Non Relativistic Limit in QFT: a Wilson Renormalization Group Approach in the Minkowski Space
We study the non relativistic limit of a model of fermions interacting through a Chern-Simons field, from a perspective that resembles the Wilson Renormalization Group (WRG) approach. The main difference of our approach to WRG is that instead of working in the Euclidean space, we directly work in Minkowski space. For the nonrelativistic limit this procedure admits a more direct interpretation of the results. As we will see, this modified WRG approach solves some difficulties, and gives a new understanding of non relativistic models that are not clear in the usual "classical" nonrelativistic (CNR) limit found in most text books. The method is general and our choice of the Chern-Simons as the gauge field is only for the sake of simplicity of illustration.
Nonrelativistic limit of the scalar Chern-Simons theory and the Aharonov-Bohm scattering
We study the nonrelativistic limit of the quantum theory of a Chern–Simons field minimally coupled to a scalar field with quartic self-interaction. The renormalization of the relativistic model, in the Coulomb gauge, is discussed. We employ a procedure to calculate scattering amplitudes for low momenta that generates their |p|/m expansion and separates the contributions coming from high and low energy intermediary states. The two-body scattering amplitude is calculated up to order p2/m2. It is shown that the existence of a critical value of the self-interaction parameter for which the two-particle scattering amplitude reduces to the Aharonov–Bohm one is a strictly nonrelativistic feature. The subdominant terms correspond to relativistic corrections to the Aharonov–Bohm scattering. A nonrelativistic reduction scheme and an effective nonrelativistic Lagrangian to account for the relativistic corrections are proposed.
Quantum description of anyons: Role of contact terms
We make an all-order analysis to establish the precise correspondence between nonrelativistic Chern-Simons quantum field theory and an appropriate first-quantized description. The physical role of the field-theoretic contact term in the context of renormalized perturbation theory is clarified through their connection to the self-adjoint extension of the Hamiltonian in the first-quantized approach. Our analysis provides a firm theoretical foundation on quantum field theories of nonrelativistic anyons. {copyright} {ital 1997} {ital The American Physical Society}