Propagators and widths of physical and purely virtual particles in a finite interval of time

We study the free and dressed propagators of physical and purely virtual particles in a finite interval of time $\tau $ and on a compact space manifold $\Omega $, using coherent states. In the free-field limit, the propagators are described by the entire function $(e^{z}-1-z)/z^{2}$, whose shape on the real axis is similar to the one of a Breit-Wigner function, with an effective width around $1/\tau $. The real part is positive, in agreement with unitarity, and remains so after including the radiative corrections, which shift the function into the physical half plane. We investigate the effects of the restriction to finite $\tau $ on the problem of unstable particles vs resonances, and show that the muon observation emerges from the right physical process, differently from what happens at $\tau =\infty $. We also study the case of purely virtual particles, and show that, if $\tau $ is small enough, there exists a situation where the geometric series of the self-energies is always convergent. The plots of the dressed propagators show testable differences: while physical particles are characterized by the usual, single peak, purely virtual particles are characterized by twin peaks.