My question arised when trying to compute the Wilson Loop of a hybrid meson. When calculating the loop one has to keep in mind the path ordering and time ordering respectively.
I have the following written
$\left( \int_0^t dt' A_0^a(t',r)T^a \right)\cdot \left( \int_t^0 dt' A_0^b(t',-r)T^b \right)$
(T are the Gell-Mann matrices and a, b denotes the adjoint colour index, A_0 is the 0th component of a gluonic field)
where time ordering has already been performed, so that the integration "downward" from t to 0 stands left of the integration "upward" from 0 to t. So far so good.
Now, since I all in all have a pretty lenghty expression I tried to shorten this term a little and therefore wanted to use the partity property of the gluon (or in my case its 0th component), namely
Performing this operation (and turning the second integral around) leads to
$\left( \int_0^t dt' A_0^a(t',r)T^a \right)\cdot \left( \int_0^t dt' A_0^b(t',r)T^b \right) = T^aT^b \cdot \int_0^tdt'dt''A_0^a(t',r)A_0^b(t'',r)$
And now my question ist: Is that step valid? The last result looks somehow fishy to me since here the integrations are no longer performed on two different spatial points r and -r but both fields are now located at r. Should therefore be somewhere a "new" time ordering added? Or is everything right and the ordering $T^aT^b$ of the Gell-Mann matrices is fixed now?