Bose and Fermi Statistics and the Regularization of the Nonrelativistic Jacobian for the Scale Anomaly

TL;DR

This paper investigates the regularization of the Jacobian under scale transformations for 2D nonrelativistic fermions and bosons with contact interactions, analyzing the resulting scale anomalies and their independence from regularization methods.

Contribution

It introduces a regularization approach for the Jacobian in nonrelativistic quantum systems and compares fermionic and bosonic cases, demonstrating anomaly cancellation and regularization independence.

Findings

Fermionic Jacobian inversion is canceled by regularization.

Results are independent of the choice of regulating function.

Comparison with effective potential methods confirms robustness.

Abstract

We regulate in Euclidean space the Jacobian under scale transformations for two-dimensional nonrelativistic fermions and bosons interacting via contact interactions and compare the resulting scaling anomalies. For fermions, Grassmannian integration inverts the Jacobian: however, this effect is cancelled by the regularization procedure and a result similar to that of bosons is attained. We show the independence of the result with respect to the regulating function, and show the robustness of our methods by comparing the procedure with an effective potential method using both cutoff and $\zeta$-function regularization.