A note on the BRST-BV Formalism Explanation of Deformation Quantization and Algebraic Index Theorem

In this note, we begin with considering the simple one-dimensional field theory describing a single free particle on manifold, and apply the BRST formalism and BV quantization to this model following the standard procedures in physics. Through calculations that, while not fully rigorous, have a more transparent physical interpretation, then subsequent geometric identifications, we recover the deformation quantization and algebraic index theorem on symplectic manifold. This note thus elucidates the physical origins and mechanisms of these mathematical results.