We explore several foundational facts about quantum field theories, starting with the well-known conversion of Schrödinger mechanics to a path integral formulation. Path integral accommodates the natural time-ordering of the operator formulation without any tweak, which leads to how the causal Green’s function replaces its classical counterpart and motivates the Wick rotation of the path integral into Euclidean time, setting the stage for the rest of the volume. Along the way, we pay some attention to the real-space version of various Green’s functions. After a brief exploration of fermionic mechanics and the following spinor fields, we turn to the symmetries and the conservation laws. Although we initially follow the standard treatise of the material, with emphasis on the role of gauging either as part of the dynamics or as an auxiliary handle, we point out how a common confusion over Noether’s energy-momentum tensor stems from an ill-conceived implementation of translations. When properly equipped with the Lie derivative, we show how the Noether procedure always reproduces the symmetric energy-momentum tensor from the metric variation and obviates the "improvement" terms.

The notion of effective action is introduced; the rest of the volume requires the concept crucially. The Wick rotation of the (effective) action is then given a new treatment in terms of how the metric is analytically continued. Spinors require a special treatment due to how the underlying Clifford algebra changes its character drastically in different signatures. Recasting the fermion action in Euclidean signature, we explore the Dirac operator and the eigenvalue problem thereof. The path integral of fermions is expressed in terms of the determinant or the Pfaffian of the Dirac operator, for which eigenvalue doubling and quadrupling in various dimensions are carefully studied. The heat kernel expansion shall be the main computational tool for the rest of the volume; a pedagogical overview is given with explicit formulae that suffice for all of the one-loop computations performed in this volume. The final section gives explicit comparisons of the path integral versus the Hamiltonian partition functions for bosonic and fermionic harmonic oscillators, in part as a way to demonstrate the equivalence and in part as a precursor for the index theorem that will enter later discussion.

In the first few sections we examine how we might define Hamiltonians which make physical sense, and we observe that the interaction picture, on which the entire approach is built, is fraught with mathematical inconsistencies. Nonetheless we proceed using it to compute the S-matrix in some of the simplest possible models. This is the heart of the theory. In a very progressive fashion we introduce the main tools, Wick’s theorem and the Feynman propagator, a very special tempered distribution. We then introduce Feynman’s diagrams. Each diagram encodes a term of a complicated calculation, and we give an algorithm to compute the value of such a diagram by a complicated integral. We pay great attention to clarify the nature and the role of the so-called symmetry factors. We then receive the bad news. As soon as the diagrams contain loops the integral giving its value has an irresistible tendency to diverge, a consequence of having attempted an ill-defined multiplication of distributions. We then show how to get a sensible physical prediction out of these infinite integrals, first in the relatively easy case of diagrams with one loop, and then in the much deeper case of diagrams with two loops, which involves a remarkable “cancellation of infinities”. We also introduce the physicist’s counter-term method to produce such cancellations.