Financial Interactions and Collective States: Part IV. Field Formalism for Dynamic Collective States and Their Transitions

In previous works (Gosselin and Lotz 2025a,b), we introduced the notion of collective states to describe capital allocation mechanisms among investors, firms, and banks across a sectoral space. A collective state is defined as a decomposition of the economy into sub-collective states---groups of sectors characterized by cross-sectoral stakes, disposable capital, and sectoral returns---which jointly determine the phase of each group. These collective states provide an intermediate-scale representation linking heterogeneous local interactions to the global organization of the economic system. These states are intrinsically unstable: small perturbations in capital or returns may propagate nonlinearly across sectors, inducing feedback effects, recompositions, and defaults. Collective states must therefore be understood as elements of a fully interconnected dynamical system. In Gosselin and Lotz (2026), we outlined the properties required of a dynamic formalism capable of capturing the multiplicity of sub-collective states, their interactions, and their transitions. We proposed a field-based framework in which a collective state is decomposed into a basis state---describing the partition of stakes among agents---and an associated fiber states---capturing a configuration of capital and returns compatible with that partition. Deviation states describe the fluctuations around collective states and are modeled as realizations of two classes of fields: basis deviation fields and fiber deviation fields. The present work develops this formalism into a full field theory of interacting deviation states. The theory explicitly describes how collective states evolve through successive transitions generated by the interactions between basis and fiber fields. For each sub-collective state, two fields encode both phase configurations and fluctuations, leading to a system composed of an infinite number of deviation fields, whose multiplicity reflects all potential group formations and transitions. The resulting framework no longer treats fluctuations and restructurations as external perturbations imposed on a stable system, but as endogenous properties of the interaction dynamics itself. We examine three complementary approaches to transitions: transition probabilities, stable-state, and operator formulations. These approaches allow sub-collective states to emerge, disappear, merge, or reorganize endogenously as a consequence of the interaction dynamics. The formalism therefore captures both continuous fluctuations and discontinuous reorganizations of the sectoral structure of the economy. We then construct an autonomous field theory for fiber deviation states, allowing multiple equilibria and internal transitions within groups. Finally, we develop the complete fibered field theory, showing that basis and fiber transitions mutually trigger one another, generating an endogenous and persistent exploration of the infinite space of collective states. The resulting system is inherently unstable, with discontinuous transitions constituting the core mechanism of its long-term evolution.