-   A related notion of “Dirac operator” $D$, in the sense that for a symbol $a \in {\mathcal{S}}_{H,c}(M)$, $[D,a]$ is (up to some lower order terms) the differential $da$ of the symbol $a$,

-   Carrying all this to the equivariant setting.

The constructions required are more tedious than the original JLO setting, however, it is much more flexible and has many advantages. As we will see, the constructions are purely algebraic, and we do not need to appeal to analytic properties of the heat equation, so that we can deal with much more general operators that Dirac type operators. In the same vein, we do not need to consider entire $(b,B)$-cochains. Thus, what we develop here holds in cases where Getzler’s rescaling does not apply.

Bimodule of Heisenberg formal symbols {#sbimod}
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We adapt the framework developed in [@Per2012] to Heisenberg calculus on foliations. When the proof of a result is deferred to this paper, this means that it applies verbatim in our case. We shall mainly focus on the changes which occur in the Heisenberg setting.\
Let $(M, {\mathcal{F}})$ be a foliated manifold of dimension $n$, where ${\mathcal{F}}\subset TM$ is the sub-bundle of rank $v$ defining the foliation, and 