Killing Tensor Quantum Numbers and Conserved Currents in Curved Space

The relationship between relativistic quantum current conservation laws in a curved-space background and the corresponding "good quantum numbers," i.e., operators that commute with the fundamental wave operator in a first-quantized field theory, is considered. It is shown that under favorable circumstances (such as vanishing Ricci curvature) the existence of such an operator for scalar fields is automatically implied by the existence of the corresponding constant for particle trajectories in the classical limit, that is to say, by the existence of a Killing vector or a "Killing tensor" in the first- and second-order cases, respectively. Thus the fourth constant of the motion for a scalar quantum field in the Kerr metric background arises automatically from the Killing tensor defining the fourth constant of the classical motion. Another application is to the Runge-Lenz constants in the nonrelativistic hydrogen atom problem. The "Schiff conjecture" concerning the relationship between classical mechanics and first-quantized field theory in connection with the equivalence principle is discussed in passing.

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