Vanishing Noether Current in Weyl Invariant Gravities

Abstract: We revisit the issue that the Noether current associated with a local scale symmtery, or equivalently the Weyl symmetry, identically vanishes. Based on only the second Noether theorem for a local symmetry, we prove that the Noether current associated with the Weyl symmetry is in general vanishing in any Weyl invariant gravitational theories in four dimensional Riemannian geometry. We also clarify the reason: The Weyl transformation is non-dynamical in the sense that it does not contain the derivative term of the transformation parameter as opposed to the conventional gauge transformation. Finally, we apply this result to a quantum theory of a general Weyl invariant gravity and derive currents associated with choral symmetry, which is the $IOSp(10|10)$ supersymmetry.