- Sun 27 January 2019
- Physics
- #green-function, #laplacian

In \(d\)-dimensional euclidean space, the laplacian operator is

The laplacian operator is the trace of the \(\partial_{a} \partial_{b}\) operator. The Green function for the laplacian operator is such that

Here \(A\) is a constant and \(\delta(x)\) is a \(d\)-dimensional Dirac delta function.

In this post I will evaluate the following matrix:

Note that \(D_{ab}\) is a symmetric matrix. The trace of \(D_{ab}\) must give the Dirac delta:

So maybe \(D_{ab}\) can be split into a traceless and non-traceless parts:

Here \(T_{ab}\) is symmetric and traceless. Maybe it can be written as

Now, dimensional analysis suggest

Here, \(B\) is a constant. Thus, apparently, you would have

The last step is to find the values of \(A\) and \(B\).