This chapter provides a self-contained introduction to the basic aspects of Quantum Mechanics, focusing on what is must for Quantum Field Theory. The notions of state space, unitary operators, self-adjoint operators, and projective representation are covered as well as Heisenberg’s uncertainty principle. A complete proof of Stone’s theorem is given, but the spectral theory is covered only at the heuristic level. We provide an introduction to Dirac’s formalism, which is almost universally used in physics literature. The time-evolution is described in both the Schrödinger and the Heisenberg picture. A complete treatment of the harmonic oscillator, providing an introduction to the fundamental idea of creation and annihilation operators concludes the chapter.

We introduce tensor products, which are the appropriate tool to construct the state space for systems that can be decomposed into non-interacting simpler components. The boson Fock space, built from such tensor products, is the appropriate state space to describe collections of any number of identical bosons. The annihilation and creation operators act on this Fock space. We then introduce the idea of quantum fields, which are operator-valued distributions, not trying yet to incorporate ideas from special relativity, and we argue that these quantum fields can be viewed as a quantized version of certain spaces of functions.