User talk:Cuzkatzimhut

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Inspiring brilliance Dick Chu (talk) 13:06, 24 December 2009 (UTC)
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Truly great work
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Einstein would admire you Dick Chu (talk) 13:06, 24 December 2009 (UTC)
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We appreciate your effort Dick Chu (talk) 13:06, 24 December 2009 (UTC)
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Your efforts motivate us Dick Chu (talk) 13:06, 24 December 2009 (UTC)
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Zohanmesser (talk) 19:57, 7 April 2009 (UTC)
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Zohanmesser (talk) 19:57, 7 April 2009 (UTC)


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Dear Cuzkatzimhut, please accept this barnstar in recognition of making over 1,000 edits to articles on English Wikipedia, and for your amazing contributions to math and science related content.

Thank you so much for all your hard work! Maryana (WMF) (talk) 21:36, 10 April 2012 (UTC)


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For all the effort you put into fixing factual and translation errors, and making the article better overall. Splendid work! M∧Ŝc2ħεИτlk 16:51, 28 October 2013 (UTC)


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Good idea with this edit: Exponential of a Pauli vector Brent Perreault (talk) 18:55, 2 November 2013 (UTC)

Perturbation methods[edit]

Dear Cuzkatzimhut, I'm very sorry for some mistakes in the page Perturbation theory but I think the original purpose is rather correct: I wish to specify that these ones are applications of analytical perturbation methods to QM, while many other fields have seen perturbation methods like neutronics basing on linear Boltzmann equation, viscoelasticity and so on. That is also the reason I have just suggested in the discussion of [Mathematical] Perturbation Theory page to solve the ambiguity on one hand by inverting the redirect with perturbation methods, and on the other hand to rename the Perturbation Theory (QM) page into Quantum perturbation Theories, (since there are more than one). Please let me know what do you think about my suggestion! 95.238.49.157 (talk) 18:11, 7 October 2014 (UTC)

It would be best if you got a WP instead of revealing your IP and subverting your purposes. I think it is a bit ambitious to change universally accepted terms, as per textbook and paper usage, starting from wikipedia, the first resort of a user. And completely unhelpful to efface the primary conceptual idea behind the methods in the opening paragraph. A reader looking for "stationary perturbation theory" will be hard-pressed looking for the relevant section, and hitting the eponymized term, R-S, you introduced. Eponymy is strongly deprecated in WP, as an indirect path to writing history, and can generate obvious tensions. In any case, I could not imagine anyone getting confused by the straw-man notional ambiguities you seem to be inventing, and so my own interest in the matter is terminated; but I have the sense the 82 page watchers might take issue with your edits, especially as you did not discuss them in the article's talk page first. Cuzkatzimhut (talk) 18:25, 7 October 2014 (UTC)
I must say first thank you for your positive politeness, I especially liked the pyrotechnic assembly "straw-man notional ambiguities": this rather seems to me an example of straw-man argument. A reader looking for "stationary perturbation theory" is not really a freshman so maybe he already knows something like the universally accepted names since at least 50 years (for example in perturbation chapter of Butkov, Mathematical physics you can see Rayleigh-Schroedinger also in the names of paragraphs: so why don't we change name to the Schroedinger equation?). Thank you also for the intimidating conclusion, it was so necessary: now these 82 page watchers are really scaring the shit out of me... 95.238.49.157 (talk) 09:11, 8 October 2014 (UTC)

Baker–Campbell–Hausdorff formula[edit]

Hi Cuzkatzimhut! I see you have been working on the BCH article. I have a couple of questions, and you might be the one to ask? I have made a few small additions (let me know if they are bad). One addition is a note box on convergence, I placed it at the first appropriate place I could see. It turned out to be the "existence" section. But as I read it more carefully, it seems that convergence is not an issue there at all. Is this guess correct?

Another thing, when I rewrote classical group, there was an addition by you that got lost. I couldn't at that moment find a place for it in the new version because I din't understand it. I meant to ask you about how to fit it in, but I forgot all about it. Do you remember what it was? Something with Moyal algebra? YohanN7 (talk) 22:15, 12 October 2014 (UTC)

Sorry I won't be helpful with the first question... The improvements look fine--even though Dirac's creation and annihilation algebra was useful long before QFT considerations emerged...Weyl's and von Neumann's work on the Heisenberg group got going in 1927, and 1930, if I recall, and they certainly relied on the degenerate baby BCH. I will keep mum on issues of convergence, as I always find workarounds for such things at the boundaries of convergence, and I fit them to circumstances... You might see Logarithm of a matrix? I cannot see a trivial nonsingular example illustrating trouble, though, i.e. well-defined exp(X) exp(Y) incapable of being expressed as exp of some Z... In fact, the QM applications deal with unbounded operators, x, ∂, etc... Weyl appreciated exponential operators such as the translation operator, exp(∂), are better behaved than ∂. It might be useful for your footnote if you found a sweet counterexample...
The sentence you deleted in March can be found by clicking on any prior version of classical group... it was "All classical Lie algebras may be fit into infinite dimensional Lie algebras, such as the Moyal algebra." The ref by Fairlie et al it had describes how all series An, Bn, Cn, and Dn, for any n can fit quite elegantly into Moyal, "the mother of all Lie algebras", and so then also the exceptionals, since they fit into these 4 series themselves. It is an aside, so I have no clue where it might belong... ideally in Lie algebra#Classification, but it was missing from there, and back-linked instead to classical groups--- which these series of Lie algebras generated... I'm not sure what is to be done now, beyond the obvious, which I don't have the time to do myself, namely defining "classical Lie algebras" in Lie algebra itself. Best, Cuzkatzimhut (talk) 00:32, 13 October 2014 (UTC)
Ah, yes, how could I have forgotten e.g Dirac's treatment of the Harmonic oscillator, etc. They created/destroyed quanta of sorts (phonons?), not particles so not QFT. How about renaming the section to Quantum mechanics and quantum field theory?
About convergence: I don't have any technical problems with it. I was mostly curious. The thing is that in the construction given in the section one deals with "formal power series", and one such that is divergent for given X and Y is still a formal power series. I think for the proof in that section, perhaps convergence issues are immaterial. Then again, perhaps not. That needs a note box of its own, and I'll could move the present one to an appropriate place. But I am uncertain. The way the intro is formulated does give the idea that the formula will hold for all X and Y. The proof, on the other hand, seems to promise that any BCH formula will either give the right answer, or the wrong answer (because X and Y are too large) but still in the Lie algebra, or diverge to infinity. Do you now understand my problem with it?
The Moyal algebra: It could surely go into Classical group. But I think it needs to be worked on. Classical group is now a rather "explanatory" article, and the statement, as it stands, is "too difficult". You would go from junior undergrad level to beginning PhD level. It needs to be fleshed out on, I'd be happy to do it, but I can't right nowFace-smile.svg. I'd need to read up.
I'll have a look at perturbation theory (quantum mechanics) tomorrow evening. It's really past bedtime. YohanN7 (talk) 02:00, 13 October 2014 (UTC)
Right, as you put it, footnotes in both cases could address the Graduate student level subtleties. All your proposals appear salutary. As said, a toy example of validity beyond the radius of convergence might be interesting... I mostly work with examples I understand, and have never run into problems using the unbounded operators x and d/dx, etc.. in these formulas... in fact that's when they prove most useful. Cuzkatzimhut (talk) 11:09, 13 October 2014 (UTC)
Come to think of it, the toy example would be the easiest thing in the world for me to produce. I have written a private little math library, and this includes matrices, including exp and log. Believe me, the radius of convergence is there for the log-series. By the way, I have reverted everything the IP (above) have produced (like 5 articles). He is not here to cooperate. I left one article about a Mexican power plant. He is obviously vandalizing there too, but it is nothing I can prove. Can you have a look? If he goes on it's time to call in an admin. YohanN7 (talk) 12:45, 13 October 2014 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Yes, such an example would probably help! (If you were ambitious, you might also finesse Stone–von Neumann theorem which utilizes BCH, essentially for â and â.) The Bologna IP might well be used by more than one person. The Laguna Verde edits, however, might not be inappropriate, since the upscaling of power units by 3 orders of magnitude seems to be bringing that plant up to standard power generation rates of nuclear plants in general. But what do I know... Cuzkatzimhut (talk) 19:21, 13 October 2014 (UTC)

