# Extended idea of diffeomorphism

1. Mar 26, 2004

### 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 dont know how to spell it in Yiddish but there is a word for it. It is high risk to contemplate extending the diffeomorphism group.

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 [Broken]

here is an earlier thread on the Fairbairn/Rovelli paper

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.

Last edited by a moderator: May 1, 2017
2. Mar 27, 2004

### 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 shouldnt 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.

Last edited: Mar 27, 2004
3. Mar 27, 2004

### 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.

Last edited: Mar 27, 2004
4. Mar 28, 2004

### 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

Last edited: Mar 28, 2004
5. Mar 28, 2004

### 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 &Sigma; 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.

6. Mar 30, 2004

### 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.

7. Mar 30, 2004

### marcus

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.

8. Mar 30, 2004

### 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

9. Mar 30, 2004

### marcus

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

10. Mar 30, 2004

### 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 [Broken]
"Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants"

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

Last edited by a moderator: May 1, 2017
11. Mar 30, 2004

### 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

12. Mar 30, 2004

### 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.

13. Mar 30, 2004

### ranyart

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 vaccum 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 definately 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.

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14. Mar 31, 2004

### marcus

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!

15. Mar 31, 2004

### marcus

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

16. Mar 31, 2004

### jeff

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 possess such a property "almost everywhere" which means everywhere except on a set of measure zero.

17. Mar 31, 2004

### jeff

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.

18. Mar 31, 2004

### marcus

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.

19. Mar 31, 2004

### jeff

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.

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

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".

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.

???

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

20. Apr 1, 2004

### ranyart

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.