Spin foam models are an attempt at a fully covariant formulation of Loop Quantum Gravity. The subject took off when the Hamiltonian constraint of Chapter 10 was developed and one tried to use it in order to define a path integral formulation of its ‘transition amplitudes’. The field has grown quite a bit since its incarnation and it almost deserves a book of its own. We will devote relatively little space to it because we focus on the most important aspect, namely its relation with the canonical formalism and the interpretation of spin foam models. For an introduction to spin foam models we recommend the really beautiful articles by Baez [671, 672] which contain an almost complete and up-to-date guide to the literature and the historical development of the subject. See also the articles by Barrett [673, 674] for the closely related subject of state sum models and the most updated review article by Perez [675] and the thesis by Oriti [676].

What follows is a structural overview of spin foam models which focuses on mediating the main ideas and the open problems in constructing spin foam models.

Heuristic motivation from the canonical framework

The prototype of spin foam models are state sum models that had been studied extensively [677–681] within the context of topological quantum field theories [682–691] long before spin foam models arose within quantum gravity. The concrete connection of state sum models with canonical quantum gravity was made by Reisenberger and Rovelli in their seminal paper [453], where they used the (Euclidean version of the) Hamiltonian constraint described in Chapter 10 in order to write down a path integral formulation of the theory.