Extended idea of diffeomorphism

[marcus]

Rovelli has introduced an extended idea of diffeomorphism which is smooth except at a finite number of points.

The group of extended (or 'almost smooth') diffeomorphisms plays a role in the Fairbairn/Rovelli paper that just came out
and also in the key chapter 6 of the new book "Quantum Gravity".

Someone I exchange emails with has been wondering what to call these
morphisms and one possibility that really ought to be considered, in my view, is to follow the example of the Skippy Peanut Butter people.

For them, peanut butter is either Smooth
or else it is smooth except at a finite number of points in which case
it is Chunky

almost smooth peanut butter is chunky

so almost smooth homeomorphisms are chunkymorphisms

----------------
Einstein's original vintage 1915 General Relativity was, it seems, "diffeomorphism invariant" meaning that if you had some matter and a geometry which was a solution to the equation then you could skootch the matter and the metric around by a smooth map and it would still be a solution.

As far as I know the original GR is not however invariant under chunkymorphisms. [correction: it seems it may be after all! one can extend the idea of a solution of the einstein equation to almost smooth metrics! see a later post in this thread] One thing about Rovelli is his audacity. I don't know how to spell it in Yiddish but there is a word for it. It is high risk to contemplate extending the diffeomorphism group.

Maybe someone will have some thoughts about this. or maybe I will have some to add later.

If you want background on chunkymorphisms (called by the correct name that Rovelli uses) its all thru Chapter 6 of the book and especially around pages 170-173 and 192

The book is online here

http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

or else look at Fairbairn/Rovelli
http://arxiv.org/gr-qc/0403047

here is an earlier thread on the Fairbairn/Rovelli paper
https://www.physicsforums.com/showthread.php?t=16144

I think there probably is a differentiable-with-finite-number-of-singularities
category, maybe someone who does category theory can look at it and see what it would be like. Fairbairn and Rovelli make a start on this. It seems potentially interesting.

 

[marcus]

As I say, the approach in the Fairbairn/Rovelli paper is audacious.
It takes risk.
The geometry of the universe is $$|s\rangle = |K,c \rangle$$
where K is a knot and c is a quantum number. Or rather, that is how the pure quantum states are written.

All the things defined in or on space are defined on this knot K
by further colorings of the links and nodes.
like particles that are constituents of matter, like stars etc.

The actual geometry is a quantum cloud of these pure states, a particular
pure state is one of a countable orthogonal basis of the hilbert space of quantum gravity (gravity = geometry, quantum gravity = quantum geometry) and a generic quantum state, a vector in that hilbert space, is a mixture of pure states----a linear combination of the basis elements.

The knot is abstract, not embedded in some prior space. It is space.
This approach connects to another branch of math called knot theory. The knots of knot theory can be described as diffeomorphism equivalence classes of networks--so this is nothing new. Why shouldn't space as far as we can see---the mostly flat gravitational field---be a knot. Or at least its quantum states be mixtures of knots.

The F/R new thing is that K is not a diff-knot but a diff*-knot.
It is an equivalence class under the operation of the chunkymorphisms, or almost smooth homeomorphisms, or "extended diffeomorphisms" of the fictional space used for purposes of definition.

They just keep the differentiable manifold in there long enough to get the intial definitions, then they mod it out.

----------
What difference does it make to use the almost smooth category?

The way R/F organize the paper they define things using the usual diffeomorphism group (as on page 4) and then
on page 8 they say go back and do it all over again with the extended diffeomorphisms.

They explain what the difference is. It gets the quantum state space to be separable.

I'm transferring their notation into PF latex. Do some more of that next post.

 

[marcus]

the 3D manifold initially used to get started with the definitions is $$\Sigma$$

We (if anyone else joins in) may need the following latex notation, which looks like what they use in the paper. so I will copy it from another thread

the space of almost smooth connections on $$\Sigma$$ is denoted $$\mathcal{A}$$

the cylindrical functions on $$\mathcal{A}$$ have an inner product $$\langle , \rangle$$ (defined on page 3) and their completion under the corresponding norm is denoted $$\mathcal{K}$$

The local SU(2) gauge invariant subspace is $$\mathcal{K}_0$$.

