Extended idea of diffeomorphism

[marcus]

this is getting repetitive

if you have some personal issue with me then write me a PM about it

if you think it is not a new mathematical topic to study]
almost smooth homeomorphisms
then show a link to a paper where they are studied
(I would be very glad to get one!)

 

[jeff]

you seem to be confusing the measure on R² with something about the Riemann integral on R¹ which is not really relevant here. this post does not seem to make sense for a ref on Lebegue measure see e.g. Halmos Measure Theory page 152[/B]

No marcus, I'm sure what I meant in my post is clear to anyone who understands the issues here. I was trying to help you by identifying a source of confusion. Don't twist my words to making it look like I'm confused since what I posted about lebesgue integration is correct and an obvious possible source of confusion based on your remarks. And btw, I have halmos and there's nothing on page 152 or on any other page that helps you.

you started this excursion into measure theory by
mentioning sets of measure zero[/B]

No marcus, you started it by claiming that the idea of functions that are smooth everywhere save for a finite number of isolated points is a new idea. Why would I have broached this issue otherwise?

I started out by asking whether studying almost smooth mappings was new or had it been done.[/B]

You claimed, as is clear for all to see, that it was in fact new.

Nobody yet has found a paper in which one studies the homeomorphisms of a manif which are smooth except on a finite set
so we must entertain the possbility that they are a new thing to study until and if someone finds a citation to a paper studying them[/B]

All things considered, this remark is bizarre.

I would hope that they have been, but I didnt find any evidence of it![/B]

Really? Then perhaps you wouldn't mind describing how you researched the subject?

it turns out to be a natural and mathematically interesting extension of the diffeomorphism group which makes a certain quantum state space separable

In your opinion, but it avoids the issue at hand.

I think the topic is now exhausted[/B]

I understand why you'd hope that I'll agree, but in my book, when someone is not only wrong, knows their wrong, but is as insulting as they are insistent, the topic isn't exhaused until that person admits their error and apologizes for taking up so much time of others who are trying to help.

 

[jeff]

if you have some personal issue with me then write me a PM about it

Trying to distract members from the substance of this debate by convincing them that this is really a personal matter shows your contempt of all members who disagree with you. You now owe the entire site an apology.

 

[marcus]

Marcus,

As far as I know the first paper about the invariants was

gr-qc/9803018

but you are right about the chunkymorphisms. They are a new invention of Rovelli. I haven't read the paper so I can not comment.

Nonunitary strikes me as probably among the most mathematically knowledgeable people occasionally posting here. His post here is about Vassiliev knot invariants---he gives a link here the first Gambini/Pullin paper using them in LQG.

I have bolded his surprising side-comment----almost smooth homeomorphisms are a new thing to study, mathematically speaking. That is as far as he knows, or anyone reading this thread knows.
Extending diffeos to have a finite set of singularities has fascinating consequences so for goodness sake if you think extended diffeos have been ever been studied (I'm talking to people who know what they are talking about) please find the paper and provide the link

(notice that nonunitary did not think they'd already been studied, which is suggestive but not conclusive)

 

[marcus]

Back on March 12, I raised the issue of whether this class of mappings has been studied or not, asked if anyone knew, and suggested that it might be a good line of research if it hadn't

The following quote from the first Fairbairn/Rovelli thread ("Separable hilbert space for LQG") provides some background on the question and asks this. See the bolded lines at the end.

https://www.physicsforums.com/showthread.php?p=161861#post161861

a new mathematical animal

there is some intriguing mathematics in the Rovelli/Fairbairn paper
(some apparently derives from talks with Alain Connes, some from a book by V.I.Arnold, some seems to be new with Rovelli.)
Among the mathematical ideas I like the "almost smooth physical fields" introduced on page 8 at the beginning of section 3, the section on the "Extended diffeomorphism group".

Here's an exerpt:

"3.1 Almost smooth physical fields

Consider a four-dimensional differentiable manifold M with topology Σ x R, as before. However, we now allow the gravitational field g to be almost smooth, as defined in the previous sections, that is: g is a continuous field which is smooth everywhere except possibly at a finite number of points, which we call the singular points of g.

Any such g can be seen as a (pointwise) limit of a sequence of smooth fields. We say that g is a solution of the Einstein equations if it is the limit of a sequence of smooth solutions of the Einstein equations. Call E* the space of such fields.

Let now φ be an invertible map from M to M such that φ and φ-1 are continuous and are infinitely differentiable everywhere except possibly at a finite number of points. The space of these maps form a group under composition, because the composition of two homeomorphisms that are smooth except at a finite number of singular points is clearly an homeomorphisms which is smooth except at a finite number of singular points. We call this group the extended diffeomorphism group and we denote it as Diff*M.

It is clear that if g ε E* then (φg) ε E* for any φ ε Diff*M. Hence Diff*M is a gauge group for the theory.

In the Hamiltonian theory, we can now take almost smooth connections A on Σ..."

In the section of Rovelli's book where it would naturally have come---around page 173 of the draft---this discussion was either omitted or implicit. I for one wanted to see it spelled out, and I've been wondering about the almost-smooth category, that now seems emerging. Perhaps someone knows of its being explored in some other context. If it isn't already explored it might be a good small research area in differential geometry, with the potential for becoming a healthy cottage industry (just a thought). Would be interesting to know if the mathematics has already been worked out.

Perhaps someone knows? It would indeed be interesting if the mathematics of these things has already been worked on!

 

[jeff]

Nonunitary strikes me as probably among the most mathematically knowledgeable people occasionally posting here. His post here is about Vassiliev knot invariants---he gives a link here the first Gambini/Pullin paper using them in LQG.

I have bolded his surprising side-comment----almost smooth homeomorphisms are a new thing to study, mathematically speaking. That is as far as he knows, or anyone reading this thread knows.
Extending diffeos to have a finite set of singularities has fascinating consequences so for goodness sake if you think extended diffeos have been ever been studied (I'm talking to people who know what they are talking about) please find the paper and provide the link

(notice that nonunitary did not think they'd already been studied, which is suggestive but not conclusive)

Although I don't begrudge you your confidence in nonunitary's opinions on mathematics, this really isn't a matter of opinion. However, eric forgy is a mathematician at MIT which has one of the strongest (actually according to "U.S. news and world report" the strongest) math department in the world. Ask him about it.

