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## 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

[itex]

\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}>

[/itex]

I need to demostrate

[itex]

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)

[/itex]

with

[itex]

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

[/itex]

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

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