User talk:Cuzkatzimhut/Archive 2

Hadamard's lemma
Hi there. I was wondering about "Hadamard's lemma" on the Baker–Campbell–Hausdorff formula page. I searched the history and see that you introduced that name. This is actually the only place I can find that gives that formula the name Hadamard. Do you have a reference for it? (I looked at the reference in that section, and don't see Hadamard anywhere.) Thanks. --MOBle (talk) 03:40, 10 August 2011 (UTC)
 * Sorry I have not found the time to do justice to your question. People do routinely refer to such conjugacies of Lie-algebraic elements (central to the study of automorphisms) as "Hadamard formulas", e.g., Proposition 5.2 on p 609 of Communications in Mathematical Physics, Volume 141, Number 3, 599-617, http://dx.doi.org/10.1007/BF02102819, Universal R-matrix for quantized (super)algebras, by S M Khoroshkin and V N Tolstoy. But, then again, some textbooks like B Wybourne's 1974 outstanding one on Classical Groups for Physicists call it "the so-called Campbell Hausdorff formula"! I did not have the patience to fuss it out of Hadamard's 1927 Course d' Analyse, but geometers call boundary variation formulas of an analogous type "Hadamard formulas". I grew up with that name, but used it for communication purposes, not a committed historical claim; I will keep the question in mind, but, indeed, Lie-algebra texts where such conjugacy formulas live and work, wisely avoid eponymy. Cuzkatzimhut (talk) 14:52, 22 August 2011 (UTC)

On spontaneous symmetry breaking
I think the term spontaneous hidden symmetry does not make sense, because we are talking about the broken symmetry. If the symmetry is always there in the equations, it means that the broken symmetry is inherent, and just amplified by the mechanism. CES1596 (talk) 17:09, 19 January 2012 (UTC)
 * Thanks for the comment. I have no strong pedagogical views on this, and you are welcome to tweak it to prevent misconceptions, provided the basic facts come across.... By that, I mean sharp contrast between the "spontaneous" word, which is the selection mechanism for the vacuum, and hence the symmetry realization mode, and the "breaking" (=hiding) mode resulting by this mechanism. The only reason one bothers to utilize the spontaneously broken mode instead of a plain asymmetric mode ("explicitly broken symmetry") is because both unbroken and SSB modes are two different realizations of the same symmetry: The currents and charges of the symmetry are conserved, the Lagrangian/Action of the theory is invariant under it, etc... in both cases. Only the specific formulas of the transformations differ between the two modes of realization of the symmetry.  (This, of course, has dramatic physical consequences, but the underlying mathematical structure ensures "good behavior" such as renormalizability, etc...) . To me, your  spontaneous hidden symmetry would be quite inoffensive, if tweaked to spontaneously hidden symmetry. That is, the energetic mechanism spontaneously shifting the symmetry of the ground state and thus the realization of the symmetry results in hiding the symmetry.  The symmetry itself is always inherent, but the "spontaneous" action drives the mechanism of flipping the realization to the Nambu-Goldstone mode, so, in effect, "hiding" it.  But, since I am not sure what part of it grates on you, I would be least qualified to express it more clearly to your satisfaction. Nevertheless, "spontaneous" refers to the cause of the hiding(=breaking), and should not be confused with it. Cuzkatzimhut (talk) 20:07, 19 January 2012 (UTC)

entropic uncertainty principle
I see you've been making a few improvements to the entropic uncertainty principle section of the uncertainty principle article. Do you know much about this subject? For more knowledgeable physicists (as opposed to first year undergraduates or the general public), this section is probably the most interesting part of the article and deserves a little TLC. I've rewritten about half the entire article in the last couple weeks, but I know less about this special topic. Here are a few issues I have noticed. Maybe you know better than I do how to fix them or improve the section in other ways.
 * The units are kind of sloppy. The position x is dimensionless, and the base of the logarithm is confusingly arbitrary (base n, then base 2, then suddenly implicitly base e).
 * Would it be helpful to show a figure with a strongly bimodal ψ and accompanying φ to illustrate how large variance doesn't necessarily mean "uncertain"?
 * The Shannon's inequality link takes me to the Hirschman uncertainty article, where Shannon's inequality is mentioned without citation or proof.
 * This isn't the von Neumann entropy, which is more familiar to the quantum mechanic. Is there a similar inequality available for mixed states?  Do you know of other generalizations involving, say, the Wigner quasi-probability distribution, either in terms of variance/covariance or entropy? Teply (talk) 00:32, 26 May 2012 (UTC)

