  We argue that recent neutron scattering measurements by Lake et. al. (Science
291, 1759 (2001)) of the spin excitation spectrum of LSCO in a magnetic field
can be understood in terms of proximity to a phase with co-existing
superconductivity and spin density wave order. We present a general theory for
such quantum transitions, and argue that their low energy spin fluctuations are
controlled by a singular correction from the superflow kinetic energy, acting
in the region outside the vortex cores. We propose numerous experimental tests
of our theory.
