Trace anomaly for non-relativistic fermions

Abstract

Abstract:We study the coupling of a 2 + 1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3 + 1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle.

Nonrelativistic trace and diffeomorphism anomalies in particle number background

“…This agrees with the results of [19]. Analogous calculations [17,18] performed in curved backgrounds without A µ lead to a result proportional to the trace anomaly of a relativistic scalar/fermion in 3 + 1 dimensions. Instead, the anomaly in A µ background is genuinely nonrelativistic, as it has no counterpart in the relativistic case.…”

“…For time independent backgrounds, the calculation of the coefficients can be found in [17]- [18] with the exception of the K 1Q i term, which was not needed neither for the scalar nor for the fermion anomaly. Here in Appendix C we calculate such a term.…”

“…Conventions on gamma matrices with lightcone indices are summarized in Appendix B and are the same used in [18]. The covariant derivative acting on fermions is…”

“…This trick is used both in the relativistic, see e.g. [48], and nonrelativistic case [18]. Specializing to a flat background geometry, i.e.…”

“…The first explicit calculation of the Schrödinger trace anomaly in NC background was performed in [17] for the scalar case and in [18] for the fermionic case. There, the heat kernel procedure was used and a NC background with vanishing particle number gauge potential was chosen.…”