Quantum asymmetry between time and space

Abstract: 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.