Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I have a severe confusion about the time-ordering operator. It is the best thing ever, I think, since it simplifies many proofs, due to the fact that operators commute (or anti-commute, but let's take bosonic operators for simplicity) under the time-ordering.

However, sometimes I feel uneasy using it, and, well, I think I have narrowed down why. I think the relevant question is: Do I have to work in a specific picture in order to time-order my objects? I seem to run into inconsistencies if I switch pictures, for example by moving from a Schrödinger picture to a Heisenberg one. Below is the simplest example I have found, which illustrate the issue.

In the usual way, the Heisenberg operator is defined as (here I assume a time-independent Hamiltonian)

$$ \hat{A}_H (t) = e^{i\hat{H} t} \hat{A}_S (t) e^{-i\hat{H} t}. $$

By taking the time ordering on both sides, I obtain

$$ \hat{A}_H (t) = T [ \hat{A}_H (t) ] = T [ e^{i\hat{H} t} \hat{A}_S (t) e^{-i\hat{H} t}] $$

And since on the right-hand side, everything commutes under the time-ordering, the exponentials cancel, and we end up with the (upsetting) equality

$$ \hat{A}_H (t) = \hat{A}_S (t) $$

My question is: where did I go wrong? I ran into the same type of inconsistencies when trying to prove some things on the Keldysh contour, but there it was way less obvious. It seems to be that time is kind of different in different pictures. Could someone confirm this, or point out some mistake I made along the path?

Cheers!

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# Confusion about time-ordering operator

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