Infrared renormalization in non-relativistic QED and scaling criticality

Abstract

We consider a spin- 1 2 electron in the framework of non-relativistic Quantum Electrodynamics (QED). Let H ( p → , σ ) denote the fiber Hamiltonian corresponding to the conserved total momentum p → ∈ R 3 of the electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by 0 < σ ≪ 1 . Ultimately, our goal is to remove the latter by taking σ ↘ 0 . We prove that there exists a constant 0 < α 0 ≪ 1 independent of σ > 0 such that for all | p → | < 1 3 and all values of the finestructure constant 0 < α < α 0 , there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies 1 < m ren ( p → , σ ) < 1 + c α , uniformly in σ ⩾ 0 , in units where the bare mass has the value 1, and we prove the existence of the renormalized mass in the limit σ ↘ 0 . Our analysis uses the isospectral renormalization group method of Bach, Fröhlich, Sigal introduced in [V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. Math. 137 (2) (1998) 299–395; V. Bach, J. Fröhlich, I.M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (1998) 205–298] and further developed in [V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal. 203 (1) (2003) 44–92; V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, The renormalized electron mass in non-relativistic QED, J. Funct. Anal. 243 (2) (2007) 426–535]. The limit σ ↘ 0 determines a scaling-critical (or endpoint type) renormalization group problem, in which the interaction is strictly marginal (of scale-independent size). A main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non-trivial scaling limit. The key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model, and a combination of the isospectral renormalization group method with the strong induction principle.