Quantum mechanics with the massless scalar field and its time-reversal relation

We study the quantum mechanics with the massless scalar field in the framework of the minimal model of the classical Schr\"odinger theory. We show that the relativistic time-reversal relation is the classical Schr\"odinger relation with the massless scalar field. This relation does not depend on the existence of the scalar field or on the time-reversal relation. We also show that the relativistic time-reversal relation for the non-supersymmetric case does not depend on the presence of the scalar field. Finally, we show that the relativistic time-reversal relation for the scalar field in the classical Schr\"odinger theory does not depend on the gauge condition, the spectral index, the amount of energy or on the time-reversal relation.