Extended State Space for describing renormalized Fock spaces in QFT

TL;DR

This paper develops a rigorous mathematical framework for non-perturbative renormalization in quantum field theory models with highly singular interactions, extending the state space to include non-square-integrable wave functions.

Contribution

It introduces extended state vector spaces over a field extension to rigorously define renormalized Hamiltonians without cutoffs in QFT models.

Findings

Defined extended state vector spaces containing non-square-integrable states.

Constructed a densely defined self-adjoint renormalized Hamiltonian.

Provided a rigorous interpretation of infinite self-energies and wave function renormalizations.

Abstract

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.