Yes, I tried to make a few changes in Hirschman uncertainty as well. But I am off on vacation, and I really don't have much time to do an adequate job, especially since it is a bit like writing on sand, with so many people with varying focus and backgrounds writing their thing. I know something on the subject, but I may have strong views on it. An entry point may be this or this. I would invite you to fix the points you raise yourself, as I can't possiby uniformize all conventions employed.... Now, to your specific points:
 * Indeed, even in this article, in the Harmonic Analysis section, and, in general, in info theory, ħ=1. Typically, this  amounts to normalizing both x and p by  √ħ  so now they are dimensionless. (Actually, in e.g. phase-space quantization, they each have been tastefully normalized out of their usual units so that each has units of √ħ); so the Robertson relation units parse out; one can then proceed to drop ħ, if desired, which information theorists almost universally do, as they rarely take the classical limit ħ=0. This schizophrenia is universal, and is even present in the (much poorer) Hirschman uncertainty article. (Incidentally, a name which I disapprove of: entropic uncertainty is more descriptive and more historically honest.) The laebile logarithm base again amounts to a normalization factor preceding the logarithm, and so a normalization for the entropy, which people are resigned to track all the time, and they miss half the time. Perhaps, at peril to sanity, you might attempt to reconcile all normalizations... In the Hirschman  uncertainty article the original authors gave up and provide both.
 * Yes, very. Ideally, the abs square of ψ and Fourier-transform φ. Again, in an ideal world, you might center the spikes symmetrically around zero by an offset a/2 on either side, and give them a width b, and  mention how Δx and  Δp are dominated by a and b, respectively, etc... the bloviation at the end of the Hirschman article... But somebody started them with Scientific-American large numbers, and I was not in a mood to make the thing more precise——and hidebound.  After all, this is an aid to the general public, not to graduate students...
 * Shannon's entropy inequality is actually mentioned there now. It is in all info theory books, and in Shannon's paper, and amounts to a two-liner constrained variation of the entropy of a distribution with fixed/constrained variance V: the maximum entropy yields a Gaussian... It's just that the Hirschman article had the facts arrayed but not connected: the Gaussian is the most random/noisy and least informative distribution of a given size/variance.
 * Your last question is important! Indeed we are talking about a classical Shannon info entropy and not a quantum von Neumann entropy. I think this point is made emphatically in the Hirschman article, but you can't emphasize it enough. The quantum vN entropy, which involves noncommuting operations, can in fact be written  in the same footing as the classical entropy, in phase-space quantization as you asked, through the Wigner Function; and is smaller (has more information,  not less) than the classical entropy, information lost as ħ=0, possibly counter-intuitively (but evident, if you consider S as a function of ħ among others, and you then drop its dependence on ħ in the classical limit.) The two papers I adduced above may illustrate the point in the all-but-trivial case of the oscillator---but you'd be surprised by how many grownups have messed that one up... really. Because I have seen so much mess and pain with quantum entropy and uncontrolled semiclassical limits of it (Wehrl entropy.... ugh...), I took the pragmatic decision to stay away from vN.  The e-print archives are teeming with entanglement entropy in black holes these days, even though most are reluctant to fix rigorous inequalities instead of back-of-the-envelope estimates.
