text:
Talk:Self-adjoint operator

Untitled
Hey there! What about having the distinction between Hermitian operators and Self adjoint operators added to this otherwise very good presentation... se for example some common lectures like http://www.hep.caltech.edu/~fcp/math/differentialEquations/linDiffEq.pdf Thanks  — Preceding unsigned comment added by 89.242.21.31 (talk) 07:30, 29 April 2012 (UTC)
 * It is already written: "Bounded symmetric operators are also called Hermitian." Boris Tsirelson (talk) 08:15, 29 April 2012 (UTC)

Notation Unclear
This is probably obvious to most experts, but I think that the notation < | > should be defined before it is used in the first section. I added a clarification tag to this. —Preceding unsigned comment added by BradLipovsky (talk • contribs) 02:12, 11 September 2010 (UTC)

Proof Needed?
A symmetric operator on a hilbert space can have no eigenvalues? Isn't 0 an eigenvalue?

 * Not in general. If it's invertible, for instance 0 is not an eigenvalue.--CSTAR 07:54, 30 December 2006 (UTC)

More general statement: Should there be a proof policy on wikipedia math articles? Or a proof tab or something for each claim? That would be nice. Maybe people should start adding proof pages or something. Discuss or copy and paste this where it's supposed to go (I don't want to look that up right now).

 * To mu knowledge there isn't a proof policy. This is supposed to be an encyclopedia, not a textbook or reference.--CSTAR 07:54, 30 December 2006 (UTC)

 * There are a couple of math wiki projects, though still in their infancy. It would be worth linking to them once they are up and running: Swim, VDASH. Drevicko (talk) 06:04, 23 March 2010 (UTC)

Quantum version
The latest anonymous "edit" introduces a so-called "quantum mechanical version" of the spectral theorem, which is the same version and unnecessarilly separates out the pure point from continuous spectrum. I have no objection to a comment that the spectral theorem is formulated in Bra-ket notation in some form, but to call it a "quantum mechanical version" is misleading.--CSTAR 20:13, 10 July 2005 (UTC)

 * Thank you for editing my "quantum mechanical version" of the spectral theorem. You are right when you say that the title was misleading.  You have replaced the term analytical function by Borel function.  The problem is that  I don't understand what is a Borel function.  In physics text books one finds analytical function and most physicists know what it is.  The physicists define $$ f(H)= f(0) + f'(0) H + {1 \over 2} f(0) H^2 + ... $$ using the Taylor expansion of f''.  Is the Borel function somewhat equivalent to this?  I also in principle agree with the fact that you suppress the summation over the discrete space but I think this is not very pedagogical for the people coming from the quantum physics or quantum chemistry community.  I am teaching to chemists.  I know what I am speaking about.  I think this is an encyclopedia and one should take care that when possible it should be readable by people who don't want to cope with the whole functional analysis but nevertheless want to understand some key concepts they are using in their own field on a daily basis.  Moreover I am persuaded you are right when you say both version of the theorem are the same.  But I am very sorry but I don't understand really why and how.  Do you think you could make some notice explaining why both version are equivallent?  Thank you very much.

As was pointed out, the power series formulation of f(H) only works in general when one has a bounded H and an analytic f. In quantum mechanics, hamiltionians are never bounded unless the state space is finite dimentional. So the two version are really NOT the same. The Borel functional calculus is honest mathematics, while the other formulation, although useful for physicists, is fundamentally incorrect. But, loosely speaking, we can see the formal resemblance between the two approaches. The projection-valued measure(PVM) in the Borel functional calculus sort of correspond to Dirac's "rank 1 projection", |x><x| over the "spectrum" to obtain the operator. While the convergence of integrals in PVM's can be defined in various topologies(e.g. the weak topology), it's never pointed out what's exactly meant but Dirac's "integral" over sometimes nonexistent "spectrum".

 * Be my guest and chuck it. A while back (a year ago?), I tried to, but kept getting reverted.--CSTAR 18:04, 29 March 2006 (UTC)

 * I need to clarify about the validity of the power series formula. There is actually an analytic functional calculus for unbounded operators, using an operatorial version of the Cauchy integral formula.

necessary?
As i know sigma-finiteness is not needed for a multiplication operator to be densely defined. Enlighten me or i will delete the remark.

 * Hmmm. I think you're right.--CSTAR 21:32, 26 February 2006 (UTC)

geometrical interpretation
The article contains a section called "geometrical interpretation". It states that
 * $$ \operatorname{G}(A^*) = (\operatorname{J}\operatorname{G}(A))^\mathrm{perp}.$$

I understand this to mean the orthogonal subspace

 * $$ \operatorname{G}(A^*)

= (\operatorname{J}\operatorname{G}(A))^\mathrm{perp} = (\operatorname{J}\operatorname{G}(A))^\perp $$

(x,y) \in H \oplus H : \langle x|A\xi \rangle = 0 \mbox{ and } \langle y|\xi \rangle = 0 \;\;\forall \xi \in H \} $$ (Wrong!)

Is this correct? If so, then I'd like to know why J was choosen to be symplectic -- i.e. why was J set up so that $$\operatorname{J}^2=-1$$ ... it seems to me that the definition would have worked equally well for a map $$(\xi, \eta) \mapsto (\eta, \xi)$$, right? I don't see the role played by the minus sign. I mean, that J is symplectic is very suggestive of something; I'd like to know why. linas 21:46, 18 March 2006 (UTC)

 * I think it does make a difference, since if (x,y) belongs to a linear space V, (x, − y) doesn't have to belong to V (consider a 45 degree line through the origin.)--CSTAR 21:55, 18 March 2006 (UTC)

 * Right. My definition is wildly flawed. Dohh. It should have read
 * $$(\operatorname{J}\operatorname{G}(A))^\perp

= \{ (x,y) \in H \oplus H : \langle (x,y)|(\xi,-A\xi) \rangle = 0 \;\;\forall \xi \in H \} $$
 * Dohhh. Thanks. I'm going to copy this into the article.linas 00:06, 19 March 2006 (UTC)

Alternate characterization
The article states:
 * The following is an alternate characterization of symmetric operator: A densely defined operator A is symmetric iff
 * $$ A \subseteq A^*.$$

What the heck does this subset notation mean?? linas 21:46, 18 March 2006 (UTC)

 * This is standard terminology for operators. One is an extension of the other.--CSTAR 21:47, 18 March 2006 (UTC)

 * Do we have an article somewhere that defines this? I'm thinking of this in terms of topology, that the one is defined in a topology that is finer than the other. But rather than guessing, I'd like to send the gentle reader to an actual definition. linas 23:09, 18 March 2006 (UTC)
 * It means the graph is a subset. -lethe talk [ +] 03:29, 19 March 2006 (UTC)

 * So should I edit the article to say "... $$ G(A) \subseteq G(A^*).$$ This is often written as $$ A \subseteq A^*$$, with the use of subset notation implicitly implying that the relation holds for the graph of the operator." I'll make this edit now; if its wrong, back it out please.  linas 05:19, 19 March 2006 (UTC)
 * That language sounds funny to me. It's not "often written as".  It's the definition of extension of operators. -lethe talk [ +] 05:23, 19 March 2006 (UTC)

 * So where is this defined? Read my first post, wherein I ask: "what the heck does this mean"? We need a definition of this notation. Anyone who does not already know the answer won't be able to glean the meaning from the current wording. Note redlinks: Graph of an operator, operator extension. linas 20:48, 19 March 2006 (UTC)

 * In the usual set-theoretic formalization of function, a function is identical to its graph .--CSTAR 20:55, 19 March 2006 (UTC)

Argh. I'm finding this conversation very irritating. The subset notation is unknown and utterly opaque to anyone coming from an engineering or physics background. I'm not convinced its all that common in the world of math that's not dealing with operator theory. I attempted to add words to define it such that an undergrad *could* understand. Lethe promptly reverted, with no explanation other than a trite complaint about the wording. Can I fix the article, already? linas 05:01, 20 March 2006 (UTC)

 * It looks OK now, mathematically, although I don't particularly like the way the inline tex looks, aesthetically.--CSTAR 05:14, 20 March 2006 (UTC)

 * I use a font size about the same size as the image fonts, and also enable MathML in my WP preferences. I have hopes for BlahTeX. linas 15:11, 20 March 2006 (UTC)

 * BTW further down, I had used \mathrm{perp} instead of \perp (maybe when I added that \perp wasn't working.) --CSTAR 05:16, 20 March 2006 (UTC)

 * Will fix shortly. linas 15:11, 20 March 2006 (UTC)

Extension of symmetric operators
I don't think this article is the right place to define an extension of an operator. It should go in unbounded operator. -lethe talk [ +] 08:51, 20 March 2006 (UTC)

