that $W({\bf{Q}}_N ,0,T_N)$ can be expressed in terms of $W({\bf{q}},{\bf{x}},T_1)$. During free-propagations we have,

$$W({\bf{q}},{\bf{x}},t)   =  W({\bf{q}},{\bf{x}} - {\bf{q}}t,0).
\label{wignerrule1}$$

With the notation $T_i^{ + / - } = T_i \pm 0^ +  $ to remove ambiguities due to delta functions, transformation of $W({\bf{q}},{\bf{x}},t)$ due to a standing wave pulse is written as:

$$W({\bf{q}},{\bf{x}},{T_i^ +  })  = \sum\limits_n {J_n (2\theta _i
\sin \frac{{{\bf{Q}}_i  \cdot {\bf{x}}}}{2}) W({\bf{q}} -
n{\bf{Q}}_i ,{\bf{x}},{T_i^ -  })}.  \label{wignerrule2}$$

### Basic displacement diagrams

![Basic displacement diagrams: a) A propagating displacement line \[Eq. (\[wignerrule1\])\]. b) A scattering vertex with an incoming and outgoing displacement line, the scattering amplitude is evaluated at the vertex position \[Eq. (\[wignerrule2\])\]. The dashed arrow represents the action of a standing wave diffraction. c) A probing vertex that stops an input displacement line \[Eq. (\[backamp2\])\]. The echo amplitude can be evaluated at the crossing between the incoming displacement line and the x-axis. The solid arrow represents the Bragg-scattering (grating echo) process. The direction of the dashed line array indicates the slope of the displacement line ${\bf x}(t)={\bf Q}t$.[]{data-label="proprulefig"}](proprules.eps "fig:"){width="5"}\

We shall consider Eqs. (\[wignerrule1\]) and (\[wignerrule2\]) in a space $({\bf q,x})$ that may be called the reciprocal phase space. From Eq. (\[wignerrule1\]), free propagation 