Invariance of the canonical quantization prescription under classical canonical transformations

Dirac's postulate of canonical quantization, [p̂_i, q̂_j] = ihδ_ij for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [pδ_i, q̂j ] = ihδ_ij must remain invariant under classical canonical transformations. This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if (P̂_i, Q̂_j) are related to (p̂_i, q̂_j) by a classical canonical transformation, then [p̂_i, q̂_j] = ihδ_ij implies [P̂_i, Q̂_j] = ihδ_ij. In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.