A nonperturbative method for solving quantum field theories in three space and one time dimensions is applied to the bound-state problem of positronium and heavy quarkonia. The model includes only one dynamical photon, i.e., the irradiation channels are closed. An integral equation of the Bethe-Salpeter type is derived, being the light-cone analogue of the Tamm-Dancoff equation, and solved numerically. The model accounts for the Bohr-Sommerfeld and Dirac physics such as hyperfine splitting including the correct retardation. Special emphasis is put on the role of the Coulomb singularity in momentum space. Numerical results for the mass spectrum and the wave functions are presented, and compared to analytical results. Agreement is found for the physical value of the coupling constant. For very large coupling constants a -0.3 discrepancies are noted and discussed.

The requirement that both the matter and the geometry of a spacetime canonically evolve together, starting and ending on shared Cauchy surfaces and independently of the intermediate foliation, leaves one with little choice for diffeomorphism-invariant gravitational dynamics that can provide consistent evolution equations to the coefficients of a given system of matter field equations. Concretely, we show how starting from any linear local matter field equations whose principal polynomial satisfies three physicality conditions, one may calculate coefficient functions which then enter an otherwise immutable set of countably many linear homogeneous partial differential equations.Any solution of these so-called gravitational closure equations then provides a Lagrangian density for any type of tensorial geometry that features ultralocally in the initially specified matter Lagrangian density. Thus the given system of matter field equations is indeed closed by the so obtained gravitational equations. In contrast to previous work, we build the theory on a suitable associated bundle encoding the canonical configuration degrees of freedom, which allows to include necessary constraints on the geometry in practically tractable fashion. By virtue of the presented mechanism, one thus can practically calculate, rather than having to postulate, the gravitational theory that is required by specific matter field dynamics. For the special case of standard model matter one obtains general relativity. a Corresponding author. Electronic address fps@aei.mpg.de 1 arXiv:1611.08878v3 [gr-qc]

We derive the gravitational dynamics of the tensorial geometry which underlies the most general linear theory of electrodynamics that features weak birefringence in vacuo. This derivation is performed by way of gravitational closure, which is a mechanism that employs the causal structure of any canonically quantizable matter dynamics on some tensorial spacetime geometry in order to derive canonical dynamics for the latter. The resulting eleven-parameter family of weak gravitational field equations allows to predict where vacuum birefringence will occur, if there is any.

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