In the mean time, I've emailed rovelli and asked him whether he views his invocation of the almost smooth category as being truly novel in the mathematical sense, as you've claimed, the issue of the possible physical implications of this well-known idea - in this paper or any other setting - being a different matter which I invite you to explore on your own dime.

 

[marcus]

OK so no one offers counterexamples or a citation to earlier work. this could mean other people also think the F/R mathematical questions are new (as they seem to me)

at least no one [correction: except Jeff! :-) ] is telling me they are old, so I will inch ahead with this a bit more

it turns out that the diffeomorphism group was the wrong group
before LQG was explored, indeed ever since GR, in other words since before 1920, people noticed that GR, the theory of spacetime geometry, was
invariant under diffeomorphisms
so if one was going to quantize GR one had to make a diffeo invariant quantum theory

but this is wrong because of what it leaves out
General Relativity is not only diffeo invariant, it is also
invariant under "extended" diffeomorphisms

(what I am jocularly calling chunkymorphisms, mappings that are allowed
to be not smooth at a finite number of points like chunky peanutbutter)

so one should try to construct a quantum theory that is not merely
diffeo invariant but is even more: invariant under this larger group
of symmetries.

I have revised what I said: Nobody except Jeff is telling me that these extended diffeomorphisms are old.
However since I first asked here around March 12 no one has come up with any prior mathematical literature about them---i.e. about extending the diffeomorphism group in this way.

(I am still hoping because it would be very helpful if the field had been surveyed and the ground dug up a little already.)

In any case there is an element of free choice what symmetry groups one uses. What is a right group and a wrong group is a matter of opinion and persuasion, not proof. In this paper we have a very persuasive argument:

the original 1915 GR was already invariant under chunkymorphisms, only we apparently didnt realize this

this involves extending the idea of what is a solution of the einstein equation to include certain kinds of limits of solutions----it really needs rigorous exploration, the extended solutions can have points where they are not smooth---the metric can have finitely many points where it fails to be infinitely differentiable.

Probably it is not a good idea to get too worked up about sporadic lack of differentiability since at the quantum level no differentiability is expected at all. Smoothness is just a macroscopic appearance that things have. If one looks microscopically it goes away. So it is mostly a matter of mathematical convenience how much differentiability one stipulates in the classical precursor.

 

[jeff]

Okay, as promised, here's carlo rovelli's response to my email along with my response:

> Hi Jeff.
> You are certainly right that Christian Fairbairn and myself have not
> been the first using the idea of maps having a property "almost
> everywhere". In fact, these maps appear very often in math.

> Just to be precise, there is a difference between "almost everywhere"
> and "on a finite number of points". As you correctly say, "almost
> everywhere" means up to a set of measure zero. In 3d space, a line, a
> surface, or an infinite number of isolated points have measure zero,
> therefore map that fails to be smooth, say, on a line, is almost
> everywhere smooth, but it is not "smooth except on a finite number of
> points".

Yes, the distinct idea of lower dimensional spaces having measure zero according to the measure defined with respect to higher dimensional spaces was also discussed.

> We needed this difference for mathematical reasons in the paper (to
> have holonomies well defined). I am quite sure that these maps as
> well have been largely used in math.

> Which forum was this discussion in?

"Physics Forums"

I also asked him to come visit, so keep those fingers crossed.

 

[Haelfix]

This argument is sort of akin to when Mathematicians discuss which integration measure is 'best' and most 'profound'. I remember I had my opinion, way back when before I became agnostic. Alas, in general, there are as many ways to formulate such a thing, as there are living mathematicians.

Usually if there is a consensus it is that it depends on what problem is being studied. Which is highly unsatisfying of course, but it seems to be the case in practise.

So too is it the case with physics IMO.

If they wish to use this extended group for their physical calculations, and they get an answer, that's fine. So long as it possesses experimental consequences, I could care less about the intuitiveness of it or not.

 

[selfAdjoint]

"But as for the nature of the heavenly spheres, that is a matter for philosopher. Astronomers rather are concerned with accounting for the motions." - Claudius Ptolemy.

 

[jeff]

This argument is sort of akin to when Mathematicians discuss which integration measure is 'best' and most 'profound'. I remember I had my opinion, way back when before I became agnostic. Alas, in general, there are as many ways to formulate such a thing, as there are living mathematicians.[/B]

Actually, the argument was about marcus's ridiculous assetion that rovelli's paper involved the introduction of a mathematical idea heretofore unknown to mathematics. Marcus is wrong, knows he's wrong, but is too petty and insecure to admit he's wrong. Thus I emailed rovelli for confirmation that marcus is wrong and then posted his reply, which says that marcus is wrong.

 

[carlo rovelli]

Hi everybody,

Thanks a lot, Jeff, for posting my mails. After you have pointed
out this discussion to me, I have a bit read through. It has
been extremely interesting to read your considerations and your
thought.

Let me add a comment about the "chunkymorphisms" (I love this
word). I am sorry the discussion was a bit inflamed, but I can
understand why. In fact, I think both are right. We asked
ourself the same question repeatedly, while working on this paper
with Winston. We introduced this group a bit forced by the
mathematical constraints, and then were quite surprised in seeing
how well it worked as an extended symmetry of general relativity.
At times I thought that this group could open a new and
interesting area of exploration. On the other hand, I am not a
mathematician. Mathematicians in general have already explored
everything, so we also answered ourselves that "we are sure
mathematicians have already considered this in one form or
another".

We certainly do not claim that we have invented a new technique
in using maps that fail to have a property on sets of measure
zero. This is all over in math. But, to be precise, we have
searched, but not found any work discussing the category of the
"chunkymorphisms" (homeomorphisms that are smooth everywhere
except on a finite number of points). If you find out about
anything of the sort, I'd be very interested to know.

So, as you see, I sort of agree with both Jeff and Marcus. In
answering to Jeff, I was careful of avoiding any claim of
priority or any claim that I know that these are "new and
important". But I do not think that Marcus has it wrong. I may
add that reading various threads, I am very impressed by Marcus'
knowledge of the problems of quantum gravity, by his
understanding and his judgement.

In fact, I am quite impressed by the entire level of your various
discussions.

