Convergent Renormalization Group Flow of Spectral Problems in Quantum Field Theory

The renormalization group (RG) flow defined by the "smooth Feshbach map" is an effective method in spectral and singular perturbation theory. In 2003, a novel operator-theoretic renormalization group method was introduced and applied to analyze a general class of Hamiltonians on Fock space by Bach, Chen, Fröhlich, and Sigal, [1]. This served as a demonstration of the potency of the smooth Feshbach map to possess a codimension-1 contractivity property. The codimension-1 contractivity ensures that the iterative application of renormalization group map generates a (time-discrete) dynamical system on small ball in a suitable Banach space of operators with a fixed point manifold of dimension one. As a fundamental aspect of the thesis, we require the coefficients to possess higher regularity compared to the assumptions in [1]. We improve the codimension-1 contractivity scheme to achieve full contractivity without any marginal directions. This allows to characterize the properties of the fixed point precisely. To this end, in Section 1.2.1, first we review the smooth Feshbach map. In Section 1.2.2, we complement the abstract framework with a concrete example, the spin-boson model, which serves to analyze spectral properties. This model involves the dynamics of a two-level atom coupled to a quantized boson field. Later in Chapter 2, we construct a Banach space of Hamiltonians and with the use of the smooth Feshbach map and the Pull-Through Formula, we express the Hamiltonian in a certain normal form and estimate its coefficients, which are called coupling functions. As a first application of the smooth Feshbach map (SFM), we eliminate the high-energy degrees of freedom of our spectral problem, Section 2.3. Then we systematize the normal-ordering. Our goal is to study Hamiltonians that drive from continuously differentiable functions on C^κ (I) with κ ⩾ 2 and obeys the L^2 -norm bound, as discussed in Chapter 3. The shift from codimension-1 contractivity to full contractivity signifies a progression toward a more comprehensive and broadly applicable understanding of contraction properties in the specified settings. This establishment allows for a deeper and more precise characterization of fixed points within the RG flow. Finally, in Chapter 4, we present the main result, which establishes the contraction property established using the smooth Feshbach map. This result leads to the formulation of a new renormalization group map that operates directly on the space of Hamiltonians. In our main theorem, Theorem 4.2.1, we present a concrete computation of certain terms that contribute to the image of Hamiltonians under the renormalization group map. The concluding section of the chapter is dedicated to the construction of the renormalization map and its contraction property.