 * I'd suggest its own unique article, e.g. extension of symmetric operators. That section and everything beyond it certainly does take the discussion of operators to a whole new niveau. linas 15:38, 20 March 2006 (UTC)
 * Well, I might agree with you on that. I think there might be enough to say about extensions to symmetric operators to merit its own article, but I'm not sure.  In any event, the definition of the closabillity of operators, and also of extension of operators, makes sense for general unbounded operators, not just symmetric operators, therefore that definition should go there.  Then we don't have to clutter this article with definitions.  As for your complaint that it's too "operator theoretic", well, the distinction between self-adjointness and symmetry of operators is itself a bit operator theoretic.  Anyone who wants to read this article will have to get at least a little of the terminology of unbounded operators. I'm going to take a look at that article now. -lethe talk [ +] 15:52, 20 March 2006 (UTC)
 * I'm looking at closed operator now, and I don't really like it. It's not really set-up to talk about general unbounded or non-everywhere-defined operators, which makes it difficult to have the definitions there that I want.  I don't think I have the energy to do a rewrite at the moment.  I'm starting to think that I'd like to see an article about linear operators on general TVSes.  It could talk about the different kinds of boundedness, and when boundedness implies continuity, and operators not everywhere-defined, graphs, extensions, and closures.  All of these make sense in general TVSes.  What do you think?  This might be nice because like eg Oleg doesn't like to have stuff about spaces that are not normable in a lot of places, but those spaces still need to have a home. -lethe talk [ +] 16:08, 20 March 2006 (UTC)
 * No reason not to. They occur in nature.--CSTAR 20:47, 20 March 2006 (UTC)
 * PS. I don't like the idea of breaking up the article. I rather like the idea of having a lot of stuff on s.a. operators in one place.--CSTAR 20:50, 20 March 2006 (UTC)

 * I'm afraid I can't add anything. My understanding of these topics is incomplete and spotty; I don't have the vision to suggest an overall structure. I know little/nothing about operator extensions beyond what this article says, and  a breif skim of some papers on rigged hilbert space and related topics. On the other hand, I do have a tried-n-true method of writing: write a few paragraphs, put them where they seem to fit best at this point in time. Write a few more. After a while, one realizes that the first few paragraphs would be better if moved somewhere else .. or expanded or deleted ... rinse, and repeat. As Charles Matthews put it, the result is "organic". linas 20:25, 20 March 2006 (UTC)

Crazy question
These days, whenever I see the words "Cayley transform" I think "Riemann surface/Fuchsian group/j-function/etc.". Can anything interesting be said along these lines? Or is this a crazy non-question? linas 15:38, 20 March 2006 (UTC)

 * Not much more than is already on the page to which it is linked (somewhere in the article). The Cayley transform is a fractional linear transformation; the namw used for operators is mostly used by analogy, although $$H \oplus H$$ does have a symplectic structure, but not too much should be made of this fact, at least not in this article.--CSTAR 20:45, 20 March 2006 (UTC)

 * OK thanks then. I've come up with a remarkable little problem, then; I dare say no more. I haven't had this much fun playing at solving things in quite a while. linas 00:11, 21 March 2006 (UTC)

Multiplication operators
This article states, in the section on the spectral theorem, that

 *  Theorem. Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.

By contrast, the article spectral theorem qualifies this further, stating that any bounded self-adjoint operator is unitarily equivalent to a multiplication operator. So, does an operator need to be bounded for this theorem to hold, or not? linas 23:05, 18 March 2006 (UTC)

 * No. It's true for arbitrary self-adjoint operators.--CSTAR 23:41, 18 March 2006 (UTC)

 * Can I get you to review the article on spectral theorem, and remove the "bounded" qualifier from it? linas 00:05, 19 March 2006 (UTC)

 * There is already a short section on the unbounded case. Isn't that good enough? --CSTAR 00:49, 19 March 2006 (UTC)

 * OK, right. I guess I skipped reading the last paragraph, which circles right back to this article. I've been feeling confused today. linas 05:07, 19 March 2006 (UTC)

Deleted graf
I deleted a paragraph which I believe used "support" incorrectly. The support of a measure, including a projection-valued measure, is the smallest closed set outside of which the measure vanishes. What the contributor meant, I am sure, was not support, but rather allude to the fact that a measure has a canonical decomposition into sum of atomic and continuous parts which are disjoint (a la Hahn decomposition), but that should be made clearer, and not use the word "support" which has a very precise meaning in analysis.--CSTAR 17:41, 2 April 2006 (UTC)

 * CSTAR Thanks. I got lazy and didn't wanna elaborate. —This unsigned comment was added by 24.155.72.152 (talk • contribs).

Functional calculus
A recent anonymous edit added the sentences:

 * For the bounded case, an alternative way of obtaining the Borel functional calculus is the following: First pass from polynomial to continuous functional calculus using the Stone-Weierstrass theorem. The use the Riesz-Markov theorem to pass from integration on continuous functions to spectral measures.

This is very misleading. To extend the polynomial functional calculus to the continuous functional calculus requires showing continuity of the mapping

 * $$p \mapsto p(T)$$

from polynomials to operators, as a function of the polynomial p in the supremum topology. This is not entirely trivial, and in fact is one stumbling block in the proof of the spectral theorem by elementary means. The proof hinges on the following fact: If T is a bounded self-adjoint operator with spectrum in the closed interval [a,b] and p is a polynomial which is non-negative on [a, b], then p(T) is a non-negative operator.

See for example S. Lang's treatment of the spectral theorem. I'm not sure of the historical facts here, but this may have been the way the spectral theorem for non-compact operators was first shown, before the Gelfand theory of commutative C*-algebras. --CSTAR 19:43, 2 April 2006 (UTC)

 * CSTAR the statement I made was correct. From the polynomial fucntional calculus one can be extended to an isometric continuous functional calculus by Stone-Weierstrass rather easily. The isometric property even holds for the Borel case, provided one is careful. See, say, Reed and Simon. This is perhaps a less deep approach than using Banach algebras, and stuff like Stone-Cech and Wiener's result are not corollaries. Also, it was sloppy of me to have said what I said about discrete spectral measures. Thanks for pointing it out. Perhaps someone can elaborate in an article about decompositon into spectral subspaces. —This unsigned comment was added by 24.155.72.152 (talk • contribs).

 * I didn't say the polynomial functional calculus couldn't be extended in this way. What I said was that this extension is not entirely trivial, because you need to establish first that the mapping is continuous (or equivalently that it is positive). Your edit omits a reference to this fact, which is why I said it was misleading (notice that I didn't say it was wrong).--CSTAR 21:17, 2 April 2006 (UTC)

 * Also could you please sign your posts on talk pages? Thanks.--CSTAR 21:19, 2 April 2006 (UTC)
 * Do so by appending ~ to your comments, for your information. -lethe talk [ +] 21:36, 2 April 2006 (UTC)

 * Previous two comments modified for additional clarity. --CSTAR 21:35, 2 April 2006 (UTC)

Ok, sorry about not signing posts. So I am 24.155.72.152 on a few of the math pages Mct mht 22:35, 2 April 2006 (UTC)

CSTAR, i still don't quite get what's nontrivial about it. the fact you cite as essential:
 * If T is a bounded self-adjoint operator with spectrum in the closed interval [a,b] and p is a polynomial which is non-negative on [a, b], then p(T) is a non-negative operator.

is actually not needed to show the functional calculus is an isometry. when you say Lang's treatment do you mean his "real and functional analysis"? will look it up when i can. Mct mht 23:11, 23 April 2006 (UTC)
 * If you assume the continuous functional calculus, then of course it's trivial. Otherwise, you need to show that any non-negative polynomial on an interval [a,b] is a sum of polynomials which when applied to T is obviously non-negative. --CSTAR 23:14, 23 April 2006 (UTC)

 * Hm, if you're saying your "you need to show..." is required to show the existence of an isometric functional calculus (which is what i thought you meant in the initial comment), i think it can be bypassed in that proof. rather it follows as a consequence. Mct mht 23:23, 23 April 2006 (UTC)
 * Hmm. I don't believe it.--CSTAR 00:10, 24 April 2006 (UTC)

CSTAR, ok, from what i can see, our disagreement is the following: let $$A \rightarrow P(A)$$ be the usual polynomial functional calculus, you're saying the spectral mapping theoremi.e. $$\sigma (\;P(A)) = P(\sigma(A))$$ is a major stumbling block en route to continuous func cal. i am saying it's rather straightforward. The following argument is from Reed and Simon:
 * Say $$\lambda \in \sigma(A)$$. Given P, we factor $$ P(x) - P(\lambda) = (x - \lambda)Q(x)$$. So $$P(A) - P(\lambda) = (A - \lambda)Q(A)$$. If $$A - \lambda$$ has no bounded inverse, neither does $$P(A) - P(\lambda)$$. This shows $$P(\sigma(A)) \subset \sigma(P(A))$$.
 * For the other inclusion, let $$\mu \in \sigma(P(A))$$. Again, we factor $$P(x) - \mu = a (x - \lambda_1) \cdots (x - \lambda_n)$$. Now $$P(A) \; - \mu$$ has no bounded inverse, that means $$A - \lambda _i$$ has no bounded inverse for some i. So we have $$P(\lambda _i) \;= \mu$$ with $$\lambda_i \in \sigma(A)$$. This proves the claim.

