B.4 The Evolution Operator

satisfies the initial condition . For finite times can be constructed explicitly by employing the SCHRÖDINGER picture

which therefore identifies

Since and do not commute with each other, the order of the operators must be carefully maintained. Equation (B.17) immediately yields several general properties of [189]

- , implying that is unitary ,
- , which shows that has the group property, and
- , implying that .

Integrating both sides of the (B.18) with respect to time with the initial condition yields

By iterating this equation repeatedly one gets

Equation (B.20) has the characteristic feature that the operator containing the latest time stands farthest to the left. At this point it is convenient to introduce the

where is the step function

(B.22) |

The second term on the right hand-side is equal to the first, which is easy to see by just redefining the integration variables , . Thus one gets

(B.23) |

Thus for the expansion of the one obtains