Classical Limit of Yukawa theory from quantum state perspective

Abstract

We derive the quantum states corresponding to classical scalar fields in the representation expanded by the eigenstates of quantum field operators. This allows us to directly observe the spatial entanglement structure of quantum states and explore the differences and relationships between quantum superposition and classical superposition. We find that if two classical fields are identical in a certain spatial region, then their corresponding quantum states have the same reduced density matrix in that region. This indicates that knowing the classical field in a local region is sufficient to derive the reduced density matrix for that region. According to the correspondence between classical quantities and quantum states, we derive the equation of motion of the classical theory from the evolution of quantum states in Yukawa theory. This leads to the relativistic classical Yukawa theory, and we further obtain the relativistic corrections to the Yukawa potential.