 * In any case, the above discussion compares entropies, and may or may not relate to uncertainty relations. The Robertson or Hirschman relations  don't really care where the probability distributions compared came from: they only contrast the shapes of prob distributions of an abs squared wavefunction and that of its Fourier-transform, the one through variances, the other through Shannon entropies. In QM,  they come from wavefunctions, in signal processing from signal amplitudes, etc. The "classical" Shannon entropy in Hirschman is a fine QM object, i.e. one need not use the Wigner function instead——in fact, the Wigner function is magically configured to yield the same answer for a pure state (once a dx or dp integral is performed, as here); for a mixed state, like the density matrix, it can also represent a mixed state, and, e.g., dp integration yields the x-dependent probability distribution for a mixed state, and dx integration yields the p-dependent probability distribution. To "finally" answer your question, no, I don't know special Robertson type relations for Wigner function marginal probabilities for mixed states, but I gather the answer is the same as for pure states. Consider diagonalizing the density matrix and you have a direct sum of pure states weighted by factors, so you could rerun the Robertson argument superselection sector by sector: e.g., take ρ=½|1><1| + ½ |2><2| with |1> and |2> Gaussian wavefunctions of widths A and B. Then,  is bounded below as for pure states, since the cross terms ( (A/B)2 plus its inverse) are bounded below by 2.  More formally, in phase space, the general mixed case is covered in the 2nd column of p 45 in this.
 * However, the ★-dominated quantum (vN) entropy with a Wigner Function will not yield a different probability distribution, it will just unzip the probability distribution, and spatchcock its components in a special way to account for QM interference phenomena. While this tells you a lot, it does not compare/contrast probability distributions, if that is the object sought. But I may be projecting too ambitious a scope on your question. Cuzkatzimhut (talk) 00:30, 29 May 2012 (UTC)
 * This is all interesting stuff. I see someone asked pretty much the same question as me at .  I also started a Wehrl entropy stub. Teply (talk) 22:49, 29 May 2012 (UTC)
 * OK, if you think so. As I harumphed above, well, the Wehrl entropy is what it is, but the public should understand it as a strange uncontrollable intermediary between the Shannon classical entropy and the quantum von Neuman entropy in phase space. For the Husimi representation, one needs to also insert the (awful) stars in the Husimi representation, to get a fully quantum vonNeuman entropy, such that it vanishes for a pure state, as illustrated for the Wigner function (Moyal representation) in one of the above references. It might inform somebody, as long as it is appreciated that some quantum information has already been sacrificed in it, but nobody can quantify how much ... More broadly, for the Husimi representation itself, more info scientists than physicists often gush at its being positive semidefinite, but they ignore the fact that it is not a simple measure in phase space to compute expectation values of observables with, as its star does not integrate out by parts. So if it is used as a plain phase-space measure, one has already employed an uncontrollable semi-classical approximation without noting or acknowledging it. All hunky-dory, if the truth is adequately demarcated, of course. I did some of that in that stub, now. However, reading the discussion on your link, I have to say, I am completely baffled as to why one is unhappy with the perfectly fine "Hirschman" entropy, to want to expand it in phase space. As I indicated, the Wigner function is configured to yield the exact same Hirschman (upon integrating out dx and dp respectively) with complete quantum information; so I don't see how measuring its lumpiness, after smoothing it out with a Gaussian, would yield anything useful, but I'll hold my breath to see their punchlines!Cuzkatzimhut (talk) 00:20, 30 May 2012 (UTC)
 * It's interesting because you have to be careful about these subtleties. I've only recently learned about the star product / Moyal bracket and how you can do a classical-->quantum mapping with it.  I'm impressed that it looks like a Poisson bracket + power series in $$\hbar$$ instead of the more familiar albeit seemingly more arbitrary association with a mix of operators, commutators and imaginary numbers.  I was wondering if there's something similar with Husimi Q (a question I recently asked math people here though maybe I should have asked instead in a physics forum).  Your updates to the Wehrl entropy article suggest that there's some family of star products for which you could possibly define some family of brackets and Liouville equations.  Is that so?  What is the non-Moyal star product in the Husimi representation?  This subject isn't covered anywhere in Wikipedia as far as I can tell. Teply (talk) 23:01, 30 May 2012 (UTC)