Hi. I checked over at Stone–von Neumann theorem. It seems ok. They use
[[P,Q]P] = [[P,Q]Q] = 0
then
e^P e^Q= e^{P+Q +\frac{1}{2} [P, Q]},
the chopped off infinite bracket series. It can be proved without BCH. The problem is probably not with such identities (even with more terms on the right). The problems come when you say
Z = \mathrm{log}(e^Xe^Y) = X + Y + \frac{1}{2}[X, Y] \cdots,
because then X and Y must be small. Both books (Brian C. Hall + Wulf Rossmann) I have that cover this point this out, and quote the numbers:
\mathrm{log}(\mathrm{exp}(A)) = A \text{ if } ||A|| < \mathrm{log} 2, \quad \mathrm{exp}(\mathrm{log} A) = A \text{ if } ||A|| < 1.
I think begin to see this a bit clearer now. YohanN7 (talk) 06:51, 14 October 2014 (UTC)
Well, one might always multiply A with a parameter s small enough to satisfy such inequalities, and ask whether the answer itself has broader validity for more general s than the actual method of calculation warranted. In practice, one hardly cares what the series does--one uses it as a crutch just to get to a closed compact expression, so all these questions of convergence are left for final cleanups. The caveat is fine, but, as in the case of generalized functions, a stout man's heart breaks bad luck... Cuzkatzimhut (talk) 10:54, 14 October 2014 (UTC)
The original problem eZ = eXeY isn't exactly linear. I agree of course. But I'm sure you agree too that an equation with a known domain of validity is incomplete without that domain of validity? You want this info before you decide how bypass it. There are other problems too. If you interpret X, Y and Z as coordinate points in a Lie algebra (exponential coordinates) of elements in the Lie group, the even if BCH in exponentiated form holds mathematically as identities in X, Y and Z, all big, these quantities are no longer coordinate points for group element. In group theory BCH is in general local.
But this brings us back to my original question. Where should this info be put? Where it presently is may be misleading. YohanN7 (talk) 16:24, 14 October 2014 (UTC)
Well, in full agreement, as usual, I'd argue it belongs very near the end, as a "fussing section", like Domain of validity. It would be useful, but, only as a caveat discussion... It would be counterproductive to send somebody away from applying the formula and its algorithmic methods before they have tried something and hit upon a problem. It is best for them to damn the torpedoes, and read up on mathematical subtleties if one came close. It is all stylistic, of course... In my own experience, I have watched people bug eye when I bring them a fish right after their suave proof of the nonexistence of fish... Cuzkatzimhut (talk) 16:38, 14 October 2014 (UTC)
I have just edited (before I saw your comment). The first nb (and the formulation just before it) is changed. I added the full thing to the matrix Lie group section (as an nb besides the equation) where it is decidedly true. YohanN7 (talk) 17:07, 14 October 2014 (UTC)
Would you perhaps say that the formula in question might still be true even if the conditions aren't met? There are certainly parts of matrix space (has to do with eigenvalues) where it would be true for arbitrarily large matrices (would be overkill to put that in). One might argue that only the means of deriving the formula fails. But I'd argue against having a philosophy of "Every formula is true until proven wrong." That would be too optimistic in my tasteFace-smile.svg YohanN7 (talk) 19:12, 14 October 2014 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────I suspect it is not worth fussing things too much. I gather we have a problem of language: What most physicists have in mind, once they hear BCH, is most certainly not Dynkin's infinite expansion, but, usually something compact like the ψ expression of section 2.1 (Magnus, Miller, etc...), whatever method one employs in evaluating it (admittedly, often series expansions). Very often, they cut corners, without losing track of the controlling essence of the problem, and take logical leaps harder to justify than to explain, on the way to a correct answer, then proven and justified in several ways. The fact is that many of these expressions are solutions to operator differential equations, and series and combinatorics need never enter, for some applications. So, typically, a derivative operator d/dx may easily be plugged into these expressions, act on fancy functions f(x), and lead to correct results, regardless of presumed intermediate steps failing or not, and fussbudgets in the audience agonizing over square integrability of the relevant expressions or not. But we are talking about hypotheticals. If you had a cogent example illustrating the need of caution and the grim consequences of insouciance, why, it would certainly be useful. Cuzkatzimhut (talk) 19:39, 14 October 2014 (UTC)

