# What substitutions are allowed within time-ordered products?

I always thought of the time-ordering in QFTs as an explicit operation. Meaning the time-ordering "operator" just takes everything I write inside it and shuffles the operators around until they are in an order given by the time label of the individual operators. However, if I view the time-ordering operator this way, it cares not only about the object/operators it acts on, but also on the explicit way I write said object. This should result in me not being allowed to substitute in other equivalent expressions for the same object inside a time-ordered product. I saw this being done several times and wonder whether maybe my view of time-ordering is incorrect.

To elaborate assume we have the time-ordered product $$T(O_1({\infty})O_2(-\infty))$$. It should also always be true that $$O(\infty)=O(-\infty)+\int_{-\infty}^\infty \frac{d}{dt}O(t')dt'$$. However, according to the logic above I would assume I can't just plug this into the time-ordered product, because it has a different explicit time-dependence. In specific derivations of the LSZ reduction formula, this is plugged in (Srednicki, my own QFT course).

A similiar problem appears in the case of Wilson loops with the path ordering, but it is identical in structure.

The situation is similiar to the difference between partial and total derivatives. When I take partial derivatives, they care about the explicit dependence on the variable and are only well defined, if I am clear on what my independent variables are. Equations, give me dependence of variables and hence I have to be careful when substituting in for variables, because the set of independent variables might change. The solution for this case is to keep straight, what the independent variables are at each step, if I do that I should be fine taking partial derivatives. I don't see a similiar solution for the time-ordering.

So far I have checked Weinberg, Peskin, Srednicki, Schwartz and several set of QFT notes, but didn't find a discussion of this issue. Any help would be greatly appreciated.