Talk:Mehler kernel

Transferring half the conversation from User talk:Cuzkatzimhut
Thanks for your comments. I know the statistical version of Mehler's formula very well - it is used in signal analysis. I will limit myself to correcting it as you say. Also the proof given of it using Fourier transform in the article is unnecessarily complicated. But I cant do anything very fast being rather unused to the Wiki markup.

I am not familiar with the physicist's version. If you know about this maybe you could supply a reference to where it is used and give more explanation of it. The transformation from one version to the other is interesting and probably means something but at present it is rather mysterious.

I think it best to keep the two version apart, as in the Mehler kernel article, but showing the connection between them. The point about the formula reducing to a delta function is a difficult one and has not been treated convincingly anywhere as far as I can see. Even Wiener's 1933 proof quoted is obscure and unconvincing. It needs some kind of continuity assumption.

Incidentally I see that Mehler did treat the multidimensional case (see the title to his paper). So this entry on 'Kibble-Slepian' extension to many dimensions does not make sense. Kibble kept to the one-dimensional case and did nothing as far as I can make out (I am unable to access the original to check). So I dont think he deserves to be called an originator; after all, the literature on this expansion goes way back to the 1930s.JFB80 (talk) 14:24, 26 July 2014 (UTC)
 * Thanks. As per top of the page, I'm replying on your page. Cuzkatzimhut (talk) 10:45, 27 July 2014 (UTC)
 * I made rather obvious oversight of forgetting to include the one-dimensional probabilites on the right hand side - it is now corrected. In the general case these one-dimensional probabilities can be different. The correct way would be to describe this general case first then say that to agree with the physics version we need the one-dimensional distributions to be the same. But before doing that I want to be certain how it ties up with the physics version about which I am uncertain because what is written does not agree with the WP article 'Propagator'. Unfortunately I am unable to check with the Pauli reference (which I am glad to see you have put in). Do you know the answer? JFB80 (talk) 11:37, 28 July 2014 (UTC)JFB80 (talk) 11:42, 28 July 2014 (UTC)
 * I changed sinh to tanh in the formula for K otherwise it does not correspond to the probability version - or perhaps it doesnt have to.JFB80 (talk) 17:52, 28 July 2014 (UTC)

Sorry, I changed it back. It does not have to "correspond"--that is the point of the "multiplying factor". The sinh(2t) is needed for satisfying the differential equation! Specifically, cosh = 1/ρ, sinh= √(1−ρ2) /ρ, and, in addition, x2--> x2/√(1−ρ2) and likewise for y. The pre-factor for the exponential is then √ρ/ √√(1−ρ2), and one keeps whatever one wishes for one's measures! There is no compelling reason why this differential equation should "please" everyone! Cuzkatzimhut (talk) 18:26, 28 July 2014 (UTC)
 * So you are saying that what is said in the article - that K is a Mehler kernel - is not actually true. It doesnt look right to me because if there were an exact correspondence with the probability kernel the semigroup property would follow easily. Now it does not. JFB80 (talk) 17:44, 29 July 2014 (UTC)
 * Yes its ok with me to merge Kibble-Slepian with Mehler. Actually it gives zero information to people like me unable to access either Kibble or Slepian. Doesnt really matter because the generalization is evident and doesnt need a reference except that without one it would be 'original research'.JFB80 (talk) — Preceding undated comment added 17:53, 29 July 2014 (UTC)

I'm not sure what you are implying. As a fundamental solution, which I checked and any reader could, it is transitive, so, of course,  ∫dy  K(x,y;t) K(y,z;T) = K(x,z;t+T), no? I suspect your artifice of using two symbols, p and K is appropriate and removes any misunderstanding. To reassure you, note what happens to, e.g. Φ ≡ exp(ct) φ which satisfies the same eqn as   φ, except shifted by −cΦ. So the kernel for that equation will be  exp(ct)K, so, then, K multiplied by a different normalization in t, or ρ, but also obeying a semigroup property, as linear exponents add. This could allow p and K to satisfy slightly different differential equations, if they were normalized differently.
 * As for Kibble and Slepian, I'm starting the formal rigmarole Merge of merging pages on that wiki...  it is a lot of fuss, but maybe worth it. Cuzkatzimhut (talk) 18:31, 29 July 2014 (UTC) OK, completed pestiferous merge and deleted superfluous content. You may choose whether to call the generalization KS or not... Your call---I have no opinion on names... I note some authors do talk about a KS formula, cf. the extant references, rightly or wrongly. Cuzkatzimhut (talk) 15:41, 24 August 2014 (UTC)

Multiplying by an exponential would be ok but not by the square root of a sinh function. JFB80 (talk) 09:07, 30 July 2014 (UTC)
 * In the imaginary time case, cf (13), (18) of the Condon article referenced Online. But the above transitive statement, ∫dy K(x,y;t) K(y,z;T) = K(x,z;t+T), can also be checked directly... In fact, it could probably be provided in the article, unless you disagreed with it.  Cuzkatzimhut (talk) 10:33, 30 July 2014 (UTC)

Merger proposal
I propose that Kibble–Slepian formula be merged into Mehler kernel. I insist that the content in the extant KS stub can best be explained in the context of this, the  Mehler article, which is of a reasonable size that the merging influx of KS will not cause any problems, as far as article size or undue weight is concerned. Cuzkatzimhut (talk) 20:43, 29 July 2014 (UTC)
 * For myself and many others the present Kibble-Slepian article is completely uninformative because it gives no description whatsoever of its subject. It only gives references which will not be accessible to everyone and so it goes against the principle of Wikipedia being a source open to all. Consequently I think such articles should be disallowed. The article seems in addition to be factually incorrect because, as far as I can make out from seeing a very brief abstract, Kibble did not discuss the higher dimensional generalization he is said to have done.JFB80 (talk) 18:06, 2 August 2014 (UTC)
 * Indeed, The KS stub is totally useless, except for the two references, already included here. I am quite inexpert on these people and their work, but, in physics, the extension to multidimensional oscillators is straightforward, and a reader would snicker at the pompous need to actually eponymize arbitrary-dimensional generalizations! Cuzkatzimhut (talk) 21:13, 2 August 2014 (UTC)

Physics version
The equation does not describe the quantum harmonic oscillator! The Schrödinger equation has a complex unit i in front of the time derivative. Unsigned by User: 217.149.175.74.


 * Take imaginary times and compare to the standard propagator through the elementary Schrödinger_equation?Cuzkatzimhut (talk) 10:44, 10 June 2020 (UTC)