LOOP CALCULATIONS IN QUANTUM-MECHANICAL NONLINEAR SIGMA-MODELS WITH FERMIONS AND APPLICATIONS TO ANOMALIES

We construct the path integral for one-dimensional non-linear sigma mo dels, starting from a given Hamiltonian operator and states in a Hilbe rt space. By explicit evaluation of the discretized propagators and ve rtices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems, are now extended to fermionic systems. We then generalize the work of Alvarez-Gaume and Witten by developing a framework to compute anomali es of an n-dimensional quantum field theory by evaluating perturbative ly a corresponding quantum mechanical path integral. Finally, we apply this formalism to various chiral and trace anomalies, and solve a ser ies of technical problems: (i) the correct treatment of Majorana fermi ons in path integrals with coherent states (the methods of fermion dou bling and fermion halving yield equivalent results when used in applic ations to anomalies), (ii) a complete path integral treatment of the g host sector of chiral Yang-Mills anomalies, (iii) a complete path inte gral treatment of trace anomalies, (iv) the supersymmetric extension o f the Van Vleck determinant, and (v) a derivation of the spin-3/2 Jaco bian of Alvarez-Gaume and Witten for Lorentz anomalies.