I have a simple technical problem. I'm following a paper [Shore, G. Ann Phys.(adsbygoogle = window.adsbygoogle || []).push({}); 137, 262-305 (1981)], and I am unable to show averysimple identity for the non-abelian fluctuation operator (eq 4.37):

[tex]D_\mu\left[-D^2\delta_{\mu\nu}+D_\mu D_\nu-2F_{\mu\nu}\right]\,\phi=-(D_\mu F_{\mu\nu})\,\phi[/tex] , (typo fixed)

where [itex]\phi[/itex] is a test function and [itex](F_{\mu\nu})^{ab}\equiv gf^{abc}F_{\mu\nu}^{c}=[D_\mu,\,D_\nu][/itex], and hence [itex]D_\mu F_{\mu\nu}=D^2D_\nu-D_\mu D_\nu D_\mu[/itex] (color indices suppressed). So far, I have worked on the LHS, and I'm almost there:

[tex]\text{LHS}=(-D_\nu D^2+D^2 D_\nu-2D_\mu F_{\mu\nu})\phi[/itex]

[tex]\phantom{LHS}=(-\underline{D_\mu D_\nu D_\mu}-[D_\nu,\,D_\mu]D_\mu+\underline{D^2D_\nu}-2D_\mu F_{\mu\nu})\phi[/tex]

combine underlined terms using identity stated above

[tex]=(-[D_\nu,\,D_\mu]D_\mu+D_\mu F_{\mu\nu}-2D_\mu F_{\mu\nu})\phi[/tex]

then first term is [itex]-[D_\nu,\,D_\mu]D_\mu=+F_{\mu\nu}D_\mu[/itex], and 2nd and 3rd terms add

[tex]=(F_{\mu\nu}D_\mu-D_\mu F_{\mu\nu})\phi[/tex]

Finally, use product rule in 2nd term: [itex]D_\mu(fg)=(D_\mu f)g+f\partial_\mu g[/itex].

[tex]=F_{\mu\nu}(\partial+A)_\mu\phi-(D_\mu F_{\mu\nu})\phi-F_{\mu\nu}\,\partial_\mu\phi[/tex]

to get

[tex]=F_{\mu\nu} A_\mu \phi-(D_\mu F_{\mu\nu})\phi[/tex].

This isalmostequal to RHS, except for that stupid [itex]A_\mu[/itex] term. How the hell do I get rid of it?!?

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# Spin-1 operator identity

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