Lowest-order pion-graviton scattering is calculated using the canonical quantization of general relativity. The process is examined in order to investigate if the canonical quantization produces Lorentz-invariant and gauge-invariant $S$-matrix elements, and also if the $S$ matrix is invariant under a specific class of $q$-number coordinate transformations. While the classical theory automatically possesses the above invariances, they are seen not to be a priori guaranteed in the quantum theory. The $S$-matrix elements calculated, however, correctly possess the above invariances.

• Received 20 August 1970

DOI:https://doi.org/10.1103/PhysRevD.3.1283