Since when isn't providing references enough? The BCH article is within the scope of both physics and math. But all right, I'll produce a set of three matrices from which you cannot solve for Z by taking logarithms. YohanN7 (talk) 19:57, 14 October 2014 (UTC)
Well, I suspect the casual reader will not run to the references... He or she is here precisely in avoidance of hitting the books. Yes, if you have an example of a well defined exp(X) and a well defined exp(Y) which, however, multiply to exp(X) exp(Y) not expressible as an exponential of something, that could scare the reader to be more careful. Going back to what most physicists understand by such a formula, they basically understand it as a fussy group multiplication law in the same parameterization... something like this for SU(2). If they can formally sum a series to get there, they'd do it, but, more often than not, they'd seek the full closed answer without the guileful "benefit" of series expansions. Cuzkatzimhut (talk) 20:58, 14 October 2014 (UTC)
An example such that there is no Z in the Lie algebra for which eZ = eXeY for X, Y in the Lie algebra can be found here: (formula S(8)) Representation theory of the Lorentz group#Non-surjectiveness of exponential mapping. (All exponentials lie in the group, which here is SL(2, C)) This is much much much deeper than what we are discussing.
What we have at hand is that if ||A - I|| > 1 for some matrix A, then log(A) is, in general, not defined. The derivation of the formula in the article (matrix section) is based on taking the log of both sides. I can't see why you insist, though I provided references, that the burden of proof that it doesn't hold for large matrices in general lies on me (no worries, I am having fun actually, haven't programmed for a whileFace-smile.svg). It may hold, though its derivation doesn't, but then that should be referenced. I think it would be quite sensational if it did, and ought to be known by now.
I am working on an example from SU(2). I don't know if I can prove that the formula doesn't hold for it, but it is probably easier to prove that (provided true) than that it, in fact, does hold (if that is the case). To be clear, I am talking about the power series expansion in the matrix section. This seems much like analytic continuation. Sure, things exist and are well defined more everywhere, but you can't use the same expression everywhere. YohanN7 (talk) 21:58, 14 October 2014 (UTC)
All is fine and stable: I don't want to appear to be giving you more work! I suspect the proverbial intelligent, well-meaning reader will automatically take care that things computed are well defined, and, as I mumbled early on, if not, try to fix them. Cuzkatzimhut (talk) 22:50, 14 October 2014 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── (EC) Just to make sure we are on the same page: When I say that log(A) doesn't exist, I do not mean that there is no matrix X(whether in the Lie algebra or not) such that eX = A. One could interpret your last post such that you were under the impression that I meant that. Also, the article Logarithm of a matrix (I din't read it until now) puts this a bit loosely, and doesn't even give the formula below. The definition of log is

\mathrm{log}(A) = \sum_{k = 1}^\infty \frac{(-1)^{k-1}}{k}(A-1)^k.

It is this formula that enters into the formulas involving BCH. It is, in general, divergent when

||A - 1|| \ge 1,

Surely you agree that formulas based on this can't be expected to have general validity, at least not a priori, and not without proof or reference? YohanN7 (talk) 23:35, 14 October 2014 (UTC)

Yes, indeed, I have been strenuously avoiding your metaphorical "page". I understand that, as a series, you may define the log through the mercator series, routinely, and if it works, fine. But my point is that if one can avoid taking the log, by hook or crook, one should at least try. So the starting point Z=log( expX expY) is fine if the log is well defined; but, if not, there is no reason to throw up one's hands, if one can still try to find some plausible Z. A series expansion is only a means of getting to the answer, not the answer itself, nor even the best way to get to it, much of the time. That's why I prefer Magnus's treatment. But I think much of what's on the Wiki page is sound and can rest in peace, by now. Cuzkatzimhut (talk) 00:25, 15 October 2014 (UTC)
Brilliant! Indeed, the formulas value is mostly theoretical, and our article says so. It also warns about convergence problems in one place and provides limits that guarantees convergence in another. In view of what I found on Googling on the net, while eating a delicious late nigh dinner, that is quite appropriate. It seems like the convergence issue is still these days a subject of active research. And yes, Magnus's name appeared. I'd like to get my hands on his books/papers. YohanN7 (talk) 01:07, 15 October 2014 (UTC)
The doi in ref 11?
An aside on your neat example (S8), not a part of our discussion above: The pathology you mention is but a self-evident singularity of the general case, call it “analytic continuation, if you wish: Since [H,X]= 2X, the generic BCH as per section 2.1 is just exp() exp(bX) = exp (aH + 2ab X/(1−exp(−2a)) ), except when it is not defined, as in the case a = iπ , b=−1, a remarkably isolated singular point, when one has the (S8) pathology, exp( iπ Η) exp(− X)= exp(− X) exp( iπΗ) . it is actually less reader-unfriendly if I take b=1 right at the start and −a=iθ and then take the inverse of the above exp(-θ Η) exp(X) to obtain your expression, exp(−X) exp(iθΗ) = exp (iθH + 2iθ X/(1−exp(2)) ), easier to survey perturbatively in θ... Now, why should I, in good faith, prevent anyone from considering the case θ= 3π/2 after this singularity? But this is just conversation with you, of no consequence to the WP article or footnoting... Cuzkatzimhut (talk) 01:26, 15 October 2014 (UTC)
Thank you for explaining this. At the time I wrote around (S8) I had, of course, no idea that there was a connection in this direction. What is happening is that things that aren't supposed to commute actually commute? It is apparent in (S8) but I never noticed it. By your last remark, do you mean this: Things go nicely for 0 ≤ θ < π but blow up at θ = π. This little explosion should not prevent us from pursuing π < θ < 2π. Tell me if I have understood this correctly.
The doi in ref 11 looks interesting. YohanN7 (talk) 18:11, 15 October 2014 (UTC)
Indeed, yes, yes and yes. exp( iπ Η) is proportional to the identity and commutative with everything. But you actually don't need it, as taking the inverse of the first formulation yields the friendlier identity exp(−X) exp(iθΗ) = exp (iθH + 2iθ X/(1−exp(2)) ). And, yes, all holds for 0 ≤ θ < π and is deprived of meaning at θ= π. I have not investigated larger θs, but why shouldn't one do so? Haven't seen any signs of second sheet or anything. For instance, for θ= 3π/2 we have
e^{\frac{3\pi i}{2} (X+H)}= -i (H+X). Cuzkatzimhut (talk) 18:57, 15 October 2014 (UTC)


