Quantum asymmetry between time and space
Joan A. Vaccaro
Published 20 January 2016.DOI: 10.1098/rspa.2015.0670
Sketches illustrate the translation of wave functions along (a) the x-axis and (b) the time axis. In (a), the wave functions represent the position eigenket |x〉 and an arbitrary state |χ〉 and the translation is by a distance δx. In (b), the wave function represents the state |f〉 and the translation is by an interval t.
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, then the formalism treats time and space symmetrically such that states of matter are localized 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, then the same sum over paths formalism yields states that are localized in space and distributed without bound over time, creating an asymmetry between time and space. Moreover, the states satisfy an equation of motion (the Schrödinger 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.
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