from the above follows the isometric property. it is perhaps good to compare with Lang's treatment you mentioned. i will look it up or perhaps you will tell me. Mct mht 02:58, 24 April 2006 (UTC)

 * OK, then how do you prove that IF A is a non-negative operator such that P(σ(A)) is contained in the interval [0,1] THEN P(A) has norm at most 1? At some point you are going to have to address the issue of the relation between spectral radius and norm. You haven't yet, by your argument. The crux of the standard Gelfand theorem as found in most books on C*-algebras is basically dealing with that technical point. You can avoid the Gelfand theorem by using the polynomial argument I mentioned.  I believe that constituted von Neumann's proof of the spectral theorem for bounded self-adjoint operators.--CSTAR 03:10, 24 April 2006 (UTC)

 * Heh, ok, interesting. the isometry is shown as follows, for self adjoint A we have that its operator norm and spectral radius coincide. Also, $$\| T \|^2 = \| T^* T\|$$ in general, so

\| P(A) \| ^2 = \| (P(A))^* P(A) \| = \| ({\bar P} P)(A) \| = \sup_{ \mu \in \sigma(({\bar P} P)(A)) } |\mu| = \sup _{\lambda \in \sigma(A)} | {\bar P}P (\lambda) | = ( \sup _{\lambda \in \sigma(A)} |P (\lambda) | )^2 $$
 * Mct mht 03:28, 24 April 2006 (UTC)

 * Nope. If you look carefully, you are using the spectral radius formula to equate norm with sup of the spectral values. That is true for C*-algebras, but is equivalent to the Gelfand isomorphism theorem.--CSTAR 03:33, 24 April 2006 (UTC)

 * ok, i said that before laying out the argument, that for self adjoint operators, spectral radius is operator norm. and the spectral radius formula can be proven by invoking the Laurent series for resolvents, without using the C* algebra formulation. Mct mht 03:44, 24 April 2006 (UTC)

 * Fine but that's the crux of the standard C*-theoretic proof of the Gelfand isomorphism!--CSTAR 03:46, 24 April 2006 (UTC)

 * it's very informative discussion. thanks. Mct mht 03:53, 24 April 2006 (UTC)

partial isometry
"partially defined isometry" is perculiar terminology. is it used at all in literature? as pointed out later in the same paragraph (extension of symmetric operators), one customarily extend by continuity and then by zero to get something defined everywhere. "unique partially defined linear operator", also rather uncommon language, should be taken to mean the continuous extension is unique anyhow. i didn't change it since i didn't see anything incorrect, just somewhat unusual and perhaps unnecessary. Mct mht
 * Why is this a problem? Partially defined function is a perfectly good concept and standard terminology.--CSTAR 23:00, 23 April 2006 (UTC)

 * If L[y] and G[y] are both self-adjoint operators having the same spectrum my question is are they related by a linear transform?.--85.85.100.144 15:11, 12 January 2007 (UTC)
 * Do you mean if they are unitarily equivalent or if not at least similar? The answer is no to both questions, because spectrum (or more strongly spectral measure) does not deal with spectral multiplicity.--CSTAR 16:24, 12 January 2007 (UTC)

operator integral
CSTAR actually, you can. it's not trivial but it's been done. i just thought it should be mentioned Mct mht 23:26, 23 April 2006 (UTC)
 * above comment is re possibility of convergence of operator integral in strong operator topology. Mct mht 23:28, 23 April 2006 (UTC)
 * OK I believe you, but it's news to me.--CSTAR 23:30, 23 April 2006 (UTC)
 * Could you provide a reference for how this is done? I'm still pretty skeptical.--CSTAR 13:17, 24 April 2006 (UTC)

 * CSTAR, ok, it is rather obscure. i haven't found an explicit reference to a paper. it's done using vector valued measures, according to Reed and Simon. convergence in the weak operator topology looks to be the most convenient and common approach taken. since all the well-known texts seem to take that approach, i am going to remove the SOT note. Mct mht 01:19, 27 April 2006 (UTC)

 * This is an ancient post, but I would like to mention that I am taking a graduate course where we are currently discussing operator integrals in the strong topology. Maybe screwy things can happen in the cases we haven't covered yet though.--91.11.110.253 (talk) 22:05, 13 January 2012 (UTC)

Bounded adjoint
No, an unbounded densely defined operator can have bounded adjoint. It can have even trivial adjoint, that is A^*={(0,0)}. Scineram (talk) 16:58, 2 November 2008 (UTC)

 * the statement only assumes a given operator A has a bounded adjoint B, in the sense that  =  for all x, y. this would in fact imply what you're claiming above. Mct mht (talk) 20:29, 2 November 2008 (UTC)

 * this is pretty laughable. the statement is a elementary one: if an operator, with no a priori assumptions on boundedness, has a bounded adjoint, then it is bounded. in particular, unbounded operators cannot have a bounded adjoint. someone else wanna step in here? Mct mht (talk) 15:12, 4 November 2008 (UTC)

 * Mct mht, what exactly is this "theorem of Hellinger-Toeplitz type" that you're using?

 * After looking through Reed & Simon, Section VIII.8, I believe the situation is as follows (I don't know that much functional analysis though, so I may well be wrong). There is no theorem that says that if a densely-defined operator A has a bounded adjoint, then A itself is bounded. Indeed, it's possible for a densely-defined operator A to have an adjoint A* whose domain D(A*) consists of only the zero element in the Hilbert space. However, there is a theorem that if an everywhere-defined operator A has a bounded adjoint, then A itself is bounded. From that theorem it follows that if an everywhere-defined operator A is unbounded, its adjoint is unbounded. Thus, it looks to me that the paragraph that you are arguing about needs to make the additional assumption that the operator is defined on the whole space. -- Jitse Niesen (talk) 16:00, 4 November 2008 (UTC)

 * yes, of course one needs to assume in the statement that the operator in question is defined everywhere. this is analogous to the fact that a symmetric (everywhere defined) operator must be bounded (this is essentially the Hellinger-Toeplitz theorem). Mct mht (talk) 22:37, 4 November 2008 (UTC)

 * That is still false. An everywhere defined unbounded operator can have bounded adjoint too.Scineram (talk) 05:52, 5 November 2008 (UTC)

 * I don't understand that. Could you point out where this is explained, or give an example?
 * My reasoning is as follows. Suppose T is a linear operator on a Hilbert space, defined on the whole space. Then T bounded implies T* bounded. Thus, T* bounded implies T** = T bounded. Thus, T unbounded implies T* unbounded. So what's wrong here? -- Jitse Niesen (talk) 13:24, 5 November 2008 (UTC)

 * nothing is wrong with it. the statement that "T bounded implies T* bounded" is due to the closed graph theorem, just as in Hellinger-Toeplitz theorem. he's been wasting people's time all along. Mct mht (talk) 22:59, 5 November 2008 (UTC)

 * The problem is there is no second adjoint. It the adjoint is not densely defined, as is the case with {(0,0)} of course, than you cannot use closed graph theorem.Scineram (talk) 20:34, 6 November 2008 (UTC)

pretty obvious that Scineram is talking nonsense. statement will be put back. Mct mht (talk) 06:13, 8 November 2008 (UTC)

Definition of a symmetric operator on a topological vector space
Drevicko (talk) 06:51, 23 March 2010 (UTC) In Section 1 "Symmetric operators" starts with by defining a symmetric operator A on a Hilbert space then on a topological vector space the definition
 * $$ \langle Ax \mid y \rangle = \lang x \mid Ay \rang $$

makes sense in a Hilbert space, as the inner product gives a natural mapping to the spaces dual but not in a topological vector space. Here we have no natural map to the dual space, so $$ \mid Ay \rang $$ makes no sense : Ay is in the dual space E∗ but a "ket" resides in E, not it's dual. If we want to use the bra-ket notation, we should write
 * $$ \langle Ax \mid y \rangle = \lang Ay \mid x \rang $$

which adds another element of confusion, as we would expect to conjugate.

