The double copy effective action: a quantum (chromodynamics) approach to space-time

Conventional Lagrangian formulations of gauge and gravity theories emphasize compactness and off-shell symmetry. This often obscures the structure of on-shell physical observables. In this work, we present a constructive framework that elevates gauge-invariant scattering amplitudes to the defining data for quantum field theory actions, including effective field theories. Focusing on double-copy theories, we promote color-dual amplitude numerators to quantum operators. This enables the systematic identification of novel local operator content at each multiplicity and the construction of double-copy-compatible actions. By applying this framework to the well-established double-copy relationship between Einstein gravity and Yang-Mills theory, which holds for all-multiplicity tree-level amplitudes, we demonstrate a systematic path to constructing the operator expansion of $\sqrt{-g}R$ from factorized gauge-theory components. This clarifies how gravitational interactions can be understood as emerging from simpler gauge-theoretic structures at the action level. This formalism extends color-kinematics duality from amplitude data to operator constructions, naturally realizing the double copy at the level of actions and asymptotic quantum states. We illustrate the method with Yang-Mills theory, Einstein gravity, and its application to generating higher-derivative operators inspired by Z-theory and open superstring amplitudes. This work provides a concrete bridge between structured amplitudes and effective actions, offering a physically grounded alternative to traditional EFT basis-building. It reveals at the operator level deep structural connections between gauge theory and gravity (connections long recognized in scattering amplitudes) from fundamental interactions to their quantum state descriptions and higher-derivative extensions.