S-matrix positivity without Lorentz invariance: a case study

We investigate the relationships between classical observables in cosmology and the classical limit of quantum scattering amplitudes. We first look at the relation between Bogoliubov transformations and the notion of classical limit. Then, we compute the cosmological redshift for a particle in a cosmological background and the emitted gravitational waveform from a quantum field theory basis and its classical limit. We observe that there is no interpretation for the geodesic redshift purely in terms of on-shell amplitudes in flat space, given that the classical limit of a scalar 2-point vanishes when considering an FRW background with two asymptotically flat in and out regions. We resolve this apparent conundrum and recover the correct observable by showing that the action of Hermitian operators differs between the in and out regions, unlike standard approaches in flat spacetime. We then show that radiation reaction corrections to the redshift enter already at order G. Furthermore, we demonstrate that the emitted waveform can be represented solely in terms of an on-shell 3-point amplitude in flat space without energy conservation, providing a closed formula for the waveform in an impulsive FRW.

Cosmology and the classical limit of the S-matrix

Aoki,

Cristofoli

2024

Non-analyticity of the S-matrix with spontaneously broken Lorentz invariance

Creminelli,

Delladio,

Janssen

et al. 2024

We study the S-matrix of Goldstones in the renormalizable theory of a U(1) complex scalar at finite charge, i.e. in a state that breaks Lorentz invariance. The theory is weakly coupled so that this S-matrix exists at all energies. Unlike the Lorentz invariant case, the resulting S-matrix is not analytic in the exchanged (complexified) four-momentum. The non-analyticities stem from the LSZ reduction formula, as a consequence of the energy-dependent mixing between the radial and Goldstone modes.

Positivity bounds on electromagnetic properties of media

Creminelli,

Janssen,

Salehian

et al. 2024

We study the constraints imposed on the electromagnetic response of general media by microcausality (commutators of local fields vanish outside the light cone) and positivity of the imaginary parts (the medium can only absorb energy from the external field). The equations of motion for the average electromagnetic field in a medium — the macroscopic Maxwell equations — can be derived from the in-in effective action and the effect of the medium is encoded in the electric and magnetic permeabilities ε(ω, |k|) and μ(ω, |k|). Microcausality implies analyticity of the retarded Green’s functions when the imaginary part of the 4-vector (ω, k) lies in forward light cone. With appropriate assumptions about the behavior of the medium at high frequencies one derives dispersion relations, originally studied by Leontovich. In the case of dielectrics these relations, combined with the positivity of the imaginary parts, imply bounds on the low-energy values of the response, ε(0, 0) and μ(0, 0). In particular the quantities ε(0, 0) – 1 and ε(0, 0) – 1/μ(0, 0) are constrained to be positive and equal to integrals over the imaginary parts of the response. We discuss various improvements of these bounds in the case of non-relativistic media and with additional assumptions about the UV behavior.