Loop calculations in quantum-mechanical non-linear sigma models sigma models with fermions and applications to anomalies

TL;DR

This paper develops a path integral framework for one-dimensional non-linear sigma models with fermions, enabling the calculation of quantum anomalies and addressing technical issues like Majorana fermions and ghost sectors.

Contribution

It extends the path integral formalism to fermionic systems and generalizes anomaly computations to higher-dimensional quantum field theories.

Findings

Derived correct Feynman rules for fermionic sigma models

Developed a framework for computing anomalies via quantum mechanics

Resolved technical issues in path integrals for fermions and ghosts

Abstract

We construct the path integral for one-dimensional non-linear sigma models, starting from a given Hamiltonian operator and states in a Hilbert space. By explicit evaluation of the discretized propagators and vertices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems \cite{paper1}, are now extended to fermionic systems. We then generalize the work of Alvarez-Gaum\'e and Witten \cite{alwi} by developing a framework to compute anomalies of an $n$-dimensional quantum field theory by evaluating perturbatively a corresponding quantum mechanical path integral. Finally, we apply this formalism to various chiral and trace anomalies, and solve a series of technical problems: $(i)$ the correct treatment of Majorana fermions in path integrals with coherent states (the methods of fermion doubling and fermion halving yield equivalent results when used in applications to anomalies), $(ii)$ a complete path integral treatment of the ghost sector of chiral Yang-Mills anomalies, $(iii)$ a complete path integral treatment of trace anomalies, $(iv)$ the supersymmetric extension of the Van Vleck determinant, and $(v)$ a derivation of the spin-$3\over 2$ Jacobian of Alvarez-Gaum\'{e} and Witten for Lorentz anomalies.