Thanks for the interest and good luck to all of you. Take care,
carlo

 

[ranyart]

I have revised what I said: Nobody except Jeff is telling me that these extended diffeomorphisms are old.
However since I first asked here around March 12 no one has come up with any prior mathematical literature about them---i.e. about extending the diffeomorphism group in this way.

(I am still hoping because it would be very helpful if the field had been surveyed and the ground dug up a little already.)

In any case there is an element of free choice what symmetry groups one uses. What is a right group and a wrong group is a matter of opinion and persuasion, not proof. In this paper we have a very persuasive argument:

the original 1915 GR was already invariant under chunkymorphisms, only we apparently didnt realize this

this involves extending the idea of what is a solution of the einstein equation to include certain kinds of limits of solutions----it really needs rigorous exploration, the extended solutions can have points where they are not smooth---the metric can have finitely many points where it fails to be infinitely differentiable.

Probably it is not a good idea to get too worked up about sporadic lack of differentiability since at the quantum level no differentiability is expected at all. Smoothness is just a macroscopic appearance that things have. If one looks microscopically it goes away. So it is mostly a matter of mathematical convenience how much differentiability one stipulates in the classical precursor.

Einstein made the Question of "CHOICE" a very important factor, if one was to read his book titled:Out Of My Later Years (a deliberate reference to the content of his thoughts, made to Future Generations who may still be around to read and learn), then careful examination of a number of aspects pertaining to what you are currently debating, become VERY..VERY intuitive and important.

The simplicity of what Einstein was really working on in his later life becomes apparent, contained not in some scientific journal or published paper..but tucked away in a 'popular book'

May I suggest you read the book Marcus, putting emphasis on your current postings.

 

[ranyart]

We certainly do not claim that we have invented a new technique
in using maps that fail to have a property on sets of measure
zero. This is all over in math. But, to be precise, we have
searched, but not found any work discussing the category of the
"chunkymorphisms" (homeomorphisms that are smooth everywhere
except on a finite number of points). If you find out about
anything of the sort, I'd be very interested to know.

carlo

There are certain 'homeomorphisms' for Entangled 'virtual' transformations. The 'image' here:http://groups.msn.com/Youcanseehomefromhere/tempusfugititalsodrags.msnw?action=ShowPhoto&PhotoID=29

where the 2-D mirrored sphere meets 3-D mirror plane surface correspond to Quasi-normal modes.

Viewed head-on from a direction of observation, the reflections of the web-cam for instance can only be viewed from a 3-Dimensional frame. If one was to sqeeze into the space where the area of 3-D sphere contacts the 2-D plane, then the area spectrum exchange's from 3-D >>2-D. Zooming in from a 3-D world we reach a point of 'directional-paramiterization' that will 'fix' the 2-D points of origin to be External, or the 2-D plane will allways surround the smallest possible 3-D space, contrary to finding any sub-space within 3-D space that is 2-D (Quarks are the smallest fractional components of any 'real' 3-D space) Matter and Space have limits that do not correspond by Equivilence.

The choice of how one performs dimensional correspondence will produce inequalities. The inequality of a 3-D matter and a 2-D space ensures that one can only perform certain functions going from 3-D Geometrics down to a certain limits.

Homeomorphisms of Spheres 3-D , and planes 2-D are like the images linked above, where the String theorists choose to look out from less than 3-D area's(inside-looking-out).

Imagine being inside a 2-D area looking out into a 3-D world?..:http://groups.msn.com/Youcanseehomefromhere/consciouswaves.msnw?action=ShowPhoto&PhotoID=59

Forever stuck in a frozen non-commutative and directionally fixed frame!..like the mirror's surface

 

[jeff]

Hi again,

I was a bit puzzled by this...

I am very impressed by Marcus' knowledge of the problems of quantum gravity, by his understanding and his judgement..[/B]

The thing is that it was the following posts I emailed you about:

it turns out that the diffeomorphism group was the wrong group
before LQG was explored, indeed ever since GR, in other words since before 1920, people noticed that GR, the theory of spacetime geometry, was invariant under diffeomorphisms
so if one was going to quantize GR one had to make a diffeo invariant quantum theory

but this is wrong because of what it leaves out
General Relativity is not only diffeo invariant, it is also
invariant under "extended" diffeomorphisms

this for me is the most remarkable bit of information that has come along here in some time
almost smooth mappings----diffeomorphisms except at a finite number of points----have not been studied before

do they suffer from some terrible pathology or something? what is wrong? this does not correspond with how I think of 20th century mathematics going into everything under the sun with almost obsessive thoroughness. how did they overlook chunkymorphisms

the idea of a diffeo which is allowed to be unsmooth at a few points
is a very simple idea
maybe it is so simple that no one thought it could have any worthwhile consequence

Marcus never did own up to these remarks, so I guess I don't really get what you meant, either by the above comment, or by your remark that we were both right. Marcus simply twisted things around, as he always does when he gets caught with his pants down.

I then got this email from carlo:

Jeff,
you are both extreemely smart guys, and you both hate been found wrong
on anything.

In discussing science, we all make all sorts of mistakes. The best
scientists are the ones that do not focus on others' mistake, but
focus on the interesting things that others say. If you start telling
somebody: "you are wrong, you are wrong, you are wrong", the only
result is that he freezes and becomes aggressive. Never do that. You
loos a good opportunity to discuss, and the only advantage that you
get is a feeling of superiority that is useless and just makes the
others dislike you.

We all make mistakes. I do all the time. You posted the line:
"The idea of mappings having a property save possibly at isolated
points is an important idea that was introduced into functional
analysis long ago. Such functions are said to possesses such a property
"almost everywhere" which means everywhere except on a set of measure
zero.
A student writing this is an exam of mine would fail. This is badly
wrong. But there is no reason of shame of course. It is the sort of
mistakes we do all the time.

When I wrote in the forum that Marcus has good understanding, I was
not thinking about this in specific. I was thinking about the many
comments of him I have found here and there in the forum.

But in the text my Marcus you forwarded to me, there is nothing really
wrong. Maybe the tone is a bit over-enthusiastic. But technically
speaking, it is all correct. So, technically speaking, he was right.