Perhaps we should instead write something like:
 * $$ (Ax)(y) = (Ay)(x) $$

or
 * $$ \langle x \mid y \rangle_A = \lang y \mid x \rang_A $$

It would also be good to add a reference to Bra-ket_notation

I came here to make a similar point. The notation
 * $$ \langle Ax \mid y \rangle = \langle x \mid Ay \rangle $$

doesn't actually seem to make sense, since the stuff inside the $$\mid\rangle$$ in Dirac notation is just a label - so to me $$\langle A x \mid$$ just looks like a bra called $$Ax$$. I think what's really meant is
 * $$ \langle x \mid A \mid y \rangle = \langle y \mid A\mid x \rangle. $$

That would define a symmetric operator (which is what that part of the article seems to be talking about), but of course, a self-adjoint one would be defined by
 * $$ \langle x \mid A \mid y \rangle = (\langle y \mid A\mid x \rangle)^*, $$

where $${}^*$$ is the complex conjugate. The article seems to be confusing symmetric operators with self-adjoint ones - but I'm not really sure what it's getting at with the whole "partially defined" thing, and I may just be mis-interpreting the notation, so I'm hesitant to change it. Nathaniel Virgo (talk) 12:13, 25 June 2010 (UTC)

(later:) Oh, I get it - the author isn't intending this to be interpreted as Dirac notation at all - (s)he's just using $$\langle a \mid b \rangle$$ to represent an inner product between vectors $$a$$ and $$b$$. Since this confused both myself and Drevicko I suggest it be changed to the more usual $$\langle a, b \rangle$$. If no-one objects I'll go ahead and do this myself at a later date. Nathaniel Virgo (talk) 22:48, 26 June 2010 (UTC)

Symmetric != Hermitian
Is it just me or is this article confusing symmetric operators with Hermitian ones? It even goes so far as to say that they are synonymous, and fails to mention complex conjugates. This seems odd to me since for matrices, "adjoint" means "take the transpose, and also take the complex conjugate of each entry", hence a "self-adjoint" matrix would be a Hermitian rather than a symmetric one.

Am I missing some subtle point, or should the article be changed? Nathaniel Virgo (talk) 22:53, 26 June 2010 (UTC)

 * The article is fine in this point the way it is, but I am not sure how best to explain. A common definition for symmetric operators in a complex Hilbert Space is that a operator A is called symmetric if  is real for all x in the Hilbert space. This is merely a convention, but it is called symmetric because it implies that you can move an operator from the right side of an inner product to the other without changing it's value. It is slightly unfortunate however that this doesn't concide with the common definition/notation for operators on finite dimensional spaces. One thing to note is that for real Hilbert spaces, the same definition doesn't suffice. And actually the article doesn't cover this, but as soon as one allows operators to be defined only on a subset of the Hilbert space, then self adjoint and symmetric cease to be the same thing. In the case of operators defined on a dense subset of the entire space, for example, there are many examples of symmetric operators which are not self-adjoint.--91.11.110.253 (talk) 21:57, 13 January 2012 (UTC)

Edit war, why?
About the ongoing edit war, , , , I fail to see, what is the fuss? All I see in the article (rather than its wikitext) is, few tiny distinctions in vertical spacing. Could the combatants explain here, what is the essense of the conflict? Boris Tsirelson (talk) 07:15, 21 April 2018 (UTC)

Hermitian operators
The section on Hermitian operators restricts the definition to "bounded operators". But, especially in quantum mechanics and quantum field theory, where this terminology is used, most operators under consideration are not bounded operators. Physicists often don't bother with boundednes and use the terminology Hermitian anyway. The preceding part of the article discusses unbounded operators, so is there a reason for the qualifier here? I feel it might confuse an uninitiated reader. Aaroneisenberg (talk) 09:30, 2 December 2021 (UTC)

Agreed. But what section is this in? I can't find the "section on Hermitian operators" in this article. StrokeOfMidnight (talk) 00:37, 3 December 2021 (UTC)

title: Talk:Self-adjoint operator


text:
User:Faust000~enwiki

Hello!!

fr:Utilisateur:Faust000

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Faust000~enwiki


text: Talk:International Space University

title: Talk:International Space University


text:
Aviv String Quartet

The Aviv String Quartet was founded in Israel in 1997, and has performed internationally for over 15 years. Among the quartet's various awards are the DaimlerChrysler Grand Prize at the 3rd Melbourne International Chamber Music Competition and top prizes at the Bordeaux and Schubert Competitions. The Aviv Quartet has recorded 3 discs of Shostakovich on Dalia Classics (available on iTunes and emusic) and 2 Naxos discs of Hoffmeister and Schulhoff. Recent highlights include a complete Shostakovich cycle at the 2007 Verbier Festival, debuts at the Zurich Tonhalle and Brussels Palais des Beaux-Arts and return visits to Wigmore Hall.

Members

 * Violin I:   Anna Göckel
 * Violin II:  Philippe Villafranca
 * Viola:      Noemie Bialobroda
 * Cello:      Daniel Mitnitsky

Former Members

 * Violin I:   Sergey Ostrovsky (1997-2023)
 * Violin II:  Evgenia Epshtein (1997-2005)
 * Viola:      Timur Yakubov (2011-2015)
 * Viola:      Shuli Waterman (1997-2009)
 * Viola:      Nathan Braude (2009-2011)
 * Cello:      Iris Jortner (1997-2002)
 * Cello:      Rachel Mercer (2002-2010)

title: Aviv String Quartet


text:
Tweet

Tweet often refers to:

 * Tweet (bird call), a type of bird vocalization
 * Tweet (social media), a post on the social media platform X (formerly Twitter)

Tweet may also refer to:

People

 * Tweet (singer) (born 1971), American R&B and soul singer-songwriter
 * Jonathan Tweet, game designer

Other uses

 * Samuel Tweet, a character of the British actor and comedian Freddie Davies (born 1937)
 * The Tweets, artists' name under which "The Birdie Song" charted in the UK in 1981
 * Cessna T-37 Tweet, a twin-engine United States trainer-attack type aircraft

title: Tweet


text:
Talk:Medion

Hm... Is it right to mention the email address of the manager? --Stebbiv 20:23, 10 July 2005 (UTC) no! removing it... UkPaolo 11:33, 16 August 2005 (UTC)

In the German version of this page, it is stated that Medion is withdrawing from America, because of lack of success, ("wobei der Rückzug aus dem amerikanischen Markt, mangels Erfolg, bereits begonnen hat.") rather than starting out there. User: Malo Hautus

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 14:54, 9 November 2007 (UTC)

aldi
there is no mention of this being an aldi exclusive brand, although in the usa you only see medion at aldi —Preceding unsigned comment added by 98.253.171.104 (talk) 20:37, 21 May 2011 (UTC)

The "customer service" section
I deleted it, it seemed highly non-NPOV, and was unsourced.

83.28.194.210 (talk) 17:59, 21 August 2014 (UTC)

Super Billing Computers?
The article states that "Medion products in Australia[4] and the United States are available exclusively at Aldi[5] and Super Billing Computers"

... however, I cannot find any source for Super Billing Computers having ever existed as a meaningful entity, let alone them selling Medion products. There seems to have been a small computer shop by that name in Australia at some point, but it seems very unlikely that Medion would have an exclusive distributor agreement with them. And while the "Aldi" claim is sourced, the "Super Billing computers" claim is not.

Is this just some very old spam/vandalism? Joepie91 (talk) 23:44, 3 July 2023 (UTC)

title: Talk:Medion


text:
McLennan Community College

McLennan Community College (MCC) is a public community college in Waco, Texas. It opened in 1965 and now serves about 9,000 students and has more than 700 employees. It is governed by a board of trustees elected from single-member districts in the county.

Campus
McLennan is one of three colleges in Waco along with Baylor University and Texas State Technical College. McLennan Community College is located on 200 acre on the north side of town near the Bosque River and Cameron Park. The college also owns Highlander Ranch, a 200 acre farm about 5 mi from the main campus.

The campus was recently expanded to include three new buildings, which were built with funding from a bond passed by voters in November 2006: the Michaelis Academic Center, New Science Building and Emergency Services Education Center.

MCC's campus also includes the Bosque River Stage, a 530-seat amphitheater located along the banks of the Bosque River. This outdoor venue was renovated in 2001 to expand the stage and update the lighting, sound and concessions area.

Organization and administration
As defined by the Texas Legislature, the official service area of McLennan Community College consists of the following:
 * All of McLennan and Falls Counties, and
 * The Calvert and Bremond school districts.

Academics
MCC offers associate degrees in arts and sciences for students who want to transfer to other colleges and universities. The school also has training programs — associate degrees in applied science and certificates — for students who want to enter the workforce. The college also offers continuing education courses for community members.

The University Center at MCC is a partnership with several public universities that offer degree programs on MCC's campus. It offers students the opportunity to earn affordable bachelor's, master's, and doctoral degrees without leaving central Texas. Partner universities are Midwestern State University, Tarleton State University, University of Texas Medical Branch, The University of Texas at Brownsville, and Texas Tech University.

