# Second-order functional dervative of the Yang-Mills action by DeWitt

DeWitt calculated a second order functional dervative of a Yang-Mills action in Table I p. 1201 in his paper PR 162 (1967) 1195. The action is the usual one,

$$S=-(1/4)\int d^4x F_{a\mu\nu}F^{a\mu\nu}.$$

with $$F^a_{\mu\nu}=A^a_{\nu,\mu}-A^a_{\mu,\nu}+f^a_{bc}A^b_\mu A^c_\nu.$$ Here, my notation is slightly different from his. $a,b,c,\dots$ are the internal indices, raised and lower by the Cartan metric $\gamma_{ab}$ and its inverse, and $\mu,\nu,\dots$ are the spacetime indices. He then displayed the second order functional dervative,

$$\delta^2S/\delta A^a_\mu(x)\delta A^b_\nu(y)=\gamma_{ab}\eta^{\mu\nu}\delta^{(4)}(x-y)_{;\sigma}^{\phantom{aa}\sigma}-\gamma_{ab}\eta^{\sigma\nu}\delta^{(4)}(x-y)^{\phantom{a}\mu}_{;\phantom{aa}\sigma}+f^d_{ac}F^{c\mu}_{\phantom{cc\mu}\sigma}\gamma_{db}\eta^{\sigma\nu}\delta^{(4)}(x-y).$$

Then, I tried to reproduce the above result. I did the following calculation. First, take the variation of the action, which takes the familiar form,

$$\delta S=\int d^4x(\partial_\nu F_a^{\phantom{a}\nu\mu}+f^c_{ba}A^b_\nu F_c^{\phantom{c}\nu\mu})\delta A^a_\nu.$$

All fields in the integrand are evaluated at $x$. Now, take the second order variation, but before that, I insert a delta function into the above integral as follows,

$$\delta S=\int d^4xd^4y(\partial_\nu F_a^{\phantom{a}\nu\mu}+f^c_{ba}A^b_\nu F_c^{\phantom{c}\nu\mu})\delta^{(4)}(x-y)\delta A^a_\nu(y).$$

In the above expression, all field inside of the brackets are evaluated at $x$, and the partial dervatives are acting on $x$, too. Now, we can take the second order variation, omitting terms proportional to $\delta^2A^a_\nu(y)$ and integrating by parts,

$$\delta^2S=\int d^4xd^4y[\delta A^\mu_a\delta A^a_\mu\partial_\sigma\partial^\sigma\delta^{(4)}(x-y)-\delta A_a^\nu\delta A_\mu^a\partial^\mu\partial_\nu\delta^{(4)}(x-y)-f_a^{bc}A_c^\mu\partial_\nu\delta^{(4)}(x-y)\delta A^\mu_b \delta A^a_\mu-f_a^{bc}A^\nu_b\partial_\nu\delta^{(4)}(x-y)\delta A^\mu_c\delta A^a_\mu-f^c_{ba}A^b_\nu\partial^\nu\delta^{(4)}(x-y)\delta A^\nu_c\delta A^a_\mu+f^c_{ba}A^b_\nu\partial^\mu\delta^{(4)}(x-y)\delta A^\nu_c\delta A^a_\mu+(f_a^{bc}\partial_\nu A^\nu_b\delta A^\mu_c\delta A^a_\mu+f_a^{bc}\partial_\nu A^\mu_c\delta A^\nu_b\delta A^a_\mu-f^c_{ba}\partial^\nu A^b_\nu\delta A_c^\mu\delta A^a_\mu+f^c_{ba}\partial^\mu A^b_\nu\delta A^\nu_c\delta A^a_\mu+f^c_{ba}f_c^{de}A^b_\nu A^\mu_e\delta A^\nu_d\delta A^a_\mu+f^c_{ba}f_c^{de}A^b_\nu A^\nu_d\delta A^\mu_e\delta A^a_\mu+f^c_{ba}F_c^{\phantom{c}\nu\mu}\delta A^b_\nu \delta A^a_\mu)\delta^{(4)}(x-y)].$$

Now, you can read off $\delta^2S/\delta A_\mu^a(x)\delta A_\nu^b(y)$, which I will not display because of its long length. But we can easily read the term containing the field strength, which is

$$f^c_{ba}F_c^{\phantom{c}\nu\mu}\delta^{(4)}(x-y).$$

This is unfortunately not the same as the last term in the 2nd equation.

What is going wrong with my calculation? Am I in the right track?