User talk:Headbomb/Archives/2012/September

Last post

OK, this is the last of what I wanted to say on the topic. Let U(theta) = exp (i sigma dot theta) as we've been discussing. Let R(theta)=exp(L dot theta) where the L are the purely real 3x3 generators of rotations; I'd mentioned them previously, they obey [L_i, L_j] = e_ijk L_k just like the 2x2 mattrices, except they are real. Then

1. R(theta) describes a rotation in 3D space
2. R is orthogonal, that is, R RR^Transpose = 1 = R^Transpose R
3. R is an element of the orthogonol group SO(3)
4. For all real 3D vectors x, we have: sigma dot R(theta) dot x = U^dagger (theta/ 2) sigma dot x U(theta/2)

That is, for every rotation in real 3D space, there are two rotations in SU(2) that correspond to it. In particular, a rotation by 4pi in SU(2) is a rotation of just 2pi in our 3D space.

By now, I hope the notation is obvious: where to write a vector symbol, when to write a dot product, what is being multiplied by what, what each of these symbols mean.

Oh, I can't help myself. One more remark, then. The last relation, #4 above, quasi-generalizes to other things. In particular, there is something similar connecting together the group SL(2,C) or 2x2 complex matricies of unit determinant, to the Lorentz group SO(3,1) of boosts and rotations in flat space-time. If you gauge this, so that its a local symmetry, not a global symmetry, you get general relativity. Some/many people like to study general relativity with spinors i.e. with SL(2,C), instead of with SO(3,1). So now you have a foot in the door into that whole area. linas (talk) 22:30, 28 August 2012 (UTC)

This isn't related to what you just said above, but I just think I've demystified everything that was confusing me about everything. Still not 100% sure about the terminology, but here goes.

• A lie algebra is a set of elements that follows some rules of the type $\left[L_{i},L_{j}\right]=\alpha _{ijk}L_{k}$, with the structure constants $\alpha _{ijk}$ being specific to that algebra.

• The various L can be just about anything (2x2 matrices, 3x3 matrices, vectors, whatever), as long as their commutation relations obey the rules that define the algebra.
• When expressed in it's 2-dimensional (fundamental) representation, the elements of the lie algebra su(2) are the 2x2 Pauli matrices, which are closely related to the spin operators.

$\sigma _{1}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right);\sigma _{2}=\left({\begin{matrix}0&-i\\i&0\end{matrix}}\right);\sigma _{3}=\left({\begin{matrix}1&0\\0&-1\end{matrix}}\right);$

• When expressed in it's 3-dimensional representation, the elements of the lie albegra are the 3x3 matrices which obey the same commutation relations as the 2x2 Pauli matrices.

$\tau _{1}={\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}};\tau _{2}={\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}};\tau _{3}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}$

• However since 3 > 2, the 3-dimensional representation of su(2) is not the fundamental representation.

• A Lie group is generated by taking an exponential mapping with the elements of it's associted Lie algebra.

• For the SU(2) Lie group, you take the exponential mapping with elements of the Lie algebra su(2) i.e. $U=\exp(i\sigma \cdot \theta )$. But $U=\exp(i\tau \cdot \theta )$ would work just as well.

• When talking about e.g. $\mathbf {2} \otimes \mathbf {2} \otimes \mathbf {2}$ stuff, the spin operators are just expressed with bigger matrices. But these matrices are still elements of su(2), as they obey the same commutation relations as before. These bigger matrices, when working in an appropriate basis, are block-diagonal (irreducible matrix). These blocks are themselves irreducible representations of the elements of the lie algebra su(2), and if you so choose, you could generate the SU(2) group by exponential mapping with either the big 8x8 matrix, or any of its diagonal blocks.

• $\mathbf {2} \otimes \mathbf {2} \otimes \mathbf {2} =\mathbf {4} \oplus \mathbf {2} \oplus \mathbf {2}$ refers to the size of the spin matrices in the fundamental representation (left side) and the size of the irreducible blocks of the big 8x8 matrix (right side).

If I got this right, or if I'm very close to being right [please feel free to correct the terminology if it's wrong], then I will finally have my foot in the door. I need to restrain myself from screaming of joy, mostly because I don't want to celebrate too early, because I very well could be wrong. But I'm just ecstatic at the possibility of understanding something about this stuff for the first time in well over 5 years of attempts. Headbomb {talk / contribs / physics / books} 04:26, 29 August 2012 (UTC)

Yes, almost everything is completely right, with only some minor confusion about i, and about h-bar. So, suppose one defines M_j = h-bar L_j. What is the commutation relation for M ? See? and if K_j = -iL_j then the commutation relations for K_j are .... So for su(2), we can make the structure constants purely real, they are the Levi-Civita symbol. To make them purely real, we really should work with (-i sigma), and not sigma. I think this confusion about i lead to the one incorrect thing I see above: its tau_2 above: there is no i in it, all three tau's can be written to be purely real. The commutation relation is purely real, the structure constants are purely real. If you squint at exactly the right angle, you will notice that each matrix tau is a single slice of the structure constants. In fact, this is generically true: for *any* Lie algebra with structure constants $f_{abc}$, the adjoint representation is given by matricies $M_{a}$ whose matrix elements are precisely $[M_{a}]_{bc}=f_{abc}$. This is by definition the adjoint rep. linas (talk) 17:15, 29 August 2012 (UTC)