Sports
The school has several student organizations and athletic teams. The mascot is a Scottish Highlander; the men's athletic teams are referred as the Highlanders, and the women's teams are the Highlassies. MCC's athletics program includes men's and women's basketball, baseball, softball, and men's and women's golf. There is also a dance team

Notable alumni

 * Jay Buhner, professional baseball player
 * Ruthie Foster, singer-songwriter of blues and folk music
 * Sean Henn, professional baseball player
 * Vinnie Johnson, professional basketball player
 * Danny Kaspar, college basketball coach
 * Pat Listach, professional baseball player
 * Sean Lowe, professional baseball player
 * Chris Martin, professional baseball player
 * Craig McMurtry, professional baseball player
 * Ryan Merritt, professional baseball player
 * Rodrick Monroe, professional football player
 * Logan Ondrusek, professional baseball player
 * Ken Patterson, professional baseball player
 * Leonard Perry, college basketball coach
 * Richie Ramsay, professional golfer
 * Jason Scobie, professional baseball player
 * Members of the Sons of the Desert, late 1990s country music group
 * Dave van Horn, professional baseball player and college baseball coach
 * Sam Worthen, professional basketball player

title: McLennan Community College


text:
Talk:Induced gamma emission

Ok, looking good.
The move is done, and I've tracked down and fixed anyone linking to ballotechnics that actually meant IGE (not easy!). I think I may have killed one redir in the process though, but IIRC it was unused anyway (still looking for it).

I have also started the cleanup, changing refs IN the article from ballotechnics to IGE. I am about 1/4 done, but I have to go so feel free to jump in. I have removed a small snippet from the intro for now:

and was originally concerned with ballotechnic nuclear reactions. The term 'ballotechnics' is attributed to Samuel Cohen, an early proponent. The concept is that there are nuclear reaction that produce gamma rays, and no alpha or beta particles. And it is proposed that this is the result of a high-energy nuclear isomer changing into a low-energy nuclear isomer, (i.e. with no change of mass number). Such high-energy isomers would include both spin isomers and shape isomers.

Maury 17:00, 30 August 2006 (UTC)

 * The "what links here" toolbox item is handy for tracking down links to "ballotechnics". A brief look showed cases that were ambiguous (mostly these just look like they just need wording tweaks to clarify that the term was being mis-applied). Thanks for taking the effort to do all of this! --Christopher Thomas 19:52, 30 August 2006 (UTC)

Trim50 (talk) 06:01, 14 December 2013 (UTC) An interesting patent published last month. "HAFNIUM TURBINE ENGINE AND METHOD OF OPERATION" http://www.freepatentsonline.com/20130300120.pdf

Wow
Well I don't know about you guys, but after basically adding an intro and moving stuff around I think the article actually seems pretty amazingly good now! Maury 17:56, 30 August 2006 (UTC)

 * I've also re-written the Ballotechnics article. See what you think. Maury 18:22, 30 August 2006 (UTC)

Inconvenient Truth
Absolutely splendid job handling the mechanics and titling of the page. Unfortunately you got the physics completely wrong. Please try to keep an open mind about this for a bit because it has nothing to do with Science politics, just established concepts in the laser field in which I published probably about 100 reviewed articles by now.

There is a tremendous difference between the production of photons by: stimulated emission, fluorescent emission, and Raman emission. The most convenient insight is given at Raman scattering down that page at the subheading "Distinction with fluorescence." Someone did a very good job in that section. "Stimulated emission" insists upon coherence between input and output photons. Gamma rays are photons and you are not going to get coherence at the gamma level of energies in this century. That would require a gamma ray laser with an unthinkable coherence length. However, the term "Gamma ray laser" has been useful but the thinking there is for a superradiant device for which the coherence length is so short that the device would be an intense directional source of gamma rays with little coherence between photons.

Induced gamma emission covers all of these analogs of laser-like processes for the gamma ray region of wavelengths. So please, give me a little time to try to correct the physics to be reasonable correct at the lay level of quantum electronics, ie laser physics.

--Drac2000 20:58, 30 August 2006 (UTC)

exoergic. (n.d.). WordNet® 2.0. Retrieved August 30, 2006, from Dictionary.com website: http://dictionary.reference.com/search?q=exoergic&x=23&y=14 Uncommon but legitimate terms, endoergic and exoergic.

--Drac2000 21:45, 30 August 2006 (UTC)

Pausing for now to see to what extent the big amount of work documenting and referencing material in the new first paragraph is treated constructively. Again, congratulations to you two, Maury and Christopher for all of the dedicated effort on the other aspects of rectifying the old Ballotechnics mess !

--Drac2000 23:03, 30 August 2006 (UTC)

 * Most of the experiment citations should frankly be moved to the "experiments" section, if they aren't already covered. The introduction also completely fails to mention with the Collins citations that the results are disputed.
 * Christopher, it seems that you have some sort of serious bias. You are asking for a disclaimer on work that was published and settled for 18 years simply on the basis of the name of one of the authors. It certainly seems you are implying that if Collins does an experiment in 2000 that remains controversial for 6 years (not unusual for an important experiment) then you want to have disclaimers stuck on everything he ever did. That is not the way science works. Try to think about NPOV.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * Also, are you sure your description of the effect is accurate?
 * Yes, you seem to have preconceived ideas that IGE is narrowly constrained to the "Collins effect" or somesuch. You would understand better if you approached with a more open mind. I guess you might accept that there are still things for you to learn. There is a lot of information in the references, links and further references contained within. I hope that you would enjoy taking a look to learn more.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * Fluoresence, as I realize you're aware, involves exciting from the ground state to a state that decays to a metastable state. Decays from the metastable state give the (relatively)
 * Relative to what, typical atomic or nuclear transition rates?
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * long-lived light emission. My understanding was that the process discussed
 * Discussed where, why? Before the move from Ballotechnics to Induced gamma emission, I explained that IGE was a field, Isomers were an example, and the controversial part was the application using Hf-178m2. Say again IGE => Exoergic IGE => Hf178m2 triggering, still controversial. There is also IGE => Endoergic IGE => In115, settled 18 years ago. Also IGE => Exoergic IGE => Ta180m, useless but independently confirmed for 8 years.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * was more or less the inverse of this: a long-lived metastable nuclear isomer is perturbed into a shorter-lived state with comparable energy, which promptly decays (as it's either not metastable or has a much shorter lifetime). The reason you get net energy release out for the "gamma ray laser" application was that you already had a population inversion,
 * population inversion has nothing to do with anything with which we have been concerned so far. If you are interested in learning as opposed to "spinning" the topic, that is probably IGE => endoergic IGE => lifetime inversion in three levels, so far unexplored.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * not that something particularly special was going on. Regarding Prof. Collins' claims, I am uncertain as to where the "resonance" mentioned in the introductory text comes into play. My impression is that Collins claims to accomplish the perturbation using x-rays of an energy much lower than the actual perturbation transition energy, but the actual mechanism proposed doesn't seem to be explained anywhere in this article. --Christopher Thomas 23:29, 30 August 2006 (UTC)
 * Best to get the field defined first and explain some things to help lay people.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * Both the beauty and the challenge of the Wikipedia is the democracy that allows those who do not know what they do not know to override expertise. So if you want to spin this topic against Prof. Collins more than you want to develop clarity and insight into the topic, you can do it. My further efforts, if any, are going to be motivated by what happens next. I have no problem if you produce a page on Induced gamma emission that will be perceived as ridiculous in the laser community, but together we could do something that would be a credit to Wikipedia. Your move, Christopher.
 * --Drac2000 00:48, 31 August 2006 (UTC)

 * Afterthought ! You realize that I have looked only at the first paragraph on this page of Induced gamma emission. There is much more work to be done on the rest of the page, if you want to join me in producing a quality article throughout.
 * --Drac2000 00:59, 31 August 2006 (UTC)

What a mess!

 * We agree completely, but as explained six lines up, I only tried to improve the first introductory paragraph to see if it were possible. Please look at what I started with yesterday. Contrary to promises, after I helped in dividing up the abominable "Ballotechnics" page I found that the part that was moved over here and was supposed to be preserved as it was instead had been completely rewritten by someone knowing absolutely nothing about quantum electronics or nuclear physics - and without any discussion.
 * --Drac2000 15:37, 31 August 2006 (UTC)

OK, I have a PhD in theoretical nuclear physics
 * Perfectly wonderful news. I have over 100 published articles in laser physics topics. We should be able to do something together, if personalities permit.
 * --Drac2000 15:37, 31 August 2006 (UTC)

, and I am having trouble understanding this article:
 * What the heck is exoergic and endoergic? The article on flourescence fails to even mention "exoergic flourescence" or "endoergic flourescence".
 * Fluorescence starts with fluor... not flour....I am hoping it was a minor typo. The terms were defined with links about 18 lines down from the start of the subsection "Inconvenient truth."
 * --Drac2000 15:37, 31 August 2006 (UTC)

 * The article on flourescence is a mess, and could use a lot of love and care.
 * Want to try it together, the "together" being the important part?
 * --Drac2000 15:37, 31 August 2006 (UTC)

 * What the heck is a "resonant process"?
 * A "resonant transition" is one that jumps directly between an excited state and the ground state: a "nonresonant transition is one which occurs between an excited state and another [lower] excited state (which is of course not the ground state).Delphwhite (talk) 00:37, 15 December 2013 (UTC)

 * The intro says "its like flourescnece". The first section says "its like lasing". So which is it?
 * That's part of the mess that I have not even tried to touch yet.
 * Drac2000 15:37, 31 August 2006 (UTC)

 * The first section states: and then coaxed to release this energy as gamma photons through the application of an external light source: Oh really? I'm gonna apply a few 1eV photons to induce a nucleus to emit multi KeV or MeV gammas? I don't think so ...
 * ...sequence of papers that raised concerns...' Huh? why would this "raise a concern?" OK, so I read a bit further in, and I see why, but its clear that these concerns should therefore be mentioned in the intro, which is currently a word-salad of jargon.

