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 symmetry is essential in the determination of particle properties. It reveals quantities that are conserved in collisions or decays. It also constrains the mathematical formulation of theories. This chapter introduces these concepts and explains how the notion of symmetry is implemented in quantum mechanics. It reviews the quantities conserved in particle collisions or decays: energy-momentum and total angular momentum, and also the internal symmetries, such as parity, charge conjugation, baryon and lepton numbers.