I Fermionic Field Time Ordering: Understanding the Time Ordered Contraction

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Hello,

I am struggling to see why for a fermionic field $\psi$, one has the time ordered contraction $<0|T(\psi(x)\psi(y))|0>$. Could someone offer an outline/hints to see this please? Thanks!
 
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Essentially, it's to preserve causality. You have a time-ordering operator so that ##\mathcal{O}(t_2)## is applied after ##\mathcal{O}(t_1)##. Remember that ##\psi## is a time-dependent operator in this case, and unless otherwise stated, ##x^0## needn't equal ##y^0##.
 
Yes, thanks for this. Is there a way to see this mathematically based on the commutation rules?
 
Can you expand on your question? One can define time-ordering for either fermionic or bosonic fields, or one can consider other notions of ordering (anti-time ordering, retarded, advanced, and more complicated combinations if you're using a formalism with multiple time contours). Are you maybe asking how time-ordering shows up in specific places?
 
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