Guhh. Please clean this up. linas 04:00, 31 August 2006 (UTC)
 * Maybe, if you will help as opposed to criticise about issues you did not take the time and effort to look 6 lines up to solve. It is hard work and no appreciation.
 * --Drac2000 15:37, 31 August 2006 (UTC)

 * Yeah, Drac, too much detail in the intro! I welcome the changes to make the intro factually correct (my bad for not being ready in the first place), but remember, it's the intro, and needs to be short, sweet, and simple. If IGE is a fluoresence-like process, the intro should simply state that and move on.
 * But now I have to stop and think about what's going on. Fluoresence generally involves a downshifting of wavelength (right?), so how is this process similar, when it is creating gammas from x-rays? That doesn't sound like fluoresence to me.
 * Why not? Any nuclear database will show many cases of gammas emitted with energies smaller than the characteristic x-rays from the same material.
 * --Drac2000 15:37, 31 August 2006 (UTC)

I'm not questioning the physics (yet), but if there is a proposed mechanism here, as others have pointed out, it desperately needs to be added. Just don't put it in the intro!
 * And could you lay off the "open mind" comments? Do you have any idea how insulting they are? Maury 13:58, 31 August 2006 (UTC)
 * In spades, I had just been dealing with Christopher as you see above.
 * --Drac2000 15:37, 31 August 2006 (UTC)

Do we go forward?
Do we? --Drac2000 15:37, 31 August 2006 (UTC)

Only the intro paragraph has been worked upon. Length and treatment of references conform to those of the page Ballotechnics from which this page was split under the agreement that both daughter pages would preserve consensus reached before the split. --Drac2000 17:12, 31 August 2006 (UTC)

 * Absolutely. The remaining problems are my using the laser as a metaphor, which I guess is just "wrong" and should be removed outright, and the lengthy intro which is simply too detailed. Split that up, separate it out, and I think it's in good shape. Maury 18:39, 31 August 2006 (UTC)
 * Maury, Appreciate your continued interest and glad you expressed an understandable human concern about how we should realize how insulting our statements can sound. Thanks for asserting that very needed comment. Now, I ask for your reconsideration of a modestly different point of view, namely mine.
 * So far I do not see the difference (but remain willing to try to see it) between the size, complexity and reference with structure between the intro of Induced gamma emission and the intro of Ballotechnics.
 * We are also challenged with 71.133.205.75 who continues to change statements without one word of participation in these discussions here. He has some sort of distinction because he is on some kind of "Special page" that has been around only a month. Do you know the significance of such an arrangement. If he is taking over, we are wasting our time seeking consensus. Can you suggest or recommend something? I would have no problem working with anyone interested, but such heavy-handed exercise of "privilege" is another insulting aspect of this page.
 * As I see it, the challenge is that IGE is an interdisciplinary effort between nuclear physics and quantum electronics. It is lightly treated formally in nuclear and discussed in these terms of quantum electronics that need more explaining. It started in 1939 as references show and that surprises most people and since there were no lasers then, it would have been "sensitized fluorescence." That's passe now, but what is relevant is that there is and was no controversy until recently when the weapons gurus tried to hijack it for war or peace, as those partisans divided up. The positive approach that I am recommending is in the next bullet.
 * Please could we finish an introduction that lays the foundation for understanding of all of the non-controversial parts. When we reach consensus that it tells the story of what the field is about then we could usefully consider how to trim it down or split it or whatever, but doing so from an informed point upon which we have arrived at consensus?
 * I am willing to try a little more while ignoring 71.133.205.75 to see if we can reach an informed consensus upon the non-controversial part of the science, if you will help.
 * --Drac2000 20:22, 31 August 2006 (UTC)

 * My mistake. I have a "Special page" just like Superman71.133.205.75. He's nothing special, just does not know how to work with people in a collegial fashion. Can be dealt with later if we can just get the field defined.
 * --Drac2000 20:38, 31 August 2006 (UTC)

 * Sometimes (rarely) I get optimistic. Let's cool it a bit and see if I can come up with a visual that would help with the non-controversial part. I think there is an opportunity to connect with nuclear reactions as well.
 * --Drac2000 20:54, 31 August 2006 (UTC)

"# So far I do not see the difference (but remain willing to try to see it) between the size, complexity and reference with structure between the intro of Induced gamma emission and the intro of Ballotechnics."

Well for one, the intro to ballotechnics is 50 words, while the intro to this article is 212. A good intro should include a definition of the term, where the term is used, and perhaps a comparison to similar topics. Things not to include in the intro are, well, pretty much everything else. Certainly a list of experiments does not belong, while at the same time the intro still fails to clearly describe what the topic is - which is the whole point. It also uses terminology that no one here has ever heard of, so that's pretty much a great definition of something that shouldn't be there. Here, try this on for size:

In physics, induced gamma emission (IGE) refers to the process of fluorescent emission of gamma rays from excited nuclei, or nuclear isomers. It is an analog of conventional fluorescence, which is defined in terms of photons of light and excited states of atoms or molecules. In the case of IGE, nuclear isomers can store significant amounts of excitation energy for times long enough for it to serve as "nuclear fluorescent" material.

THAT is an intro.

Maury 21:59, 31 August 2006 (UTC)

 * And an excellent one, at that. English was never one of my strong subjects. Also the clean-up on the article is very helpful.
 * --Drac2000 22:15, 31 August 2006 (UTC)

Wow - once again
Maury, Splendid improvement I think. If you feel the example of the 5 fluorescent isomers should come later, please change it as you judge best. I got used to those concrete examples being right up front and thought it made the concepts more comfortably tangible. --Drac2000 22:37, 31 August 2006 (UTC)

Tough job, but none of the history up to 1999 as written should be controversial. Maury, I am very interested in your judgement and upgrading of this History part. --Drac2000 00:40, 1 September 2006 (UTC)

 * This work helped tremendously! I think even I am starting to understand it now (no mean feat, let me assure you). Maury 12:37, 1 September 2006 (UTC)

 * Yes, intro currently looks good. Sorry for being overly critical above, got off on a bad start. But perhaps its easier if I play role of critic for a little longer. So:

 * Intro lists five isomers, but the history section promptly mentions a sixth (indium).
 * Made small addition and believe problem fixed. However cost is word count increase to 118 in Intro. Problem? Not for me, anyway. --Drac2000 17:57, 1 September 2006 (UTC)
 * History should explicitly state that In is indium (I had to guess).
 * Done --Drac2000 17:57, 1 September 2006 (UTC)
 * The commentary about "usually overlooked" raises an eyebrow: what's the point of raising questions in the readers mind?
 * Anyone taking the link to Nuclear reaction will not find gamma-gamma' reactions.--Drac2000 17:57, 1 September 2006 (UTC)
 * Don't use the condensed notation. Use A+B arrow C+D, this will aid in understanding when non-experts read this.
 * '''Let's save for later, I copied this below under "Pending"
 * Indicate energy of gamma-prime and gamma, so we don't have arguments about whether gamma's are x-rays or not. I assume the reaction that reaction is prompt; I presume the incoming gamma is an MeV or so, the outgoing gamma is 100KeV or so.
 * Hope figure fixes it?
 * Not quite clear what Pontecorvo and Lazard did: Is the news that they created an isomer? That they created the isomer using only gammas? That they discovered something with a half-life of hours?
 * Hope figure fixes it?
 * Indicate that there are two reactions: one to create the isomer, and the second being the decay of the isomer. The second is the actual "fluorescence".
 * Hope figure fixes it?
 * According to the indium article, the isomer decays by beta to tin, and not by gamma, as the fluorescence label suggests ... so this is confusing.
 * Link in 3rd line of intro shows an IT (isomeric transition) from isomer as indicated in schematic drawing of that data.
 * Need a better link to resonant process than resonance. I still have no clue what a "resonant process" is. OK, I'm guessing spectral line? As in, a sharp emission line/absorption line? This would explain the funny talk about energy/momentum conservation that currently seems to confuse things.
 * The paragraph that starts with "There is little conceptual difference..." finally made the light bulb go off in my head. If I understand this correctly, the reaction that is actually of interest is the time-reversed reaction: taking the indium isomer, shining a low-energy x-ray on it, thus popping it into an excited state where it promptly decays to a ground state by emission of a high-energy x-ray. Right? That's the point of this, right? If so, this somehow needs to made clearer, earlier, cause I had to concentrate before that lightbulb went off.
 * Hope figure fixes it?
 * Bold inline comments this section added by --Drac2000 20:40, 1 September 2006 (UTC)

