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Introduction

Virtual particles are particles which doesn't obey the energy–momentum relation: \begin{align}E^2=(mc^2)^2+(pc)^2\end{align}This is possible if it happens short enough time. Often Heisenberg's uncertainty principle is used to determite relation for that: \begin{align}\Delta E \Delta t \geq \frac{\hbar}{2}\end{align} when searching information about virtual particles, it is done by just simply turning "≥" sign. We can borrow the energy ∆E to create a virtual particle from the vacuum, as long as we give it back within a time ∆t ≈ ℏ/∆E.

Energy operator, which is bounded below, doesn't satisfy canonical commutation rule. And time isn't quantum observable at all and there is no Hermitean operator whose eigenvalues were the time of the system.

Question

My question is how is turning the sign "≥" actually justified to use to determine time which is allowed to be off mass shell in Quantum Mechanics or Quantum Field Theory? (What is idea behind turning sign and why we can do it). Is there other method to derive the relation between mass and range than Heisenberg's uncertainty principle?

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