TeX in Pauli Matrices article[edit]

(Sorry, I'm new here and still learning wikipedian conventions) Why's the TeX problematic? I thought it was much more readable than the makeshift caret-as-a-hat. Twilightrook (talk) 00:40, 16 October 2014 (UTC)

it is a long discussion, here. Depending on missing MathJax implementation, plain text in many, if not most, browsers can look like a ransom letter, with mismatched font sizes, text lines shifted, etc.. Contributors sometimes unjustifiably assume TeX is easier, and use it in a hurry, but the math templates are there for a reason, and TeX is deprecated. If the caret offended your browser (it is almost certainly a font selection issue in your WP preferences) I will format the non-template one. Cuzkatzimhut (talk) 11:00, 16 October 2014 (UTC)

Full proof (outline)?[edit]

Should we go for a (condensed) full proof of BCH (Dynkins formula)? It would take two more regular-sized sections. One for the differential of exp and one for the proof. YohanN7 (talk) 16:53, 16 October 2014 (UTC)

I'm not sure it is a good idea, but if you have the gumption... Any reader intrigued enough by it ought to hit the books, knowing it exists. Personally, I never use Dynkin's formula, as it is very unfriendly: it is basically an algorithm which produces "just so" terms, without illuminating the functional flows involved. I rather prefer the algorithms of section 2.1, and Magnus' or Miller's short and sweet proofs cannot be beat. (They are summarized in the External Source Crib sheet linked.) But they are not provided here for the same reason. As you pointed out, there are two issues involved: one the general principle that the exponent is in the Lie algebra, taken care of by Friedrich's theorem, and two the explicit form of the group composition law involved, for which Dynkin's algorithm is not as efficient as the others. But, then again, neither I nor you are the intended audience, and I am not good at simulating the mind of the reader, so let me not be discouraging there... Cuzkatzimhut (talk) 17:04, 16 October 2014 (UTC)
I am actually neutral myself. The differential of exp (dexp) is missing altogether Wikipedia from what I can see. So, I though about writing that as a standalone article. Then it struck me that, equipped with that, it would be an easy matter (well, due to others it is fairly short and easy to understandFace-smile.svg) to prove Dynkin's formula, and it could all be done within this article. But, as I said, I'm neutral.
What would you say about dexp only as a section? I don't know if it has any practical use in this direction. It is certainly used in proofs. Or should it go into its own article? YohanN7 (talk) 17:35, 16 October 2014 (UTC)
A yes, Duhamel's formula, the mainstay of all these algorithms. To my taste, it could either belong, without proof, which would be trivial, in a footnote here; but, as your inspired suggestion puts it, better in a separate short stub (of your making!). The reason is that it is used by itself all the time in the Dyson expansion, in perturbation theory in QFT, in lattice gauge theory, etc., so it is a much broader tool than mere BCH.Cuzkatzimhut (talk) 18:39, 16 October 2014 (UTC)
Here:User:YohanN7/Derivative of the exponential map. I just submitted it. Happy editing! YohanN7 (talk) 14:26, 17 October 2014 (UTC)

History[edit]

I found this in a paper "Lie algebra" (almost a complete book) by Shlomo Sternberg that I'm reading right now:

6. The formula is named after three mathematicians, Campbell, Baker, and Hausdorff. But this is a misnomer. Substantially earlier than the works of any of these three, there appeared a paper by Friedrich Schur, “Neue Begruendung der Theorie der endlichen Transformationsgruppen,” Mathematische Annalen 35 (1890), 161-197. Schur writes down, as convergent power series, the composition law for a Lie group in terms of ”canonical coordinates”, i.e., in terms of linear coordinates on the Lie algebra. He writes down recursive relations for the coefficients, obtaining a version of the formulas we will give below. I am indebted to Prof. Schmid for this reference.

It seems well-referenced enough to deserve mention. Still, it isn't present in the typical literature. YohanN7 (talk) 17:59, 16 October 2014 (UTC)

If you wish, why not, as long as it is concise. Indeed, Wilfried Schmid mentions it in his "Poincare and Lie Groups" article, Bull Amer Math Soc, 6 No 2, March 1982, but he also mentions Lie himself thought such hidebound and pedantic... (I downloaded Schur's paper from the Springer site, and I fear I have to agree with Lie: it is so turgid that, even if the result is there, I just don't want to know where...must be near the Bernoulli generator on page 270.) As always, if you were up to it, extra small facts could not hurt. The important facts, however, needed in the history, is that Campbell did all the heavy lifting with commutators early on, and then Poincare. In an imaginary universe, I'd skip Baker who didn't do that much that was original or salutary, and call it the CPH formula... just kidding! But Sternberg has the guts and taste to call it CBH! Cuzkatzimhut (talk) 18:39, 16 October 2014 (UTC)

PS. I see my favorite version in (1.2) of Sternberg's notes (also possibly worth linking in the article?). You might be interested to know that in Varadarajan's book, the formula comes out of lectures by Bargmann he attended at PU. Cuzkatzimhut (talk) 19:07, 16 October 2014 (UTC)

