Abstract:
In this doctoral thesis, we develop and investigate new mathematical tools that are intended to allow for a rigorous description of non-perturbative quantum field theory (QFT) dynamics. Here, the term QFT is to be understood as describing a quantum system with particle creation and annihilation that can, but does not need to, comply with special relativity. The tools aim at cases where a formal Hamiltonian exists but is ill-defined.
The first investigated tool is an axiomatic setting called hypersurface evolution, which has recently been introduced by Lienert and Tumulka. One may view it as an alternative to the well-established Wightman and the Haag-Kastler axiom systems, as it sets up a framework for relativistic non-perturbative QFT dynamics. In contrast to these two systems, the hypersurface evolution setting works in the Schrödinger picture, instead of the Heisenberg picture. The state of the system is described by a family of vectors $ \Psi_\Sigma $, one for each Cauchy surface $ \Sigma $. This situation is similar to a QFT description suggested by Tomonaga and Schwinger, which works via Cauchy surface-dependent vectors $ \Psi_\Sigma $ in the interaction picture.
The hypersurface evolution setting is at a comparably early stage of development. We further refine it in this thesis and briefly discuss, how it might be modified and related to existing axiomatic frameworks in non-perturbative QFT.
It is a peculiarity of this setting, that the Born rule for all $ \Psi_\Sigma $ cannot simply be postulated, but must be proven as a theorem: Provided that the Born rule holds on a certain subset of all Cauchy surfaces $ \Sigma $ (e.g., only on flat $ \Sigma $), one may reconstruct detection probabilities on the set of all Cauchy surfaces. These reconstructed probabilities may or may not coincide with those predicted by the Born rule. In this dissertation, we prove that for certain reconstructions, both expressions indeed agree.
The main set of new tools, which is introduced in this thesis, is given within the ``extended state space'' (ESS) framework. We provide a construction scheme for vector spaces that allow for a rigorous treatment of certain infinite quantities, which appear in formal calculations concerning the cutoff-free renormalization of QFT models. Among these spaces, there are two extensions $ \overline{\mathscr{F}}, \overline{\mathscr{F}}_{\ex} $ of a dense subspace of Fock space, which allow for a rigorous description of ``virtual particle states''. The scheme has been inspired by a recently developed cutoff-free non-perturbative renormalization technique called ``interior-boundary conditions'' (IBC), as well as the cutoff-free perturbative Epstein-Glaser renormalization.
We present two concrete constructions following this scheme: One of them is designed to allow for a non-perturbative renormalization in polaron models, and the second is adapted to a treatment of Bogoliubov transformations.
For the first construction, we prove that a cutoff-free non-perturbative renormalization is indeed possible for $ M $ resting fermions linearly coupled to a boson field (i.e., several coupled Van Hove models).
The second construction is used to implement certain Bogoliubov transformations in an extended sense, although they violate the Shale-Stinespring condition and thus cannot be implemented on Fock space.
For Bogoliubov transformations, we also investigate a Fock space extension framework given by von Neumann's infinite tensor product (ITP) space $ \widehat{\sH} $. Here, we prove that certain Bogoliubov transformations violating the Shale-Stinespring condition can nevertheless be implemented using $ \widehat{\sH} $. We then provide examples, where a successful diagonalization of quadratic Hamiltonians is possible by a Bogoliubov transformation, implemented using $ \overline{\mathscr{F}} $ or $ \widehat{\sH} $, that violates the Shale-Stinespring condition.