  An asymmetry exists between time and space in the sense that physical systems
inevitably evolve over time whereas there is no corresponding ubiquitous
translation over space. The asymmetry, which is presumed to be elemental, is
represented by equations of motion and conservation laws that operate
differently over time and space. If, however, the asymmetry was found to be due
to deeper causes, this conventional view of time evolution would need
reworking. Here we show, using a sum-over-paths formalism, that a violation of
time reversal (T) symmetry might be such a cause. If T symmetry is obeyed, the
formalism treats time and space symmetrically such that states of matter are
localised both in space and in time. In this case, equations of motion and
conservation laws are undefined or inapplicable. However if T symmetry is
violated, the same sum over paths formalism yields states that are localised in
space and distributed without bound over time, creating an asymmetry between
time and space. Moreover, the states satisfy an equation of motion (the
Schr\"{o}dinger equation) and conservation laws apply. This suggests that the
time-space asymmetry is not elemental as currently presumed, and that T
violation may have a deep connection with time evolution.