This probably means it's worthwhile to invest in Varadarajan's book. My understanding (the little I have) comes mostly from the matrix Lie group point of view. It's time to go deeper, and Sternberg's paper seems perfect for an introduction to the manifold side of affairs. His equation (1.11) is used (and his paper referenced) in the new article draft. YohanN7 (talk) 14:26, 17 October 2014 (UTC)
Yes, Varadarajan's book is fine, but basically Sternberg's book does the same, and especially Willard Miller's---my own source of wisdom, cf. the Crib notes, etc... Your proposed article looks good. I have not tracked down the attribution to Duhamel, but you may have noticed top-appearing google results calling it that. Perhaps you want to stick in a footnote to the effect that the proof is evident from the definition of the exponential as a large N limit of the monomial, exp(X(t)) = (1+X(t)/N)N; that is, differentiation leads to a sum of N terms, going to an integral, etc... Further, I have been stumbling on your function φ or its multiplicative inverse, ψ, the generating function of Bernoulli numbers, all my life, but have never seen a "name" for it... does it have one? Finally, I suspect Magnus expansion is better than the wikilink to the Dyson series. Cuzkatzimhut (talk) 14:46, 17 October 2014 (UTC)
I did the same thing, i.e. Googeled, found the same thing and was content with that, though a proper reference would be much better. About your baby proof, one needs to assume that X(t) does not commute with its derivative, right? Otherwise the middle factor (with the Φ(-adX) reduces to 1 for me - as it should I guess. It doesn't seem entirely trivial to get to the right expression. YohanN7 (talk) 16:33, 17 October 2014 (UTC)
Yes, of course, it is only when X(t) fails to commute with its derivative when the derivative of the exponential is nontrivial... In that case it would be just the conventional scalar result for commutative quantities! Cuzkatzimhut (talk) 16:38, 17 October 2014 (UTC) All I am saying is, loosely, ∂ (1+X(t)/N)N =∑m=1N (1+X(t)/N)N-m X' (1+X(t)/N)m /N so in the limit ∂ exp(X) = ∫ds exp((1-s)X) X' exp(sX) . The rest is dressing up ... Cuzkatzimhut (talk) 16:46, 17 October 2014 (UTC)
I'm at the middle equation in my sandbox. I was worried that I had to "manually" commute the derivative to the right keeping track of everything. Now I see how the integral relieves me of that. (The order in which things are done is kind of importantFace-smile.svg)
By the way, does the popup citations work as they should in the draft for you? They don't for me, I. e. I don't get a popup, but the right thing happens when I click them. YohanN7 (talk) 16:54, 17 October 2014 (UTC)
Yes they work. By the way, the short hyphen you use in B-C-H wikilinks needs to be long, otherwise all these links go there through redirects. Also, the adjoin action link ends up on a disambiguation page... I'd use the algebra adjoint as opposed to the group one, since that's what you end up working with, anyway. I added W Schmied's free historical ref in the BCH article---maybe it belongs here too? Cuzkatzimhut (talk) 17:02, 17 October 2014 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── The baby proof is in place. It should be put in a note box, but this is problematic for me to test since I have this popup problem when i work in my user-space. YohanN7 (talk) 18:09, 17 October 2014 (UTC)

I put the central eqn in an eqnbox: I hope it is OK. Should I stick the baby proof in a collapsible box? I don't know their formal names. Cuzkatzimhut (talk) 19:02, 17 October 2014 (UTC)
I think it looks okay now. Thanks. I had though of a regular popup, but this is just as fine I guess. I'll move the page to the "creation area". It is already submitted, but it says in the top that it should perhaps be moved, so I'll do that. It might speed up the process. YohanN7 (talk) 19:32, 17 October 2014 (UTC)
The old link still works. YohanN7 (talk) 19:35, 17 October 2014 (UTC)

Article in place[edit]

Red Wine Glass.jpg Derivative of the exponential map
Thank you so much for all inspiration, guidance and hands on editing.
Getting so much help from someone truly brilliant in the field, and other fields too,
I strongly suspect on good grounds, the result can't become any other than good.
But I am overwhelmed over that kind of caliber barnstar. Thank you again.
I hope the year of the wine is right. Chateau Cheval Blanc 1996
YohanN7 (talk) 00:59, 4 December 2014 (UTC)

Thanks. But, having checked on the internet the price a bottle of this fetches at auction, I am obliged to confess I have found clearance discount bin "poor cousins" of it for $4.99 on occasion (luck; Chile, Portugal, Romania) which were actually good--with food. So, then, arguably the price equivalent of 50 bottles of those: At the rate of a bottle a week, that would amount to a year's supply... a heady supply, no? To think that my extremely distant (if not imagined) ancestors would trade wine to (alcoholic) Etruscan kings for slaves... getting heady already without the wine, there! Thanks again, Cuzkatzimhut (talk) 01:26, 4 December 2014 (UTC)

On a more serious note, I tried, and failed, to imagine alternate titles that some WP reader would try to find this type of stuff through, in a search, so as to fashion Redirects to this page... But with no luck... I cannot replicate the language one would use to search for it. In the past, I have seen things like that referred to as "Feynman's identity", which is a terrible cop-out... Perhaps Rossmann has pithy expressions on such? Anyway, I also stuck wikilinks to it in far-flung articles with clear conceptual links, but with no obvious handles to steer interested readers there. Cuzkatzimhut (talk) 02:04, 4 December 2014 (UTC)

dexp of course!
I was just about to ask someone how to create redirects. YohanN7 (talk) 03:19, 4 December 2014 (UTC)
And dexp it is.
Rossmann's book b t w was very well received. Anthony Knapp wrote a long very positive review, and even created an errata for parts of it because there were millions of small typo-like errors. The second edition is better in that respect, now only thousands of small errors remain. In spite of it being a senior undergrad/beginning grad level book in a mature subject it had novel approaches to proofs, especially of the Lie correspondence. YohanN7 (talk) 09:16, 4 December 2014 (UTC)

S-matrix#Definition in quantum field theory[edit]

The talk about different Hilbert spaces for in and out states has been nagging me for some time. The QFT texts I have seen never expresses the possibility, and Weinberg's volume 1 (I have that one) is very clear, and emphasizes that all states inhabit the same Hilbert space.

I don't mean to burden you with anything here, but cold you at least say yes/no/maybe? YohanN7 (talk) 16:27, 5 December 2014 (UTC)

Oh, I really don't know, I shrug off any difficulties, and always agree with Weinberg, and "damn the torpedoes" (as RPF used to tell us). The torpedoes are normally obsessed on only by Haag's theorem types that I never run into.... (I'd rather attend a lecture proving the non-existence of fish. [1])
That is, different complete sets of states sharing the same Hilbert space, with S connecting them! I assume that, to do anything useful, some type of overlap of in and out states exists, whether nicely formulated or not, and all practical problems produce meaningful answers when transitions from in to out states are computed, and measured in our real world. As a rule, you know when you have an answer that makes sense... A friend of mine from graduate school kept emphasizing that "The theory is always smarter than you!". Cuzkatzimhut (talk) 16:44, 5 December 2014 (UTC)
I thought I said I don't want to burden you Thanks, now the article makes a little more sense. YohanN7 (talk) 22:06, 5 December 2014 (UTC)

