Time evolution and Feynman Integrals

[clasicoun]

Homework Statement

With the time evolution time operator, where there is time dependent hamiltonian, show the new form of the feynman propagator between two states. Consider the Weyl Integral.

2. Equations
from

$\newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \newcommand{\braket}[2]{{<\!\!{#1|#2}\!\!>}} \newcommand{\braketop}[3]{{<\!\!{#1|\hat{#2}|#3}\!\!>}} \braket{q_{f};t_{f}}{q_{0};t_{0}} = <q_{f};t_{f} |T \left[ e^\left(- \frac{i}{ \hbar} \int ^{t_{f}}_{t_{0}} H(t)dt\right) \right] |q_{0};t_{0}>$

I need to demostrate
$K(q_{f};t_{f};q_{0};t_{0})= \int\int D[q(t)] D\left [\frac{p(t)}{2\pi\hbar}\right] e^\left(- \frac{i}{ \hbar} I[q(t), p(t)]dt\right)$

with

$I= \int ^{t_{f}}_{t_{0}} \left(pq^{.} - H_{w}(q,p:t) \right)dt$

Hw is Weyl tranform of Hamiltonian.

The Attempt at a Solution

I could not put equations in right way. In the equation for this exercise, I begin with the amplitude transition between two states in Heisenberg Picture. then I express it with Schrodinger picture using time evolution operator. In the operator I have time integral of Hamiltonian, and I have a Time ordering operator. the things is I need to express the exponential like a product of n exponentials and then introducing completeness relation, this is a standard way. I mean I want to have a partition in the time but I do not know how I can do it with the integral. I thought in express time exponential like a sum, but if I do it then I will not have product exponential.

Other way is using composition properties for evolution time operator, but I am not sure if I can do it, The time ordering Operator is a problem for me. Is possible that I divide the integral in many integral, then I have a exponential with many integrals. But I do not know if I need to express it via Campbell Baker Hausdorf Identity.

I appreciate any path for this path integral

 

[dextercioby]

Please, rewrite your post and format the LaTex code correctly.