But you were also right in saying: "well, wait a minute, this sort of
things are done in math, and tune down his enthousiasm". In doing
so, you made a small technical mistake, confusing "alomost everywhere"
with "in a finite number of points". so, he focused in pointing out
your mistake and you focused in pointing out his exageration. this
leads nowhere. I insisted that I think that you were both right. I
myself told myself precisely those same sentences in thinking about
that. Namely "This is new and great", then "no, it is just like doing
almost everywhere, and then "but not really, because it is actually
different", and concluded "maybe it is new and interesting maybe not",
and "I wander what is in the math literature on this group". This
were my thights, and this is precisely the exchange that you and
Marcus had.

I think this is really great. I suppose that both of you are
far younger and know physics and math far less than me, therefore if,
with less tools and less experience, you are capable of arguing so
correctly about a topic, this means that you must be very brilliant.

Why wasting your brilliant mind in sterile polemics? find a way to
transform your excanges in something useful for both of you.
collaboration and friendship brings you very far. competition and
desire to be the smarter one leads nowere and riuns your life.

If lee smolin and myself had started telling each other "you are
wrong", at every step, and trying to outsmart each other, there would
be no LQG today. we ignored any tension and focused on being friend
and learn from each other. and we did good physics. you can do quite
good physics as well, if you want, with your intelligence. use it for
he best.

ciao
carlo

This was my response:

Hi carlo,

You pointed out that I said that

> "The idea of mappings having a property save possibly at isolated
> points is an important idea that was introduced into functional
> analysis long ago. Such functions are said to possesses such a property
> "almost everywhere" which means everywhere except on a set of measure
> zero.
> A student writing this is an exam of mine would fail. This is badly
> wrong.
>you made a small technical mistake, confusing "alomost everywhere"
> with "in a finite number of points".

There's nothing in my posts that indicates I was unaware that sets of measure zero can contain more than just isolated points. That's why I pointedly avoided saying - and I actually made this point quite explicitly in my posts - that "almost smooth" functions are the only examples of functions that are smooth almost everywhere, which of course would be wrong, as you mention. However, you did define almost smooth functions to be smooth everywhere except for a finite number of isolated points, i.e., on a set of measure zero. In the books on real or functional analysis that I studied, functions that have a property everywhere except on a set of measure zero - which includes sets containing only a finite number of isolated points - are said to be smooth almost everywhere.

> But in the text my Marcus you forwarded to me, there is nothing really
> wrong. Maybe the tone is a bit over-enthusiastic. But technically
> speaking, it is all correct. So, technically speaking, he was right.

As you've seen, marcus stated without qualification or equivocation that it's simply wrong to think of diffeomorphisms as the gauge group of GR and that the almost smooth category is a heretofore unknown concept in mathematics. If you agree that whether advertising speculation as fact should be dismissed as over-enthusiasm depends on the situation - and I don't see why you wouldn't - then I really don't think it's fair to fault me as you have since it doesn't take much to get people at these online forums to take posts like marcus's at face value, and there's nobody at PF whose mislead other members as much as he has.

Marcus habitually freezes out and defames anyone who directly challenges him, especially on the subject of LQG. I've tried many, many times to smooth things over with him and stop him from bullying people, but it's like he has no conscience or something.

Anyway, have no doubt that I appreciate very much your wasting time with me on this. I really am sorry about this whole damn thing.

Jeff.

Final email from Carlo:

Okay,
I understand. Fine. Sorry if it sounded too much against you.
I did not mean so. Take care and good luck for everyhthing.
ciao
Carlo

My conclusion

I think because of carlo's use of the word "Okay", and the absence of any indication of continued disagreement on the facts, it's reasonable to assume that he realizes now that I was right all along and that he blundered when he posted admiration for marcus and slapped me in the face. If you want to respond to this carlo go right ahead, after all you're a member, but I've got to tell ya, you disappointed the hell out of me. I have to wonder, was any of this a strings versus lqg thing?

 

[marcus]

It sounds like Rovelli has said his say and wished us well and left.

I want to reiterate the following. This is my personal opinion and is one of the things I find most surprising and remarkable in this context:

Originally Posted by marcus
----quote-----
this for me is the most remarkable bit of information that has come along here in some time
almost smooth mappings----diffeomorphisms except at a finite number of points----have not been studied before

do they suffer from some terrible pathology or something? what is wrong? this does not correspond with how I think of 20th century mathematics going into everything under the sun with almost obsessive thoroughness. how did they overlook chunkymorphisms


the idea of a diffeo which is allowed to be unsmooth at a few points
is a very simple idea
maybe it is so simple that no one thought it could have any worthwhile consequence
-----end quote--------

I am not talking about the general idea (very common, almost a cliche)
of studying functions with Property A except on a set of type B, an exceptional set. There is no useful conclusion I know of, at that level, because what happens depends so much on the specific A and B you chose.
Functions that are bounded except on a set of measure zero are a different kettle of fish from matices which are invertible except on a set of measure zero (or a discrete set or a submanif of lower dimension etc etc).

What I asked about back on March 12 here at PF was specifically about diffeos smooth except at a finite exceptional set.
It is an extremely simple idea (almost a Columbus egg, why didnt you or I think to study it?) partly because 20th c mathematicians have studied almost every thing like Proprty A except on exceptions B that you can think of.
So why wouldn't they have already studied this?
But they appear not to have done so!

Here's from my March 12 post about this (the day after their paper appeared on arxiv):

... wanted to see it spelled out, and I've been wondering about the almost-smooth category, that now seems emerging. Perhaps someone knows of its being explored in some other context. If it isn't already explored it might be a good small research area in differential geometry, with the potential for becoming a healthy cottage industry (just a thought). Would be interesting to know if the mathematics has already been worked out.

So since March 12 I am calling attention to this. To paraphrase, I have been saying: is this new? does anyone know of a paper studying this? (namely smooth except for finite) and since I can find nothing by search and nobody comes up with any paper I am taking the step of saying that (unlikely and surprising as this seems) yes this is a new type of function to study

furthermore it is important because these functions make a key hilbertspace in quantum gravity have countable (instead of uncountable) dimension.