←outdenting. Yes, one has to be careful: I almost blew the vN entropy, skipping a crucial Husimi ★ there at first, which integrates out in the Moyal prescription, but not in the Husimi one—I caught it in time. The systematic correction to PBs with powers of ħ is what charmed Moyal in the first place, and what makes this phase-space formulation good for considering classical limits, instead of the large quantum number stunts with coherent states that most people use. Note Liouvile's theorem (incompressibility of phase fluid) Fails in QM: these very corrections violate it!

These structures are identical for all prescriptions and their equivalent ★s, albeit not as simple/pretty as for the Weyl/Moyal one (the cartesian coordinate system of the lot, so to speak). So, absolutely, as for all such equivalent prescriptions, for the Husimi prescription and its ★, look at the Example in section 0.13 of this update. The Husimi ★-product is given there, its equation of motion, etc...(in fact, its oscillator ★genstates are even simpler than those for the Moyal case!)I stuck it into your Wehrl stub, but it looks oracular as it stands.

But the crucial inability of that ★-product to be integrated out inside a phase-space integral is dealt with in section 0.13 and on a footnote on p 9. It is a very important footnote, reminding one that expectation values of observables need such a ★ in the integrals, otherwise one has made an uncontrolled approximation; and electrical engineers if not a fair fraction of physicists, completely miss its import! It would be too messy to dump technical detail in wikipedia, but the 5 dozen people who love phase-space quantization know all about it. The systematic equivalence and translation of all these prescriptions was worked out by L Cohen, but the wikipedia page the Cohen classification scheme redirects to, Bilinear time–frequency distribution, is magnificently bad and useless... Cuzkatzimhut (talk) 23:55, 30 May 2012 (UTC)


 * See section 1.5.3 of, on which I based an answer to the forum question. Anyway, the authors there and elsewhere make it clear that you end up with an extra $$\ln{(\delta x \delta p / \hbar)}$$ term hanging around when you're careful with units and binning.  These conference notes might be around the right pace to incorporate into the article, along with maybe some of the angular stuff. Teply (talk) 19:00, 18 August 2012 (UTC)
 * Thanks, I was not aware of the chapter, but it looks like a nice exposition of their paper on the Hirshman uncertainty, not quite a joint phase-space treatment, unless I missed something... In 1.5.3, they basically repackage their x and k probabilities into an object that looks like the Mehta distribution, with an all-crucial self-duality constraint that makes it non-negative in general. So, basically, by dint of the normalization of the probablilities, ρ(x), σ(p), of the  normalized mutually Fourier conjugate wavefunctions, they notice ∫dx ρ lnρ + ∫dp σ lnσ  =  ∫dxdp  ρσ  (lnρ + lnσ) = ∫dxdp(ρσ) ln(ρσ), and that's that... ρ(x)σ(p) is now a "joint" phase space distribution... It may not matter that its square root coincides with the Mehta, up to an irrelevant phase exp(ipx),  as none of the ★s or properties of the Mehta are used... I have not yet seen an obvious generalization to all mixed states.... The extra binning offset you mention is the standard "quantum offset", e.g. eqn (7) of the conf paper, and more extensively in, I'd mentioned back then, which Wehrl has described well in his review, and plumbs the essence of the classical limit.  Also see followup of eqn (14) & (19) in that conference discussion. Basically, gaussians are as classical and uninformative (random) as you can get... But, ultimately, the Hirschman inequalities really look at the classical, Shannon, entropy of quantum systems, and not the lower vN entropies, as I had suggested... that is, to my knowledge, introducing ★s as Lubos threatens is silly. So, a bona-fide quantum distributions in phase-space to quantify uncertainties is still an open problem (which may not need to be solved?) Your answer looks sound, although I'd personally shout that this has nothing to do with vN entropies (or Wigner functions, except that you can derive probability densities in x and p from Wigner functions, if you wished to, instead of working with wave-functions!) In any case, I always urge people to test new ideas of this type with both the ground state, and the first excited state, and then their mixed superposition, of the SHO, and get numerical answers. Looks like B-B&R followed the lead of the above conf refs and did half of that.Cuzkatzimhut (talk) 21:43, 18 August 2012 (UTC)

