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Consider for instance the QED Lagrangian. Is it possible to compute the rate of virtual electron-positron creation from the vacuum?

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If you want to consider this in terms of QED, then it is easy. Virtual pair production-annihilation is described by a vacuum bubble diagram where two propagators a contracted into a loop and there are no external legs. These kinds of diagrams are usually thrown away when we consider the partition function of the theory. And there is a good reason for that, as the matrix element of this diagram diverges at momentum $0$ running around the loop. So you might say that the rate of this process is infinite, whatever this means.

And if you are going to ask from the position of conventional QM, then your question is badly posed as vacuum production (i.e., non-conservation of energy) is defined only with respect to repeated measurements of the system which will "define" the rate.

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