The spin networks are SU(2) invariant so they belong to $$\mathcal{K}_0$$ and indeed (by the Peter-Weyl theorem) span. The spin networks are taken as a basis and the subspace consisting of their finite linear combinations is denoted $$\mathcal{S}$$

Any element of $$\mathcal{S}$$
can be viewed as a linear functional on $$\mathcal{S}$$
by means of the inner product $$\langle \Psi,\Psi' \rangle$$

F/R define $$\mathcal{S}'$$ as the algebraic dual given the topology of pointwise convergence----essentially comprised of infinite sequences of elements of $$\mathcal{S}$$
which converge pointwise

Fairbairn/Rovelli call the gauge group Diff* for "extended diffeomorphisms" of $$\Sigma$$,
that is the almost smooth homeomorphisms of $$\Sigma$$

on page 4 they refer to this gelfand triple
$$\mathcal{S} \subset \mathcal{K}_0 \subset \mathcal{S}'$$

they define a projection onto the almost-smooth-invariant states
$$P_{\text{diff}}:\mathcal{S} \rightarrow \mathcal{S}'$$

$$(P_{\text{diff}}\Psi)(\Psi') = \sum_{\Psi'' = \phi\Psi} \langle \Psi'',\Psi' \rangle$$

the sum is over all states Ψ" for which there exists an
almost smooth homeomorphism φ taking Ψ' to Ψ"

I guess the point here is that there is a subspace of
$$\mathcal{S}'$$ consisting of those states which are invariant under almost smooth homeomorphisms (I considered calling them "Q-morphisms", Q for quasi-smooth, and then decided chunkymorphisms was better.) These would have to be linear functionals, real members of $$\mathcal{S}'$$ not just
members of $$\mathcal{S}$$ moonlighting as members of the dual.

So what we are interested in is a projection from $$\mathcal{S}'$$ into that subspace----the chunky-invariant states, or the "extended diffeomorphism"-invariant states, or the "almost smooth homeomorphism"-invariant states. how dreadful to have as-yet-unsettled nomenclature.

Anyway that is what $$P_\text{diff}}$$ is, the projection into the subspace of invariant states.

And finally F/R define $$\mathcal{H}_{\text{diff}}$$
which is essentially the image of that projection
for historical reasons it is written with subscript "diff" because it is the hilbertspace of SU(2) and diffeo invariant states----except that now the diffeos are "extended" so they can have a finite number of singularities. And this is a familiar notation for the kinematic state space of LQG.

$$\mathcal{H}_{\text{diff}}$$ inherits the inner product
and they write it various ways, as in equation (9) on page 4 and
as in equation (12) for the case where the arguments were originally spin network states.

 

[marcus]

a theorem ripe for the plucking

To recap:
In the notation of the Fairbairn/Rovelli paper, a pure state of the geometry of the universe is $$|s\rangle = |K,c \rangle$$where K is a knot and c is a discrete quantum number, which takes care of the coloring of the links and nodes of the knot.
All the things defined in or on space are defined on this knot K
by further colorings of the links and nodes. Particles and fields are
supposed to be defined on K that way.
The actual geometry is a quantum mix of these pure states, which form a countable orthogonal basis of the hilbert space of quantum gravity.

Because the paper is seminal it is a possible place to hunt for theorems and counterexamples. One might publish a short paper by proving or disproving a sentence or two on page 8, or generalizing something they say is true in the limited context they need it, but which might also be true more generally.

Let's look at the last paragraph of page 8, the couple of lines that go
"Notice that...because it is the product of the holonomies..."
They didnt spell out a proof and we are free to doubt it and either spell out a proof or find a counterexample. It might be enough for someone to get a short paper. Or it might not. Have to see. My feeling is there is something nice there about the topology to be put on the space of connections, the closure in that topology, getting extra (almost-smooth) connections and so forth.
Let's take a look

 

[marcus]

The metrics g are allowed to be almost smooth
and the definition of an almost smooth gravitational field g being a
solution of the Einstein equation is that it is the limit of
a sequence of smooth g_n which themselves are solutions of
the equation.

the chunkymorphisms on the basic manifold are a gauge group for the
almost smooth gravitational fields g. So far this is just classical GR.
A theorem just went by.

that was the next to the last paragraph of F/R paper. It is a cool theorem
which somebody could prove dotting the eyes and crossing the tees. It says that although Einstein did not know it his theory was not just diffeomorphism invariant it is, at the classical level, chunkymorphism invariant.

