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


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?


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