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.
Impact of relativity on particle localizability and ground state entanglement
Can a relativistic quantum field theory be consistently described as a theory of localizable particles? There are many known issues with such a description, indicating an answer in the negative. In this paper, we examine these obstructions by tracing how they (partially) subside in passing to an approximation of ordinary quantum mechanics in the non-relativistic regime. We undertake a recovery of the characteristic features of non-relativistic quantum mechanics beyond simply the Schrödinger equation. We find that once this is achieved, there are persisting issues in the localizability of particle states. A major focus is on the lingering discrepancy between two different localization schemes in quantum field theory. The non-relativistic approximation of the quantum field theory is achieved by introducing an ultraviolet cutoff set by the Compton scale. The other main undertaking of this paper is to quantify the fate of ground state entanglement and the Unruh effect in the non-relativistic regime. Observing that the Unruh temperature vanishes in the naive limit as the speed of light is taken to infinity motivates the question: is the Unruh effect relativistic? It happens that this is closely related to the former issues, as ground state entanglement causes obstructions to localizability through the Reeh–Schlieder theorem.
Gauge noninvariance and parity nonconservation of three-dimensional fermions
The effective gauge field action due to an odd number of fermion species in three-dimensional $\mathrm{SU}(N)$ gauge theories is shown to change by $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}|n|$ under a homotopically nontrivial gauge transformation with winding number $n$. Gauge invariance can be restored by use of Pauli-Villars regularization, which, however, introduces parity nonconservation in the form of a parity-nonconserving, topological term in the effective action.