Question

I was reading about Feynman rules for scalar field in ϕ4 theory in section 4.6, pages 113-114 of Peskin & Schroeder, and, calculating amplitudes for processes, the authors show that Feynman propagators Δ(0) appear as multiplicative factors. Until that moment I thought that Feynman propagators Δ(x−y) were only defined for x≠y, and, in fact, if the formula Δ(x−y)=∫+[infinity]−[infinity]d4p(2π)4e−ip(x−y)p2−m2+iϵ(1) is valid for x=y, then Δ(0)=∫+[infinity]−[infinity]d4p(2π)41p2−m2+iϵ(2) would diverge (?).

Also, it is written at some point in Peskin, discussing feynman diagrams, that a diagram, obtained from the expansion of the Green function for the process considered, needs to be amputated before it is used to calculate amplitudes. For "amputated" the authors mean that every external line should be removed together with "bulb" which represent, precisely, propagators which starts and end at the same vertex, i.e multiplicative terms built like Δ(0)
. My question is: does it makes sense to define Δ(0)=1, so that even this Feynman rule makes sense (the authors don't give convincing reasons to drop Δ(0))? It could be the case, being the the propagator interpreted as the amplitude for a particle to move from a point to another, so that Δ(0)=1
would mean the particle has a probability of 1 to move from a point to the same point.

Edit: Peskin (and many other authors) explicitly defines the propagator (1)
only for different times (formula 2.60
in Peskin & Schroeder), i.e. x0 or x0>y0. It is true that, for x→y, Δ(x−y)→[infinity], but still Δ(0) is not defined.

Answer 1

The Feynman propagator Δ(0) is not defined in quantum field theory due to divergence at x = y. Regularization or renormalization techniques handle such infinities in calculations, and defining Δ(0) as 1 would be unconventional and not justified in the standard framework. The issue addresses challenges in QED faced by Nobel laureates Richard Feynman, Julian Schwinger, and S.I. Tomonaga.

Step-by-step explanation:

The question raises the issue of defining the Feynman propagator Δ(0) within the context of scalar field theory in Φ4 theory, particularly in Feynman diagram calculations. According to Peskin & Schroeder, the propagator is defined only for different times and diverges as x approaches y. The concept of amputation in Feynman diagrams refers to removing external lines and associated propagators that begin and end at the same vertex. Defining Δ(0) as 1 might seem a solution to ease the interpretation of particle movement, suggesting a unit probability of a particle remaining at the same point.

However, in quantum field theory, Δ(0) is usually not defined due to the divergence at x = y, and a regularization technique or renormalization is used to handle such infinities. Assigning a finite value like 1 to Δ(0) would be unconventional and not justified within the standard framework of quantum field theory. Instead, the role of such propagators is managed through regularization processes that ensure the physical observables remain finite and meaningful. Richard Feynman, Julian Schwinger, and S.I. Tomonaga's foundational work in quantum electrodynamics (QED) involved dealing with such infinities, which earned them the 1965 Nobel Prize.