Microlocal techniques for perturbative algebraic quantum field theory in the functional formalism

In this thesis, we study the functional formalism for perturbative Algebraic Quantum
Field Theory (pAQFT), with a focus on microcausal functionals. These functionals
are required to have a restricted singular structure, that ensures that the Poisson
bracket and the quantum ⋆-product can be rigorously defined on them. However,
we identify several inconsistencies in their treatment in the literature. The most
significant of these is the fact that the Poisson bracket of two microcausal functionals
can fail to be continuous, implying that the Poisson bracket does not close on this
class of functionals.

A secondary goal of this thesis is the construction of tools that allow one to
prove homotopical statements within the functional formalism, in the context of the
Batalin-Vilkovisky (BV) formalism. The main result we present here is a prescription
for lifting a retract of field complexes to a retract of the functionals on those
complexes, as an extension of the results presented in [52]. However, these new tools
do not extend to graded microcausal functionals, as demonstrated by an explicit
counterexample. This presents a significant obstruction to proving the time-slice
axiom for these functionals, raising doubts about their ability to properly account
for local dynamics.

To address these problems, we introduce the class of equicausal functionals.
These functionals address the identified inconsistencies by satisfying a stricter set of
microlocal conditions. We show that equicausal functionals are closed under both
the homotopical tools we define and the ⋆-product. The introduction of equicausal
functionals enables us to recover several key results previously inaccessible due to
the limitations of the microcausal framework, thereby reinforcing their significance
for both the BV-formalism and pAQFT as a whole.