Perturbation theory (quantum mechanics)[edit]

The persistent attacks at this article by an anonymous guy from MIT started from Marco Frasca's blog and now is reiterating here. You can check the IP address to verify it comes from MIT. I think a better way to stop them is to stop the IP itself as this guy does not seem to be too much skilled on computer science.--Pra1998 (talk) 20:42, 10 December 2014 (UTC)

Thanks. As I indicated on my message on his talk page User talk:18.62.31.139 several MIT IP asses have been used, and it would be messy to be playing "whack-a-mole" with all of them and punishing potentially other users of them collaterally. I am not really interested in the subject of his beef, as it is barely intelligent, but editorial procedure must be followed. He has been put on notice, so he understands all future moves from an IP would be treated as plain vandalism. You may transfer this entire section/discussion to the talk page of the article. Cuzkatzimhut (talk) 21:02, 10 December 2014 (UTC)

A question about in-line TEX on Wikiversity (NOT Wikpedia)[edit]

I am trying to write educational physics materials on Wikiversity, something I'm sure you appreciate, given all the well-intentioned efforts to insert mini-tutorials into Wikipedia. One issue that comes up on Wikipedia is the use of inline-TEX, which I use to compensate for the fact that equation numbers don't work well in wikitext (i.e., since I can't refer to previous equations by number I must describe them with words). In-line TEX is also fast to write, an important consideration for the highly underdeveloped Wikiversity. In other words, rough looking (but accurate) prose on Wikiversity is better than no prose. I understand from a recent comment you made, that some browsers poorly display inline-TEX. So I have two questions:

  1. Do these inline-TEX incompatible browsers render the prose unreadable, or merely ugly?
  2. Are these inline-TEX incompatible browsers obsolete (i.e., being phased out), or are they "the wave of the future"?

--guyvan52 (talk) 15:46, 18 December 2014 (UTC)

Apologies for my technical ignorance... I don't know what templates or workarounds Wikiversity employs, in contrast to Wikipedia... If it used all the WP gadgets, however, it could also use numbered formulas, as exemplified, e.g. in Derivative of the exponential map here.
Now, concerning in-line TeX, it is merely Uuuuugly, and slow (since graphics have to be imported and spatchcocked in like a ransom letter, differing in size, line justification, etc, from the normal text). If you emailed me, I could, privately, email you back screenshots of the look and feel. But it is certainly readable---there are no strictly inline TeX incompatible browsers (but those are not obsolete, they are just not Microsoft!). In WP it is deprecated, since it is meant to be read from tablets, iPhones, old browsers, etc... It is meant to be universal. Unfortunately, the math templates or in-line HTML symbols rely on font symbol collections of different browsers that now suffer by flakey collections of fonts/symbols, sometimes missing from several operating systems.
To experience some issues yourself, you might go to your WP Preferences>Appearance ...section Math , and click alternate boxes on several options , PNG, MathJax, etc... to see how inline TeX or HTML or formulas get to look... The problem there is that you may optimize these settings for one machine, but not for 4-5 that some use in the course of their day, and you readers of course.
HTML is "faintly" preferable in WP, as more universal, but there is divergence of opinion there, so maybe you wish to research the issue further. However, if your math symbols in HTML or templates look ugly (check them vs the left column of List of mathematical symbols), you might go to your WP preferences and chose suitable fonts, etc... I believe the future will include increasingly richer font/sets of glyphs etc... in most systems, even though I am a bit clueless about iPhone OSs... Cuzkatzimhut (talk) 16:21, 18 December 2014 (UTC)
MathJax is supposed to be the wave of the future, but we are definitely not there yet. It is meant to address inline TeX ugliness issues. Cuzkatzimhut (talk) 16:28, 18 December 2014 (UTC)
Thanks for the info. FYI, there is another reason for using inline-Tex. This Equation sheet is intended to be printed out as a pdf file for students to use as they take a first-year college physics course. Since each professor emphasizes different topics, it is important that this sheet be editable. (Wikiversity encourages parallel pages, even poorly written ones if they are done by students.) The use of inline text permits the equations to be presented in the most compact form possible. The intent of the equation sheet is that students studying for an exam not bother with memorizing equations; my best students know the material so well that they almost never refer to the equation sheet. But having it allows them to not worry about whether or not to memorize an equation.--guyvan52 (talk) 16:33, 18 December 2014 (UTC)

No argument there! I am a great fan of inline formulas. Indeed, for wikimedia my preferences of Math rendering were set to PNG, which made your page look Terrible, indeed, distracting (on a Yosemite OS Safari browser), but MLL etc makes it look tasteful. In that sense the future will improve compatibility. But doesn't Wikiversity support WP math templates? PS: I just checked it does! Cuzkatzimhut (talk) 16:44, 18 December 2014 (UTC)

Yes, MML almost gets it "right" on Firefox/XP too. By the way, does anyone know how to produce \hat f in HTML? The obvious template (hat) does something completely different. But wait...; û, ..., ^f..., damned! YohanN7 (talk) 17:14, 18 December 2014 (UTC)

ƒ̂ , , , f , f , ƒ̂ , and sundry reminders i have jumbled in my sandbox.... Cuzkatzimhut (talk) 17:22, 18 December 2014 (UTC)

BUT look at what happens to the above under continuous magnification, positive And negative.... Cuzkatzimhut (talk) 17:43, 18 December 2014 (UTC) There is also the world of stuff ... Cuzkatzimhut (talk) 19:28, 18 December 2014 (UTC)

These look pretty good; ĝ, ĥ, but, alas, an f with a circumflex is what we need the most (Fourier transform). YohanN7 (talk) 21:27, 18 December 2014 (UTC)

VB update[edit]

To let you know, the "last resort" regarding VB didn't respond. (The first two were very helpful and quick to respond.) As a parenthetical remark, I find some copyright laws around pretty absurd. It isn't exactly like VB himself is allowed to object to having his picture giving glory our articles. YohanN7 (talk) 23:27, 28 December 2014 (UTC)

Bummer... I am not surprised, in my cynical take of this lawyer-infested world... It is indeed bizarre to not have free access to what any google Image searcher could instantly access themselves! I have had a similar confusion. Achilles Papapetrou is missing a pic, but its russian analog, ru:Папапетру, Ахиллес has had less squeamish editors, who somehow stuck that one in... If only I knew how to transfer/ wikimedia-move/ transclude it to the english version... If I were up to squabbling, I might try a fast one with VB, hoping for the best.... Cuzkatzimhut (talk) 00:18, 29 December 2014 (UTC)

