Reconstructing configurational Hamiltonians from Fock space

We present a unified operator-based framework for reconstructing configuration-space Hamiltonians directly from their Fock-space formulations. This approach clarifies how local dynamics and geometric structure emerge from global quantum operator algebras. Beginning with elementary systems such as the harmonic oscillator, we extend the method to scalar and vector fields in both flat and curved spacetimes. By systematically inverting creation and annihilation operators, we recover local expressions involving canonical fields and their conjugate momenta. Applications include free and interacting scalar fields, Abelian and non-Abelian gauge theories, and massless spin-2 graviton modes. We further explore the Sachdev-Ye-Kitaev (SYK) and double-scaled SYK models, showing how bilocal collective fields and emergent infrared geometries arise from quartic Fock-space interactions. This reconstruction framework reveals a deep structural unity across quantum mechanics, field theory, gauge dynamics, and holography, and opens new avenues for studying locality, emergent geometry, and duality in quantum systems.