  We prove spectral optical identities in quantum field theories of physical
particles (defined by the Feynman $i\epsilon $ prescription) and purely virtual
particles (defined by the fakeon prescription). The identities are derived by
means of purely algebraic operations and hold for every (multi)threshold
separately and for arbitrary frequencies. Their major significance is that they
offer a deeper understanding on the problem of unitarity in quantum field
theory. In particular, they apply to "skeleton" diagrams, before integrating on
the space components of the loop momenta and the phase spaces. In turn, the
skeleton diagrams obey a spectral optical theorem, which gives the usual
optical theorem for amplitudes, once the integrals on the space components of
the loop momenta and the phase spaces are restored. The fakeon
prescription/projection is implemented by dropping the thresholds that involve
fakeon frequencies. We give examples at one loop (bubble, triangle, box,
pentagon and hexagon), two loops (triangle with "diagonal", box with diagonal)
and arbitrarily many loops. We also derive formulas for the loop integrals with
fakeons and relate them to the known formulas for the loop integrals with
physical particles.