It always was, we just did not notice it. Can something that simple be true well prove it or find a counterexample.

Now the next paragraph, the last paragraph on page 8, is even more interesting because it gets down to the spatial slice Σ and the
almost-smooth connections A defined on it.

These are limits of the smooth connections A_n that come from the smooth metrics g_n.
The holonomy of a connection A over a path is well-defined even if the path goes thru singularities of A because one just breaks up the path by introducing nodes at the singularities and then on a particular segment of the path one takes the limit of the holonomies of the A_n on that segment.

Now I am expanding some on those two lines of that paragraph at the end of page 8, but I am still not expanding very much or getting to the point of being rigorous.

I think there might be things that could go wrong and more exact definitions of where you get the sequences and how you take the limits and more of a proof is needed and it looks a bit intriguing. But also attractively simple.

Has this all been gone over before in some other context. I did not see it or hear of it and they do not cite a reference here. Maybe not.

 

[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 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.

 

[marcus]

To recap:
In the notation of the Fairbairn/Rovelli paper, a pure state of the geometry of the universe is $$|s\rangle = |K,c \rangle$$where K is a knot and c is a discrete quantum number, which takes care of the coloring of the links and nodes of the knot.
All the things defined in or on space are defined on this knot K
by further colorings of the links and nodes. Particles and fields are
supposed to be defined on K that way.
The actual geometry is a quantum mix of these pure states, which form a countable orthogonal basis of the hilbert space of quantum gravity...

in conventional knot theory (e.g. Jones) the knots do not have nodes

in that case the diffeomorphism group is good enough
and you can get the abstract knots by taking diffeo equivalence classes
of embedded knots

two node-less knots are the same abstract one if one can be mooshed into the other by a smooth mapping

but introducing nodes makes knots different

knots-with-nodes are a different theory which Vaughn Jones and the others apparently did not do yet (or did they does anyone know? i did not hear of it)

how can such simple extensions be new mathematics? am I missing something obvious?

knots-with-nodes are different because as Fairbairn and Rovelli point out you cannot just use the diffeomorphism group you have to allow finitely many
singularities to take care of high-valence nodes

they cite the book by the catastrophe theory guy V.I.Arnold
to the effect that high-valence nodes are stiff and they show why
and they show a revelatory picture Figure 1, on page 9.

 

[nonunitary]

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

 

[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

well you are one poster here who is worth his weight in gold
thankyou

thumbs up for Gambini and Pullin for being ahead of the curve

 

[marcus]

in the generally very good bibliography of Rovelli in Quantum Gravity
I did not find a reference to Gambini/Pullin about Vassiliev invariants
and knots-with-intersections
maybe this is an oversight
or maybe it can be added to the Fairbairn/Rovelli paper's bibliography

but sometimes I miss things

does anyone know the name of the G/P paper?

Ah! here it is in arxiv

http://arxiv.org/gr-qc/9909063
"Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants"

and two followups
http://arxiv.org/gr-qc/9911009
http://arxiv.org/gr-qc/9911010

 

[marcus]

drat! I was over-optimistic. Gambini and Pullin do not seem to be
talking about what I was hoping they would.
Still no prior research seems to connect with the rovelline
extended diffeomorphisms and abstract knots-with-nodes
or at least I do not see how it connects

 

[nonunitary]

Marcus,

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

gr-qc/9803018

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

 

[ranyart]

Rovelli has introduced an extended idea of diffeomorphism which is smooth except at a finite number of points.

The group of extended (or 'almost smooth') diffeomorphisms plays a role in the Fairbairn/Rovelli paper that just came out
and also in the key chapter 6 of the new book "Quantum Gravity".

Someone I exchange emails with has been wondering what to call these
morphisms and one possibility that really ought to be considered, in my view, is to follow the example of the Skippy Peanut Butter people.