PS: The B-B&R k-x sandwich corresponds to the Mehta prescription, Moyal's (5.5), in phase-space quantization (but their self-duality constraint basically unzips the complexity out of the joint distribution). Archeological aside: It was first stumbled upon, to my knowledge, by Terletsky in 1937--go to the end of the paper--  and then by Blokhintsev in  1940. Qua reps of the vNeumann entropy, they need a Mehta *-product, assuming you symmetrize in x and k, but it may give you the Hirshman result fast... Try gnd and 1st excited state.....Cuzkatzimhut (talk) 21:57, 18 August 2012 (UTC)
 * If you've got something worth shouting, I'd suggest doing it yourself by commenting on the stack exchange. Your talk page isn't exactly the most public arena.  B-B&R don't go through and explicitly show the final result, though if I understand their notation it's something like $$2H^{(x,k)}>1-\ln{2}-\ln{(\delta x \delta k)}$$ where $$H^{(x,k)}=-\int f(x,k) \ln{|f(x,k)|} \, dx dk$$, I think?  If f is the Mehta-Rivier representation (or close to it), then you should be able to rewrite it as some convolution with the Wigner function, right?  You're more the expert on that.  It would look messy, but it would accomplish the program goal. Teply (talk) 23:25, 18 August 2012 (UTC)
 * Alas! I don't want to engage in public arguments on a non-starter, let alone waste shouting... The all-important constraint is the self-duality (1.22) of theirs. You are absolutely right that there is an invertible Cohen transformation from Wigner to Mehta, something like (5.5) in Moyal's original paper? but I don't see the point of doing this work, and seeing how (1.22) mutates, etc... and all for what? to just re-express the standard Hirschman uncertainty so it looks "unified"?  My original feeling was that you don't get past Hirschman by looking at the quantum vN entropy... you may express it in the Mehta prescription, with the Mehta ★s all over the place, but this is not what BB&R do... they drop any and all ★s (an uncontrollable semiclassical approximation), impose the self-duality constraint, and reproduce the Hirschman result...if someone felt fulfilled by looking at joint distributions, I should not stand in their way... If something informative came out that way, I should not stand in its way, either... But... why all this? Cuzkatzimhut (talk) 00:35, 19 August 2012 (UTC)

PS In fact, looking at Migdal's question again, I'm not quite sure what he is asking: 1) can you re-express vN's entropy in phase space? (obviously yes, by starring the Wiggies, as Lubos reminds him, and you find in the literature;positivity issues are pointless for *-functions defined through series.) 2) can you improve on Hirschman's inequality by using something like that? (my bet is not, and BB&R fail to change my mind.) But... am I parsing the question right?? Cuzkatzimhut (talk) 01:12, 19 August 2012 (UTC)
 * I think he's brainstorming for a way to improve on the Hirschman inequality slightly. Looking at his comment on Stefan's answer, I get the feeling he's trying to adapt something like Wehrl entropy to do this.  The dissatisfaction with the Wehrl version is that it's weaker than the usual Hirschman inequality because of the smearing.  He was hoping if there's something analogous to Wehrl entropy for Wigner (or some altogether new entropy definition), then it could at least reproduce the Hirschman inequality and possibly do a little better.  If I'm understanding correctly, the Mehta version is faithfully reproducing the Hirschman inequality with no smearing.  Maybe in some other representation (which?), it's possible to do better. Just speculation, and way beyond my mathematical prowess.  And as you say, it's not obvious to me that there is an improvement to be made using the approach. Teply (talk) 01:52, 19 August 2012 (UTC)
 * Beyond me, too--i don't want to choose between patience or ability...I haven't fully bought the structure of their constraint (1.22) in selecting mixed states that would do the trick. Does their portentous "This rearrangement is not so trivial when we extend it to the mixed states" promise a result, or pose an open question?? In any case, PM's vision looks too deep: Hirschman uses classical (Shannon) information theory to probe the uncertainty, but somehow PM wishes to stretch it to some type of quantum information. There definitely is lots in quantum informatics I have not learned...  Cuzkatzimhut (talk) 11:16, 19 August 2012 (UTC)

Wigner quasiprobability distribution
Hi. Thanks for the corrections. If I am not mistaken, Berezin discusses the phase-space path integral for the amplitude, whereas Marinov discusses the phase-space path integral for the Wigner function. If I'm right, then quoting Berezin in this article is not entirely appropriate. Ciao, Taulalai (talk) 08:30, 25 September 2012 (UTC)
 * The picture and techniques are there, though, and a step and a half get you there. Of course, Marinov was hardly uninspired by his old collaborator's setup. In any case, Sharan is incontestable. My sense is that this is not the place to parse out historical precedence: Anyone interested in actual evolution of WFs can hit the professional literature and verify most of these setups are purely formal and do not contribute much to the bread and butter pursuit of actual answers, as we speak. If you felt strongly, however, you might drop Berezin, but Sharan surely belongs. Your ambition in appreciating Marinov's approach could best express itself in a thorough rewriting of the quantum characteristics article, which is so substandard... Do you feel up to blending/grafting Marinov in there?Cuzkatzimhut (talk) 12:37, 25 September 2012 (UTC)
 * Beresin’s paper does not fit the topic. This paper could be mentioned in Path integral formulation. I have not found the path integral for the Wigner function in Sharan's paper. The article on quantum characteristics is too long and overloaded with formulas. I can provide small remarks only.Taulalai (talk) 09:05, 29 September 2012 (UTC)

