Diagrammar of physical and fake particles and spectral optical theorem

Abstract

We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman iϵ 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.

Purely Virtual Particles in Quantum Gravity, Inflationary Cosmology and Collider Physics

“…It is the "peak uncertainty", which measures the impossibility of experimentally approaching the fakeon too closely. The fakeon projection is compatible with unitarity order by order (and diagram by diagram) in the perturbative expansion [1].…”

“…and the proof to all orders can be found in ref. [1]. The threshold decomposition and the fakeon diagrammatics are compatible with gauge invariance and general covariance, through the WTST identities [31][32][33][34].…”

“…The simplest way to introduce fakeons is by means of the diagrammatics developed in ref. [1], which is useful for physical particles as well. It is based on the threshold decomposition of ordinary (cut and uncut) diagrams and the suppression of all the thresholds that involve fakeon frequencies.…”

“…The diagrammatics of fakeons [1] is obtained from the usual one by means of surgical operations that selectively remove degrees of freedom at all energies and preserve the optical theorem. The only requirement is that fake particles should be massive and nontachyonic.…”

“…We stress that the fakeon diagrammatics is relatively straightforward, to the extent that it can be implemented in software like FeynCalc, FormCalc, LoopTools and Package-X [6][7][8][9][10][11][12] and used to work out physical predictions. For proofs to all orders, see [1,13]. It is also possible [14,15] to avoid certain troubles of the Lee-Wick models [16][17][18][19][20][21] by switching to theories of particles and fakeons.…”