Why Is Fourier Transform Used in Diagonalizing Operators in Mean Field Theory?

[tirrel]

Hi...

I hope somebody can help me...

Studying mean field theory in a passage it was necessary to calcolate the inverse of this operator defined on Z^2:

$A(I,K)=-J\sum_e \delta(I,K-e)+1/(\beta)*\delta(I,K)$

where I,K pass all ZxZ and the sum on $e$ is a sum on the for basis vectors e_1,e_2,...,e_4. $\delta(A,B)$ is the usual delta function. $J$ and $\beta$ are constants.

well my book tries to compute $A'(q,p)$ as the discrete time Fourier transform of $A(I,K)$... then finds a certain function $g$ which respects this equation $A'(q,p)*g(q,p)=\delta(q-p)$ and anti-transforms it, pretending thus to find an integral representation of the inverse matrix...

unluckily I don't see why this passage is true... does anybody can help me?

 

[tirrel]

I apologize... but I don't know how to write better the formulas...

anyway, pheraps this topic had to be written on the section about field theory... sorry... I'm a newbye!

 

[Gokul43201]

Replace the $ signs with tex tags (or itex for inline typesetting).

Example (click to see code): $$A(I,K)=-J\sum_e \delta(I,K-e)+1/(\beta)*\delta(I,K)$$

PS: Also, when asking questions about a specific text, it may be useful to cite the text and the page/chapter where the passage is found.

 

[tirrel]

thank you Gokul...

I found this passage in the pdf file of a teacher of an other university, but unluckily it is not written in english...

Anyway similar questions (more or less) arise here:

-Itzykson Drouffe... "Statysitical field theory", pag. 128. How can I pass from formula 59 to formula 60?

- G.Parisi... "Statystical field theory", chap.3 (mean field)... at the beginning of the chapter (haven't got the book with me right now!) there is written exactly the same operator I wrote in the first message;

 

[tirrel]

Ok... I guess I solved... thank u for the attention... if anyone is interested, I'll post what I've understood...

 

[lbrits]

The general story is as follows. You can think of your operator as a matrix where the real space coordinates are it's indices. To find the inverse of that matrix is in general difficult. But the inverse of a diagonal matrix is of course easy (simply 1/every component along the diagonal).

To take a general matrix and render it as a diagonal matrix is the process of diagonalization. But diagonalization is simply "picking a basis" in which the matrix is diagonal, i.e., finding it's eigenvectors and eigenvalues.

Fourier transforming is simply "picking a basis" and writing the object as a linear combination in that basis, and usually operators that are translation invariant in position space become diagonal in Fourier space.

I hope this has clarified things. I don't know the exact problem you are looking at, but this is a fairly general concept.