In a recent work \cite{Shaikh:2026nsi}, it was shown that for a non-relativistic particle, the operator equations of motion can be cast in the same differential form as Newton's equation of motion. This establishes a correspondence between quantum operator dynamics and classical mechanics at the level of the equations of motion. In this work, we investigate whether this correspondence extends to quantum field theory and linearized gravity. We explicitly show that the Heisenberg equations of motion for quantum field operators reproduce the corresponding classical equations of motion for a broad class of field theories. The Klein-Gordon equation, Maxwell's equations, Yang-Mills equations, and linearized Einstein equations are shown to arise at the operator level without taking the limit $\hbar\rightarrow0$ . We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.In a recent work \cite{Shaikh:2026nsi}, it was shown that for a non-relativistic particle, the operator equations of motion can be cast in the same differential form as Newton's equation of motion. This establishes a correspondence between quantum operator dynamics and classical mechanics at the level of the equations of motion. In this work, we investigate whether this correspondence extends to quantum field theory and linearized gravity. We explicitly show that the Heisenberg equations of motion for quantum field operators reproduce the corresponding classical equations of motion for a broad class of field theories. The Klein-Gordon equation, Maxwell's equations, Yang-Mills equations, and linearized Einstein equations are shown to arise at the operator level without taking the limit $\hbar\rightarrow0$. We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.

gr-qcCross-listedarXiv:2609.25360Sep 23, 2026

Relativistic Equations in Quantum Mechanics

Eugene Paul Wigner

Springer eBooksJan 1, 19732 citations

On classical and quantum relativistic dynamics

F. Reuse

Foundations of PhysicsDec 1, 197918 citations

Quantum Classical Correspondence of Dirac Equation

Pranaav Sinz

Abstract: it has been shown that unphysical results of quantum classical correspondence pertaining to Dirac equation have their roots in unjustified calculation of commutation relation involving 4 × 4 component operators acting on single component non normalized wave function. Zitterbewegung like unphysical phenomena have been completely eliminated as the treatment establishes a seamless connection between operators and observables both in relativistic and non relativistic limits. Intrinsic nature of spin angular momentum has been found questionable.Abstract: it has been shown that unphysical results of quantum classical correspondence pertaining to Dirac equation have their roots in unjustified calculation of commutation relation involving 4 × 4 component operators acting on single component non normalized wave function. Zitterbewegung like unphysical phenomena have been completely eliminated as the treatment establishes a seamless connection between operators and observables both in relativistic and non relativistic limits. Intrinsic nature of spin angular momentum has been found questionable.

Nov 1, 20110 citations

The correspondence considerations in relativistic quantum mechanics

Andrzej Herdegen

The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).

Jan 1, 19840 citations

Relativistic Invariance of Quantum-Mechanical Equations

Eugene Paul Wigner

Springer eBooksJan 1, 19972 citations

\hbar\rightarrow0

. We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.