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Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should $T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$ return $T[c_{t_2,\downarrow}^\dagger c_{t_1,\uparrow}]$ or $-T[c_{t_2,\downarrow}^\dagger c_{t_1,\uparrow}]$ if $t_2 > t_1$?

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It should. For any pair of fermionic variables it is natural to consider their anticommutator (I guess it is possible in some cases to build a theory with different fermionic fields that commute with each other, like take a tensor product of any two theories, but its ugly), the only problem is whether it is zero or not. $T$ assumes it to be zero. You should look at the definitions and motivations in order to understand why it is so. –  Peter Kravchuk Apr 28 '13 at 21:50
See also Wikipedia. –  Qmechanic Apr 28 '13 at 21:51

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