We report a rigorous quantization of topological quantum mechanics on \(\mathbb {R}_{\geqslant 0}\) and \(\textbf{I}= [0, 1]\) in the perturbative BV-BFV formalism. Costello’s homotopic renormalization is extended and incorporated in our construction. As a consequence, we obtain an algebraic characterization of the solutions to the modified quantum master equation. In addition, BV quantization of the same model studied in previous work (Wang and Yan 2022) is derived from the BV-BFV quantization, leading to a comparison between two different frameworks in the study of quantum field theories on manifolds with boundaries.
