Euclidean Quantum Field Theory in One Axiom

For a given dimension, there is a well-known monoidal category of compact manifolds with Riemannian metric and boundary (possibly empty). The defining operation is the disjoint union and the empty set the neutral object. The category is self-adjoint in the sense that boundary components can be moved between source and target. An equally well-known monoidal category is given by the linear maps of real Hilbert spaces, with tensor products as defining operation and the real numbers as neutral object. This category is self-dual with respect to the move of tensor factors between source and targets. A unitary euclidean quantum field theory is a functor from the former to the latter category. Everything should be smooth and continuous. To obtain non-unitary theories, one must generalize the Hilbert spaces to self-dual Banach spaces. Examples will be given and it will be explained how theories like the standard model fit in this framework.