# HaMiDeW coefficients - recursive calculation of the coincidence limits

In his book Aspects of Quantum Field Theory in Curved Spacetime Stephen Fulling calculates the coincidence limit $$[a_1]$$ and gives an idea of how $$[a_n]$$ with $$2 ≤ n$$ can be found recursively.

Since the calculation of $$[a_2]$$ is much more laborious than the calculation of $$[a_1]$$, I hope that it can be found in the literature. Some references would be welcome.

• I’ve deleted my answer. I hope that someone else can help you more than I did. Commented Mar 16, 2023 at 21:26
• @Filippo there is a book by SA Fulling where the procedure you are interested in is explained in some details. It is entitled Aspects of QFT in curved spacetime. Commented Mar 16, 2023 at 21:37
• By the way, I suspect that the root of our argument over my now-deleted answer is that your paper (which I could not see) factors out $\mathscr D^{1/2}$ whereas my answer factored out $\Delta^{1/2}$. If you had provided sufficient information in your question, I would not have wasted an hour writing an answer that did not match what your paper does. Commented Mar 16, 2023 at 21:38
• cambridge.org/core/books/… Commented Mar 16, 2023 at 21:43
• @Ghoster I am sorry. But both my paper and your paper say that $a_0$ is defined by $\langle\mathrm d\sigma,\nabla a_0\rangle=0$ and $[a_0]=1$ and I don't think that $a_0\equiv 1$ is a solution. Here is another argument, that you may not like: In the language of mathematicians, we assume that we are given a vector bundle $E\to M$ and a generalized Laplacian $F\colon\Gamma(M,E)\to\Gamma(M,E)$ (there exist $\nabla$ and $P$ s.t. $F=\nabla^*\nabla+P$). Then it doesn't make sense to assume that $x\mapsto a_0(x,x')\in L(E_{x'},E_x)$ (defined on a neighborhood of $x'$) is constant. Commented Mar 16, 2023 at 22:04