For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

In 2017, Lienert and Tumulka proved Born’s rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Born’s rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces, satisfying that there is no interaction faster than light and no propagation faster than light. Here, we prove Born’s rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $$\Sigma $$ Σ , then the observed particle configuration on $$\Sigma $$ Σ is a random variable with distribution density $$|\Psi _\Sigma |^2$$ | Ψ Σ | 2 , suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant.

In quantum field theory (QFT) models, it often seems natural to use, instead of wave functions from Fock space, wave functions that are not square-integrable and have pre-factors involving divergent integrals (known as infinite wave function renormalizations). We provide here a rigorous implementation of this approach. Specifically, we define a certain space of divergent integrals, a field extension eRen of the complex numbers containing exponentials of such integrals, and certain vector spaces F ⊂ F ex of wave functions (not necessarily square-integrable) with pre-factors from eRen. These spaces extend a dense subspace of Fock space F . We apply this construction to a class of non-relativistic QFT models with Yukawalike interaction, including the Van Hove model, Nelson's model, the Gross and the Fröhlich polaron. The formal (UV-divergent) Hamiltonian H 0 + A † + A together with an infinite self-energy renormalization E ∞ and a mass renormalization operator δm is rigorously implemented on F . We construct a Weyl algebra over eRen that can be represented by dressing transformations on a subspace of F ex . One such dressing transformation is used to pull the Hamiltonian back to Fock space, giving it physical meaning. Self-adjointness of the pulled-back Hamiltonian is proven for a dense Fock space domain. This Hamiltonian involves pairwise interaction potentials and does not suffer from a flat mass shell. For the polaron, the dynamics generated by it differ from those obtained by cutoff renormalization.