Superconductive or superfluid condensation in curved spacetime

Abstract

We provide a proof of unitarity for quantum field theory in a general spacetime. Our argument expresses the Bogoliubov transformations in terms of a unitary squeezing operator relating the initial and final Hilbert spaces. The $S$-matrix in curved spacetime is thus the product of the squeezing operator and the $S$-matrix in the out-Hilbert space (typically Minkowski). Since both factors are unitary, their product is unitary. It follows that the Bogoliubov in-vacuum is described by a BCS-like state (Bardeen--Cooper--Schrieffer): (i) for fermions, it is exactly the BCS state, but with electrons and positrons in place of electrons with opposite spin; (ii) for bosons, it is the Bose--Einstein condensate (BEC) superfluid ground state. Thus, gravity, or an accelerating force, creates from the vacuum a many-particle system unstable towards forming a new ground state of Cooper pairs. Technically, gravity converts the vacuum into an electron-positron condensate, with respect to which quantum field theory evolves unitarily. By reverse engineering, we reconstruct the effective Hamiltonian of QFT in a black hole (or Rindler) background, together with a simple formula for the mass gap function, which grows approximately linearly with the temperature. Hence, electrons and positrons condense into a superconducting state at increasingly higher temperatures as the black hole mass decreases. Unitarity and the particle interpretation are preserved at every stage of evaporation. The Hawking state is exactly a BCS state. Finally, we compute the entanglement entropy and derive the area law.