A while back I posted a question: http://mathoverflow.net/questions/30632/theoretical-physics-why-not-just-r4 looking for references, either for or against a particular hunch of mine. The responses were good, although (perhaps somewhat predictably, given the vagueness of my hunch) were not as clear cut as I might have hoped them to be.

Imagine my surprise when, yesterday, one of the authors of the very book Steve Huntsman identified and I have been trawling London's libraries for two weeks to find (now full preview on google books- hooray!) turned up on the thread and proffered his paper on exactly what I was looking for.

I, sadly, am not yet at a level where I could digest an average mathematical paper in less than 4 days solid work- and am reluctant to spend that time if the paper is not worth it. As such, I tend to go on recommendations only and rely heavily on the opinions of a reasoned source (on MO, this usually comes in the form of a comment from a high rep user).

Now, since his arrival, Torsten's response has garnered 3 down votes and has been left untouched by 4 people who voted up the question. At first gance this would not seem to commend the paper to me, but without comments- I am rather left in the dark as to their motivation.

It seems overkill, and perhaps a little offensive to ask this as a front page MO question, so here it is on meta:
>Do these voters know something I don't?
>Is there, for example, a well-known blog-post taking the paper in question apart?
>Is this simply a matter of a small amount of users voting without investigating further?

And:
>More generally, how can I make sure references I'm not sure about are seen by the right people?

Torsten's answer automatically bumped the question, but that hasn't done much in terms of commenting- and I'm sure a separate MO question/ comment on my original post would probably be either rude in itself or would not be successful in soliciting comments (the potential respondents would fear, themselves, sounding rude). I am almost certain frankly, that this meta thread is probably the wrong thing to do, but I had run out of ideas- what should I have done instead?

If you google the authors, their papers, books, not much comes up in terms of discussion of their work. In fact Googling the authors' names, the #1 hit is Math Overflow, ahead of things like their book, personal blogs and such.

I suggest practicing "scanning" papers. It will save you a lot of time and energy in the long run. Look not just at the details, but the author's style, hyperbolic statements, gaps in reasoning and such. For example, in one of Torsten's papers on the arXiv I believe I came across three statements of the form "matter is X" where X ranged over at least three distinct objects. Quotes from their blog: "The Brieskorn sphere (2,5,7) is the cosmos" "two Poincaré spheres - is the origin of dark energy" "Matter can be represented by an energy density T != 0 or by a non-trivial differential structure DT of an empty spacetime M."

For me, this is a combination of too vague and too hyperbolic to get my attention.

The new work contains errors, too....

@Tom: the short of it is, unfortunately, you are stuck with what you have.

Do these voters know something I don't?

The FAQ and the Downvote mechanism both tell people to leave a comment about why they voted something down. So does common courtesey (don't just criticize without being constructive). But some people just cannot be bothered, I guess.

More generally, how can I make sure references I'm not sure about are seen by the right people?

Again, you can't. Beyond tagging your question appropriately (which you already did), the nature of MO is such that you cannot demand of other people. If you really want comments by particular experts, I suggest you contact them privately via e-mail.

@Jeremy:

> "it's pretty clear "where" matter "comes from"!"

Really?

@Jeremy--Not to beat a horse already killed in many other places, but as for the topics you list starting with "what happens when you add in GR": while they are interesting, their relevance to the origin of matter is far from clear.

[deleted rant]

Hopefully, a constructive comment on why exotic smoothness != matter.

Let $M$ be a smooth closed simply connected 4-manifold, and $M'$ be an exotic copy of $M$. There exists an Akbulut cork, a compact contractible 4-submanifold $W \subset M$ with complement $N$ and an involution $f : \partial W \rightarrow \partial W$ such that $M = N \cup_{id} W$ and $M' = N \cup_{f} W$. [arXiv:0807.4248v1].

A world line of a particle is an embedding $g: \mathbb{R} \rightarrow M$ that, observation implies (think a rock on a table), need not lie in a compact set, but the Akbulut cork lies in a compact set.

@Ryan Budney

"scanning" papers and googling blogs is that a new kind of scientific methodology? All your arguments against our paper - which you have not read - are more concerned with the authors person and style then the subject of the paper. If you have any arguments with regards to the content of the paper I will try my best to answer your critique. But excuse me, I will not respond to comments which are only based on "scanning" and arguments ad hominem.

