  Quantum fields are well known to violate the weak energy condition of general
relativity: the renormalised energy density at any given point is unbounded
from below as a function of the quantum state. By contrast, for the scalar and
electromagnetic fields it has been shown that weighted averages of the energy
density along timelike curves satisfy `quantum weak energy inequalities'
(QWEIs) which constitute lower bounds on these quantities. Previously, Dirac
QWEIs have been obtained only for massless fields in two-dimensional
spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields
of mass $m\ge 0$ on general four-dimensional globally hyperbolic spacetimes,
averaging along arbitrary smooth timelike curves with respect to any of a large
class of smooth compactly supported positive weights. Our proof makes essential
use of the microlocal characterisation of the class of Hadamard states, for
which the energy density may be defined by point-splitting.
