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Background-Field Formalism in Quantum Systems

The background-field formalism is used extensively in fundamental approaches to QCD to explore hadronic matrix elements of various currents. While the lattice QCD approach is formulated in the fully-interacting Hilbert space, which includes both QCD and background field interactions, the QCD sum-rule formalism is traditionally developed in the pure-QCD Hilbert space. The latter approach encounters difficulties with excited state contaminations which are not exponentially suppressed and requires extrapolations to isolate the desired physics. Proponents of the pure-QCD Hilbert space formalism used in the QCD sum-rule approach have criticized the lattice QCD approach as neglecting important physics. In this letter, the equivalence of the two approaches is established and the flaws in the former criticisms are resolved. Finally, the application of the fully-interacting Hilbert space formalism to the study of electromagnetic polarizabilities is outlined.