# Time ordering identities in integration over gluon fields

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

$$A_0(-r)=-A_0(r)$$.

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?