# 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. Mar 16 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. Mar 16 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. Mar 16 at 21:38
• cambridge.org/core/books/… Mar 16 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. Mar 16 at 22:04