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In GSW's string theory book, p.43, he derived and demonstrated the Ward identity. Start from the M-point function,

[tex]A_{\mu_1m_2\cdots\mu_M}(k_1,k_2,\cdots,k_M) = \int d^4x_1\cdots d^4x_M\,e^{ik_i\cdot x_i}\,\left\langle T\big[J_{\mu_1}(x_1)J_{\mu_2}(x_2)\cdots J_{\mu_M}(x_M)\big]\right\rangle[/tex]

where [tex]\langle\cdots\rangle[/tex] is the vacuum expectation value. If we have a gauge symmetry, such that the current is conserved, we have

[tex]\left\langle T\big[\partial_{\mu_1}J^{\mu_1}(x_1)J_{\mu_2}(x_2)\cdots J_{\mu_M}(x_M)\big]\right\rangle = 0[/tex]

The key point(which I don't quite understand) is, if the currents commute, we can move the derivative [tex]\partial_{\mu_1}[/tex] outside the time ordering, hence

[tex]0 = \partial_{\mu_1}\left\langle T\big[J_{\mu_1}(x_1)J_{\mu_2}(x_2)\cdots J_{\mu_M}(x_M)\big]\right\rangle\quad\cdots(*)[/tex]

so that, with (*), we can easily derive,

[tex]k^{\mu_1}A_{\mu_1\mu_2\cdots\mu_M}(k_1,k_2\cdots,k_M) = 0[/tex], i.e. the Ward identity.

And subsequently he said, "In the nonabelian case the structure of Ward identities is more complicated. Roughly speaking, the electromagnetic current is replaced by Yang-Mills currents [tex]J^a_\mu[/tex], which, while still conserved, do not commute with each other. Hence, in trying to remove the derivative from inside the T-product, one picks up extra terms involving involving equal-time commutators."

What I don't understand is,

(1) [tex]J^a_\mu[/tex] are just composed of the matter fields, say [tex]\psi[/tex], [tex]J^a_\mu[/tex] have nothing to do with the generators, why are they related to the nonabelian nature of the gauge group?

(2) I guess it's due to that [tex]J^a_\mu[/tex] are composed of the matter field and the canonical momentum, so that to change the order of [tex]J^a_\mu[/tex], we have "equal-time" commutators. But this is the feature of both abelian and nonabelian gauge theory. why we only have this problem is abelian gauge theory?

(3) It not clear to me why we have to change the order of currents after moving derivative to the outside? Can anyone give me an example of how to do this?

Thanks so much for any instructions!

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# Derivative inside, outside time ordering

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