 

[lethe]

This was my response:

Hi carlo,

You pointed out that I said that

> "The idea of mappings having a property save possibly at isolated
> points is an important idea that was introduced into functional
> analysis long ago. Such functions are said to possesses such a property
> "almost everywhere" which means everywhere except on a set of measure
> zero.
> A student writing this is an exam of mine would fail. This is badly
> wrong.
>you made a small technical mistake, confusing "alomost everywhere"
> with "in a finite number of points".

There's nothing in my posts that indicates I was unaware that sets of measure zero can contain more than just isolated points.

Jeff, in your initial post, which Rovelli cites, it sounds to me like you are saying that sets of isolated points are sets of measure zero. you did not in that post say that the notion of ioslated points and measure zero are equivalent, and i don't know why marcus would think that you did.

I agree with you that although sets of measure zero are not the same thing as sets of isolated points, the latter is example of the former. all sets of isolated points have Lebesgue measure zero.

However, in a later post, you make this statement:

...but sets of measure zero ... by definition contain only isolated points.

This statement is clearly mistaken. the definition of measure zero is not the same as the definition of isolated points. a set of measure zero need not contain only isolated points. indeed, it was at this point that i posted a counterexample to your statement: the xy-plane in R3. Here is a set which contains many points which are not isolated, and yet is measure zero.

perhaps you misspoke with your statement, and did not mean to imply that the property of "containing only isolated points" is the definition of measure zero sets, but rather just provided an example of a certain class of measure zero sets. but this is not what your wording indicated.

it is therefore not surprising that Rovelli, coming to the thread later, and perhaps influenced by marcus, might assume that you are confused about the difference between these two types of sets.

I think because of carlo's use of the word "Okay", and the absence of any indication of continued disagreement on the facts, it's reasonable to assume that he realizes now that I was right all along and that he blundered when he posted admiration for marcus and slapped me in the face.

i think a more reasonable assumption would be that Rovelli simply wasn't interested in continuing the dialogue.

 

[marcus]

Lethe this sounds like a fair and unpolemical thing to say and I am
grateful to you for making the point.
I also am longing for us to stop bickering personalities and focus on the mathematics

the question that fascinates me is
Are these mappings a new thing to study?
Are homeomorphisms smooth except on finite exceptions already studied or not. There must be a lot to find out about them.
They have the incredible property of chopping an uncountable infinity down to countable. There are probably a whole bunch of theorems to prove about these things. Does anybody know a paper that begins the process of finding out about this class of functions? I have done some searching but can't find anything

I hope we can keep it about the mathematics or the physics (no sharp distinction I can see) and muffle the vituperation

 

[eforgy]

Piecewise Smooth Manifolds and Diffeomorphisms

Hello,

There is an entire field of mathematics studying "piecewise smooth manifolds" and their respective diffeomorphisms, which are precisely homeomorphisms that are smooth except on a set of a measure zero. As has been pointed out (and is probably obvious to all concerned), sets of measure zero are not confined to isolated points. Depending on your measure, even isolated points might have nonzero measure.

It has been a while since I had any references on this material in my hands, but I believe that piecewise diffeomorphisms are probably general enough to handle the case where they are smooth except at isolated points.

Anyway, this might help you in your literature search. I am also of the opinion that there is nothing new here, and it will just be a matter of finding the correct references (which may use completely different terminology).


Eric

PS: With a little more restraint it would have been very nice to have Rovelli become a regular around here. I know I could definitely benefit from his insight. I also agree with lethe's assessment that his "Okay" simply means he lost interest in discussing the topic with us.

PPS: Isn't a black hole essentially a manifold with a point removed? In this case, GR is already known to handle diffeomorphisms everywhere except at isolated points because of its ability to deal with black holes. I'm probably missing something.

 

[marcus]

Hello,

There is an entire field of mathematics studying "piecewise smooth manifolds" and their respective diffeomorphisms, which are precisely homeomorphisms that are smooth except on a set of a measure zero. ...

... but I believe that piecewise diffeomorphisms are probably general enough to handle the case where they are smooth except at isolated points.

Anyway, this might help you in your literature search. I am also of the opinion that there is nothing new here,
...

PPS: Isn't a black hole essentially a manifold with a point removed? In this case, GR is already known to handle diffeomorphisms everywhere except at isolated points because of its ability to deal with black holes. I'm probably missing something.

Hi Eric thanks for the helpful post. I'm familiar with piecewise smooth and piecewise linear.

Indeed Fairbairn/Rovelli mention some research along the same lines as theirs which used piecewise linear IIRC. They cite research by Zapata which I think uses piecewise approach.

[edit: yes on page 8 "Another possibility, investigated by Jose Zapata, is to start from a piecewise linear manifold..."]

but I believe there is a non-trivial difference between finite exceptional set and piecewise

that is, I disagree with you that piecewise is general enough to include
finite----this however is a possible theorem for someone to prove! you think yes and I think no (except in dimension 1 or less) and a grad student could settle it by a counterexample or some small theorem! delightful.

I am very happy you have the opinion that "there is nothing new here" because my intuitive feeling is that there is and we are clearly going to see
whether or not there is---so having a difference of opinion in mathematical judgment makes it more interesting

PPS. I also was thinking about this black hole thing that you bring up.
it is very interesting. I do not see an obvious answer right away.

 

[jeff]

...you are saying that sets of isolated points are sets of measure zero.

Yep.

you did not in that post say that the notion of ioslated points and measure zero are equivalent, and i don't know why marcus would think that you did.

To be fair to marcus, there actually was a good reason why he thought this and it was my post that created the confusion. I cleared things up thusly.

Note that in my post,

"sets of measure zero - which can be finite or countably infinite - by definition contain only isolated points. It's only connected sets of points that can have nonzero measure: isolated points, and thus sets of isolated points since measures are additive, have measure zero simply because they have no measurable extension"

I carefully specified that "It's only connected sets of points that can have nonzero measure" as opposed to something like "connected sets of points can only have nonzero measure". Thus it should be reasonably clear that where I uncarefully used the phrase "can be", I probably intended "when", and this is in fact the case.

I suppose you could say if he'd read it more carefully, he might have said it sounds like I'm contradicting myself, which would've been true, but really, it was just an accident, as I've indicated, and I do understand measure theory.

I agree with you that although sets of measure zero are not the same thing as sets of isolated points, the latter is example of the former. all sets of isolated points have Lebesgue measure zero.[/B]

Yep.