@Jeremy

I understand, you do not like our proposed model - thats ok. You criticize there are "no real" new ideas in it - thats also true: the first words in the abstract are "Cliffords hypothesis is investigated". The paper is also not a mathematical one - and all mathematics in it is made by the great ones. The paper discussed only one little idea: that the desire of some physicists to understand matter geometrically could be given a mathematical formulation by the theory of exotic smooth 4-manifolds. Both, "geometrical matter" and "exotic smoothness" are not new but the combination: matter = casson handle is new. It takes us 15 years to come to this idea and thats why it was very impressing and fascinating to us as we read Toms statement: "I'm not saying it's going to be electron= Casson handle, but it must be worth at least looking." (@Tom: It would be very nice if you could explain how you find this idea, which takes us so long)

Ok, Jeremy I hope you understand that we do not try to think about just another new exotic explanation for matter, but try to understand matter as a geometrical property (a intrinsic property - not an additional entity) of space and spacetime. It is ok when you do not like this idea and you are not willing to discuss with us. But you call our arguments non sequiturs and this is not a fair critique. If you really has found errors please explain it in a constructive and definite from. You can do it here or sent me an email.

@Kelly Davis

I am not the expert but the Akbultut cork represents the smoothness structure not very well. It is the involution \tau - gluing the boundary of the cork in M - which determines the smoothness structure. This gluing is in the neighborhood N(A) of the cork - and N(A) is a open set. This region of N(A) were the gluing \tau happens can be described by a Casson handle - and the discs, representing spinors in our model, are located in the Casson handle.

Hi Ryan,

sorry if I misunderstood the intention of your comment.

Regarding the compactness question (and sorry if I not so precise like a mathematician, I am only a physicist): At first I can say, that we noted in the discussion of the paper: "It is to note, that this result is only based on the structure of the Casson-handle and do not use any topological properties of the 4-manifolds. Thus one can easily generalize the model also to non-compact 4-manifold especially to the exotic R4." But I will not avoid your right question. In the model the particles are not point particles thus the world "line" is a 4-manifold. The particles are represented by spinors (immersions) or by 3-dim knot-complements $H(K)$. The World "line" is a 4-dim cobordism $W$ between two 3-manifolds $H(K,t_0), H(K,t_1)$. $W$ is in $N(A)$ and $N(A)$ is open. For simplicity I consider only the trivial case $W = H(K)\times I$. Would you agree, that if $N(A)$ is open one can consider $I = [0,1)$ ? Thus $H(K,0)$ is the particle at the start and $H(K,1)$ is the particle at infinite time (if one reparameterize the open time interval $I$ ).

sorry if I not so precise like a mathematician, I am only a physicist

Disclaimer: I am by training a physicist: both my BSc and PhD degrees are in Physics.

I do not wish to comment about the "only a physicist" here, about which a lot of ink could be spilled.

Instead I would like to voice a strong objection to the idea that being a theoretical/mathematical physicist gives you license for doing sloppy mathematics. If one publishes papers in mathematical journals (by which I include the major mathematical physics journals such as Communications in Mathematical Physics, the Journal of Mathematical Physics, Advances in Theoretical and Mathematical Physics,...), participates in mathematical fora (such as MO) or otherwise tries to engage in meaningful communication with mathematicians, one ought to abide by the rules of mathematical communication. By this I do not mean necessarily copying the mathematical style of "Definition, Lemma, Proposition, Theorem, Corollary" (although that may often be appropriate), but certainly striving for precision when making mathematical statements.

I believe that as evinced by the last couple of decades, there can be very meaningful communication between physicists and mathematicians, but this is something which some people (of either persuasion) have yet to embrace. Statements like the one I quoted do not help: they perpetuate the negative (and increasingly false) impression of theoretical/mathematical physicists as sloppy mathematicians.

@Tom

Thanks for the view in your thoughts. It is still a fascinating coincidence.

@José

Thanks for your comment - you are right it would be the best way of communication. But in the most cases it seems that the intention of physicists and mathematics are very different and so the way of speaking and thinking. For a physicists mathematics is a rich wonderland providing structures - structures which can be used to build models which may describe the world. In this view the properties of the structures are important, not were they come from or how to prove propositions about them. And in this way the language about the mathematical things becomes less precise. But you are right if one want to speak about mathematics one has to use its language. This is not easy, but I will try my best.

@Ryan

Yes the slogan "matter = Casson handle" is only a slogan. The easiest way to get a better understanding is to read the paper but I will try to give a short answer. In my comment I do not say "particles are 4-manifolds", I say that particles in our model are spinors (immersions) or 3-dim manifolds (Knot-complements).

