Lie group and unitary irreducible representations: Quantization on positive real plane ℝ+2

This article discusses the application of Lie groups to analyze the symmetries of a physical system, specifically focusing on the behavior of a quantum particle on the positive real plane ℝ+2. Group theory provides a framework to explore self-adjoint operators through the identification of irreducible and unitary representations. The non-selfadjoint nature of the momentum oper-ators ^^i for positive real plane where x, y>0 has motivated the study of this system. This work extends a quantization example from the positive real line ℝ+ to pedagogically approach the connection between Lie groups and representations within the context of the quantization approach. Canonical group quantization is used to identify the group that describes the phase space for the ℝ+2 is ℝ2 ⋊ℝ+2. The operators acting on ℝ+2 are modifications of operators on the real plane ℝ2. The work proceeds to generalize the previous discussion, exploring the general case, namely GL+(2, ℝ). In conclusion, we approach ℝ+2 and GL+(2, ℝ) systems by considering the action of a Lie group on the phase space of the systems, and their representations, which contributes to a clearer understanding of the quantized description of physical systems, particularly affine quantum gravity.