Hi, I have a bugging-me question
I have been away from editing for a while. I was looking at a definition of "quantum" somewhere, one with which I rather disagreed, and it reminded me of an assertion made by one of the people with (if I remember correctly) graduate physics credentials. I have been trying to find the conversation by going through old discussion pages, but so far with no results. I thought you might either remember the discussion or have something to say on the subject yourself.

As best I can reconstruct it the claim it was that since there can be no fractional quantum of action, anything such as what appears to be a continuous spectum must in fact have gaps in it, but because of the extremely small numerical value of the Planck constant these gaps would be too small to be detected or observed. I hope the original poster will forgive me if I have turned something that was originally correct into nonsense. The only things that remain clearly in my mind after a year or two interval is that an assertion that argued from the Planck constant and my mental image as I read what he said, which is of a segment of a spectrum of some sort but interrupted periodically by tiny gaps.

Thanks.P0M (talk) 01:16, 3 July 2013 (UTC)


 * Apologies, I was not part of that converation, so I do not recall it! You do seem to be outlining the basic idea behind quantization, e.g. of energy levels, etc. Gaps become insignificant at high quantum number states, so the respective quantities appear continuous, macroscopiclly. But QM is the study of these gaps. Perhaps classical limit or Correspondence_principle might come close to addressing your question. Cuzkatzimhut (talk) 02:50, 3 July 2013 (UTC)


 * Thank you. Your way of putting it avoids the problem I had with the original conversation. P0M (talk) 15:45, 3 July 2013 (UTC)

Abel function: Quick question
Regarding your recent edit summary on the page functional square root - are you saying that the Abel function article is a disaster, or the Abel function itself, or both? Also, is Schröder's equation the superior way of finding the "iteration orbits" of functions (not to claim I understand anything about the subject at all)?

Very nice contributions; I've seen your edits before. You may well be one of the best contributors on Wikipedia, in terms of quality. Ginsuloft (talk) 15:12, 5 August 2013 (UTC)


 * Apologies for my fractured syntax; yes, the Abel function stub is a disaster, and frankly, hardly adds anything to the Abel equation one. Ideally it should be gone, or merged, but I don't want to fuss. Thanks for your compliments. Indeed, Schröder's equation  is the way to produce iteration orbits, cf iterated function, but some mathematicians prefer the almost equivalent Abel equation, contingently on the actual nature of the solutions. Schroeder, Boettcher, and Abel are always in the toolbox, and one uses the one that suits one best. It might be clear from my edits which one I have found helpful in the past myself! Cuzkatzimhut (talk) 15:24, 5 August 2013 (UTC)

Pauli–Lubanski pseudovector: Who exactly is Lubanski?
As you added the reference:



do you know who he is? It would be nice to have an article on him, over several months of searching, I have only been able to make a very crude and probably inaccurate (or completely wrong...) draft: User:Maschen/Lubanski. Diacritics etc. in names will be fixed later.

Your help would be appreciated. Thanks in advance, M&and;Ŝc2ħεИτlk 10:05, 13 October 2013 (UTC)


 * Sorry, I have only heard of him in connection to that paper.... And the Polish WP doesn't have him. Your draft looks neat and ready for a WP stub, and, once there, someone could add to the knowledge stack! I'll keep my eyes open. Cuzkatzimhut (talk) 10:55, 13 October 2013 (UTC)


 * I'm not confident the draft is accurate, but as you say, if this enters mainspace it allows others to look at it. Cheers, M&and;Ŝc2ħεИτlk 12:20, 13 October 2013 (UTC)

New article: Myron Mathisson
Just a pointer that we now have an English article on him, cf talk:Myron Mathisson. If you have edits to make, by all means don't hold back. Best, M&and;Ŝc2ħεИτlk 10:09, 27 October 2013 (UTC)
 * I linked the obituary by PAMD, but know too little about him to be useful right now, trying to migration manage a new mac. No, translation from pl is not plagiarism, and overlaps its superior references quite completely. You are raising the academies of the worthy dead quite tellingly! Best, Cuzkatzimhut (talk) 10:52, 27 October 2013 (UTC)


 * Thanks, as always! ^_^ M&and;Ŝc2ħεИτlk 10:58, 27 October 2013 (UTC)

We're not done yet.