1. "Particles = spinors fields fulfilling the Dirac eq." this is the classical pysical interpretation (Dirac)
2. The Casson Handle is generated by a process which tries to embed a Disc $D^2$ in a 4-Manifold $M$. This map is in general a immersion with self-intersections. The Casson-process eliminates this self-intersections by iteratively execute the immersion (1. stage: immersion$D^2 \to M$, $n_1$ singular points, blow up each point to a disc (Whitney trick), 2.stage: $n_1$ immersions of the new discs, ... )
3. A immersion $D^2 \to M$ can be described by a Weierstraß representation or equivalently by a spinor representation. That means the immersion is equivalent to a spinor if this spinor-map fulfills a geometrical condition - this condition is the Dirac eq.
4. Casson-handle is build by a iterative execution of immersions. The immersions can be described by spinors fulfilling the Dirac eq. The Dirac spinors are particles. => Casson handle are build by classical Dirac spinor fields (= particles)

If you do this (section 4 of the paper), you find that the support of the spinor field is a knotted soild torus $T(K) = K \times D^2$ (the knot K is the boundary of the immersed disc (S^1 \to K)). You can think T(K) within a $S^3 = T(K) \cup H(K)$ with the knot-complement $H(K)$. The complement H(K) is unambiguous related to T (K) by the 3-sphere S 3 and thus also determined by the spinor map and the complement operation. See footnote page 11: There is a corresponding spinor on H(K) and one can also interpret this one as the particle.

Rosé,

I think it would help if you used the equality symbol more literally when talking with mathematicians, because it seems clear the objects you relate via an equality symbol are never equal, nor even equivalent in any sense. I do know physics of a variety of flavours, although it's been a while since I've thought much about any kind of quantum mechanics.

For me it's not my modest physics background that's the issue when reading your paper, it's the seeming disregard with which you use rather basic terminology like "equal" that's the source of the problem. It seems like you want to say two things are related when you say they're equal or that there is a process that takes you from A to B. These are very different things to a mathematician. When I read a paper and come across an error like this it's a "show stopper" -- unless there's something else compelling me to continue reading, I put the paper down and do not come back.

Saying the much weaker "there is an equivalence" would amount to saying there is a process that takes you back from B to A. A hypothetical that illustrates the issue: which combination of Casson handles describes a hydrogen atom? I'm not asking for you to provide a description -- the point is you seem to be going only one way. The lack of a well-defined reverse would indicate your equality signs don't hold up to any scrutiny. Moreover, if the reverse is defined only up to a messy equivalence (ambiguity), that would also indicate perhaps this relationship you hope for isn't so useful.

Anyhow, I'll stop posting here. I'm not convinced this is really a physics-math cultural divide, as I have read many physicists and by and large I never come across these issues. The only community in physics where I come across issues like these tends to be string theory, which I believe not to be representative of physics norms.

It's not always that I come across these issues in string theory, but the only time they've come up seems to be in string theory. This is perhaps just the random sequence of events in my life -- one of my former office-mates referees a good deal of string theory papers and he sometimes sought my advice on technical matters to do with manifolds and algebraic topology. All the other physics papers I read are usually published in strong journals, or in books, frequently considered classics. I suppose those experiences could lead me astray.

At the end some remarks from me....

@Tom:
Have fun with the paper. In any case of a question or assistence in 4-manifold theory, please contact me directly. Sometimes one haven't to read all papers....

@Ryan:
I like your work about the topology of the knot space using the JSJ decomposition. It will be one part of our later work where we will analyze the knot complements. The old work was about cosmology, i.e. one has three sorts of matter/energy: baryonic matter and radiation, dark matter and dark energy. That is the reason, why we identify matter with three different objects.
But I understand that you don't like the idea...
Last thing: I found in your blog some information about Stoimenow. We studied together knot theory in a 3-years seminar presenting talks to each other. If you meet them please send greetings from Torsten Asselmeyer (my old name).

@Jeremy:
What you tell about matter is all contained in standard textbooks except for supersymmetry (not confirmed yet). Secondly I know the ideas from 60s - 80s. I had the honor to speak with some of these persons directly. Especially Wheelers ideas were very important having a similar goal then we have. All the rest of your reply are purely polemic. We don't know what matter is about. We can describe matter by fields but not more.

@Kelly:
Helge explained everything correctly. I will add that the neighborhood N(A) of the cork is a ribbon R^4, i.e. an exotic R^4 constructed from the failure of the smooth h cobordism theorem in dimension 4. But secondly, there are also curves of infinite length lying in some compact set (the Peano curve for instance). So, your argument don't work, sorry.