However, in a later post, you make this statement:

This statement is clearly mistaken. the definition of measure zero is not the same as the definition of isolated points. a set of measure zero need not contain only isolated points.[/B]

Remarkably, this claim of mine is actually true! The classic example is the "cantor" set which is both uncountable (Edit: as opposed to countable, which isolated points are) and of measure zero. This is one of a number of bizarre results that almost made me drop physics and go into mathematics.

indeed, it was at this point that i posted a counterexample to your statement: the xy-plane in R3. Here is a set which contains many points which are not isolated, and yet is measure zero.[/B]

Right, the measure of lower dimensional subspaces is zero when that measure is the one defined with respect to the enveloping space of higher dimension. Okay, I understand now why you posted this. I indicated my agreement with it, but pointed out that this wasn't what marcus and I were discussing.

perhaps you misspoke with your statement, and did not mean to imply that the property of "containing only isolated points" is the definition of measure zero sets, but rather just provided an example of a certain class of measure zero sets. but this is not what your wording indicated.[/B]

I'm just sorry for my part - which is nearly all - in causing the confusion.

it is therefore not surprising that Rovelli, coming to the thread later, and perhaps influenced by marcus, might assume that you are confused about the difference between these two types of sets.[/B]

I think that's what actually happened.

i think a more reasonable assumption would be that Rovelli simply wasn't interested in continuing the dialogue.

You could very easily be right and I wouldn't bet against it. The thing is that the curt tone is at odds with the basically conciliatory language, and this is not an uncommon way for people to acquiesce when after a long battle they realize they can no longer deny that the other guy had a point after all, but don't want to take the discussion to the "penalty phase". But again, you could very well be right. Anyway, I found this whole experience to be extremely embarrasing.

 

[marcus]

...


Remarkably, this claim of mine is actually true! The classic example is the "cantor" set which is both uncountable (as opposed to countable, which means the points are isolated) and of measure zero. This is one of a number of bizarre results that almost made me drop physics and go into mathematics.

you are saying that a countable set must consist of isolated points
I assume you mean a countable subset of the reals in the ordinary topology
the rationals are a countable set not consisting of isolated points

you say "countable, which means the points are isolated"
that is, you say countable implies isolated
but the rational numbers would seem to afford a counterexample

this basic measure theory and pointset topology stuff is a bit off topic, we could start a tutorial thread in the math forum if anyone wants, definition of what an isolated point is etc

 

[marcus]

Lethe said

However, in a later post, you make this statement:

--------------
Originally Posted by jeff
...but sets of measure zero ... by definition contain only isolated points.
--------------


This statement is clearly mistaken. the definition of measure zero is not the same as the definition of isolated points. a set of measure zero need not contain only isolated points. indeed, it was at this point that i posted a counterexample to your statement: the xy-plane in R3. Here is a set which contains many points which are not isolated, and yet is measure zero.

Lethe was right, sets of measure zero do not (by definition or by any chain of reasoning) consist solely of isolated points----in the real line with the usual type of measure. One might be able to construct unfamiliar sets with unfamiliar measures in which measure zero implies isolated, but it isn't true in ordinary settings

However jeff replied
"remarkably, this claim of mine is actually true" or words to that effect
(that is measure zero implies isolated)
and then went over to the thing quoted above namely
countable implies isolated
but AFAIK neither implication is right

it's distracting to have basic misconceptions of pointset top and meas theory come in when the topic is a new development in quantum gravity
anybody want to start a tutorial thread in math forum?

 

[jeff]

Lethe said
it's distracting to have basic misconceptions of pointset top and meas theory come in when the topic is a new development in quantum gravity

Actually, I'm pretty sure you know it's common for long threads to veer off topic, so the only reason you posted this remark about "basic misconceptions" and "distraction" was to provoke me.

Let me respond by pointing out that it in fact truly is distracting to have claims that lqg proves that previous conceptions of general relativity are in fact wrong when the topic is a new development in quantum gravity, but refuse to admit that such a prepostorous statement is wrong and then after wasting everyone's time with it, not apologize for the remark. In fact such behaviour has a habit of doing more than just distracting, it poisons the entire thread. There's a difference between making a careless error and intentionally misrepresenting the facts.

Anyway marcus, I don't know why you continue to post that I claim zero measure means isolated points. It's strange that someone who posted an objection to the raising of an off topic subject would repeat the same wrong statement over and over on said subject. On the other hand, yes, in the usual metric topologies on the reals, every neighbourhood of a rational contains rationals.

 

[lethe]

you are saying that a countable set must consist of isolated points
I assume you mean a countable subset of the reals in the ordinary topology
the rationals are a countable set not consisting of isolated points

i think what measure you put on a space is independent of what topology you put on the space. so the topology is not relevant

 

[marcus]

Split between Ashtekar and Rovelli's approaches

Ashtekar and Lewandowski just posted a 125 page survey paper on Loop Quantum Gravity

"Background Independent Quantum Gravity"
http://arxiv.org/gr-qc/0404018


"extended diffeomorphisms" in Rovelli's sense of
finite exceptional set were nowhere mentioned.
Rovelli's book was not cited (the paper version is not
out yet, although a draft is online
and has been cited considerably by other authors)

the closest Ashtekar and Lewandowski come is on page
56, where they consider backing off C^\infty
to C^0
that is going from infinitely differentiable to continuous,
from diffeomorphisms to homeomorphisms

in paragraph (iii) of page 56 they briefly consider that
this would get the hilbert space to be separable
but mere continuous maps do "not even have a well-defined
action on the phase space"

for some reason it did not occur to A and L to consider maps
infinitely differentiable (together with the inverse) except at a finite set
or else they considered it and found it so little appealing that it
did not deserve comment. I find this fascinating.

It prompts me, because of my and other's high regard for Ashtekar,
to rethink the whole business about chunkymorphisms to see
what might be wrong with it!

On the other hand there is often a substantial lagtime in getting out long survey articles like this, so it might be that "extended diffeos" are so new
that they didnt yet show up on Ashtekar's radar. Hard to know what to think about this.