 * Maybe this time, I was more helpful? linas 04:19, 1 September 2006 (UTC)

 * Absolutely, Yes. ! I find such truly constructive criticisms to be extremely helpful. Also, it shows that we will have a lot more discussions to do (exciting in my opinion) and still being inexperienced with the technicalities of Wikipedia processes, I am worried about warnings I keep getting that this page is too long. I am going to start by trying to delete a lot of the discussion and contention about the Ballotechnics part that applied to what was left on that page when we moved over here. If anyone feels we lost something in doing so, please put back those parts and do something else to cope with the length warnings on this talk page.
 * --Drac2000 15:02, 1 September 2006 (UTC)

 * Completed process by archiving using
 * How to archive a talk page

 * Have to stop and contribute to earning a living for a bit. I did change the title of the history section to bracket the major period of study and conceptualization over which there was no lingering controversy. I am hoping it might help deal with the upset over later events that began (conveniently) in the 61st year, 2000. Welcome your critiques on that idea. More work here later today.
 * --Drac2000 15:47, 1 September 2006 (UTC)

Suggesting framework; lasing.
OK, the below is my guess, emphasize "guess", as to what the main points are, in describing IGE.

1) Nuclear isomers exist.

2) Certain nuclear isomers can be created with the following process:

 * N + g --> M + x

where N is a nucleaus in the ground state,

 * g is a gamma ray, say a 1MeV gamma ray
 * M is the excited nucleus aka isomer
 * x is another gamma-ray (aka x-ray), say about 100KeV

3) The above reaction is "instantaneous", in that x is emitted within picoseconds after absorption of g.

3a) the state M is metastable, i.e. will decay but is long-lived

4) This reaction is "resonant" in that it involves only photons, and thus kind-of resembles normal emmission/absorption-line type atomic physics with photons.

In short, one takes a nucleus, pumped some energy into it, and the resulting isomer is stable over relatively long periods of time. It will eventually decay, by alpha, beta or gamma radiation: in this sense, anything that is radioactive is "kind of like" fluorescence. By restricting the analogy to nuclear isomers only, and nuclear isomers that decay only by gamma emission, it is even more "kind of like" fluorescence. I think the analogy stands.

OK next step:

5) All quantum processes are time-reversible.

6) Thus, we can consider the reaction:

 * M + x --> N + g

We take an isomer, shine a 100KeV photon on it, and, whammo: a very energetic 1MeV photon pops out. Yes, this looks like upconversion.

7) Some lasers, e.g. dye lasers, made with fluorescent dyes, use a similar process (???), with eV scales insteade of MeV scales, and orbital electrons instead of nuclei. That is, you set up a population of M, pump it with low-frequency x, and whammo, you lase out a bunch of high-energy g, finishing with a completely depleted, ground state dye N that remains. But I know just about nothing about lasers, so this is speculation. I do know that some lasers, e.g nitrogen lasers, have such a short lasing length that no mirrors are needed. I have no clue what sort of density one needs to get the population inversion.

I suggest the above, if correct, should provide the technical expository framework for explaining this stuff in understandable terms. linas 17:39, 1 September 2006 (UTC)

 * Close, and terrific approach, but needs some "tuning". Our considerations crossed in Wiki-space. Please see inline changes I made that are found above in bold, just to help them be found. Short break, then back to concentrated effort. Many thanks.
 * --Drac2000 17:49, 1 September 2006 (UTC)

Illustration
Added picture, hopefully worth 1000 words, as the saying goes. It is an excellent point attempting to contrast time-reversed processes as recommended above, but this illustration shows the complications that must be addressed. I am hoping the illustration, drawn just for this article will make the job easier. Let's see.... --Drac2000 20:20, 1 September 2006 (UTC)

Pending concerns

 * Don't use the condensed notation. Use A+B arrow C+D, this will aid in understanding when non-experts read this.

 * Only needs to be done once, to introduce the shorter notation.

 * The graphic looks great, its exactly what was needed.

 * The intro still lists five elements, and indium is not one of them. A sentence should be added to explain why indium is not one of the five. The intro should also include mention of the applications, singling out the high-energy density battery, and the gamma ray laser. The intro should also mention the controversy. This probably means that the technical content in the current intro gets moved to a section called "technical overview". linas 22:00, 2 September 2006 (UTC)

Non-controversial version
The version of this time and date contains what I offer as being the non-controversial aspects of Induced gamma emission. As recommended somewhere in the Wikipedia protocols I saw where we were asked to concentrate upon non-controversial "cores" of a subject first and then consider linking to another subpage of "status" and "contrast of perceptions." I really think this will be the only way to get something credible. Please let's make this version into concensus without adding here any of the old "fighting statements." In my view anything beyond what is in the article at this time is "research", unsettled and supposed to be excluded from Wikipedia. In any case I strongly recommend that we get this pre-2000 work into consensus if you feel it is not acceptable as is, before enlarging the scope to catch more trouble. --Drac2000 23:39, 1 September 2006 (UTC)

Some clean-up added. --Drac2000 13:26, 2 September 2006 (UTC)

 * Looks to be in fine shape now. Good job! As a controversial topic, I suppose someone will show up next month or next year and insist on something; all you can do now is wait to see what happens. linas 21:48, 2 September 2006 (UTC)
 * Linas, many thanks again for for essential and constructive work. Having reflected upon matters some more and looked at WP:SP I think they make a good point. Indeed, the page here still needs the touches you wisely recommended that are still pending, but I think to protect what we have already, we need to put the "lightening rod" up somewhere else to try to develop a subpage that deals with IGE after 1999 when thing became so impassioned. A Google search on "Imaginary weapons" will show just how unfortunate things have become. Though opinions seem to be smoothing somewhat, unspent passions need an outlet and have to be considered as we try to write something about the last 6 years of IGE. Perceptions remain very time-dependent and so the last 6 years will be best treated as a subpage.
 * --Drac2000 13:21, 3 September 2006 (UTC)

Gamma ray laser
So .. out of idle curiosity ... how well are the cross-sections known? Has anyone done the calculations for ASE, is it known what sort of densities are required? How feasible is this? I mean, aside from the obvious risks of a criticality accident or outright explosion of the thing, it seems like a theoretically straightforward thing to do, once the various cross-sections, etc. are known. Right? What's the blocker? linas 22:11, 2 September 2006 (UTC)
 * Great point. Currently studying the excited states involved in the IGE cascades looking for lifetime inversion.
 * --Drac2000 13:21, 3 September 2006 (UTC)

Hafnium-178m2 triggering: controversy after 1999
'''A 6 year controversy has raged over the results of using Hafnium-178m2 in IGE experiments. Attempting to conform to WP:SP while developing a second page reporting the last 6 years of IGE, the prototype of that second page and its discussion have been moved. Please take the link User:Drac2000/Induced gamma emission: Hafnium-178m2 and participate in the development of the second page of IGE.''' (Emphasis added --Drac2000 20:18, 13 September 2006 (UTC))

--Drac2000 13:34, 3 September 2006 (UTC)

 * Woo hoo! I see now why its controversial! Anyway, I think that the above should be moved into the main article namespace, under the title of IGE controversy or Hafnium controversy or Hafnium bomb, and just let the general public have at it. You may want to personally invite some of the previous participants in these discussions (e.g Chris Thomas) to review. linas 14:12, 6 September 2006 (UTC)
 * Please see reply and talk continuation at User_talk:Drac2000/Induced_gamma_emission:_Hafnium-178m2. Everyone is invited.
 * --Drac2000 16:40, 6 September 2006 (UTC)

NPOV tag
It's obvious from the talk page that this has been a controversial article. The present state of the article is ridiculously credulous.--76.81.160.198 (talk) 06:22, 21 December 2007 (UTC)
 * Unsubstantiated use of the perjorative ridiculously credible because of unspecified obvious attributes constitutes vandalism, not collegial improvement.
 * Drac2000 (talk) 14:26, 19 February 2008 (UTC)

false picture
Info: this image: Image:HafniumMetalUSGOV.jpg isn't hafnium. It look similar to germanium. I think it was a confusion from the author. --Alchemist-hp (talk) 21:56, 20 June 2009 (UTC)

