# quantum fluctuations and the virtual particles

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?

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".