In the interaction picture, a HamiltonianH, can be split into a free part H0 and an interacting partVS(t) as H = H0 + VS(t).
The potential in the interacting picture is
where is time-independent and is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, stands for in what follows.
In the interaction picture, the evolution operator U is defined by the equation:
This is sometimes called the Dyson operator.
The evolution operator forms a unitary group with respect to the time parameter. It has the group properties:
and from these is possible to derive the time evolution equation of the propagator:[4]
In the interaction picture, the Hamiltonian is the same as the interaction potential and thus the equation can also be written in the interaction picture as
An iterative solution of the Volterra equation above leads to the following Neumann series:
Here, , and so the fields are time-ordered. It is useful to introduce an operator , called the time-ordering operator, and to define
The limits of the integration can be simplified. In general, given some symmetric function one may define the integrals
and
The region of integration of the second integral can be broken in sub-regions, defined by . Due to the symmetry of , the integral in each of these sub-regions is the same and equal to by definition. It follows that
Applied to the previous identity, this gives
Summing up all the terms, the Dyson series is obtained. It is a simplified version of the Neumann series above and which includes the time ordered products; it is the path-ordered exponential:[5]
This result is also called Dyson's formula.[6] The group laws can be derived from this formula.
Application on state vectors
The state vector at time can be expressed in terms of the state vector at time , for as
The inner product of an initial state at with a final state at in the Schrödinger picture, for is:
The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:[7]
Note that the time ordering was reversed in the scalar product.
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