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.
Highlights
Unitarity is a key requirement to claim that a quantum field theory has chances to be fundamental, together with locality and renormalizability
We prove spectral optical identities in quantum field theories of physical particles and purely virtual particles
We show that unitarity in quantum field theory can be reduced to a set of algebraic identities, which do not require to integrate on the space components of the loop momenta, or the phase spaces in cut diagrams, and hold for each physical threshold separately
Summary
Unitarity is a key requirement to claim that a quantum field theory has chances to be fundamental, together with locality and renormalizability. We show that unitarity in quantum field theory can be reduced to a set of algebraic identities, which do not require to integrate on the space components of the loop momenta, or the phase spaces in cut diagrams, and hold for each physical threshold separately. Virtual particles [22], or “fakeons”, can be used to formulate a consistent theory of quantum gravity [7], which is experimentally testable thanks to its predictions in inflationary cosmology [23] They can be used to solve the problem of ghosts in higher-derivatives theories. At two loops we study the triangle with “diagonal” and the box with diagonal These examples cover the needs of most calculations in high-energy physics phenomenology.
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