For them, peanut butter is either Smooth
or else it is smooth except at a finite number of points in which case
it is Chunky

almost smooth peanut butter is chunky

so almost smooth homeomorphisms are chunkymorphisms

----------------
Einstein's original vintage 1915 General Relativity was, it seems, "diffeomorphism invariant" meaning that if you had some matter and a geometry which was a solution to the equation then you could skootch the matter and the metric around by a smooth map and it would still be a solution.

As far as I know the original GR is not however invariant under chunkymorphisms. One thing about Rovelli is his audacity. I don't know how to spell it in Yiddish but there is a word for it. It is high risk to contemplate extending the diffeomorphism group.

Maybe someone will have some thoughts about this. or maybe I will have some to add later.

If you want background on chunkymorphisms (called by the correct name that Rovelli uses) its all thru Chapter 6 of the book and especially around pages 170-173 and 192

The book is online here

http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

or else look at Fairbairn/Rovelli
http://arxiv.org/gr-qc/0403047

I think there probably is a differentiable-with-finite-number-of-singularities
category, maybe someone who does category theory can look at it and see what it would be like. Fairbairn and Rovelli make a start on this. It seems potentially interesting.

Marcus, it is interesting that a number of people seem to be changing 'Transforming' (I use this word as a defined and precise reason!) their views on certain dimensional aspects of our Universe?

For instance here:http://arxiv.org/abs/hep-th/9805108 The similarity with Rovelli/Fairburn becomes apparent when one treats 'Our Galaxy' as a 'Baby-Universe'.

Choosing a framework of lattice space, and then to place a Knot at the 'Crunchy-Parts', is somewhat like Feynman saw, for instance here we have a field around a 'corner' ? (18) :http://mathworld.wolfram.com/ConformalMapping.html

and here if one imagines the Galaxy as being 'framed' within a Backgorund 2-dimensional vacuum field, then as Our Galaxy is not very square, it results in it being only loosely based on the field it sits within, a 3-D knot tied to a 2-D field.

There a certain changes for any 'subspace' within our Galaxy, when we examine it down to its dimensional limits, one being the smaller it gets, the more it loses its 3-Dimensionality,(Iam pretty sure that F/R understand this is so, and the 'Knot' is in effect a result of this transformation) it becomes more of a 2-D field, but not entirely 'similar' to the 2-D field external to our Galaxy.

Space of 2-Dimensions, cannot be transformed or morphed from within a 3-D space,this is to say that all of LQGists are conceptually aware that starting from within a Three-Dimensional Background, and working their way down to a 'subspace' within this 3-D background, can only end up at a 'singular' fragmentation of the smallest possible 3-D bit?..this fetches us to the Quark component of the very structure contained. It is well known that Quarks cannot be 'broken' or 'Un-tied' or 'Untangled' or 'Seperated' from the space they exist within, namely 3-D space!

Suffice to say Einstein most definitely knew this, its in his literature its just that nobody else spotted it!..well actually one can look back in hindsight and state that the E-P-R is an exercise in 3-Dimensional transformations, aimed specifically at the Quantum Theorists at the time, if one was to study Jung Philosophy, one can attribute the Symmetry between E-P-R and Bohr-Hiesenburg-Shroedinger as the opposing team players ;)

The really interesting thing is that when one transforms from a Space that is 2-D background, its just imagining the starting point!..which is where Einstein really exelled.

 

[marcus]

Marcus, it is interesting that a number of people seem to be changing 'Transforming' (I use this word as a defined and precise reason!) their views on certain dimensional aspects of our Universe?

For instance here:http://arxiv.org/abs/hep-th/9805108 The similarity with Rovelli/Fairburn becomes apparent when one treats 'Our Galaxy' as a 'Baby-Universe'.

Hello Ranyart, I am glad you were able to resume and hope all's well with the Moorglade (which for some reason I picture as a boat although I've never heard you say anything about her, maybe a barge or sailboat equiped to live in)

You mentioned Renate Loll's
"Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change"
http://arxiv.org/abs/hep-th/9805108

I admire her papers, generally speaking, but am not sure I have seen this one and will have to check. Since you said Jung I will say Rorschach. Some of Renate's 1+1D gravity pictures actually resemble Rorschach bilaterally symmetric inkblots and are visually interesting. This reminds me that theories of the universe, cosmological models, sometimes work as a kind of Rorschach inkblot and get people to read into them what was already in some corner of the person's imagination. I suppose that is one reason that the field of cosmology is so exciting and why people are excited by images such as big bang. Well I had better not drift off onto some other topic!

