  The complete lack of theoretical understanding of the quantum critical states
found in the heavy fermion metals and the normal states of the high-T$_c$
superconductors is routed in deep fundamental problem of condensed matter
physics: the infamous minus signs associated with Fermi-Dirac statistics render
the path integral non-probabilistic and do not allow to establish a connection
with critical phenomena in classical systems. Using Ceperley's constrained
path-integral formalism we demonstrate that the workings of scale invariance
and Fermi-Dirac statistics can be reconciled. The latter is self-consistently
translated into a geometrical constraint structure. We prove that this "nodal
hypersurface" encodes the scales of the Fermi liquid and turns fractal when the
system becomes quantum critical. To illustrate this we calculate nodal surfaces
and electron momentum distributions of Feynman backflow wave functions and
indeed find that with increasing backflow strength the quasiparticle mass
gradually increases, to diverge when the nodal structure becomes fractal. Such
a collapse of a Fermi liquid at a critical point has been observed in the
heavy-fermion intermetallics in a spectacular fashion.
