Xavier Calmet, Stephen D. H. Hsu
We demonstrate the existence of quantum hair in electrodynamics and gravity using effective action techniques. In the case of electrodynamics we use the Euler-Heisenberg effective action while in the case of quantum gravity we use the unique effective action. We give a general formulation of these effects which applies to both theories and discuss analogies and differences between them. Furthermore, we present a QED analog to black hole evaporation. Spontaneous pair production in the external field of a ball of charge is analogous to Hawking radiation from black holes. Assuming spherical symmetry, the Gauss law prevents the external field from depending on the density profile of the ball. Quantum corrections violate these expectations, showing that quantum radiation can encode classically forbidden information about the source.
We found it interesting that quantum hair can already be found using the familiar Euler-Heisenberg effective action, which results from integrating out the electron in QED.
The paper also contains a general argument for why solutions to the semiclassical field equations resulting from the effective action (both in gravity and QED) carry more information about the state of the source than in classical physics.
From the Conclusions:
The quantum effective actions for both electrodynamics and gravity lead to field equations which couple a compact source (charge current or energy-momentum tensor) to external fields (electromagnetic or graviton field) in a manner which, generically, leads to quantum memory and quantum hair effects. External solutions of the field equations deviate, due to quantum corrections, from the familiar classical forms that satisfy the Gauss law. As a specific consequence, more information about the interior source configuration is encoded in the external field than in the classical theory.
As specific applications, we considered semiclassical sources (large black hole, macroscopic charge distribution), which allowed us to solve the quantum corrected field equations by expanding around a classical solution. However, fully quantum statements regarding quantum hair are also possible, which do not, for example, require a semiclassical source. In [1–3] it was shown that the quantum state of a compact source (e.g., in an energy eigenstate or superposition thereof) determines certain aspects of the quantum state of its external field. In principle, measurements of the external fields can fully determine the interior state of a black hole.