In the '60s and '70s, Nelson proved that the Markov property for Euclidean random fields, such as the Gaussian Free Field, is sufficient to reconstruct quantum fields on Minkowski space. Despite overwhelming success in 2d to analyse non-gaussian fields, this approach is notoriously difficult to carry out in 3d. Softer methods exist, but they often give an implicit description of fundamental objects -- such as the Hamiltonian of the theory.  

I will talk about joint work with Nikolay Barashkov where we give the first proof of the Markov property for one the simplest 3d non-gaussian models -- the $\varphi^4_3$ model on cylinders. Along the way, we establish a stronger property that is a toy version of Segal's axioms, allowing us to glue different $\varphi^4_3$ models by integrating along an appropriate boundary measure. As an application, we prove novel fundamental spectral properties of the $\varphi^4_3$ Hamiltonian.