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Sep 5, 2019 at 8:23 comment added P. U. After some investigations I realized you were right with the definition of the time-ordered GF. I edited my answer but now I only have an idea left. Sorry for the confusion...
Sep 5, 2019 at 8:23 history edited P. U. CC BY-SA 4.0
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Sep 4, 2019 at 23:16 comment added user1830663 See new edit. I tried to make my notation even more precise...
Sep 4, 2019 at 17:34 comment added user1830663 Every text or notes that I have dont include commutator in the first "=" sign for the time ordered GF. I am confused by your definitions.
Sep 4, 2019 at 6:02 comment added P. U. This will will give you only the retarded GF in the end because the term with theta(negative time) will vanish. You can see this procedure in my answer above.
Sep 4, 2019 at 5:57 comment added P. U. You still need to include the commutator after the first "=" sign.
Sep 3, 2019 at 22:00 comment added user1830663 I have indeed made several typos. I updated my question. Sorry for the confusion.
Sep 3, 2019 at 6:47 history answered P. U. CC BY-SA 4.0