Kasia Rejzner

Algebraic Approach to Quantum Theory

Before entering the realm of the quantum theory of fields, let’s have a look at something simpler and better understood, namely quantum mechanics (QM). To prepare the ground for what follows, we will present an abstract formulation of QM and discuss how it relates to the more standard Dirac–von Neumann axioms (Dirac, The principles of quantum mechanics, 1930, [Dir30], Neumann, Mathematische grundlagen der quantenmechanik, 1932, [vN32]).

The Quantum Sine-Gordon Model in Perturbative AQFT

We study the Sine-Gordon model with Minkowski signature in the framework of perturbative algebraic quantum field theory. We calculate the vertex operator algebra braiding property. We prove that in the finite regime of the model, the expectation value—with respect to the vacuum or a Hadamard state—of the Epstein Glaser S-matrix and the interacting current or the field respectively converge, both given as formal power series.

Effective Quantum Gravity

The functional approach to pQFT together with the BV framework introduced in Chap. 7 has been successfully applied to gauge theories [FR12b, FR12a] and can also be used in quantization of theories where the local symmetries involve transformation of spacetime points.

Interaction and Renormalization of the Scalar Field Theory

In the previous chapter we have covered the quantization of free theories (quadratic actions); now is the time to introduce the interactions. This is where we have to start working perturbatively. The ultimate goal of AQFT is to be able to construct interacting models in 4 spacetime dimensions non-perturbatively, but at the moment no such models are known. The perturbative approach, on the other hand, has proven to be successful in describing many phenomena in particle physics, so it is worthwhile to try to understand its mathematical foundations. It turns out that a careful analysis of the problem and employing some tools from functional analysis allow us to avoid dealing with ill defined “divergent” expressions, as is often done in physics textbooks.