The effect of gravitational tidal forces on renormalized quantum fields

Abstract

The effect of gravitational tidal forces on renormalized quantum fields propagating in curved spacetime is investigated and a generalisation of the optical theorem to curved spacetime is proved. In the case of QED, the interaction of tidal forces with the vacuum polarization cloud of virtual e + e − pairs dressing the renormalized photon has been shown to produce several novel phenomena. In particular, the photon field amplitude can locally increase as well as decrease, corresponding to a negative imaginary part of the refractive index, in apparent violation of unitarity and the optical theorem. Below threshold decays into e + e − pairs may also occur. In this paper, these issues are studied from the point of view of a non-equilibrium initial-value problem, with the field evolution from an initial null surface being calculated for physically distinct initial conditions and for both scalar field theories and QED. It is shown how a generalised version of the optical theorem, valid in curved spacetime, allows a local increase in amplitude while maintaining consistency with unitarity. The picture emerges of the field being dressed and undressed as it propagates through curved spacetime, with the local gravitational tidal forces determining the degree of dressing and hence the amplitude of the renormalized quantum field. These effects are illustrated with many examples, including a description of the undressing of a photon in the vicinity of a black hole singularity.

Causality violation, gravitational shockwaves and UV completion

“…This decreases once the focal point is passed and manifests a real-time wavefunction renormalization of the photon field which we can interpret as an increased dressing of the photon by the virtual e + e − cloud [10]. Figure 12.…”

“…Indeed, the conventional flat-space KramersKrönig relation, with the usual analytic properties of the relevant Green functions, would imply that the UV theory necessarily inherits the causal problems of the low-energy theory. However, in our previous work [5,10], we have shown how the novel analytic structure induced by geometric properties of the curved spacetime background imply a re-interpretation of the usual flat-space dispersion relations, with important consequences for causality and the optical theorem. In another paper in this series [29], we return to these issues and present a new analysis of dispersion relations for QFT in curved spacetime.…”

“…As demonstrated in our previous investigations of the realisation of causality in curved spacetime [5][6][7][8][9][10], in order to verify that causality is respected we need to demonstrate that the phase velocity, which may be superluminal for low frequencies, is equal to 1 in the high-frequency limit [5,[11][12][13]. This implies constraints on the phase shift of the photon modes as they scatter from the shockwave.…”

“…One of the main insights of our previous work [5][6][7][8][9][10], is that, as long as m σ (the transverse curvature scale), we may approximate the geometry in the vicinity of the chosen ray with its associated Penrose limit geometry. This is illustrated in figure 8.…”

“…In particular, ref. [10] gives a careful derivation of the solution in terms of an initial value problem, evaluated using the correct causal propagators. The field equation is…”