time dependence", for example, due to its dependence on an applied external time-varying electric field. Another instance of explicit time dependence may occur when AS(t) is a density matrix (see below). Hamiltonian operator For the operator itself, the interaction picture and Schrödinger picture coincide: This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called without ambiguity. For the perturbation Hamiltonian , however, where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [H1,S, H0,S] = 0. It is possible to obtain the interaction picture for a time-dependent Hamiltonian H0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H0,S(t), or more explicitly with a time-ordered exponential integral. Density matrix The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let and be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability to be in the physical state |ψn⟩, then Time-evolution Time-evolution of states Transforming the Schrödinger equation into the interaction picture gives which states that in the interaction picture, a quantum state is evolved by the interaction part of the Hamiltonian as expressed in the interaction picture. A proof is given in Fetter and Walecka. Time-evolution of operators If the operator AS is time-independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for AI(t) is given by In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian . Time-evolution of the density matrix The evolution of the density matrix in the interaction picture is in consistency with the Schrödinger equation in the interaction picture. Expectation values For a general operator , the expectation value in the interaction picture is given by Using the density-matrix expression for expectation value, we will get Schwinger–Tomonaga equation The term interaction representation was invented by Schwinger. In this new mixed representation the state vector is no longer constant in general, but it is constant if there is no coupling between fields. The change of representation leads directly to the Tomonaga–Schwinger equation: Where the Hamiltonian in this case is the QED interaction Hamiltonian, but it can also be a generic interaction, and is a space like surface that is passing through