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In the introduction of chapter-12 of “An Introduction to Quantum Field Theory” by Peskin and Schroeder I encountered this line: “The quantum fluctuatuations at arbitrarily short distances appear in Feynman diagram computations as virtual quanta with arbitrarily high momenta.” My questions are:

  1. What is the mathematical (and physical) meaning of quantum fluctuations of a quantum field $\phi(x)$ at a point x?

  2. How the quantum fluctuations are related to the virtual particles? And how do the quantum fluctuations make their appearance in Feynman diagrams?

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I think you should not take the Peskin and Schröder quote too seriously.

They are likely just using the Fourier relationship "short distance <-> high momenta" and the idea that there are propagators $\langle \phi^2 \rangle$ (which are the "fluctuation"/variance of $\phi$, see this post) associated to the Feynman lines of virtual particles, so "small distance fluctuations" correspond to "high momentum virtual particles".

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  • $\begingroup$ Probably connected with the renormalization group concept à la Wilson he explains later in that same chapter (if I recall right), where the momentum space propagator is separated in fast and slow contributions. You only need slow contributions for large distance behaviour and vice versa. $\endgroup$ – giulio bullsaver May 14 '15 at 16:19

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