The hafnium image now on the main page is an exceptionally well photographed piece of a Van Arkel crystal bar - my compliments! The previous image Image:HafniumMetalUSGOV.jpg is of a small piece broken off a hafnium Van Arkel bar that is poorly illuminated and out of focus (doesn't really look like hafnium). -- Delphwhite (talk) 19:24, 21 September 2011 (UTC)

Platinum-186m does not exist...
This isomer of platinum is not listed in the element's article, nor could I find any credible source for it. Isomers of platinum are 193m, 185m and 197m. Please check... —Preceding unsigned comment added by Rnbc (talk • contribs) 13:42, 6 August 2010 (UTC)

I agree--there is no Pt186m. That was my comment as well. Charlie Moquin 24May2011 —Preceding unsigned comment added by 68.236.71.92 (talk) 22:03, 24 May 2011 (UTC)

Long living isomers:

title: Talk:Induced gamma emission


text:
Talk:Ilario Pantano

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Ilario Pantano. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20060326201820/http://newyorkmetro.com/nymetro/news/people/features/11774/ to http://newyorkmetro.com/nymetro/news/people/features/11774
 * Added tag to http://upload.wikimedia.org/wikipedia/en/0/00/House_Resoultion_167.pdf

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 04:40, 10 September 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Ilario Pantano. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20100218151329/http://www.pantanoforcongress.com/ to http://pantanoforcongress.com/

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 07:37, 7 November 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 2 external links on Ilario Pantano. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added tag to http://foxnewsinsider.com/2011/12/15/marine-ilario-pantano-exonerated-of-murdering-iraqi-prisoners-runs-for-congress-in-north-carolina/
 * Added archive https://web.archive.org/web/20131004224210/http://projects.newsobserver.com/under_the_dome/pantano_gets_veterans_post_in_mccrory_administration to http://projects.newsobserver.com/under_the_dome/pantano_gets_veterans_post_in_mccrory_administration
 * Added archive https://web.archive.org/web/20120308055852/http://www.marinecorpstimes.com/news/2007/05/marine_marsoc_website_070509/ to http://www.marinecorpstimes.com/news/2007/05/marine_marsoc_website_070509/
 * Added tag to http://admin.washingtontimes.com/news/2011/nov/30/for-marine-a-rush-to-judgment-and-belated-vindicat/?page=all

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 22:31, 11 November 2017 (UTC)

title: Talk:Ilario Pantano


text:
Wikipedia:Articles for deletion/Little april

 This page is an archive of the proposed deletion of the article below. Further comments should be made on the appropriate discussion page (such as the article's talk page or on a Votes for Undeletion nomination). No further edits should be made to this page.

The result of the debate was

Speedied   — Gwalla | Talk 00:09, 11 July 2005 (UTC)

Little april
Porn ad. Joyous  (talk) 20:27, July 10, 2005 (UTC)
 * Delete. I'm fairly sure this sort of thing falls under Speedy Deletion. Gwk 20:46, 10 July 2005 (UTC)
 * Delete: pr0ncruft. Wile E. Heresiarch 20:48, 10 July 2005 (UTC)
 * Delete. Non-notable. Advert. -- BD2412 talk 20:54, July 10, 2005 (UTC)
 * Delete. nn and ad. Gwk may be right, so it could go to speedy if need-be. -mysekurity 21:26, 10 July 2005 (UTC)
 * Speedied. Advert, just an excuse for an external link.   — Gwalla | Talk 00:09, 11 July 2005 (UTC)
 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in an undeletion request). No further edits should be made to this page.


text:
João Magueijo

João Magueijo (born 1967) is a Portuguese cosmologist and professor in theoretical physics at Imperial College London. He is a pioneer of the varying speed of light (VSL) theory.

Education and career
João Magueijo studied physics at the University of Lisbon. He undertook graduate work and Ph.D. at Cambridge University. He was awarded a research fellowship at St John's College, Cambridge. He has been a faculty member at Princeton and Cambridge and is currently a professor at Imperial College London where he teaches undergraduates General Relativity and postgraduates Advanced General Relativity.

In 1998, Magueijo teamed with Andreas Albrecht to work on the varying speed of light (VSL) theory of cosmology, which proposes that the speed of light was up to $3 km/s$ in the early universe. This would explain the horizon problem (since distant regions of the expanding universe would have had time to interact and homogenize their properties) and is presented as an alternative to the more mainstream theory of cosmic inflation.

Magueijo discusses his personal struggles pursuing VSL in his 2003 book, Faster Than The Speed of Light, The Story of a Scientific Speculation. He was associated with a misunderstanding over priority concerning VSL with John Moffat. He was also the host of the Science Channel special João Magueijo's Big Bang, which premiered on 13 May 2008.

In 2009, he published A Brilliant Darkness, an account of the life and science of vanished physicist Ettore Majorana.

In 2014, he published Bifes Mal Passados, describing his observations on the United Kingdom. In this book, Magueijo described British culture as one of the "most rotten societies in the world", and wrote that "When you visit English homes, or the toilets at schools or in student lodgings, they are all so disgusting that even my grandmother's poultry cage is cleaner". British people were described as "unrestrained wild beasts". The book was only published in Portugal, where it sold 20,000 copies.

João Magueijo has been interviewed by Morgan Freeman in season 2 of Through the Wormhole.

Books

 * Bifes Mal Passados: Passeios e outras catástrofes por terras de Sua Majestade, Gradiva, 2014, ISBN 978-989-616-583-3
 * A Brilliant Darkness: The Extraordinary Life and Disappearance of Ettore Majorana, the Troubled Genius of the Nuclear Age, Basic Books, 2009/2010, ISBN 978-0-465-00903-9
 * Faster than the Speed of Light: The Story of a Scientific Speculation, Basic Books, 2003, ISBN 978-07382-0525-0

title: João Magueijo


text:
Cross Rhythms City Radio

Cross Rhythms City Radio is a community radio station broadcasting to Stoke-on-Trent and Newcastle-under-Lyme, Staffordshire, on 101.8FM. The station is music-driven with contemporary pop, rock, urban music and dance. It is underpinned by Christian values but driven by local issues.

In addition to regular programming with community groups such as the council, police, health, education and employment agencies, the radio daily interacts with local businesses, arts and culture groups, local events and charities, theatres, ethnic-focused organisations and many others.

The station was one of the original 15 to be granted pilot licences for a new form of local radio, then known as Access Radio. Following several extensions of the pilot scheme (now renamed Community Radio), a full five-year licence was granted in June 2005. Cross Rhythms City Radio was the first FM Christian station in the UK to be granted a full licence.

Cross Rhythms City Radio is operated by Cross Rhythms, a media organisation whose aim is to impact youth and the wider community for good through FM radio, contemporary Christian music and a globally influential website.

Presenters
Jonathan Bellamy

Chris Cole

Deanna Fletcher. Sophia Barrett, Paul Reid (Audacious Radio Show)

Steve Gambill

Jan Husband

Jenny Hutchinson

Chip Kendall

Chris Mountford

Shell Perris

Lee Plummer

Bernard Reilly

Straff

DJ Zorro

title: Cross Rhythms City Radio


text:
Stained Glass, Soma Fountains

Stained Glass, Soma Fountains is a compilation album by the Legendary Pink Dots, issued in 1997 on Soleilmoon Records. The album covers several tracks from the group's early career; some were previously released on small and independent labels, while others were private demos.

It was originally intended for these songs to be included as bonus tracks on Soleilmoon's CD reissues of the early Legendary Pink Dots albums, but contractual reasons prevented this.

"April's Song" dates from 1980, when the group was still known as One Day.

Track listing

 * 1) A Pale Green Introduction/Love on a Pale Green Postage Stamp
 * 2) Premonition 12
 * 3) Mmmmmmmmmmmm ... (first version)
 * 4) The Lifesucker
 * 5) The Divorce
 * 6) Jungle (first version)
 * 7) The Haunted Supermarket
 * 8) No Bell No Prize
 * 9) Plague 2
 * 10) Find The Lady
 * 11) Judith A Hum/Praum Naizh/A Pale Green Sequel
 * 12) Suicide Pact
 * 13) Waiting For The Call/You 'n' Me
 * 14) April's Song
 * 15) Frosty
 * 16) Defeated
 * 17) A Message From Our Sponsor (first version)
 * 18) Ice Baby Cometh
 * 19) Hanging Gardens (Flow Motion version)
 * 20) Jack/Thursday Night Fever #1/Die With Your Eyes On/Opus Dei
 * 21) Premonition 18 (part one)
 * 22) Premonition 18 (part two)

Personnel

 * Edward Ka-Spel: keyboards, vocal
 * The Silverman (Phil Knight): keyboards

title: Stained Glass, Soma Fountains