algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
A central result in the construction of perturbative quantum field theories via the method of causal perturbation theory is called the main theorem of perturbative renormalization theory. This says that any two renormalization schemes, hence any two solutions to the inductive construction of the S-matrix $V \mapsto S(V)$ for interaction terms $V$, are related by a unique transformation $Z \colon V \mapsto V'$
The collection of these operations $Z$ forms a group called the Stückelberg-Peterson renormalization group. Hence the space of renormalization schemes is a torsor over this group.
This theorem is a mathematical reflection of the idea that renormalization is about regarding a perturbative quantum field theory with interaction $V$ as a effective field theory at some energy scale and then discovering that at higher energy there is a more fundamental interaction $Z(V)$ which effectively looks like $V$ at lower energy.
The theorem is originally due to
In various variants it has been (re-)proved in the following articles:
G. Pinter, section 6 of The Action Principle in Epstein Glaser Renormalization and Renormalization of the S-Matrix in $\phi^4$-Theory, Annalen Phys. 10 (2001) 333-363 (arXiv:hep-th/9911063)
Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, section 4.1 of Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics (arXiv:0901.2038)
Michael Dütsch, Klaus Fredenhagen, section 4.2 of Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity, Rev. Math. Phys. 16, 1291 (2004) (arXiv:hep-th/0403213)
Stefan Hollands, theorem 2 in Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20:1033-1172, 2008 (arXiv:0705.3340)
Review is in