Heisenberg picture

In the Heisenberg picture, states are fixed and an observable evolves asFor a time-independent Schrödinger-picture observable,

For an operator $O_H=U^{†}{O}_{S}U$ in the Heisenberg picture, differentiating the evolution operators gives $d{O}_H/dt=i[H_H,O_H]+U^{†}(\partial O_S/\partial t)U$ in units $\hslash =1$. With no explicit operator time dependence, the second term vanishes. Applying this commutator identity to the canonical momentum and real scalar field gives their operator field equations.