The mathematical formulations of the fundamental interactions in nature are all expressed in terms of fields classified by their mass and spin. The structure of two of the most successful theories in physics, General Relativity and the Standard Model of particle physics, can both be derived more or less uniquely from fields with fixed mass and spin, guided by theoretical consistency conditions. The Standard Model contains well-understood interactions of particles with spins less than two, while theories involving spins higher than two are generally problematic. Theories of spin-2 fields are much less explored, and in particular, any consistent multiplet structure, crucial for the Standard Model, is practically unknown.

Even without considering multiplet structures, classical theories with interacting spin-2 fields generically propagate additional pathological fields, so-called Boulware–Deser ghosts, rendering such theories physically unacceptable. Until recently, only a non-derivative interaction between two spin-2 fields, called bimetric theory, had been proven consistent. While the bimetric interaction can to some extent be trivially extended to more than two fields, genuine ghost-free multi-field interactions have remained elusive.

In this thesis, we derive necessary conditions on ghost-free non-derivative interactions between multiple spin-2 fields. With a suitable parametrisation of the fields, the ghost mode can be isolated, allowing one to derive necessary conditions for it to be non-propagating. These conditions essentially determine the form of the interaction potential uniquely, up to a set of free constant parameters. The admissible class contains all previously known ghost-free theories, but is not itself ghost-free in general. Imposing an additional necessary condition for ghost-freedom on these parameters further restricts the interaction to the Hassan–Schmidt–May theory and trivial extensions thereof. Thus, much as for the lower-spin theories in the Standard Model, the requirement of theoretical consistency, here the absence of ghosts, essentially singles out the admissible theory. We then show that this theory possesses the full set of constraints needed to eliminate all ghost modes, establishing it as the most general known ghost-free multi-gravity interaction. Finally, we derive the mass spectrum, finding one massless and multiple massive spin-2 fields with a non-tachyonic spectrum.