  In this brief, the spontaneous symmetry breaking (SSB) of the $\varphi^4$
theory in phase space, is studied. This phase space results from the
appropriate system of Poincare maps, produced in both the Minkowski and the
Euclidean time. The importance of discretization in the creation of phase
space, is highlighted. A series of interesting, novel, unknown behaviors are
reported for the first time; among them the most characteristic is the change
in stability. In specific, the stable fixed points of the $\varphi^4$ potential
appear as unstable ones, in phase space. Additionally, in the Euclidean-time
phase space a unique instability in the position of the critical point, can be
created. This instability is further proposed to host tachyonic field in
Euclidean space.
