content_image/331663.jpg content_image/331664.jpg content_image/331665.jpg content_image/331662.jpg content_image/331661.jpg content_image/331660.jpg content_image/331659.jpg content_image/331658.jpg content_image/331657.jpg content_image/331656.jpg content_image/331655.jpg
ouville operators as follows: 

<img src='content_image/331663.jpg'>

<img src='content_image/331664.jpg'>

<img src='content_image/331665.jpg'>

where 

In L$_{A}$ there appears an operator with the form 

<img src='content_image/331662.jpg'>

<img src='content_image/331661.jpg'>

<img src='content_image/331660.jpg'>

<img src='content_image/331659.jpg'>

<img src='content_image/331658.jpg'>

where ( k, i ) = ( χ, ξ ) for L$_{A}$ . The action of the propagator associated with this operator on p$_{i}$ is derived in Appendix A, and is given by 

<img src='content_image/331657.jpg'>

The apparently singular function 

<img src='content_image/331656.jpg'>

is in fact well behaved as p$_{k}$ → 0, and can be expanded in a Maclaurin series to suitably high order [37]. In our implementation we used an eighth order expansion. 

The propagators for the BKNH dynamics can now be defined as 

<img src='content_image/331655.jpg'>

