In the previous two talks of this series (on May 24 and June 28), it was discussed how quantum transition amplitudes can be obtained through the imposition of suitable first class constraints. In the case of a finite-dimensional linear system, this perspective provides an unambiguous prescription to derive the correct amplitude.

The key to lift these results to the infinite-dimensional case is to understand an infinite-dimensional linear theory (aka. free field theory) as a consistent collection of finite-dimensional ones:

• first, quantum states and amplitudes on the boundaries of finite spacetime regions are constructed for arbitrary truncations; consistency here take the form of coarse-graining relations from finer truncations to coarser ones;
• second, the thus obtained theories over different regions are shown to be consistent under gluing of regions; the applicable gluing rules are a straightforward adaptation of the GBF axioms.

We show how to associate natural boundary Hilbert spaces to the quantisation of classical systems in bounded regions that can be understood as composite boundary-bulk systems. As an example, we will show in detail how one can Fock quantise a linear field-particle interacting system consisting of a string with two point masses attached at the ends. We shall see that, although the quantised system cannot be immediately decomposed as a tensor product of 'masses' and 'string' Hilbert spaces, there is a natural Hilbert space that can be associated to the boundary with the aid of so-called trace operators from PDE theory.

Let G be a simply-connected, compact, simple Lie group (such as the special unitary groups SU(n)), and let M be a 4-dimensional, compact and oriented smooth manifold. It is a standard fact that the space of equivalence classes of principal G-bundles P -> M is parametrized by a single topological invariant, the so-called "topological charge" in the physics jargon, which corresponds to an integer.

This talk will begin with a brief introduction to the previously mentioned notions, and then I will describe work in progress with José A. Zapata, on the reconstruction of such topological invariants in the context of extended lattice gauge fields, a formalism that we have introduced to enhance the discretization mechanism of gauge theories, in such a way that each ELG field corresponds to a certain homotopy class of gauge fields in the continuum. The reconstruction is motivated by analogous ideas in 2-dimensional abelian topological gauge theory.

At the end of the 1980s the concept of topological quantum field theory (TQFT), motivated originally from physics and due to Witten, Atiyah, Segal and others, originated a revolution in algebraic topology, knot theory and other areas. In particular, TQFTs serve in the construction of invariants of manifolds and of knots.

I discuss a new class of topological quantum field theories that I call "positive TQFTs". In these the objects associated to hypersurfaces are ordered vector spaces and the morphisms associated to cobordisms are completely positive linear maps. I describe a functorial construction that converts "ordinary" TQFTs into positive TQFTs by "taking the modulus square". An attractive feature of positive TQFTs is that they provide an inherent means of dealing with infinities. This means in particular that one can work with infinite dimensional vector spaces in a well defined way, rather than only with finite dimensional ones as is customary. One could hope that this may lead to new invariants of manifolds for example.

Instead of formulating the states of a Quantum Field Theory (QFT) as vectors in a single large Hilbert space (or, more generally, density matrices thereon), it has been proposed by Kijowski to construct them as *consistent* families of *partial* density matrices, the latter being defined over *small* 'building block' Hilbert spaces. In this picture, each small Hilbert space can be physically interpreted as extracting from the full theory specific degrees of freedom (aka. 'coarse-graining' the continuous theory). The resulting quantum state spaces tend to be more robust than those obtained via ordinary Fock-like constructions: they allow to bypass vacuum ambiguities, facilitate the systematic construction of semi-classical states and support a wide range of dynamics.

I will give an introduction to this framework, illustrating it on the example of polymer quantization (as used in isotropic LQC). In particular, I will discuss how the thus obtained quantum state space relates to the Schrödinger representation and the Bohr compactification one respectively, and I will comment on how these techniques could help deepen the LQC/LQG connection (in particular at the dynamical level).

Anti-de Sitter is not a globally hyperbolic spacetime. When studying a field theory in anti-de Sitter, one needs an appropriate choice of boundary conditions at the conformal boundary such that the classical field equation is well-posed. Moreover, at the level of the standard formulation of quantum field theory, the existence of physical quantum states, the so-called Hadamard states, is only guaranteed (and defined) on globally hyperbolic spacetimes.

In this talk, I discuss the classical and quantum theory of a real, massive scalar field in the Poincaré patch of the (d+1)-dimensional anti-de Sitter spacetime, with arbitrary, Robin boundary conditions (including the commonly used Dirichlet boundary conditions). We show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to this system. As a consequence, we are able to generalize the definition of Hadamard states and construct a global algebra of Wick polynomials.

Dynamical reduction models have been proposed in (non-relativistic) quantum mechanics to deal with the so-called measurement problem by modifying the Schrödinger evolution equation and including non-unitary terms, which induce the 'spontaneous collapse' of the wave-function of a system. In this talk, we shall show some recent developments in generalising these models to quantum field theory in curved spacetimes. An important obstruction in this generalisation is the possibility that an initial Hadamard state of a free theory could evolve into a non-Hadamard one, rendering the model unacceptable. We show that, for a large class of models, this problem does not arise and the Hadamard property is preserved. Based on: [arXiv:1708.09371 [gr-qc]].

The polymer harmonic oscillator is a toy model mimicking some of the techniques used in LQG. Among other features, it is a non-regular representation of the Weyl algebra. This makes unclear what the symmetry group of this model is. In this talk, we will show the first steps to answer this question.