  We derive exact physical consequences of particle-hole symmetry of the
$\nu=1/2$ state of electrons in a strong magnetic field. We show that if the
symmetry is not spontaneously broken, the Hall conductivity and the
susceptibility satisfy an exact relationship, valid at any wave numbers and any
frequencies much below the cyclotron frequency. The relationship holds for
clean systems and also for systems with statistically particle-hole symmetric
disorder. We work out the constraints this relationship imposes on the theory
of the Dirac composite fermion. We also argue that that the exact relationship
is violated in the Halperin-Lee-Read (HLR) field theory and present an explicit
calculation within a Galilean invariant mean-field approximation to the HLR
theory to illustrate the breakdown.