Would have expected the idea to show up in that paragraph (iii) of page 56

 

[marcus]

wisdom in Rovelli's emails to jeff

I was just reading over some of the posts in this thread and saw this exerpt of a post by jeff, with two emails from Carlo Rovelli. On reflection it seems to me the two emails have lessons for me and others here. Parts I want to think more about are bolded. Spelling errors fixed.
---------penultimate C.R. email to jeff, quote------
Jeff,
you are both extremely smart guys, and you both hate been found wrong
on anything.

In discussing science, we all make all sorts of mistakes. The best
scientists are the ones that do not focus on others' mistake, but
focus on the interesting things that others say. If you start telling
somebody: "you are wrong, you are wrong, you are wrong", the only
result is that he freezes and becomes aggressive. Never do that. You
lose a good opportunity to discuss, and the only advantage that you
get is a feeling of superiority that is useless and just makes the
others dislike you.

We all make mistakes. I do all the time. You posted the line:
"The idea of mappings having a property save possibly at isolated
points is an important idea that was introduced into functional
analysis long ago. Such functions are said to possesses such a property
"almost everywhere" which means everywhere except on a set of measure
zero.
A student writing this is an exam of mine would fail. This is badly
wrong. But there is no reason of shame of course. It is the sort of
mistakes we do all the time.

When I wrote in the forum that Marcus has good understanding, I was
not thinking about this in specific. I was thinking about the many
comments of him I have found here and there in the forum.

But in the text my Marcus you forwarded to me, there is nothing really
wrong. Maybe the tone is a bit over-enthusiastic. But technically
speaking, it is all correct. So, technically speaking, he was right.

But you were also right in saying: "well, wait a minute, this sort of
things are done in math, and tune down his enthusiasm". In doing
so, you made a small technical mistake, confusing "almost everywhere"
with "in a finite number of points". so, he focused in pointing out
your mistake and you focused in pointing out his exageration. this
leads nowhere. I insisted that I think that you were both right. I
myself told myself precisely those same sentences in thinking about
that. Namely "This is new and great", then "no, it is just like doing
almost everywhere, and then "but not really, because it is actually
different", and concluded "maybe it is new and interesting maybe not",
and "I wonder what is in the math literature on this group". This
were my thoughts, and this is precisely the exchange that you and
Marcus had.

I think this is really great. I suppose that both of you are
far younger and know physics and math far less than me, therefore if,
with less tools and less experience, you are capable of arguing so
correctly about a topic, this means that you must be very brilliant.

Why waste your brilliant mind in sterile polemics? find a way to
transform your exchanges in something useful for both of you.
collaboration and friendship brings you very far. competition and
desire to be the smarter one leads nowere and ruins your life.

If lee smolin and myself had started telling each other "you are
wrong", at every step, and trying to outsmart each other, there would
be no LGQ today. we ignored any tension and focused on being friend
and learn from each other. and we did good physics. you can do quite
good physics as well, if you want, with your intelligence. use it for
the best.

ciao
carlo
--------end quote--------

--------jeff's response, quote-------

Hi carlo,

You pointed out that I said that

> "The idea of mappings having a property save possibly at isolated
> points is an important idea that was introduced into functional
> analysis long ago. Such functions are said to possesses such a property
> "almost everywhere" which means everywhere except on a set of measure
> zero.
> A student writing this is an exam of mine would fail. This is badly
> wrong.
>you made a small technical mistake, confusing "alomost everywhere"
> with "in a finite number of points".

There's nothing in my posts that indicates I was unaware that sets of measure zero can contain more than just isolated points. That's why I pointedly avoided saying - and I actually made this point quite explicitly in my posts - that "almost smooth" functions are the only examples of functions that are smooth almost everywhere, which of course would be wrong, as you mention. However, you did define almost smooth functions to be smooth everywhere except for a finite number of isolated points, i.e., on a set of measure zero. In the books on real or functional analysis that I studied, functions that have a property everywhere except on a set of measure zero - which includes sets containing only a finite number of isolated points - are said to be smooth almost everywhere.

> But in the text my Marcus you forwarded to me, there is nothing really
> wrong. Maybe the tone is a bit over-enthusiastic. But technically
> speaking, it is all correct. So, technically speaking, he was right.

As you've seen, marcus stated without qualification or equivocation that it's simply wrong to think of diffeomorphisms as the gauge group of GR and that the almost smooth category is a heretofore unknown concept in mathematics. If you agree that whether advertising speculation as fact should be dismissed as over-enthusiasm depends on the situation - and I don't see why you wouldn't - then I really don't think it's fair to fault me as you have since it doesn't take much to get people at these online forums to take posts like marcus's at face value, and there's nobody at PF whose mislead other members as much as he has.

Marcus habitually freezes out and defames anyone who directly challenges him, especially on the subject of LGQ. I've tried many, many times to smooth things over with him and stop him from bullying people, but it's like he has no conscience or something.

Anyway, have no doubt that I appreciate very much your wasting time with me on this. I really am sorry about this whole damn thing.

Jeff.
------end quote--------

---------last email to jeff from C.R., quote----

Okay,
I understand. Fine. Sorry if it sounded too much against you.
I did not mean so. Take care and good luck for everything.
ciao
Carlo
--------end quote--------

I guess for me the thing that came out here was this:

"...In discussing science, we all make all sorts of mistakes. The best
scientists are the ones that do not focus on others' mistake, but
focus on the interesting things that others say. If you start telling
somebody..."

much of the rest is probably just Carlo being nice and diplomatic in the situation. but this part could be generally useful to reflect on.

 

[marcus]

In mathematics the question of whether some class of objects has already been studied some or whether it is virgin territory has a very practical significance

that has to do with low-hanging fruit.

if you happen onto something that hasnt been studied there are apt to be useful basic concepts to define and comparatively easy but useful theorems to prove about it.

It appears increasingly likely that chunkies have not been studied yet.

Nonunitary says they have not been---that they are new mathematical objects---which is one piece of evidence. Also Carlo Rovelli says that he and Winston Fairbairn conducted a search for math papers about homeomorphisms smooth except for a finite set and could not find any prior research. Also I have looked and I would imagine several other people have.

Moreover I posed the question here at PF on March 12 (are these things new, does anyone know of prior research on them?) and it has been almost a month now with no one coming up with anything.

