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Clearly if we have two operators $\phi(t_1)$ and $\psi(t_2)$ and define a time ordering operator $T$ acting on operators such that $$T(\phi(t_1)\psi(t_2)):=\phi(t_1)\psi(t_2),~\text{if $t_1>t_2$ otherwise the other way around}$$ we could pretend that inside $T$ these operators commute. However I am not quite convinced that we are allowed to simplify expressions inside, as there might well be such operators $A$ and $B$ that $A(x)B(y)$ is non-zero for all $x,y$ but $B(y)A(x)$ is always zero. Thus if we choose to simplify expressions inside $T$ it may well go wrong as we end up with a null operator, and $T(0)$ certainly will not return you with a time-ordered non-zero result.

What is going on here?

Qmechanic
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Rescy_
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  • Can you give an example of a manipulation you've seen in a book, paper, or course that you think is suspect? – Andrew Dec 03 '22 at 20:54
  • Possible duplicates: https://physics.stackexchange.com/q/323801/2451 and links therein. – Qmechanic Dec 03 '22 at 21:20
  • I am not sure if it is a duplicate. I don't think you should simplify expressions inside the $T$. It is just a notation to state operators should appear in a given order on the right-hand side. If you perform the simplifications you are describing and then apply time ordering, whatever you have on the right-hand side, may not be the same as if you had applied time ordering to the original expression. In some sense, you can shift operators around, but you cannot evaluate them as you were describing. – JGBM Dec 03 '22 at 21:43

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