They did answer now, equally politely, after having searched through the boxes with VB's papers where it conceivably could have been found. Maybe I should make this my life mission? Legally obtain a photo that a 9-year old could arrange in a split-second. Sigh! YohanN7 (talk) 01:20, 2 January 2015 (UTC)

Seriously, I think an article without a good history section and a picture or two of the most prominent persons involved is lacking something. Preferably, there should be an anecdote or two as well. Just a naked publication date of the seminal paper in question feels rather dry. I have fairly recently gained access (through WP) to the natural sciences publications of the Royal Society (it's great, lots from the 30's, 40's and 50's is directly relevant for what I mostly have been writing about here), and I'll get full access to JSTOR in the days to come (and hopefully Elsevier's physical sciences later on). But I failed to apply for access to the history of science publications. I regret that. YohanN7 (talk) 01:32, 2 January 2015 (UTC)

Query about an edit[edit]

Can you clarify your reason for this edit? Of course linearity is essential to the subject of the article, but why is it relevant to the use of "powers" in reference to iterative application of the operator? I am at a loss to see any connection. The editor who uses the pseudonym "JamesBWatson" (talk) 17:08, 29 December 2014 (UTC)

OK, I should think it is self-evident: It is extremely important in my mind to contrast to iterated functional composition, which is aggressively nonlinear and thus amenable to very different techniques, and, indeed, culture. A fractional index is just a point on a continuous trajectory, and should not warrant any confusion with functional iteration flows. (Personally, I believe the bloviation on functional composition there to be unwarranted, but if it evokes something for somebody, let it be.) But it is important to remind the reader up front that this entire article is about linear transforms, and any confusion to nonlinear contexts is counterproductive, especially to computer simulators. I do not recall the history of this, but there has been a persistent quasi-vandal insisting on cross-linking functional trajectories to fractional calculus, as through the reader will get a deep idea out of this confusion, somehow. Help, if you think you could prevent the confusion.... Cuzkatzimhut (talk) 17:46, 29 December 2014 (UTC)
My curiosity brought me there, and I'm totally unfamiliar with the subject. But it struck me, there is a place or two where operator appears unqualified with linear (and I don't mean D or J). YohanN7 (talk) 18:08, 29 December 2014 (UTC)
It would be nice if you could then qualify those... As I indicated, I personally find the bogus connection to functional iteration as the source of the problem. If that parenthetical (footnote) tangential analogy went away, the issue of linear operators or not would have never arisen in this article--nobody would take the left turn. But the entire article is clearly about differential and integral style operators, not the hellish maps of functional composition--not even the much much simpler Legendre transformation, and the reader should not get ideas.... Cuzkatzimhut (talk) 18:18, 29 December 2014 (UTC)
Ok, will do. Keep an eye on your watchlist for bogus edits. As I said, I have no clue about the subject. YohanN7 (talk) 18:35, 29 December 2014 (UTC)

Another query. This one I can't figure out: Rotation matrix What is the (vector?) A in section Exponential map? Also (unrelated but related) is there a half-angle version of Rodrigues' rotation formula on matrix form? YohanN7 (talk) 23:54, 30 December 2014 (UTC)

I'm not sure where the trouble might be... Isn't A the 3-vector defined in the previous section, 9.2, with components being the three matrices Ax ...? and then applied to 9.3? (so that u is the unit vector k and \tilde{\boldsymbol{\omega}} was/is \theta \mathbf{K}.
Uhmmmm... are you sure it is unrelated? The formula in 9.3 in RotationMatrix is the same as the one in Rodrigues' rotation formula except for the half angle representation of the former. It is an inside "secret" in angular momentum that, even though integral spin reps such as the 3 dim one here, have periodicity 2π, rewriting them in terms of half the angle of rotation, with these half angles having periodicity 4π, perverse as it might look, is in fact useful, as the systematics of all reps, half And half integral spins are best systematized in terms of these half angles, so one does not flip from angles to half angles going from integral to half integral spins... It is nothing beyond plain rewriting at the level of the 9.3 formula, of course... Cuzkatzimhut (talk) 00:24, 31 December 2014 (UTC)
Sorry, this is all my mistake. I couldn't make sense of the A because it is gone in the present version from 9.2. By "unrelated" I meant that I didn't fully understand even with the A question out of the way and threw in a question about the possibility that there might be a Rodrigues' rotation formula for half-angles. I got lazy, couldn't find my old Spiegel MH with the trig-formulas. (Don't use the often enough to have them permanently in my system). But now you have answered in the affirmative. Thanks.
I hope I have time to straight this out in the article today, because now it is very hard to decipher with A gone. But it is new years eve... YohanN7 (talk) 10:28, 31 December 2014 (UTC)
I think the article may be okay now. But the tildes over u didn't make sense (I think). Happy new yearFace-smile.svg YohanN7 (talk) 11:45, 31 December 2014 (UTC)
Happy New year--to come. I am not hands-on in this, but among these 4 articles, there are all these symbols that are the same but sometimes undefined... So \tilde{\boldsymbol{\omega}} in Rotation matrix is undefined there, and only defined in the article wikilinked, K in Rodrigues formula is u.L, etc... Right now, as the reader clicks from article to article to find definitions, his/her head might spin---did I make a pun? In some cases, really all, putting the explicit expression θ u.L in the exponents for \tilde{\boldsymbol{\omega}} in the exponents might save a lot of grief...Cuzkatzimhut (talk) 12:35, 31 December 2014 (UTC)
Yep, unification of notation across is a good idea. Unable to edit for a day or two now. (I did fix Rotation matrix.) YohanN7 (talk) 13:26, 31 December 2014 (UTC)

Wavefunctions for particles with spin[edit]

Hi, excuse the favour, but user:YohanN7 and myself have had a discussion on how to clarify wave functions for particles with spin; particularly on domains and codomains, and the decomposition into a product of a space function and spin function (when possible). It seems badly covered in the literature, but it's no reason for the WP article to be just as unclear.

We have reached a conclusion, and intend to edit the actual article, but before we do a third opinion and your expertise would be valuable. I ask here given your collaboration with Yohan and because of your activity on the Pauli matrices article.

