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@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, Steve, Tom,
this is getting off-topic, even by the standards of meta.
@Scott: +1
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).
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.
Is there not a blog where this conversation could be carried out?
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.
You must be talking about a different string theory community than the one I belong to. Not all of hep-th is string theory and indeed, I would say that most papers I read on the subject, while stylistically far from the mathematical norm, do abide by and large to mathematical standards of precision. Granted, there's the occasional confusion between equality and isomorphism and perhaps in some cases the language is not very modern, but don't confuse modernity with precision. You can use local coordinates or bases or any kind of non-natural isomorphisms and still obtain a precise result.
@Ryan
I have understood your point. In my comment sentences like "Particles = spinors fields fulfilling the Dirac eq." are not statements of mathematical equality. It is a physical interpretation - a mapping between different semantic domains. In the paper the usage of "equal" between mathematical objects is correct. Of course offen the objects in question are equivalence classes because we consider differential geometrical objects up to diffeomorphisms (because in the physical view they are "equal" - both objects describe the same physics). Maybe the different meanings of "equal" (physical, mathematical) are your problem. I can put up with this and agree with you to stop the discussion here.
I've closed this thread as off-topic.
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