In this chapter we discuss quantum mechanical path integrals defined by time slicing. Our starting point is an arbitrary but fixed Hamiltonian operator Ĥ. We obtain the Feynman rules for nonlinear sigma models, first for bosonic point particles xi(t) with curved indices i = 1, …, n and then for fermionic point particles ψa(t) with flat indices a=1, …, n. In the bosonic case we first discuss in detail configuration-space path integrals, and then return to the corresponding phase-space integrals. In the fermionic case we use coherent states to define bras and kets, and we discuss the proper treatment of Majorana fermions, both in the operatorial and in the path integral approach. Finally, we compute directly the transition element 〈z|e−(β/ħ)Ĥ|y〉 to order β (two-loop order) using operator methods, and compare the answer with the results of a similar calculation based on the perturbative evaluation of the path integral with time slicing regularization. Complete agreement is found. These results were obtained in. Additional discussions are found in.

The quantum action, i.e. the action to be used in the path integral, is obtained from the quantum Hamiltonian by mathematical identities, and the quantum Hamiltonian is fixed by the quantum field theory, the anomalies of which we study in Part II of the book. Hence, there is no ambiguity in the quantum action.