Fermionic Field Time Ordering: Understanding the Time Ordered Contraction

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Discussion Overview

The discussion centers on the time ordered contraction of a fermionic field, specifically the expression <0|T(ψ(x)ψ(y))|0>. Participants explore the implications of time ordering in the context of quantum field theory, particularly regarding causality and mathematical definitions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes the importance of time ordering to preserve causality, indicating that the operator 𝓞(t₂) must be applied after 𝓞(t₁).
  • Another participant inquires about the mathematical basis for understanding time ordering in relation to commutation rules.
  • A further contribution suggests that time-ordering can be defined for both fermionic and bosonic fields, and raises the possibility of other ordering methods such as anti-time ordering and retarded or advanced forms.

Areas of Agreement / Disagreement

Participants express different aspects of the topic, but no consensus is reached regarding the specific mathematical understanding of time ordering or its implications.

Contextual Notes

The discussion does not resolve the mathematical steps involved in understanding time ordering, nor does it clarify the definitions or implications of various ordering methods mentioned.

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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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