Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional
Ψ[φ,t] encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.
We present a variational neural network approach for solving quantum field theories in the field basis, focusing on the free Klein-Gordon model formulated in momentum space. While recent studies have explored neural-network-based variational methods for scalar field theory in position space, a systematic benchmark of the analytically solvable Klein-Gordon ground state—particularly in the momentum-space field basis—has been lacking. In this work, we represent the ground-state wavefunctional as a neural network defined on a discretized set of field configurations and train it by minimizing the Hamiltonian expectation value. This framework enables direct comparison to exact analytic results for a range of key observables, including the ground-state energy, two-point correlators, expectation value of the field, and the structure of the learned wavefunctional itself. Our results provide quantitative diagnostics of accuracy and establish a validated foundation for extending neural-network wavefunctional methods to interacting field theories and position-space formulations.