I. INTRODUCTION 2 1 For isovector form factors, the restriction to quantized momentum transfer can be lifted by imposing isospin twisted boundary conditions on the quark fields [3, 4]. The only known method to handle the isoscalar contribution is to increase the lattice volume. In the current context, we focus our attention on properties of spin-half baryons in external electric fields. Neutral hadrons in electric fields have been investigated with lattice QCD using the quenched approximation at pion masses greater than 500 MeV [8, 9]. A recent calculation has explored neutron properties at lower pion masses with electrically neutral sea quarks [10]. There has also been a fully dynamical calculation for the neutron in an electric field using a pion mass of 760 MeV [11]. In this work, we treat the baryon spin in a relativistic manner, and thereby demonstrate how to determine both nucleon magnetic moments and electric polarizabilities using lattice QCD in background electric fields. In essence, our method allows one to perform the subtraction of the nucleon pole term aris- Computation of hadronic electromagnetic properties using lattice QCD can be accom- plished in at least two different ways. The current insertion method, see e.g. [2], can be used to determine hadronic matrix elements of the electromagnetic current. This method is ideal for the computation of electromagnetic form factors, but is limited in the extraction of multipole moments due to the available lattice momentum, which, for periodic boundary conditions, is quantized in units of 2 π/L , where L is the size of the lattice. 1 For current lat- tice sizes, the determination of multipole moments relies on a long extrapolation to vanishing momentum transfer. For multipole polarizabilities, the temporal extent of current lattices also makes direct computation of the Compton scattering tensor infeasible. Were lattices long enough to allow the computation of matrix elements with two current insertions, the extraction of polarizabilities would still require a long extrapolation to zero momentum. Al- ternately, hadronic properties can be determined using the background field method [5–7]. With this method, one determines lattice two-point functions in the presence of classical external fields. Observables are then determined from the variation of these hadronic corre- lators with the strength of the external field. Understanding low-energy properties of hadrons directly from QCD remains a challeng- ing endeavor. In this low-energy regime, quark and gluon interactions must be treated non- perturbatively, ultimately resulting in their confinement into hadrons. After three decades of dedicated work, lattice QCD has evolved into a tool to address quantitatively the non- perturbative dynamics underlying hadrons and their interactions, see [1] for an overview. Electromagnetic moments and multipole polarizabilities are low-energy properties of hadrons with transparent physical meaning. These properties characterize the distribution of charge and magnetism within a hadron, and the response of the charge and magnetism distributions to external fields, respectively. Low-energy properties of hadrons can be described using an effective theory of QCD, based upon treating pseudoscalar mesons as the Goldstone modes arising from spontaneous chiral symmetry breaking. A picture of hadrons emerges from chi- ral dynamics: that of a hadronic core surrounded by a pseudoscalar meson cloud. In part, the electromagnetic properties of hadrons encode the distribution of charged mesons, and the stiffness of the charged meson cloud. Chiral dynamics consequently makes predictions for the form of electromagnetic observables. Confirming these predictions both experimen- tally and from the lattice will be a milestone in our understanding of non-perturbative QCD dynamics.