes of tiny strings
> (etc.), but that knowing that fact is completely useless, because you can
> never find out enough about the state of the vacuum to make any predictions.

That's just ridiculous. If the string scale were separated from the Planck
scale by a lot - imagine near a couple of TeVs - we could actually produce
the strings, and their qualitative behavior would be independent of the
details of the compactification. Be sure that this would convince all
rationally thinking people that string theory is correct, and I don't care
about the rest (of the people).

> I understand that part perfectly. I think being predictive and being
> correct are even more independent of each other than you do.

Right. It's because you think that a theory can be correct even if it
predicts nothing, and you are wrong.

> But if the answer to that question is "almost anything can happen, those
> discrete choices cover the space of low energy theories very densely", then
> you have tortured the term "prediction" beyond it's breaking point.

This is certainly not the type of predictions that string theory is
producing - or is designed to produce. String theory (has been and) is
making bold and very specific predictions - so specific in fact that some
people are irritated by it and they prefer "theories" that don't make any
new predictions whatsoever. ;-) The predictions in the models that we
study are very quantitative, and the predictions for the real world are
bound to be equally quantitative once the correct background is found.

All the best
Lubos
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