 

[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.

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
but here in F/R paper (and earlier in rovelli's book) one sees that it makes a huge difference to a certain hilbertspace----whether the quantum state space is separable or not separable, whether a certain basis is countable or uncountably infinite

 

[jeff]

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
but here in F/R paper (and earlier in rovelli's book) one sees that it makes a huge difference to a certain hilbertspace----whether the quantum state space is separable or not separable, whether a certain basis is countable or uncountably infinite

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.

 

[jeff]

it turns out that the diffeomorphism group...is wrong because...General Relativity is...invariant under "extended" diffeomorphisms...so one should try to construct a quantum theory that is...invariant under this larger group

Are you on drugs? Nowhere in this paper do the authors insist that their speculation is in fact a proof that what you're claiming is true. In fact, they do not claim to be certain that they completely understand the problem to begin with and mention a couple of approaches that differ from theirs.

 

[marcus]

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.

Are you on drugs? Nowhere in this paper do the authors insist that their speculation is in fact a proof that what you're claiming is true. In fact, they do not claim to be certain that they completely understand the problem to begin with and mention a couple of approaches that differ from theirs.

Did anyone besides me notice the statements here?

The post appears to conflate the well-known idea (in functional analysis) of "almost everywhere" (except on a set of measure zero) with another idea namely "except possibly at isolated points"

these are not equivalent (as undergrads learn in 2nd or 3rd year IIRC)

and likewise conflate it with yet a third notion: "except on a finite set".

Sets of measure zero are not necessarily or even typically finite sets nor do they typically consist of isolated points, although the reverse is often the case.

Rovelli's idea of extending the diffeomorphisms does appear to be novel
and it is certainly not the same as talking about functions which are infinitely differentiable except on a set of measure zero----a different kettle of fish!

As for Fairbairn/Rovelli's paper the issue is not "proving" a mathematical fact but arguing persuasively (or not) that the extended diffeomorphisms are the right choice.
This is outside the realm of proofs and theorems. It would not be appropriate or good form for them to express absolute certainty. But they argue (I think) persuasively that it is the right choice.

this is not in reply to the previous post but about the larger issue. I think they make a good case that at the very minimum one should certainly not
automatically assume that the (unextended) diffeomorphisms are the right group of symmetries for quantum gravity. It is worth considering that they may, in fact, be too restricted.

 

[jeff]

Sets of measure zero are not necessarily or even typically finite sets nor do they typically consist of isolated points, although the reverse is often the case.[/B]

I'm sorry marcus, but 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.

Rovelli's idea of extending the diffeomorphisms does appear to be novel[/B]

I'm sorry marcus, but as I said, rovelli has not turned the world of mathematics on it's ear with this idea.

...and it is certainly not the same as talking about functions which are infinitely differentiable except on a set of measure zero----a different kettle of fish![/B]

Again, I'm sorry marcus, but the fact that the property here is that of infinite differentiability doesn't make this a "different kettle of fish".

As for Fairbairn/Rovelli's paper the issue is not "proving" a mathematical fact but arguing persuasively (or not) that the extended diffeomorphisms are the right choice.
This is outside the realm of proofs and theorems. It would not be appropriate or good form for them to express absolute certainty. But they argue (I think) persuasively that it is the right choice.[/B]

Again, are you on drugs? As is clear from our posts, it was you who was making inappropriately categorical statements, not them, which is precisely the point I was making. Stop trying to twist things around.

this is not in reply to the previous post but about the larger issue.[/B]

?

I think they make a good case that at the very minimum one should certainly not
automatically assume that the (unextended) diffeomorphisms are the right group of symmetries for quantum gravity. It is worth considering that they may, in fact, be too restricted.

Sticking to posting level-headed remarks like this is the best way of keeping me out of "your" threads.