Thanks! ^_^ M∧Ŝc2ħεИτlk 17:35, 12 January 2015 (UTC)

Humph... Nothing in the Pauli matrix article should warrant an opinion on this... And I really fear I cannot reconstruct the whole conversation with its winding turns and twists in a short time... So, if you are just asking whether the green hidebox is fine, yes, sure, it's fine---I might have said "complex two-vectors (two-spinors)" instead of "complex vectors", myself. But I don't want to get involved into subtle discussions of when the space part and the spin part factor out / decouple / ignore each other:⊗.... These are all, after all, special limits/rewrites of the fully coupled relativistic Dirac spinor... quite well described in JJ Sakurai's Advanced Quantum Mechanics, actually. (In any case I linked L&L online at the Pauli article, but now i see you already did that too!) The space part and the spin part start coupling in atomic physics spin-orbit coupling, i.e., the term σ⋅L (a Pauli vector!) in the Hamiltonian, where that level of accuracy is required, tells you the space part and the spin part "talk" to each other and coordinate to reach eigenfunctions of the full hamiltonian including that term---otherwise, they'd not talk to each other:⊗. Did I come close to grasping what is troubling one? Cuzkatzimhut (talk) 20:18, 12 January 2015 (UTC)
Yes... Thanks for reminding about spin-orbit coupling, now the condition is clearer (any coordinate-dependent entity coupled to the spin operator prevents the factorization, a.k.a any coordinate-dependent Pauli vector in the Hamiltonian), and for the pointer to Sakuari (which I used to have access to at uni, but not anymore).
You're confirming exactly what the sources say, that the wavefunction can be written as a complex vector, which is fine, but Yohan is saying it's just a complex number as a function of spin and space and time, or a complex vector with space and time dependence only. This is the core of the discussion.
Finally... reference to Pauli matrix since you were discussing higher spins on the talk page there.
I'll not clutter your page further. Cheers! M∧Ŝc2ħεИτlk 21:02, 12 January 2015 (UTC)
Yohan is saying that the wave function as a function on coordinate space and spin z-component is complex valued, not vector valued. If you lump together these values for different z-components of spin for the same spacetime point, then you can organize them in a column vector if you want to. This is something different. You confuse
\Psi(\ldots, 1/2) \in \mathbb{C}
with
 \left(\begin{matrix}\Psi(\ldots, 1/2)\\0 \end{matrix}\right) \in \mathbb{C}^2.
Furthermore,
 \left(\begin{matrix}\Psi(\ldots, 1/2)\\\Psi(\ldots, -1/2) \end{matrix}\right) \in \mathbb{C}^2
is the wave function evaluated at two points in the domain. When sloppy (standard) you might write
 \Psi(\ldots) = \left(\begin{matrix}\psi_{1/2}(\ldots)\\\psi_{-1/2}(\ldots) \end{matrix}\right) \in \mathbb{C}^2.
(no spin arguments on the left, but present as an index on the right) and call it vector-valued. This latter is not going to help the mathematically inclined who have been asking about domain and range. YohanN7 (talk) 21:44, 12 January 2015 (UTC)
Here is what I just said (or at least meant), which echos what you wrote here and here:
"a complex number as a function of spin and space and time" \Psi(\ldots, 1/2) \in \mathbb{C}
"a complex vector with space and time dependence only"  \left(\begin{matrix}\Psi(\ldots, 1/2)\\0 \end{matrix}\right) \in \mathbb{C}^2.
so how does that mean I confused the two in reply to Cuzkatzimhut? And please, let's keep this on talk:wave function, the post above was a simple question. M∧Ŝc2ħεИτlk 21:58, 12 January 2015 (UTC)

Oh, dear, as I said, I am very bad in replicating what others thought, when, etc.. in a convoluted appeals case! I seem to be agreeing with everyone, which might mean that I am missing a point, or not... In any case, I completely agree with the "sloppy"

 \Psi(\ldots) = \left(\begin{matrix}\psi_{1/2}(\ldots)\\\psi_{-1/2}(\ldots) \end{matrix}\right) \in \mathbb{C}^2,

above, and hardly imagine it could confuse anyone. I am completely confused about any (presumed??) practical difference (skipping t) between (x,i) ↦ c and x ↦ (c1, c2) for i a bimodal variable, =1,2 and c's complex numbers... Is there any? Cuzkatzimhut (talk) 23:36, 12 January 2015 (UTC) Trip cancelled. Thanks to Maschen for move of higher spin matrices. To my surprise (well, perhaps not really, given the caliber of the guy...), Willard Gibbs was teaching the Pauli vector composition formula to his students in 1884, "essentially", before Pauli matrices, before Engø, of course, and without the benefit of quaternions, even... I added the ref, but not the link to the free copy of the IInd volume of his collected works on Google books. The things one learns at the end of the day...Cuzkatzimhut (talk) 20:16, 13 January 2015 (UTC)

A content question on Compton scattering images[edit]

Hi - I have an issue with a number of images pertaining to Compton scattering. If you think you have any insights, please visit:

commons:User_talk:Guy_vandegrift#A_physics_question_about_compton_scattering_images

Thanks - --guyvan52 (talk) 17:37, 18 February 2015 (UTC)

Apologies, no time, no interest. Cuzkatzimhut (talk) 18:09, 18 February 2015 (UTC) But it is certainly the case that the angle between the scattered electron and photon is not π/2 in general. For, if they were, look at the awful constraint m=ν' (in natural units, ħ = c = 1) this would produce, when you equate momenta in both directions.... Cuzkatzimhut (talk) 20:15, 18 February 2015 (UTC)
I will take your brief statement as confirmation that my drawing is correct. Thanks. I will worry about it no more. Keep busy! --guyvan52 (talk) 20:25, 18 February 2015 (UTC)
Yes. In point of fact, the exact expression for the electron angle φ as a function of the outgoing photon angle θ that is often used in undergraduate lab courses (and can be found in Adrian Melissinos' book) is, again in natural units, cotφ = (1+ν/m) tan(θ/2). You may thus "dial" yourself a φ to be distinctly different than the π/2 complement of θ by choosing a suitable incoming photon frequency. Cuzkatzimhut (talk) 01:41, 19 February 2015 (UTC)