Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities

TL;DR

This paper explores a nonperturbative approach to calculating transition probabilities in quantum field theory using a mathematical framework where distributions can be multiplied, enabling meaningful computations of probabilities despite the divergence of perturbation series.

Contribution

It introduces a nonperturbative method for computing transition probabilities in QFT within a distribution multiplication framework, addressing issues with divergent series.

Findings

Nonperturbative calculations yield probabilities between 0 and 1.

Perturbation series do not converge or make sense.

Numerical methods are feasible for simplified models.

Abstract

In a mathematical context in which one can multiply distributions the "`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field Theory makes sense mathematically, which can be understood a priori from the fact the so called "`infinite quantities"' make sense unambiguously (but are not classical real numbers). The perturbation series does not make sense. A novelty appears when one starts to compute the transition probabilities. The transition probabilities have to be computed in a nonperturbative way which, at least in simplified mathematical examples (even those looking like nonrenormalizable series), gives real values between 0 and 1 capable to represent probabilities. However these calculations should be done numerically and we have only been able to compute simplified mathematical examples due to the fact these calculations appear very demanding in the physically significant situation with an infinite dimensional Fock space and the QFT operators.