 

[ranyart]

Hello Ranyart, I am glad you were able to resume and hope all's well with the Moorglade (which for some reason I picture as a boat although I've never heard you say anything about her, maybe a barge or sailboat equiped to live in)

You mentioned Renate Loll's
"Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change"
http://arxiv.org/abs/hep-th/9805108

I admire her papers, generally speaking, but am not sure I have seen this one and will have to check. Since you said Jung I will say Rorschach. Some of Renate's 1+1D gravity pictures actually resemble Rorschach bilaterally symmetric inkblots and are visually interesting. This reminds me that theories of the universe, cosmological models, sometimes work as a kind of Rorschach inkblot and get people to read into them what was already in some corner of the person's imagination. I suppose that is one reason that the field of cosmology is so exciting and why people are excited by images such as big bang. Well I had better not drift off onto some other topic!

Hi Marcus, I like your reference to Rorschach!

Just want to clarify The Moorglade?


In music there comes a story by Jon Anderson which he places onto Vynyl record called :Olias of Sunhillow. Regarded by some as a Timeless Masterpiece of musical and spiritual enlightenment.

Just a link for soundbites:


The Moorglade is a ship?..yes in a way! its a figment of Jon Andersons Imagination, let's call it a Thought-Wave, in that it transports its passengers (Ranyart-Olias-Qoquaq) through the Universe in a quest to 'save the Universe' from evils?

Not (Going for the One) explaining any more than this, woven into all of Jon Andersons music-art-writings, is a background of journey's into various domains of Life and Experience.

Needless to say that Jon Anderson gives out the 'Song of Life' through his Music!

 

[marcus]

I'm sorry marcus, but 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.
.

sets of measure zero can be uncountably infinite and may contain no isolated points at all

it's trivial to give an example of a set which is connected,
contains uncountably many points, none of which are isolated, and which]
has Lebesgue measure zero

Halmos Measure Theory would be a good source, I guess, for learning about sets of measure zero.

No need to be always saying "Im sorry". You just need to brush up on undergraduate math and mind your own business more.

 

[jeff]

sets of measure zero can be uncountably infinite and may contain no isolated points at all.[/B]

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.

But the original statement nonetheless still correctly makes the point I intended: that mathematicians would recognize what rovelli is doing as using functions that are smooth almost everywhere since they're smooth everywhere except on a set of measure zero.

As far as your suggestion that I learn some undergraduate mathematics (and, to be fair, my suggestions that you learn some undergraduate physics), I think it's time for us to bury the hatchet, and I'm pretty sure the rest of the forum feels the same way that I do. I therefore will not be the first of us after this point in time to react to any of the other's posts in a way that is disrespectful or rude. However, I do give you credit for at least responding to my posts (as opposed to ignoring them or pointedly addressing them to people other than the person who posted it - which is one way of being disrespectful or rude - and hence provoking a similarly disrespectul or rude reaction). Can we not express our disagreements, especially about lqg, with suitable civility? I just honestly see no good reason why this kind of acrimony must continue, it's just completely ridiculous.

 

[Stingray]

I have to agree with Jeff on this. A set of measure zero in physics almost always means a countable number of points. Yes, there are uncountable sets of Lebesgue measure zero, but I've never seen Cantor sets come up in a physical situation (I imagine the discreteness of quantum geometry would exclude them, but this is a wildly speculative guess!). And maybe the Lebesgue measure is not the appropriate one here.

 

[marcus]

I have to agree with Jeff on this. A set of measure zero in physics almost always means a countable number of points. Yes, there are uncountable sets of Lebesgue measure zero, but I've never seen Cantor sets come up in a physical situation (I imagine the discreteness of quantum geometry would exclude them, but this is a wildly speculative guess!). And maybe the Lebesgue measure is not the appropriate one here.

but physics often uses ordinary euclidean space Rⁿ
and Lebesgue measure is a common measure to use on it
even consider just n = 2
(ordinary integration on the plane, basically)
the x-axis has measure zero
but is uncountable, connected, with no isolated points

 

[lethe]

i am sure everyone here knows this already, but a submanifold of Rn of dimension strictly less than n is of measure zero in Rn. for example, the xy-plane has measure zero in R3.

 

[marcus]

i am sure everyone here knows this already, but a submanifold of Rn of dimension strictly less than n is of measure zero in Rn. for example, the xy-plane has measure zero in R3.

