On the unitary transformation between non-quasifree and quasifree state spaces and its application to quantum field theory on curved spacetimes

TL;DR

This paper develops a method to calculate a unitary transformation linking non-quasifree and quasifree states in quantum field theory on curved spacetimes, crucial for understanding scattering processes and particle interpretations.

Contribution

It introduces a novel approach using $igstar$-calculus and extended Feynman graphs to explicitly compute the unitary transformation between state spaces in curved spacetime QFT.

Findings

Derived the unitary transformation relating non-quasifree and quasifree states.

Showed that the outgoing field is free but generally not quasifree in non-stationary spacetimes.

Confirmed that interacting Wightman functions satisfy key axioms of quantum field theory.

Abstract

Using $\star$-calculus on the dual of the Borchers-Uhlmann algebra endowed with a combinatorial co-product, we develop a method to calculate a unitary transformation relating the GNS representations of a non-quasifree and a quasifree state of the free hermitian scalar field. The motivation for such an analysis and a further result is the fact that a unitary transformation of this kind arises naturally in scattering theory on non-stationary backgrounds. Indeed, employing the perturbation theory of the Yang-Feldman equations with a free CCR field in a quasifree state as an initial condition and making use of extended Feynman graphs, we are able to calculate the Wightman functions of the interacting and outgoing fields in a $\phi^p$-theory on arbitrary curved spacetimes. A further examination then reveals two major features of the aforementioned theory: firstly, the interacting Wightman functions fulfil the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity, and locality. Secondly, the outgoing field is free and fulfils the CCR, but is in general not in a quasifree state in the case of a non-stationary spacetime. In order to obtain a sensible particle picture for the outgoing field and, hence, a description of the scattering process in terms of particles (in asymptotically flat spacetimes), it is thus necessary to compute a unitary transformation of the abovementioned type.