Oh, and I see one more "minor" error. The confusion about i lead to something incorrect. When tau is finally written correctly, one gets that the group elements are $R(\theta )=exp(\tau \cdot \theta )$ and there is no i in the eqn. This has a weird side effect: the group generated by tau is not SU(2), but is SO(3), which is almost SU(2) but not quite. Remember above, where we deduce that SU(2) is exactly the same as a 3-sphere? Now, imagine taking a line through the origin of the 3-sphere. It intersects the 3-sphere at two polar-opposite points. Let us now insist that these two points are "the same". We do this not just for one point, but for all of them: all opposites are identified. One gets a new object, which is still a manifold, and in fact, its locally smooth and differentiable. But the stiching-together of opposites means its not a sphere any more. The construction is called a projective space. Turns out the projective 3-sphere is exactly SO(3). This is why they say: "SU(2) is a double covering of SO(3)". So, locally, SU(2) and SO(3) look exactly alike (they are symmetric, homogenous...) but globally, they differ, with SO(3) being this weird stitch-up, mashup of a sphere. Note that technically SO(3) is still a manifold: it is perfectly smooth, infinitely differentiable at all points, etc. Just the global structure is whacked. The entire field of mathematics called homology is to try to figure out how globally-whacked different manifolds can be (even when locally they are just real euclidean space). Oh, and Morse theory is homology applied to classical mechanics. So we are all mathematical physicists now! linas (talk) 17:32, 29 August 2012 (UTC)

In "last post" above, please ponder point #4. In particular, note that it holds for both U and -U -- eqn 4 holds for two different U's. These two U's correspond to the polar opposites which you can easily see by writing U=a+i tau b so that -U corresponds to $(-a,-b_{1},-b_{2},-b_{3})$ which gives exactly the same rotation as $(a,b_{1},b_{2},b_{3})$. These two polar opposites on the sphere are the same rotation in SO(3). linas (talk) 18:09, 29 August 2012 (UTC)

See also real projective space and spin group. Turns out all SO(N)'s have a double cover, and the double cover is called a spin group! Who knew? linas (talk) 18:16, 29 August 2012 (UTC)

Why lookit, the article in spin group even explains that SL(2,C) is isomorphic to Spin(3,1) i.e. it represents the relativistically covariant spinors in 3+1 dimensional spacetime of physics. I'm guessing that the Lorentz group SO(3,1) is a double-cover of Spin(3,1) more ore less exactly as explained above. So everything you just learned here generalizes in may ways. Just be careful with factors of i and minus signs, which can fuck things up but good. So, if you want a brand-new minus-sign-induced headache, the pin groups are the spin groups, but without an S ... the name is an inside joke, just like SO(N) and O(N) are related up to a +/-1, or the difference between SU(N) and U(N). linas (talk) 18:26, 29 August 2012 (UTC)

Well I should have prefaced my claims with "true within factors of i and hbar". The only thing I now need to figure out is why do people saying something is the representation of the Lie groups SU(2)/SU(3)/SU(6), when you seem to hint that it's a representation of su(2)/su(3)/su(6). Is that an abuse of language, or do groups have representations? Headbomb {talk / contribs / physics / books} 19:18, 29 August 2012 (UTC)

Groups have representations too. So the spin-1 representation of the group SU(2) is the set of real, orthogonal 3x3 rotation matricies -- which just happens to be called SO(3). In much of physics literature, there is a lot of 'abuse of language', and a general failure to distinguish the group from the algebra. This is because the two are intimately related, and its supposed to be "obvious" from context which is meant (and sometimes both are meant). By contrast, mathematicians will almost never fail to make this distinction. (One the other hand, they will have other abuses, e.g. since SU(2)=S_3, they'll say one when they're talking about the other, because after all, they're the same thing).

All algebraic structures have representations. Maybe not as matrices, but as something. although, e.g. all graphs can be represented by matrices. All hypergraphs can be represented as graphs. In set theory, all relations can be represented as sets. Most things in math can be represented as sets... Historically, representation theory is devoted almost entirely to groups, algebras, rings, modules, but not other things but is has now expanded to just about everything. (I stand corrected by the linked WP article). linas (talk) 19:18, 1 September 2012 (UTC)

How to report an act of vandalism

The recent edit by 14.139.43.12 for Kapil Muni Tiwary was, I think, a blatant act of vandalism. I have undone the edit. As you had shown some interest in the article recently, I would like to seek your guidance as to how and to whom to report for the same. Arunbandana (talk • contribs) 16:23, 29 August 2012 (UTC)