Right lethe! It seemed pretty obvious to me too!

 

[marcus]

it is an ancient time-honored practice to study functions which have some
behavior or other "almost everywhere" or "except on a set of measure zero"
and when I was in school the phrase was even used jokingly as a metaphor.

But I do not remember ever hearing of someone studying the class of functions that are C^oo except on a set of measure zero.
that would be very weird ( I don't see either math or physics sense to studying such things)

for example the rational numbers Q are dense in R¹
and yet they have measure zero
so you are trying to contemplate a function which is infinitely differentiable except at points corresponding to rational numbers

however the class of mappings which Rovelli proposes we look at is not this
and it does (surprisingly enough, to me) make sense:
bijections which are infinitely differentiable except at a finite
set of points

no one has come up with a paper in which this class of mappings has been studied. I hope someone will! But so far the little evidence we have is that
(even though the class is sensible and simple to define) there are no papers written about it. so I refer to it as novel

 

[jeff]

i am sure everyone here knows this already, but a submanifold of Rn of dimension strictly less than n is of measure zero in Rn. for example, the xy-plane has measure zero in R3.

Yes, (n-1)-dimensional spaces have zero measure with respect to n-dimensional measures or higher.

Right lethe! It seemed pretty obvious to me too!

But lethe's point, though correct, isn't relevant here, and I think you know that. This is why he prefaced his post with

i am sure everyone here knows this already

...integration on the plane...the x-axis has [Lebesgue] measure zero...[/B]

Perhaps you've "conflated" lebesgue measure with lebesgue integration, since in extending riemann integration to this more generally applicable kind of integration, we divide not the domain (the "x-axis" when the region is R²) but the range into progressively finer pieces. But this doesn't mean that the domain has vanishing lebesgue measure.

If you are claiming that you meant zero measure in the sense posted by lethe, than the lebesgue measure of intervals in the range of a function would also be zero.

But really, this is just a distraction from the point, which is that rovelli hasn't introduced a new mathematical idea in this paper and in fact it's been around for a long time.

 

[marcus]

Perhaps you've "conflated" lebesgue measure with lebesgue integration, since in extending riemann integration to this more generally applicable kind of integration, we divide not the domain (the "x-axis" when the region is R²) but the range into progressively finer pieces. But this doesn't mean that the domain has vanishing lebesgue measure. Anyhow, none of this is relevant to the main point which is that rovelli is using functions that are smooth almost everywhere and hence nothing new.

this post does not seem to make sense
for a ref on Lebegue measure see e.g. Halmos Measure Theory page 152
you seem to be confusing the measure on R² with
something about the Riemann integral on R¹ which is not really relevant here. you started this excursion into measure theory by
mentioning sets of measure zero and I think the topic is now exhausted
(at least for this thread which is not about measuretheory) so let's move on

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

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

it turns out to be a natural and mathematically interesting extension of the diffeomorphism group which makes a certain quantum state space separable
(countable instead of uncountable dimensioned)

 

[marcus]

Yes, (n-1)-dimensional spaces have zero measure with respect to n-dimensional measures or higher.

----
see my 7:50 AM post. Lethe's 7:54 AM post echos the point
-------




But lethe's point, though correct, isn't relevant here, and I think you know that.

--------
it was my point originally and it is relevant
--------


Perhaps you've "conflated" lebesgue measure with lebesgue integration, since in extending riemann integration to this more generally applicable kind of integration, we divide not the domain (the "x-axis" when the region is R²) but the range into progressively finer pieces. But this doesn't mean that the domain has vanishing lebesgue measure.

----------
no I am not conflating anything
------------

If you are claiming that you meant zero measure in the sense posted by lethe, than the lebesgue measure of intervals in the range of a function would also be zero.

----------
I am not "claiming" I made the point earlier and Lethe corroborated it.
riemann integration (range intervals) is not the topic of discussion
why do you return to it? why mention range intervals?
---------------

But really, this is just a distraction from the point, which is that rovelli hasn't introduced a new mathematical idea in this paper and in fact it's been around for a long time.

-------
If the idea has been around for a long time (to study almost smooth homeomorphisms) then show a paper

 

[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!