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Abstract

We discuss three instances where zero-energy or soft modes appear in quantum field theory. First, we examine massless fermions in a 2+1 dimensional system with a spatial boundary, specifically graphene in half-space. Two boundary conditions and their interplay with the discrete and continuous symmetries of the system are analyzed. For doubled fermions, we identify a special case that respects CPT symmetry but breaks Lorentz and conformal symmetry, featuring fermion zero mode edge states. These edge states lead to unconventional representations of scale, phase, and translation symmetries, and enforcing symmetry constraints results in edge ferromagnetism. Second, we investigate the infrared structure of a massless scalar theory coupled to fermions. We demonstrate the existence of a field theory containing massless scalar particles that mirrors the infrared structure of quantum electrodynamics and perturbative quantum gravity but lacks gauge invariance, internal symmetries, or apparent asymptotic symmetry. Unlike soft photons and gravitons, soft scalars do not decouple from dressed states and are generally produced during interactions of hard dressed particles, though their entanglement is minimal. Lastly, we develop a novel method to calculate changes in an operator’s expectation value at asymptotic times, relevant to gravitational wave observations, by exploiting its soft limit. We derive a formula for asymptotic in-in observables from the soft limit of five-point amputated response functions. Using this, we re-derive the KMOC formulas for linear impulse and radiated momentum during scattering and provide an unambiguous calculation of radiated angular momentum at leading order. We introduce a causal method of computing classical observables using the Schwinger-Keldysh formalism.