In this paper, we propose a new method for constructing approximate value functions for nonlinear optimal regulator problems with infinite time horizons. We consider a canonical quantization, which is an operation to change classical mass systems to quantum mass systems. First, we apply the canonical quantization to mass systems equivalent to the optimal regulator systems. Second, we construct approximate optimal regulators by using the canonical quantized mass systems. Third, we clarify the conditions under which approximate solutions converge to the strict solutions. In addition, we demonstrate some examples with one-dimensional systems.