Newton-Cartan limit of Klein-Gordon AQFT: gravitational atoms and the loss of vacuum entanglement
We take the non-relativistic limit $c\to\infty$ of the free Klein-Gordon field on static spacetimes at three levels: one-particle resolvents, quasi-free states and the time-zero net. After subtraction of the rest energy, the one-particle Hamiltonian is an explicit function of a Schrödinger-type operator whose potential on the Schwarzschild exterior is exactly Newtonian. On regular static stars it converges to the Newtonian Schrödinger operator in norm-resolvent sense at rate $c^{-2}$, with convergence of eigenvalues and an explicit first post-Newtonian correction. On the Schwarzschild exterior the Boulware one-particle spectrum has no eigenvalues for any $c$; the convergence is strong but not in norm, the limit selects the Friedrichs realisation of the gravitational hydrogen atom, and its bound states emerge by spectral concentration. The same holds on the Reissner-Nordström exterior. In all cases the rescaled two-point functions converge to those of the Schrödinger-Fock vacuum. The hydrogenic states are limits of resonances: in each partial wave there is exactly one near each hydrogenic level, its real part carries the first post-Newtonian correction, and its width, proportional to $c^{-(4l+3)}$, is the one found for gravitational atoms by matched asymptotics. At the level of nets the time-zero Weyl algebra is the same for every $c$, and the vacuum and the dynamics converge on it, yet for smooth metrics the algebra of each bounded region with smooth boundary is a type III$_1$ factor at finite $c$ and a type I factor with a pure product vacuum in the limit. What collapses is the entanglement of the vacuum; thermal states in flat space and on stars survive the limit. With positivity of the energy imposed modulo the central charge, the axioms of Galilean nets hold in the limit for every static star and for Schwarzschild, and the vacuum is not separating.