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We provide a diagrammatic formulation of perturbative quantum field theory ina finite interval of time $\tau $, on a compact space manifold $\Omega $. Weexplain how to compute the evolution operator $U(t_{\text{f}},t_{\text{i}})$between the initial time $t_{\text{i}}$ and the final time$t_{\text{f}}=t_{\text{i}}+\tau $, study unitarity and renormalizability, andshow how to include purely virtual particles, by rendering some physicalparticles (and all the ghosts, if present) purely virtual. The details aboutthe restriction to finite $\tau $ and compact $\Omega $ are moved away from theinternal sectors of the diagrams (apart from the discretization of thethree-momenta), and coded into external sources. So doing, the diagrams are assimilar as possible to the usual $S$ matrix diagrams, and most known theoremsextend straightforwardly. Unitarity is studied by means of the spectral opticalidentities, and the diagrammatic version of the identity$U^{\dag}(t_{\text{f}},t_{\text{i}})U(t_{\text{f}},t_{\text{i}})=1$. Thedimensional regularization is extended to finite $\tau $ and compact $\Omega$,and used to prove, under general assumptions, that renormalizability holdswhenever it holds at $\tau =\infty $, $\Omega =\mathbb{R}^{3}$. Purely virtualparticles are introduced by removing the on-shell contributions of somephysical particles, and the ghosts, from the core diagrams, and trivializingtheir initial and final conditions. The resulting evolution operator$U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ is unitary, but does not satisfy themore general identity$U_{\text{ph}}(t_{3},t_{2})U_{\text{ph}}(t_{2},t_{1})=U_{\text{ph}}(t_{3},t_{1})$.As a consequence, $U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ cannot be derivedfrom a Hamiltonian in a standard way, in the presence of purely virtualparticles.

the journal of high energy physics/the journal of high energy physics

Quantum field theory of physical and purely virtual particles in a  finite interval of time on a compact space manifold: diagrams, amplitudes and  unitarity