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:
What is the mathematical (and physical) meaning of quantum fluctuations of a quantum field $\phi(x)$ at a point x?
How the quantum fluctuations are related to the virtual particles? And how do the quantum fluctuations make their appearance in Feynman diagrams?