Hi,Can anyone explain why in field theory we require [f*(x),f(y)]

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SUMMARY

In quantum field theory, the commutation relation [f*(x), f(y)] = 0 for spacelike intervals x and y is essential to maintain causality, ensuring that influences do not propagate faster than light. Conversely, the expectation value <0|f*(x)f(y)|0> ≠ 0 indicates that particles cannot be confined to a single point, highlighting the nonlocality of quantum fields. This nonlocality is reconciled by the presence of antiparticles, which cancel the influences of emitted particles. Understanding these relationships is crucial for grasping the foundational principles of quantum field theory.

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  • Quantum field theory fundamentals
  • Commutation relations in quantum mechanics
  • Understanding of spacelike intervals
  • Concept of antiparticles in particle physics
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  • Study the implications of commutation relations in quantum mechanics
  • Explore the role of Green's functions in quantum field theory
  • Investigate the concept of causality in relativistic quantum mechanics
  • Learn about the significance of antiparticles and their interactions
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Physicists, students of quantum mechanics, and researchers in particle physics will benefit from this discussion, particularly those interested in the foundational aspects of quantum field theory and causality.

LearningDG
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Hi,

Can anyone explain why in field theory we require [f*(x),f(y)] = 0 for space-lkie intervals x,y; but not <0|f*(x)f(y)|0> = 0?

Thanks!
 
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LearningDG, You're right to call attention to this. It's the miracle that makes quantum field theory work! <0|f*(x)f(y)|0> ≠ 0 expresses the fact that a particle cannot be confined to a single point, that the Green's function extends outside the light cone a distance given by the Compton wavelength. And yet [f*(x),f(y)] = 0 expresses the fact that this apparent nonlocality does not destroy causality, saying that influences cannot propagate faster than light. And it's all due to the existence of antiparticles. The influence caused by emitting a particle is exactly canceled by the influence caused by the absorption of an antiparticle.
 

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