A novel class of functionals for perturbative algebraic quantum field theory
arXiv:2312.15203
The paper demonstrates that microcausal functionals, commonly used in perturbative algebraic quantum field theory, are not closed under the Peierls bracket, and introduces a new class of equicausal functionals that are closed under the bracket and star-product and satisfy the time‑slice axiom.
Abstract
Perturbative Algebraic Quantum Field Theory (pAQFT) is based upon formal power series valued in spaces of functionals. This is usually done with microcausal functionals, which are defined using microlocal analysis and motivated by propagation of singularities. In this paper, we prove that the class of microcausal functionals is not closed under the Peierls (Poisson) bracket by showing that a Peierls bracket of regular functionals can fail to be smooth. Consequently, microcausal functionals are not a suitable basis for pAQFT. To remedy this issue, we introduce the class of equicausal functionals. We show that this class contains the local functionals and that it closes under the star-product and Peierls bracket. Furthermore, we prove the time-slice axiom for equicausal functionals, using a chain homotopy. The class of microcausal functionals is not closed under this chain homotopy, which strongly suggests that the class of microcausal functionals does not fulfill the time slice axiom.
46 pages, 2 figures, to appear in Communications in Mathematical Physics