There always could be something hiding out there, the mathematics literature is huge. Also 20th century mathematicians have made a habit, almost an obsessive ritual, of studying everything they could think of including any exceptions that seemed reasonable---for every reasonable choice of A and B they are likely to have already studied functions of type A except on an exceptional set of type B. Partly it is a way of finding a research topic, a new tree to pick.

But somehow they seem to have missed the case of C-infinity except on finite.

So there's low-hanging fruit.

And Fairbairn/Rovelli point to some of the kinds of theorems,
they indicate the kinds of things that could use rigorous definitions and
checking.

My guess is that a bunch of this is now in progress at Marseille and some other French universites. Absurd for it not to be. So we should, in time, see the appearance of some papers about "almost smooth homeomorphisms" or
whatever they decide to call them----chunkies in effect.

 

[marcus]

Marcus,
If I am correct, the invariants for knots with intersections (I don´t know how many of them) are called Vassiliev invariants, and have been studied by J. Pullin and R. Gambini, even in the context of spin networks.

Nonunitary

Early in this thread Nonunitary mentioned Vassiliev invariants. This paper has references to some earlier papers involving these.

http://arxiv.org/abs/hep-th/9505027
Topological BF Theories in 3 and 4 Dimensions
Aberto S. Cattaneo, Paolo Cotta-Ramusino, Juerg Froehlich, Maurizio Martellini
J.Math.Phys. 36 (1995) 6137-6160


This paper was cited by a recent one by Buffenoir, Henneaux, Noui, Roche
http://arxiv.org/gr-qc/0404041
Hamiltonian Analysis of Plebanski Theory

 

[marcus]

the abstract for the Cattaneo et al article is
"In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2-knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals."

the reference list of the Cattaneo article is dazzling
[1] M.F.Atiyah, R.Bott The Yang-Mills Equations over Riemann Surfaces--Phil. Trans. R. Soc. Lond. A (1982) 523-615
[2] D.Altschuler, L.Freidel On Universal Vassiliev Invariants--hep-th/9403053
[3] I.Ya.Aref'eva Non-abelian Stokes formula--Teor. Math. Fiz. 43 111-116
[4] A.Ashtekar, C.Rovelli, L.Smolin Weaving a Classical Geometry with Quantum Threads--Phys. Rev. Lett. 69 (1992) 237-240
[5] R.Ball, M.L.Mehta Sequence of Invariants for knots and Links J. Physique 42 (1981) 1103-1199
[6] D.Bar-Natan Perturbative aspects of the Chern-Simons Field theory--PhD Thesis, Princeton University (1991)
[7] D.Bar-Natan On Vassiliev's Knot Invariants--Topology (to appear)
[8] D.Birmingham, M.Blau, M.Rakowski, G.Thompson Topological Field theories, Phys. Rep. 209 (1991) p.129-340
[9] B. Bruegmann, On a geometric derivation of Witten's identity for Chern-Simons theory--preprint MPI-PH-93-107, hep-th/9401055.
[10] R.Bott, C.Taubes On the self-linking of knots--J. Math. Phys. 35 (1994) 5247-5287
[11] A.S.Cattaneo Teorie topologiche di tipo BF ed invarianti dei nodi--PhD Thesis, University of Milan (February 1995)
[12] R.Capovilla, T.Jacobson, J.Dell General relativity without the metric--Phys. Rev. Lett. 63 (1989) 2325-2328
[13] A.S.Cattaneo, P.Cotta-Ramusino, M.Martellini, Three-dimensional BF Theories and the Alexander-Conway Invariant of Knots--Nucl. Phys. B 436, 355-384
[14] A.S. Cattaneo, P.Cotta-Ramusino, A. Gamba, M.Martellini The Donaldson-Witten Invariants and Pure QCD with Order and Disorder 't Hooft-like Operators--Milan University preprint (1995) IFUM 493/FT, hep-th/9502110
[15] K.T.Chen Iterated Path Integrals--Bull. Am. Math. Soc. 83 (1977) 831-879
[16] P.Cotta-Ramusino, M.Martellini, BF-theories and 2-knots--in Knots and Quantum Gravity, J.Baez (ed.), Oxford University Press, Oxford, New York (1994) 169-189
[17] P.Cotta-Ramusino, E.Guadagnini, M.Martellini, M.Mintchev, Quantum Field theory and link invariants--Nucl. Phys. B330 p. 557-574 (1990)
[18] P.De La Harpe, M.Kervaire, C.Weber On the Jones Polynomials--L'Ens. Math. 32 (1986) 271-335
[19] J.Froehlich, C.King The Chern{Simons Theory and Knot-Polynomial--Comm. Math. 126 (1989) 167-199
[20] E.Guadagnini, M.Martellini, M.Mintchev, Chern{Simons model and new relations between the HOMFLY coefficients--Phys. Lett. B228, p. 489-494 (1989)
[21] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K.C. Millet, A. Ocneanu A new polynomial invariants of Knots and Links--Bull. Am. Math. Soc. 12 (1985) 239-246
[22] V.F.R. Jones A polynomial invariant for links via Von Neumann algebras--Bull. Am. Math. Soc. 12 (1985) 103-112
[23] L.H.Kauffman On Knots Princeton University Press (1987), Princeton N.J.
[24] L.H. Kauffman The Conway Polynomial--Topology 20 (1981) 101-108
[25] M.Kontsevich Vassil'ev Knot Invariants--Adv. Sov. Math. (1992) 16 137-150
[26] W.B.R. Lickorish, K.C. Millet A polynomial invariant of Oriented links--Topology 26, p.107-141 (1987)
[27] M.Rinaldi, private communication
[28] N.Reshetikhin, V.G.Turaev Invariants of 3-Manifolds via Link Polynomials and Quantum Groups--Invent. Math. 103 (1991) 547-598
[29] A.S. Schwarz The Partition Function of a degenerate quadratic functional and the Ray-Singer invariants--Lett. Math. Phys. 2 (1978) 247-252
[30] V.G.Turaev, O.Yu.Viro State Sum Invariants of 3-Manifolds and Quantum 6j-symbols--Topology 31 (1992) 865-902
[31] E.Witten Topological Quantum Field Theory--Comm. Math. Phys. 117 (1988) 353-386
[32] E.Witten Quantum Field Theory and the Jones polynomials--Comm. Math. Phys. 121 (1989) 351-399