Reconstruction of Quantum Field Equations from Multi-Branch Hamilton–Jacobi Structures

Abstract

We develop a framework in which quantum field equations are reconstructed from a multi-branch Hamilton–Jacobi structure supplemented by density functionals on configuration space. Each branch is defined by a solution of a functional Hamilton–Jacobi equation, while the associated density encodes measure-theoretic and dynamical information. Under suitable analytical assumptions, we show that the superposition of these branchwise contributions yields a wave functional satisfying the functional Schrödinger equation. The construction is established in a mathematically controlled setting, including finite-dimensional regularizations and a continuum limit under appropriate convergence hypotheses. Explicit examples demonstrate the exact reconstruction of both free and interacting scalar quantum field equations. Extensions to fermionic and gauge fields are outlined, highlighting the role of graded structures and constraints. Beyond the reconstruction result, the framework provides a structural reinterpretation of quantum theory. Classical dynamics is encoded in the space of Hamilton–Jacobi solutions, while quantum phenomena arise from the coherent superposition of these classical configurations through density and phase. From a geometric viewpoint, the construction is naturally related to multisymplectic structures and admits a cohomological interpretation. This approach establishes a direct bridge between classical variational principles and quantum field theory, offering a unified perspective on the emergence of quantum behavior from structured ensembles of classical configurations.