I was tempted to release a direct translation from the German WP on Achilles Papapetrou into mainspace, but lots of work needs to be done and it's easier to cobble things together in a draft, even in just a few edits. If you have the inclination and time, feel free to add to User:Maschen/Achilles Papapetrou before its release into mainspace. Thanks, M&and;Ŝc2ħεИτlk 14:47, 27 October 2013 (UTC)

A barnstar for you!

 * Pleased to be appreciated. This was just a small tweak in passing, by way of linking this section to the relevant 2×2 section of the Matrix exponential, where the very same identity is derived in different language and with a somewhat different focus and logic.Cuzkatzimhut (talk) 23:14, 2 November 2013 (UTC)

Higher-dimensional gamma matrices
Just to say "thanks" for all the work you've been doing on this article. Deltahedron (talk) 17:16, 19 April 2014 (UTC)
 * Thanks for your questions.... I need such to see what is missing (a lot, obviously...). In an ideal world, one would simply refer to the primary/main articles, Gamma matrices, spinors, etc., and let the imaginative reader translate and extend. However, these articles themselves are really deficient... Ask the poor reader for the explicit Lorentz transformation of a spinor rep, and he'll walk away muttering and shaking his head in frustration. I finally got a decent link in Representations of the Lorentz group. I think I am now done, though. Cuzkatzimhut (talk) 14:32, 22 April 2014 (UTC)

Regarding an edit on BCH formula page
Hi, in |one of your edits you added a corollary that I cannot see how you obtained. In $$e^{X}Y e^{-X} = e^{\operatorname{ad} _X} Y $$ if we replace Y by exp(Y) and multiply both sides with exp(-X), we get $$\operatorname{exp}(X)\operatorname{exp}(Y) = e^{\operatorname{ad} _X} (\operatorname{exp}(Y)) \operatorname{exp}(X)$$ which seems to me very different from the stated corollary $$\operatorname{exp}(X)\operatorname{exp}(Y) = \operatorname{exp}(e^{\operatorname{ad} _X} Y) \operatorname{exp}(X)$$. Could provide some insight or a resource? Thanks! at0 08:40, 3 July 2014 (UTC)
 * Indeed, multiplication of these expressions is associative, so (exp(X) Y exp(−X))   (exp(X) Y exp(−X)) = exp(X) Y2 exp(−X), as an equivalence map. You  repeat this many times to generalize to any power of Y, and whence to any function of Y understood as a power series; and hence to the exponential function you are trying to appreciate,     exp( exp(X) Y exp(−X)) =  exp(X) exp(Y) exp(−X).  The analogous result, of course, holds for an arbitrary function, not just the exponential that puzzles you. In the article itself, the adjoint endomorphism involved is applied in Baker–Campbell–Hausdorff formula.     Cuzkatzimhut (talk) 10:51, 3 July 2014 (UTC)
 * Thank you. at0 14:42, 3 July 2014 (UTC)

Re Mehler formula
Transferred to Talk:Mehler kernel Cuzkatzimhut (talk) 20:25, 29 July 2014 (UTC)

Regarding Uncertainty Principle page
I hope this is right place to communicate with you, I am sorry If not. I have successfully published new paper on Certainty Principle Using Complex Plane, you can find it following link http://www.ijapm.org/show-45-431-1.html.

This paper proves uncertainty principle is not complete picture what we observe. I am not able to add any of my work in Wikipedia (because it will be self advertisement).

Can you please guide me how to get my work in Wikipedia ? You can see all my work in following page https://www.researchgate.net/profile/Bhushan_Poojary2?ev=hdr_xprf — Preceding unsigned comment added by 182.59.189.230 (talk) 10:33, 10 August 2014 (UTC)


 * I'm not sure why you are contacting me. Wikipedia is not a discussion forum,or a publicity machine and should not be misused as such. Try professional venues, such as Researchgate, or ArXiV, or conferences. You are very much in the wrong place. Please keep off Wikipedia. Cuzkatzimhut (talk) 11:10, 10 August 2014 (UTC)