Causality for internel vertex in Feynaman diagrams

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Discussion Overview

The discussion revolves around the implications of causality in the context of internal vertices in Feynman diagrams, particularly referencing Eq 4.44 in Peskin and Schroeder. Participants explore the relationship between time coordinates of internal vertices and external fields, as well as the interpretation of propagators in quantum field theory (QFT).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether causality requires the time coordinates of the internal vertex z to lie between those of the fields phi(x) and phi(y).
  • Another participant asserts that in vacuum QFT, time-ordered propagators do not impose constraints on the times of the internal space-time point z.
  • A further contribution suggests that if one interprets propagators as particles, it raises questions about the directionality of time in the integration of contributions from internal vertices, particularly regarding whether contributions come from all points in spacetime or only from those within a timelike cone.
  • Another participant counters that internal lines in Feynman diagrams should not be interpreted as particles, emphasizing that they are mathematical tools for expressing perturbative contributions, while only external legs represent observable particles.
  • This participant also mentions that causality and Poincare invariance are ensured through the microcausality property of field operators.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of internal vertices and the implications of causality in this context. There is no consensus on whether the time coordinates of internal vertices must adhere to specific constraints or how propagators should be understood.

Contextual Notes

Participants highlight potential ambiguities in the interpretation of propagators and the implications of time-ordering, suggesting that assumptions about particle behavior and causality may vary based on different interpretations of the mathematical framework.

dontknow
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Eq 4.44 in Peskin and Schroeder. My question is:
Does causality imply that time coordinates of z (internal vertex over which we doing the integration) should lie between the time coordinates of field phi(x) and phi(y)?
 
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The propagators in vacuum QFT are the time-ordered propagators (which in the vacuum case are the same as the Feynman propagators). Thus there's no constraint on the times of the inner space-time point ##z##.
 
sorry, I was not clear. we usually interpret propagators as particle popping out in space at some spacetime, let's suppose x and then it gets annihilated at spacetime point y. so if we follow the same interpretation here the particle which pops out at some point x can go back in time gets annihilated at z (internal vertex) and then come back to spacetime point y. while integrating this kind of propagation for all values of internal vertex, Does integration contribution comes from all points in spacetime or only those lying inside the cone (timelike) of spacetime?
Or maybe I am missing a point, does time-ordered product implies that we can go in only one direction of time?
Let me know if i am still not clear.
Thanks for answering.
 
You cannot interpret the internal lines as particles. They are just clever mathematical notations to express the contributions of the perturbative series to the S-matrix element under consideration. Only the external legs stand for observable asymptotic free particle states, and only those are interpretable as particles. The causality and Poincare invariance of the scheme is guaranteed for such observable quantities through the microcausality property of the field operators.
 
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