Dyson series ($U_I(t,t_0)$)

The Dyson series solves interaction picture evolution by keeping noncommuting Hamiltonian operators in chronological order. With $\hslash =1$, integrating $i{\partial }_t{U}_I=H_I(t)U_I$ and iterating gives

Time ordering puts later times on the left. Integrating over all time labels instead of the ordered region introduces $1/n!$. The formal scattering operator limit isAn ordinary matrix exponential suffices when all the $H_I(t)$ commute. For quantum fields, the displayed expansion requires a prescription for singular products and need not converge. ^[ref]

For bounded norm-continuous Hamiltonian operators on $[t_0,t]$, the Dyson series has an $n$-th term bounded by $M^n|t-t_0{|}^n/n!$, so it converges in operator norm. In its nested-integral form, differentiating the outer endpoint puts $H_I(t)$ on the left of the $(n-1)$-fold term. Summing yields ${\partial }_{t}U=-i{H}_I(t)U$ and $U(t_0,t_0)=I$. If $H_I$ is Hermitian, differentiating $U^{†}U$ proves unitarity. For unbounded field operators the same algebra is a formal perturbative identity unless domain and convergence assumptions or a regulator justify the steps. Evolution backward uses inverse, anti-time-ordered evolution.