Oh it's nothing requiring a "report", just undo the edit and that's pretty much all there is to it. If it's persistent vandalism, or vandalism spread across several pages, then you'll want to report the issue to WP:AIV. Headbomb {talk / contribs / physics / books} 16:25, 29 August 2012 (UTC)

Thank you for the response. Arunbandana (talk • contribs) 16:04, 2 September 2012 (UTC)

Isotopes

re Isotopes: Interestingly, the RELC of isotopes pages show a lot of activity [1]. It may be more actiove than one thinks. Now I do not have an issue with this topic (isotopes), but I am happy to have provided the list and its RELC. -DePiep (talk) 17:30, 1 September 2012 (UTC)

Journal of Scientific Exploration

I suggest checking some of the articles "peer reviewed" by Journal of Scientific Exploration. Pick your favourite fringe topic and have a look; it's a mixed bag. That is why I am doubtful if it has peer review in any meaningful sense (for fringe journals its an exceptional claim). I'd prefer a reliable source that states it is peer reviewed (and answers the question of what counts as a peer). IRWolfie- (talk) 23:32, 3 September 2012 (UTC)

Oh I'm quite aware of it and agree with everything. I'm not quite sure how best to word things, but the reality is that they probably have an existent, but shitty, peer review system. I'm certainly not married to the current version, and much preferred yours over that other guy's. Headbomb {talk / contribs / physics / books} 23:37, 3 September 2012 (UTC)

It's just that the article it redirects to describes standard academic peer review; implying it has standard peer review. This is in contrast to the sources which state it supports pseudoscience. It unduly legitimises the topic. IRWolfie- (talk) 23:49, 3 September 2012 (UTC)

Feel free to hack away. I'm just about done copy-editing the references, which should make things a lot easier. Haisch, B.; Sims, M. (2004) should be useful for self-descriptions and a general history of the journal. This would also be useful for criticism and the like. Headbomb {talk / contribs / physics / books} 23:52, 3 September 2012 (UTC)

Yeah I'll give it a stab later, cheers. IRWolfie- (talk) 09:33, 4 September 2012 (UTC)

Footnoting archival images

Hi there,

As someone active in working with citation templates, I wanted to run something by you. Recently, I have been working on a mock-up of citing historical imagery in articles with regular footnotes, rather than having to click through to the image description page. I can talk a little more about how this idea came about—several of us have been talking about it at local DC meetups recently—but I thought I would just show you first. The mock-up is at User:Dominic/Image citation, and there is an explanation of the rationale on the talk page. Before bringing it the broader community, I'm wondering if you have any thoughts on that implementation or the idea in general. The idea of changing the manual of style or WP:CITE and editing thousands of articles to add these is pretty daunting, but my eventual goal is just to make sure that this is deemed an acceptable practice that people won't revert for being non-standard. Thanks! Dominic·t 21:23, 5 September 2012 (UTC)

mr = outside citation template?

Hi, I'm sure this was an error, but if it was done via a semi-automatic process I'm worried that there might possibly have been other articles whose references were subjected to a similar treatment. I'm just letting you know, in case you think it's worth checking out. Best, Sławomir Biały (talk) 01:33, 15 September 2012 (UTC)

I was reviewing everything manually so this stuff didn't happen. I guess I missed this one or did a misclick. I don't have time to review these right now (I just moved, new job, etc...) but I'll give it a double-check when I do. Good catch. Headbomb {talk / contribs / physics / books} 14:51, 15 September 2012 (UTC)

It's not a big deal anyway, even if some tiny number of articles is affected. So I'd give it a pretty low priority. Sławomir Biały (talk) 22:46, 16 September 2012 (UTC)

cite doi and cite pmid templates

Hi, you may be interested in this discussion. Cheers! --Guillaume2303 (talk) 08:52, 25 September 2012 (UTC)

Friendly note

I have made an argument at Wikipedia:Articles for deletion/Interface: a journal for and about social movements (2nd nomination) which I base on my interpretation of a comment of yours (linked there). You may want to review it to see if you agree with how I use your comment :) --_{Piotr Konieczny aka Prokonsul Piotrus| reply here} 18:01, 28 September 2012 (UTC)

Frank Zappa project

There's has been a new Frank Zappa wikiproject proposed and I figured the person who created Zappa's book would be interested. Go here if you'd like to join! The guy who proposed the wikiproject wants at least a dozen editors to join before making it official and it currently has 11. If you know of any other wikipedia users who would be interested please do not hesitate to inform them. Thank you. --Mrmoustache14 (talk) 03:56, 30 September 2012 (UTC)

Category:Goldfinger (band)

Category:Goldfinger (band), which you created, has been nominated for possible deletion, merging, or renaming. If you would like to participate in the discussion, you are invited to add your comments at the category's entry on the Categories for discussion page. Thank you. —Justin (koavf)❤T☮C☺M☯ 18:41, 30 September 2012 (UTC)