Revisiting quantum mechanics on non-commutative space-time

We construct an effective commutative Schroolinger equation in Moyal space-time in (1 + 1)-dimension where both t and x are operator-valued and satisfy [(t) over cap, (x) over cap] = i theta. Beginning with a timereparametrised invariant form of an action we identify the actions of various space-time coordinates and their conjugate momenta on quantum states, represented by Hilbert-Schmidt operators. Since time is also regarded as a configuration space variable, we show how an 'induced' inner product can be extracted, so that an appropriate probabilistic interpretation is obtained. We then discuss several other applications of the formalism developed so far. (C) 2017 Elsevier Inc. All rights reserved.