This chapter introduces the formal second quantization method for fermions in quantum field theory, and the connection to second quantization of bosons is shown. The picture of fermions as rotations between two states is presented, which helps the reader to see where the Pauli exclusion rule comes from. Finally, Dirac’s original derivation of his equation for relativistic motion of fermions is given.

This chapter introduces the formal “second quantization” method for bosons in quantum field theory. It is shown that phonons (sound particles) and photons (light particles) are simple extensions of the physics of a spring-like oscillator. The connection of boson states to classical waves is shown in a discussion of “coherent states.”

We introduce a kinetic theory of electron transport on the nanoscale, formulated in terms of the Fock space of an open many-electron system, and the “second quantization” Hamiltonian. To model a thermal electron reservoir (e.g., a metal electrode), the Fermi–Dirac distribution is derived from the corresponding density operator. A nanoscale system, weakly coupled to the reservoir, is modeled as an impurity. When the Born–Markov and secular approximations are valid, quantum master equations are derived, showing that the impurity equilibrates with the reservoir. To account for charge transport through the impurity, as in atomic point contacts or single molecule junctions, the master equations are generalized for cases of an impurity coupled to different reservoirs at different chemical potentials/temperatures. In these cases, we show that the system reaches a nonequilibrium steady state, where current flows through the impurity. Analytic expressions are derived for this steady state in simple models.

This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction