Current Commutators and Electron Scattering at High Momentum Transfer

Sum rules for electron proton total cross sections are deduced from the vanishing of various equal time commutators; it is shown how these cross sections determine the (spin-averaged) proton expectation value of all equal time commutators of components of the electric current and time derivatives thereof. (to be submitted to Phys. Rev. Letters) * Work supported by the U. S. Atomic Energy Commission and in part by the National Science Foundation t Alfred I?. Sloan Foundation Fellow t‘tPermanent Address: Department of Physics, University of California, Los Angeles, California Electron scattering on a proton at rest is described by the cross section (electron mass = 0) d2ce13 Lr2 1 zz-dq2d v M2E2 q2 2M2EE’ 2 5 M2 ) Al&I24; + y2Al(s2,u) -q2A2(q2,v) 1 , where a= e2/4n = (137):: E(E’) = initial (final) electron energy qP=e P eb = momentum imparted to proton v = (E E’)M = -q p (p = proton momentum), and the hi are the absorbtive parts of the forward, off-shell Compton amplitudes Fi defined by TCLv (9, p) = i Jf d4x e-%’ X c ( w + Y(qpPv + q, PJ .Y2 6 1 pu F1 + (4,&, s2s,,)F2 9 cw where j ct is the electric current, and a spin average is implicit. The subscript Vf indicates the covariant time ordered product; thus T differs from the PU ordinary time ordered product by a polynomial in q if the equal-time commutator of j, and ji has a connected matrix element. Bjorlren’ has pointed out that for large q. and fixed q the coefficient of 90 -I-’ (1 20) in an expansion of TClv gives the matrix element of the equal-time