Galilean Covariance Almost Fixes the Quantum Probability Current

The continuity equation fixes only the divergence of the quantum probability current, leaving the flow of probability underdetermined. We show that Galilean boost covariance, imposed pointwise at fixed time, nearly resolves this ambiguity. For any one-body current, local or nonlocal, pure, mixed, or many-body, the boost law holds exactly when the current equals the Schrödinger current plus a linear functional of the density. Equivalently, the current is covariant under local phase changes, or its density and momentum density satisfy the canonical commutator of the nonrelativistic current algebra. Among translation-invariant currents, rotational covariance together with conservation removes the remaining density term in three or more dimensions, while conservation together with the momentum–translation relation removes it in every dimension. The same framework classifies known alternatives: Nelson and Doebner–Goldin currents obey the algebra but are not conserved; chiral currents are conserved but violate the algebra; convolution kernels violate the boost law; and a conserved boost-covariant current exists exactly when the \(f\)-sum rule holds.