Talk:Theta representation

A difference of opinion

The text currently reads

For a fixed value of τ, define a norm on entire functions of the complex plane as

$\Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy.$

User:Mathsci had modified this [1] to $\exp \left({\frac {-2\pi (x^{2}+y^{2})}{\Im \tau }}\right)$ with edit summary holomorphic Fock space - typo in original ms and User:Dapengzhang0 has just reverted [2] simply stating Undid revision 482167306 by Mathsci. Unfortunately the section is quite unsourced, and it is not clear what the "original MS" is. Is there an independent reliable source for either version, and is there reason to believe that there is a typo in the "original MS"? Deltahedron (talk) 10:32, 20 July 2014 (UTC)

The norm should be defined such that the action of the group elements $U_{\tau }(\lambda ,a,b)$ is unitary as claimed in the context. One source of it is Chapter I Section 3 of the book by David Mumford that is listed in the reference section. However, the Jacobi theta function itself unfortunately has infinite norm. This situation is similar to the fact that the sine function is not square-integrable on the reals. Dapengzhang0 (talk) 19:31, 20 July 2014 (UTC)

I don't have access to Mumford at the moment: are you saying that the formula is stated there? Assuming that is the "original MS" referred to by Mathsci, is it possible that there is a typo as alleged? Is there another source we could use to check? Deltahedron (talk) 20:35, 20 July 2014 (UTC)

Yes, that formula is explicitly stated there. You can access page 7 of that book on [books.google.com] and find the formula. I don't know what the abbreviation 'MS' is meant by User:Mathsci, but his/her modification on the norm clearly breaks the unitarity of the representation. I don't have another source. Dapengzhang0 (talk) 01:16, 21 July 2014 (UTC)

Thanks for that, I can see it too. I assume Mathsci's "MS" meant "manuscript". Deltahedron (talk) 17:58, 21 July 2014 (UTC)

The hand-wavey explanation for why its just $y^{2}$ instead of $x^{2}+y^{2}$ is that this is all happening on the upper half-plane with the hyperbolic metric. The theta functions are closely related to the modular forms. The tau is the nome (mathematics). Perhaps this article should explicitly list out those connections; they're interesting because they show up in both number theory and in AdS/CFT. The theta functions have been generalized to hyperbolic spaces in higher dimensions as well; I'm blanking on the name of one of the principal proponents. 67.198.37.16 (talk) 22:06, 1 March 2024 (UTC)