 The time-evolution unitary operator formula_9is the time-ordered exponential of an integral formula_10 (Note that if the Hamiltonian at one time commutes with the Hamiltonian at different times, then this can be simplified to formula_11)

For perturbative quantum mechanics and quantum field theory, it is often more convenient to use the interaction picture.The interaction picture operator is

formula_12

Where formula_13.Then defining formula_14 we have

formula_15

Since the time-evolution unitary operators satisfy formula_16, the above expression can be rewritten as

formula_17

or with formula_18 replaced by any time value greater than formula_19.