  In the topological phase of p-wave superconductors, zero-energy Majorana
quasi-particle excitations can be well-defined in the presence of local
density-density interactions. Here we examine this phenomenon from the
perspective of matrix representations of the commutator $\mathcal{H}
=[H,\bullet]$ ,with the aim of characterising the multi-particle content of the
many-body Majorana mode. To do this we show that, for quadratic fermionic
systems, $\mathcal{H}$ can always be decomposed into sub-blocks that act as
multi-particle generalisations of the BdG/Majorana forms that encode
single-particle excitations. In this picture, density-density like interactions
will break this exact excitation-number symmetry, coupling different sub-blocks
and lifting degeneracies so that the eigen-operators of the commutator
$\mathcal{H}$ take the form of individual eigenstate transitions $|n\rangle
\langle m|$. However, the Majorana mode is special in that zero-energy
transitions are not destroyed by local interactions and it becomes possible to
define many-body Majoranas as the odd-parity zero-energy solutions of
$\mathcal{H}$ that minimise their excitation number. This idea forms the basis
for an algorithm which is used to characterise the multi-particle excitation
content of the Majorana zero modes of the one-dimensional p-wave lattice model.
We find that the multi-particle content of the Majorana zero-mode operators is
significant even at modest interaction strengths. This has important
consequences for the stability of Majorana based qubits when they are coupled
to a heat bath. We will also discuss how these findings differ from previous
work regarding the structure of the many-body-Majorana operators and point out
that this should affect how certain experimental features are interpreted.
